University of Montana
ScholarWorks at University of Montana
Theses, Dissertations, Professional Papers
Graduate School
1993
Box-Cox transformations of a travel-cost demand
function for stream fishing in Montana
Joseph M. Holliman
The University of Montana
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Recommended Citation
Holliman, Joseph M., "Box-Cox transformations of a travel-cost demand function for stream fishing in Montana" (1993). Theses,
Dissertations, Professional Papers. Paper 8770.
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B O X - C O X T R A N S F O R M A T I O N S OF A T R A V E L - C O S T D E M A N D F U N C T I O N
F O R S T R E A M F I S H I N G IN M O N T A N A
By
Jos e p h M.
B. A.,
B.
Holliman
S. U n i v e r s i t y of Montana,
P r e s e n t e d in p a r t i a l
1985
f u l f i llment of the r e q u i r e m e n t s
for the degr ee of
M a s t e r of Ar ts
U n i v e r s i t y of M o n t a n a
1993
A p p r o v e d by
Ohaj/6man,
Board' 6f E x a m i n e r s
Dean,Graduate
School
/O . /f f 3
Date
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UMI Number: EP39571
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Ho ll i m a n ,
J o s e p h M . , M.A.
May,
1993
Economics
B O X - C O X T R A N S F O R M A T I O N S OF A T R A V E L - C O S T D E M A N D F U N C T I O N
F O R S T R E A M F I S H I N G IN M O N T A N A
(12 8 pages)
Direc t o r :
John W.
Duffield
The z o n a l - a g g r e g a t e , r e g i o n a l t r a v e l - c o s t m o d e l (TCM) has
b e c o m e a w i d e l y a c c e p t e d m e t h o d to e s t i m a t e the d e m a n d an d
ne t e c o n o m i c v a l u e for ac cess to p u b l i c l and s for
p a r t i c i p a t i o n in o u t d o o r r e c r e a t i o n a c t i v i t i e s . In its
t r a d i t i o n a l form, p e r c a p i t a tr ips are m o d e l e d as a funct i o n
of th e v a r i a b l e t r a v e l a n d time co sts a s s o c i a t e d w i t h
t r a v e l i n g to a n d fr om a r e c r e a t i o n site.
The m o d e l is also
u s e d to e v a l u a t e ch a n g e s in net e c o n o m i c v a l u e s for p u b l i c
resources
in c as es w h e n a d d i t i o n a l r e s o u r c e s are ma de
available
to th e p u b l i c or w h e n r e s o u r c e s are s u b j e c t e d to
c h a n g e s in site a t t r i butes .
The m e t h o d s u s e d to e s t i m a t e
net e c o n o m i c v a l u e s a n d d e m a n d in such cases req ui r e s a well
s p e c i f i e d m o d e l r e s u l t i n g in a c c u r a t e t r i p p r e d i c t i o n acro s s
th e s a m p l e .
S e v e r a l a l t e r n a t i v e m a t h e m a t i c a l forms of e s t i m a t e d
t r a v e l - c o s t d e m a n d f u n c t i o n s have b e e n i n v e s t i g a t e d in
a t t e m p t s to e n h a n c e t r i p p r e d i c t i o n and r e l i a b i l i t y of net
e c o n o m i c v a l u e e s t imat es.
However, no d e f i n i t i v e
c o n c l u s i o n s r e g a r d i n g th e a p p r o p r i a t e f u n c t i o n a l fo rm of the
m o d e l h a v e b e e n reached.
U s i n g two p r i o r T C M studi es on
d a t a c o l l e c t e d in 1985 by th e M o n t a n a D e p a r t m e n t of Fish,
W i l d l i f e a n d P a r k s for th e M o n t a n a c o l d - w a t e r str e a m
fisheries
as a be n c h - m a r k , this study e x a m i n e s a l t e r n a t i v e
B o x - C o x t r a n s f o r m a t i o n s of the b a s i c b i v a r i a t e d e m a n d
function.
A c o m p r e h e n s i v e s e a r c h of p l a u s i b l e f u n c t i o n a l forms
s u g g e s t s a m o d e l in w h i c h th e n atural log of p e r c a p i t a
t r i p s r e g r e s s e d on a v e r a g e r o u n d - t r i p d i s t a n c e r a i s e d to the
.172 p o w e r m a x i m i z e d th e l o g - l i k e l i h o o d f u n c t i o n w i t h
s i g n i f i c a n t v a l u e s of all e s t i m a t e d par ameters.
This m o d e l
wa s t h e n d i s c r i m i n a t e d f r o m a p r e v i o u s l y p r o p o s e d
a l t e r n a t i v e f o r m of th e d o u b l e - l o g m o d e l in w h i c h a v e r a g e
r o u n d - t r i p d i s t a n c e w as s h i f t e d o u t w a r d at all o b s e r v a t i o n s
by a c onstant.
The J - t e s t and a d j u s ted-R^ r e v e a l e d
i n c o n c l u s i v e re s u l t s as to w h i c h m odel was m o s t a p p r o p r i a t e .
It wa s c o n c l u d e d t h a t the tw o m o d e l s p r o v i d e a l t e r n a t i v e
m e a n s of d e s c r i b i n g the v a r i a t i o n in p e r c a p i t a t r i p demand.
H o w ever, a p r e v i o u s l y s u g g e s t e d t h e o r y s u p p o r t i n g the
d o u b l e - l o g m o d e l a p p e a r s to p r o v i d e g r e a t e r s upport for its
us e t h a n the m o d e l in w h i c h d i s t a n c e is r a i s e d to the .172
power.
11
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T A B L E OF C O N T E N T S
ABSTRACT
........................................................
ü
L I S T OF T A B L E S .................................................. vi
L I S T OF
I L L U S T R A T I O N S ..........................................vii
ACKNOWLEDGMENTS
................................................
ix
CHAPTER
1.
S T A T E M E N T OF THE P R O B L E M ................................ 1
O r g a n i z a t i o n .........................................
3
Th e T r a v e l Cost M o d e l ................................ 3
S i m p l i f y i n g A s s u m p t i o n s ......................
5
5
Q u a n t i t y M e a s u r e s .............................
P r i c e ...............................................7
D a t a R e q u i r e m e n t s .............................
8
E x a m p l e s of T C M A p p l i c a t i o n s
..............
9
S t a t i s t i c a l D e t e r m i n a t i o n of the F o r m of the
F i r s t - S t a g e D e m a n d F u n c t i o n ...................
11
S t u d y M e t h o d o l o g y .................................... 13
S t u d y O u t l i n e .........................................17
2.
L I T E R A T U R E R E V I E W OF M O D E L S P E C I F I C A T I O N AN D
TH E B O X - C O X T R A N S F O R M A T I O N .......................... 19
G e n e r a l E l e m e n t s of M o d e l S p e c i f i c a t i o n
F o r OLS E s t i m a t e d M o d e l s ........................20
A p p r o a c h e s to M o d e l S p e c i f i c a t i o n
..............
22
The B o x — Co x F a m i l y of P o w e r T r a n s f o r m a t i o n s
. . 26
P u r p o s e a n d A s s u m p t i o n s of th e B asic
and Extended Box-Cox Regression . . . . 2 8
M e t h o d s U s e d to E s t i m a t e a B o x - C o x
R e g r e s s i o n s ................................. 30
M e t h o d s of M o d e l d i s c r i m i n a t i o n
................. 38
The L i k e l i h o o d R a t i o T e s t ................... 38
T e s t s for N o n - n e s t e d m o d e l s ................ 40
D i a g n o s t i c T ests U se f u l fo r M o d e l
S p e c i f i c a t i o n ....................................
41
M e t h o d s for T e s t i n g N o r m a l i t y of
R e s i d u a l s .................................... 41
D e t e c t i o n of H e t e r o s c e d a s t i c i t y
.......... 41
M e t h o d s for D e t e c t i n g a n d M e a s u r i n g
the I n f l u e n c e of O u t l i e r s ................... 43
C o n c l u s i o n s an d S u m m a r y
........................... 44
111
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3.
D A T A S O U R C E S A N D D E S C R I P T I O N ......................... 4 7
D e s c r i p t i o n of th e Sit es C o m p r i s i n g the
Montana Stream Fisheries
...................... 47
D a t a S o u r c e s I n c l u d e d in the
C o l d - W a t e r St r e a m s F i s h e r i e s D a t a Ba se
. . .
51
D a t a P r e p a r a t i o n for F i r s t — S tage T C M D e m a n d
Function Estimation
........................
51
C o l l e c t i o n of S u r v e y D a t a ................... 51
A g g r e g a t i o n of the S u p p l e m e n t e d
F i s h e r i e s Surv e y
........................
54
4.
E S T I M A T I O N A N D A N A L Y S I S OF D E M A N D
F U N C T I O N S ............................................. 58
B i v a r i a t e E s t i m a t e s of Two P r e v i o u s l y
...................
60
Estimated Demand Functions
D i a g n o s t i c Test s on M o d e l s 1 a n d 2 . . . .
64
G e n e r a l M o d e l S p e c i f i c a t i o n s for th e B o x — Cox
R e g r e s s i o n .........................................72
F o r m A n a l y s i s of the B i v a r i a t e F i r s t - S t a g e
T C M D e m a n d F u n c t i o n ............................... 7 5
E s t i m a t e d M o d e l s ............................. 75
M o d e l D i s c r i m i n a t i o n .......................... 78
D i a g n o s t i c Test s on M o d e l 5
................ 80
D i s c r i m i n a t i o n of M o d e l s 2 a n d 5 ................... 83
O v e r v i e w of M o d e l s 1, 2, an d 5 ..................... 87
S c a t t e r a n d Line P l o t s ........................87
P r e d i c t i v e P o w e r ............................. 93
E l a s t i c i t i e s .................................. 93
5.
C O N C L U S I O N S ................................................. 95
S u m m a r y ................................................ 95
L i m i t a t i o n s a n d S u g g e s t i o n s Fo r
Further Research
...............................
98
H e t e r o s c e d a s t i c i t y ........................... 99
T r u n c a t e d D i s t r i b u t i o n of the
R e s i d u a l s in M o d e l 5
..................... 99
E r r o r s in M e a s u r e m e n t of the
D i s t a n c e V a r i a b l e ......................
100
O m i t t e d V a r i a b l e B i a s ...................... 101
Market Segmentation
......................
102
L I S T OF R E F E R E N C E S ............................................ 104
IV
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APPENDIX
A.
T E C H N I C A L D E S C R I P T I O N OF THE T R A V E L C O S T M O D E L
B.
T E C H N I C A L D E S C R I P T I O N OF THE J - T E S T
C.
T E C H N I C A L D E S C R I P T I O N OF D I A G N O S T I C T E S T S . . . .
.
...............
W h i t e ' s T e s t for H o m o s c e d a s t i c i t y
............
DFFITS and DFBETAS Measures
...................
M o d e l i n g th e E f f e c t s of O u t l i e r s U s i n g
D u m m y V a r i a b l e s ............................ 119
A n a l y s i s of M o d e l i n g Tr ips from A l a s k a
as a D u m m y V a r i a b l e in M o d e l s 1 a n d
2 .. .
A n a l y s i s of M o d e l i n g Tr ips from A l a s k a
......... 122
as a D u m m y V a r i a b l e in M o d e l 5
D.
S U R V E Y F O R M S ....................................... 125
V
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109
Ill
114
114
115
120
LIST OF TABLES
TABLE
1.
M o n t a n a S t r e a m Fis h e r i e s :
N o n - U n i q u e Wat e r s
2.
Montana
Unique Waters
3.
D e s c r i p t i v e S t a t i s t i c s for the V a r i a b l e s
S p e c i f i e d in the M o d e l s P r e s e n t e d
in C h a p t e r 4 ........................................... 57
4.
B i v a r i a t e S p e c i f i c a t i o n s of M o d e l s S p e c i f i e d
by D u f f i e l d et al (1987) an d D u f f i e l d (1988)
5.
S t r e a m Fis h e r i e s :
. . . .
.............
49
50
. . 63
P l a u s i b l e F o r m s of E q u a t i o n (2) O t h e r t h a n the
D o u b l e — log. S e m i - l o g an d L i n e a r forms
..........
73
6.
Estimated Bivariate First-Stage TCM Demand
F u n c t i o n s ............................................. 77
7.
L i k e l i h o o d R a t i o T e s t s of B i v a r i a t e Models:
w i t h M o d e l 3 . a. as the General,
U n r e s t r i c t e d C a s e ............
79
8.
R e s u l t s of J— T e s t for D i s c r i m i n a t i o n B e t w e e n
M o d e l s 2 a n d 5 ........................................ 84
9.
E s t i m a t e d O w n - P r i c e E l a s t i c i t i e s of D e m a n d
for M o d e l s 1, 2, a n d 5 ............................... 94
10.
P o s s i b l e O u t c o m e s of The J - T e s t .................... 113
11.
E s t i m a t e s of M o d e l s 1 an d 2 w i t h A l a s k a
T r i p s M o d e l e d as an I n t e r a c t i o n
Parameter Dummy Variable
......................
121
E s t i m a t e s of M o d e l 5 wi t h A l a s k a Trips
M o d e l e d as an I n t e r a c t i o n P a r a m e t e r
Dummy Variable with Round-Trip Distance
124
12.
VI
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. . .
LIST OF ILLUSTRATIONS
FIGURE
1.
N o r m a l P r o b a b i l i t y (P— P) Plo t of S t a n d a r d i z e d
R e s i d u a l s fo r
Model 1
.............................. 67
2.
N o r m a l P r o b a b i l i t y (P— P) Plot of S t a n d a r d i z e d
R e s i d u a l s for
Model 2
.............................. 67
3.
S c a t t e r p l o t of t h e R e s i d u a l s an d P r e d i c t e d
N a t u r a l - L o g o f P e r C a p i t a Trip s P r o d u c e d
Model 1
by
68
4.
S c a t t e r p l o t of th e R e s i d u a l s an d P r e d i c t e d
N a t u r a l - L o g of P e r C a p i t a Trips P r o d u c e d by
Model 2
................................................ 68
5.
S c a t t e r p l o t of t h e R e s i d u a l s P r o d u c e d by M odel
1 a n d the N a t u r a l — log of A v e r a g e R o u n d - t r i p
D i s t a n c e ................................................ 69
6.
S c a t t e r p l o t of th e R e s i d u a l s P r o d u c e d by Model
2 a n d th e N a t u r a l — log of S h i f t e d A v e r a g e
Round-trip Distance
................................
69
7.
N o r m a l P r o b a b i l i t y (P— P ) Plot of the
S t a n d a r d i z e d R e s i d u a l s for
M o d e l 5, T a b l e 6 ...................................... 81
8.
S t a n d a r d i z e d S c a t t e r p l o t of p r e d i c t e d Pe r
C a p i t a T r ips a n d R e s i d u a l s for
M o d e l 5, T a b l e 6 ...................................... 81
9.
S t a n d a r d i z e d S c a t t e r p l o t of R o u n d - T r i p D i s t a n c e
T r a n s f o r m e d b y th e B o x - C o x P a r a m e t e r
l i s t e d for M o d e l 5, T a b l e 6 w i t h the
r e s i d u a l s f r o m th e same m o d e l ...................... 83
10.
11.
S c a t t e r p l o t of t h e n a t u r a l - l o g of p e r - c a p i t a
t r i p s w i t h the n a t u r a l - l o g of a v e r a g e
round-trip distance
...............................
89
S c a t t e r p l o t of t h e n a t u r a l - l o g of p e r - c a p i t a
t r i p s w i t h t h e n a t u r a l - l o g of a v e r a g e
r o u n d - t r i p d i s t a n c e p l u s 6 4 ......................... 90
Vll
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12.
S c a t t e r p l o t of th e n a t u r a l - l o g of p e r - c a p i t a
trips with average round-trip distance
r a i s e d to th e .172 p o w e r ............................ 91
VIXI
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ACKNOWLEDGEMENTS
P r i m a r y r e c o g n i t i o n goe s to John W.
practical
values
i n s t r u c t i o n on d i f f e r e n t
of o u t d o o r
D u f f i e l d for his
a spects of m e a s u r i n g
r e c r e a t i o n d u r i n g the p r o c e s s of
p e r f o r m i n g the M o n t a n a B i o e c o n o m i c
studies.
t o w a r d f o r m u l a t i o n of t h i s p r o j e c t
an d f e e d b a c k on c ertain
issues
served most useful
pr oject.
His co mments
in l i m i t i n g the scope of the
I a l s o r e c o g n i z e Dr.
Duffield's
c o n t i n u e d su pp ort
t o w a r d c o m p l e t i o n of this paper.
I also t h a n k the b a l a n c e of the r e v i e w comm i t t e e
including Michael
D a v i d A.
H.
Pat t e r s o n .
Kupil ik ,
D o u g l a s R. Dalenberg,
and
T h e i r d e v o t i o n to the ard u o u s t a s k of
r e v i e w i n g an d c o m m e n t i n g on key t o p i c s an d the drafts of
th is p a p e r hav e p r o v e n m o s t v aluable.
to M a r k D.
Partridge
for
R e c o g n i t i o n al so goes
f e e d b a c k on e c o n o m e t r i c issues,
C h r i s t o p h e r J. N e h e r for a s s i s t a n c e w i t h the data u s e d in
th is
s tudy an d i n f o r m a t i o n
the d a t a base,
Becky Hanway
and Denise Peterson
Final ly,
H o l l i m a n an d th e
their
regarding previous
fo r a s s i s t a n c e w i t h WORDPERFECT,
fo r editi ng.
I t h a n k my parents ,
James D.
f a c u l t y of th e D e p a r t m e n t
su p p o r t a n d b e l i e f
an d M a r y H e l e n
of E c o n o m i c s
for
in m y a b i l i t i e s to succeed.
P a r t i c u l a r r e c o g n i t i o n goes to Dr.
continual
a n a lysis of
K ay C. U n g e r for he r
i n s p i r a t i o n a n d i n s t r u c t i o n of e c o n o m i c theory.
ix
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CHAPTER 1
S T A T E M E N T OF TH E P R O B L E M
The p u r p o s e
functional
of this p a p e r is to e x a m i n e the
f o r m of a s t a t i s t i c a l l y estimated, (first-stage)
travel-cost model
stream fishing
(TCM)
demand function
in M o n t a n a d u r i n g
1985.
for c o l d - w a t e r
The T C M is a w i d e l y
a c c e p t e d a p p r o a c h to e s t i m a t i n g the d e m a n d for a n d net
e c o n o m i c v a l u e of n o n —m a r k e t
re so urces.
A p p l i c a t i o n s of the
T C M a p p r o a c h to r e s o u r c e v a l u a t i o n p r o b l e m s
a n d a i r q u a l i t y to
a c t i v i t i e s .^
to s p e c i f i c
se v e r a l
However,
c o n s u m p t i v e use r e c r e a t i o n
the m o s t
common applications pertain
recreation activities
s ite s
ar e p r o v i d e d by g o v e r n m e n t a l
(see,
e.g.,
McConnell
g enerally not
packages",
available
r ange f r o m w a t e r
(1985)).
sold in a m a r k e t
in w h i c h n a t u r a l
entities
as p u b l i c good s
S ince suc h a c t i v i t i e s
s p e c i f i c a l l y as
no d i r e c t l y o b s e r v a b l e m a r k e t p r i c e s
from which
TCM exploits
resource
economic values
"recreation
are
can be derived.
th e a b i l i t y to o b s e r v e act u a l
are
Thus,
expenditures
a s s o c i a t e d w i t h o u t d o o r r e c r e a t i o n trips w h i c h serve as the
^ Wals h, et. al. (1988) i d e n t i f i e s sever al a c t i v i t i e s
s u c h as c a m p i n g a n d s w i m m i n g (e.g., Su the rland, 1980)),
p i c n i c k i n g , hiking, a n d h u n t i n g (e.g., M a r t i n et. al., 1974
(1981)), f i s h i n g (e.g., S o r g et. al., 1985), a n d w i l d e r n e s s
u s e (e.g.. S m i t h a n d Kopp, 1980).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2
price proxy
for t he d e m a n d function.
Economic theory provides
mathematical
for m of a d e m a n d f u n c t i o n
Koutsoyiannis
Accordingly,
(1977)
there
s p e c i f y i n g th e
other than
f o r m of the
first-stage TCM demand function
Thus,
(see,
Smith
an d M c C o n n e l l
functional
r e s e a r c h e r s have i d e n t i f i e d the
form u s i n g s t a t i s t i c a l
(1975),
(1985)) .
Zeimer,
Musser,
inference
a n d Hill
f o r m of the f i r s t - s t a g e T C M d e m a n d f u n c t i o n
The a p p r o a c h uses
fu nct ion .
rivers an d stre ams d u r i n g 1985.
statistical
This
a n d one of tw o
infe r e n c e to s p e c i f y th e
Brooks
1992)
has
a n d P a rks
(1987)
previous
s u b s e q u e n t a n a l y s e s of the zonal
(DFWP)
and Duffield
analysis
f o c u s e d on th e
p e r f o r m e d by Duffield,
(1988).
Duffield's
of F i s h
Loomis,
and
(1988 an d
of the s t r e a m f i s h e r i e s dat a b a s e
f o r m of the d e m a n d f u n c t i o n w i t h r e s p e c t
to t he p r i c e v a r i a b l e .
l i m i t e d to t he
form
study is a r e e x a m i n a t i o n of the
a g g r e g a t e d d a t a c o l l e c t e d by the M o n t a n a D e p a r t m e n t
Wildlife
(1980),
The centr al t o pic of th is p a p e r is
fo r f i s h i n g on M o n t a n a ' s
original
for
r e c o g n i t i o n of the invers e r e l a t i o n s h i p b e t w e e n
functional
of t h i s
(1979)).
is no a p r i o r i t h e o r e t i c a l b a s i s
appropriate
the
{see e.g.,
a nd Ru s s e l l an d W i l k i n s o n
p r i c e a n d q u a n t i t y .^
e.g..
little g u i d a n c e on the
The a n a l y s i s
in this
stud y is
s a m e pr oblem .
^ A s d i s c u s s e d below, one m e t h o d of e s t i m a t i n g the
e c o n o m i c v a l u e of a c c e s s to a r e c r e a t i o n site u sing T C M is
b a s e d on s t a t i s t i c a l l y e s t i m a t i n g a f i r s t — stage d e m a n d
funct i o n .
This f u n c t i o n is th en u s e d to d e r i v e a s e c o n d s t a g e f u n c t i o n f r o m w h i c h c o n s u m e r s urplus is computed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
Organization
The b a l a n c e of this
First,
a general
a n d i n t u i t i v e d e s c r i p t i o n of the
travel-cost model
is pr ovided.^
r e v i e w of th e l i t e r a t u r e
of th e
f o r m of th e
e sti m a t e d .
Thi s
c h a p t e r is o r g a n i z e d as follows.
is f o l l o w e d by a
in w h i c h s t a t i s t i c a l
determination
f i r s t - s t a g e d e m a n d f u n c t i o n has b e e n
s e c t i o n d e s c r i b e s the m e t h o d s u s e d in this
p a p e r to d e t e r m i n e th e
f u n ctional
first-stage demand f u n c t i o n .
b r i e f o v e r v i e w of the
e s t i m a t i o n u s i n g th e
Finally,
This
zonal
an o u t l i n e
fo rm of the b i v a r i a t e
I n c l u d e d in this
sec tion is a
f i n dings of p r e v i o u s T C M d e m a n d
same data bas e u s e d in this
study.
of the r e m a i n i n g c h a p t e r s of the p a p e r
is p r e s e n t e d .
Th e T r a v e l — C o s t M o d e l
A w i d e v a r i e t y of t r a v e l - c o s t m o d e l s h a v e b e e n
d e v e l o p e d a n d are c l a s s i f i e d by W a r d an d D u f f i e l d
conventional
t r a v e l - c o s t models,
h e d o n i c models .
on t he
single equation
the general
m ode ls.
However,
random utility models
the a naly s i s
(per c a p i t a visits)
a g g r e g a t e d by o r i g i n
as
and
in th is p a p e r fo c u s e s
zonal a g g r e g a t e T C M w h i c h falls u n d e r
c l a s s i f i c a t i o n of t r a d i t i o n a l
In th is model,
(1992)
travel-cost
r e c r e a t i o n d e m a n d or p a r t i c i p a t i o n
for a site or region of sites,
zone,
is s t a t i s t i c a l l y
^ R e a d e r s are r e f e r r e d to A p p e n d i x A
t e c h n i c a l d e s c r i p t i o n of the TCM.
e s t i m a t e d as a
for a m o r e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
f u n c t i o n of th e p r i c e
travel
time
expenditures
(opportunity)
specific activity
t he
demand
income,
for e a c h trip.
Average variable
i n c u r r e d to m a k e a tr ip an d foreg one
c os ts
a s s o c i a t e d w i t h eac h trip for a
serv e as a p r o x y
for price.
Variables
in
f u n c t i o n o t h e r t h a n p r i c e t y p i c a l l y inc lud e
p r i c e of subs t i t u t e s ,
demographic variables
s o c i o - e c o n o m i c an d/or
a n d site attributes.
F r o m this
f u n c t i o n a s e c o n d - s t a g e d e m a n d r e l a t i o n can be d e r i v e d on
the a s s u m p t i o n t h a t
access prices
travel
in the
costs.
a b o v e th e
as the net
r e c r e a t i o n i s t s w o u l d r e s p o n d to site
same w a y as they r espond to v a r y i n g
Th e a r e a u n d e r the
costs
actually
simu l a t e d d e m a n d f u n c t i o n
i n c u r r e d to make a t r i p is d e f i n e d
economic value
s u r v e y e d users h o l d for
m a i n t a i n i n g a c c e s s to th e r e c r e a t i o n sites u n d e r study.^
This m e t h o d r e q u i r e s th e m o d e l
across
the
sample.
The
to a c c u r a t e l y p r e d i c t trips
f i r s t - s t a g e dema n d f u n c t i o n can be
i n t e g r a t e d to d e t e r m i n e ne t e c o n o m i c value
(1983)
in D u f f i e l d et al.
represent Marshallian
reasonable proxy
(1987)).
consumer
for net a m o u n t
w i l l i n g to p a y to m a i n t a i n
(Menz an d W i l t o n
The resul t i n g val u e s
surplus w h i c h serve as a
r e c r e a t i o n i s t s w o u l d be
a c c e s s to a site ^ (see.
Just
et
See Dwyer, Kelly, a n d B owes (1977) for an
e x p l a n a t i o n of t he m e c h a n i c s of the travel cost model.
^ A l t h o u g h T C M can be u s e d to m e a s u r e net e c o n o m i c
v a l u e s for e n v i r o n m e n t a l c o n c e r n s o t h e r t h a n to m a i n t a i n
a c c e s s to a r e c r e a t i o n site, the focus of this study is to
l i m i t th e d e m a n d s t u d y to th e acc ess issue.
It s h o u l d als o
b e n o t e d t h a t d e m a n d a n d e c o n o m i c val u e s e s t i m a t e d w i t h the
z o n a l T C M p e r t a i n to c o n s u m p t i v e use value.
Other economic
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
al.
(1982)).
E a c h face t of this p r o c e s s
is s u m m a r i z e d
below.
Simplifying A s s u m p t i o n s .
aggregate travel
characteristics
homogeneous
cost model,
To s i m p lify the
economic
of r e c r e a t i o n i s t s
zonal
and d e m o g r a p h i c
are a s s u m e d to be
a c r o s s a n d r e p r e s e n t a t i v e of e a c h or igin zone's
population.
Furthermore,
on ly d a t a
a n d s i n g l e p u r p o s e t r i p s are
for s i n g l e - d e s t i n a t i o n
i n c l u d e d in the data set.
This
m i t i g a t e s p r o b l e m s w i t h a l l o c a t i n g costs among d e s t i n a t i o n s
a n d a c t i v itie s.
each
The a m o u n t of t i m e r e c r e a t i o n i s t s
site is als o a s s u m e d to be h o m o g e n e o u s
recreationists
in e a c h o r i g i n
the o p p o r t u n i t y
homogeneous
w i t h i n the
zone.
cost of t r a v e l
for all v i s i t o r s
at the
McConnell
It is also a s s u m e d that
time is c o n s t a n t or at least
to the o b s e r v e d site or sites
G e n e r a l l y q u a n t i t y has b e e n
s p e c i f i e d by tw o d i f f e r e n t m easu r e s :
u s e r days,
for all
study region.
Quantity M e a s u r e s .
hours
s pend at
site.
User-days,
has a l s o b e e n u s e d
(1975)
is i n d e p e n d e n t
maximization
argues,
of t r a v e l
trips t a k e n and u s e r —
a l i n e a r t r a n s f o r m a t i o n of
(McConnell
(1975)).
t h a t the m a r g i n a l
costs.
However,
cost of u s e r - d a y s
S i n c e the u t i l i t y
f r a m e w o r k u n d e r l y i n g th e tra v e l cost m e t h o d is
v a l u e s p l a c e d on an e n v i r o n m e n t a l r e s o u r c e i n clude option,
b e q u e s t , a n d e x i s t e n c e valu es.
T h e s e v a l u e s p e r t a i n to the
o p t i o n to us e a r e s o u r c e s o m e t i m e in the future, the v a l u e
a s s o c i a t e d w i t h p r o v i d i n g th e r e s o u r c e s as an e n d o w m e n t to
f u t u r e g e n e r a t i o n s , a n d th e v a l u e of k n o w i n g that the
r e s o u r c e exists, r e s p e c t i v e l y .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
b a s e d on the
quantity,
fit
in u t i l i t y
he a r g u e s
in this
user-day
change
that the d e m a n d for u s e r - d a y s does not
framew ork .
That
is i n d e p e n d e n t
He a l s o a r g u e s
is,
of the
demanded.
This
th e m a r g i n a l
of t r a v e l
t h a t con sumers'
knowledge
s u b j e c t to a change in
cost s once a trip is made.
s urplus
r e l a t i o n s h i p of u n i t s
compu t a t i o n s
costs and quantity
f r a m e w o r k in w h i c h the dema n d funct ion
is s i m i l a r to a d e r i v e d d e m a n d for t r i p s .
Ward and Duffield
(1992)
in t h i s
Specifically,
n o t e th at t he b asis
for the travel
f r a m e w o r k is the w e a k c o m p l e m e n t a r i t y
between marketed commodities
f r o m a site,
require
is c o n s i s t e n t w i t h the t h e o r y of TCM in a
household production
cost model
cost of a
among other
s e r v i c e s p r o v i d e d by the
into the production
a s s o c i a t e d w i t h trav el to an d
inputs,
site.
function
an d the n o n - m a r k e t e d
Thus,
trav el
is an input
for the p r o v i s i o n of r e c r e a t i o n
services.
Walsh
(1985)
notes
us e r - d a y ,
in t e r m s
different
recreationists
can be p r o b l e m a t i c w h e n
s p e n d d i s p a r a t e am oun ts of time
intended purpose
t h a t w h e n the
recreationists,
a p r e c i s e d e f i n i t i o n of a
of p e r s o n hours,
p a r t i c i p a t i n g in th e
suggests
that
of the trip.
He
l e n g t h of stay is s imilar for all
p e r c a p i t a t rips
is a s u i t a b l e m e a s u r e of
quantity.
P e r c a p i t a t r ips
this
study.
primarily
is th e q u a n t i t y m e a s u r e a d o p t e d in
The d e c i s i o n to use t h i s
d e f i n i t i o n is b a s e d
on m a i n t a i n i n g c o n s i s t e n c y w i t h th e two p r e v i o u s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7
d ema n d studies
n o t e d above.
u s i n g th e M o n t a n a
B a s e d on th e a b o v e
capita trips
also appears
stream fisheries
literature
to be the m o s t
data base,
r e v i e w p er
l ogical m e a s u r e m e n t
of q u a n t i t y .
Price.
trips
is n o n - c o m m e r c i a l
out-of-pocket
marginal
travel
costs)
an d at th e
exchange
trip,
S i n c e the m a r k e t
in nature,
is p r o x i e d by
(short-run
a n d the o p p o r t u n i t y co st of ti me en route to
site.
Price
in this m o d e l
cost)
on a v e r a g e w i t h i n the
for p r i c e
variation across
r e p r e s e n t s the rate of
for p r o d u c i n g the r e c r e a t i o n
sa mp l e u s e d for e a c h study.
is a d v a n t a g e o u s
sin ce it all ows p r i c e
o r i g i n — d e s t i n a t i o n p a i r s to be o b s e r v e d
cross— sectionally.
variation
price
and visitation expenditures
(or t h e m a r g i n a l
Th is p r o x y
for o u t d o o r rec r e a t i o n
in thi s
Furthermore,
there
is u s u a l l y m o r e
s u r r o g a t e p r i c e m e a s u r e t h a n pric e
v a r i a t i o n g e n e r a t e d in c o m m e r c i a l m a r k e t s
(Burt an d B r e w e r
(1971) ) .
A t th e minim um,
costs
travel
of o p e r a t i n g a v e h i c l e
and Brewer
Department
However,
(1971)).
This
accurately
c os ts
e.g.,
WRC
(1983)
and Burt
can be o b t a i n e d from
s t a t i s t i c s p u b l i s h e d annually.
suggested that
c o s t s p e r c e i v e d by the
of-pocket
(see,
i nclude the v a r i a b l e
information
of T r a n s p o r t a t i o n
it has b e e n
cost s
r e p o r t e d costs or t h ose
surveyed recreationists
as the o u t -
a s s o c i a t e d w i t h t a k i n g the t r i p more
represent true travel
cost s
(Duf fie ld et ai.
(1987) ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
The p r i c e p r o x y a l s o i n c l u d e s
of t i m e
in t r a n s i t
definitive
two methods
wa y of m e a s u r i n g this
(1987))
recreationists'
time by o n e - h a l f
W h i l e no
These methods vary between
w i l l i n g n e s s to pa y to reduce
(see e.g.
D u f f i e l d et al.
to d e t e r m i n i n g the v a l u e b a s e d on a r atio of the
estimated travel
travel
site.
cost has b e e n developed,
ar e w i d e l y acce pted.
d e t e r m i n i n g the
the t r a v e l
to a n d f r o m the
the o p p o r t u n i t y cost
time
cost a n d c o m b i n e d a v e r a g e
(McConnell
an d S t r a n d
(1981)).
a d o p t e d b y th e W a t e r R e s o u r c e C o u n c i l
on a n a l y s i s b y C e s a r i o
(1976)
measures
of t i m e as o n e - t h i r d the w a g e
fam ily in com e an d
rate
(WRC
On e m e t h o d
(1983))
and b a s e d
the o p p o r t u n i t y cost
for adults.
O n e — fourth
o f this v a l u e w o u l d r e p r e s e n t the o p p o r t u n i t y cost of time
for c h i l d r e n T h e d a t a u s e d b y D u f f i e l d et al.
travel
and time
survey
s u p p l e m e n t i n g the M o n t a n a
Mail
Survey
in this
is
(1989)),
Trip-weighted,
s p e c i f i e d as the p r i c e p r o x y
f unc tion.
The
sa me p r a c t i c e
Data R e q u i r e m e n t s .
Statewide Angler Pressure
the p r i m a r y d a t a so urce u s e d
average
zone
for a s i n g l e
round-trip distance
in t h e i r
first-stage demand
is u s e d in this
The t r a d i t i o n a l
H o t t e l l i n g - C l a w s o n - K n e t c h T C M uses
origin
for the
cost m e a s u r e s was g a t h e r e d in a s e p a r a t e
( M c Farlan d
study.
(1987)
study.
(zonal)
sur vey d a t a a g g r e g a t e d by
site or a c o l l e c t i o n
of sites
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
9
s i m i l a r a c t i v i t i e s w i t h i n a r e g i o n .^
from users
whose primary purpose
particular
site was to p a r t i c i p a t e
recreational
activity,
Da ta are g a t h e r e d
for t a k i n g t h e i r t r i p to a
in a si ngle
consumptive
such as h u n t i n g or fishing.
Although
the T C M has b e e n d e s c r i b e d as m e a s u r i n g the d e m a n d and
benefits
general
a s s o c i a t e d w i t h an entire
agreement
r e c r e a t i o n trip,
in the l i t e r a t u r e that the m o d e l
l i m i t e d to m e a s u r i n g such d e m a n d a n d b e n e f i t s
t here is
s h o u l d be
for the site
a n d s i n g l e p u r p o s e trip s as n o t e d abo ve
(Dwyer et al.
(1977)).
s u g g e s t e d to
E v e n t h o u g h a m e t h o d has b e e n
rationally
allocate travel
clustered destinations
general
consensus
consistently
costs to several
(Haspel a nd J o h n s o n
is th at
and destinations.
w i t h the
sole
(1982)),
for tr ips t a k e n
Thus,
are
in ten t to v i s i t a single
site for the p u r p o s e
environmental
(in this
case
i n c l u d e d in the analysis.
E x a m p l e s of T C M a p p l i c a t i o n s .
estimate net
for sever al
onl y da ta for trips m a d e
of p a r t i c i p a t i n g in the a c t i v i t y u n d e r study
fishing)
the
it is to o d i f f i c u l t to
a l l o c a t e the costs
purposes
closely
economic values
qualities
T C M m a y be u s e d to
associated with
^ for s p e c i f i c
status qu o site
recreation
® T w o o t h e r m e t h o d s rely on o b s e r v a t i o n s of i n d i v i d u a l
r e c r e a t i o n i s t s ; the q u a n t i t y v a r i a b l e in t h e s e m o d e l s is the
n u m b e r of t r i p s p e r r e c r e a t i o n i s t o v e r a season, (see, e.g..
Brown, Sorhus, Chou- Yan g, a n d R i c h a r d s (1983)).
^ " Status qu o e n v i r o n m e n t a l q u a l i t y " is d e f i n e d her e
as t h e e n v i r o n m e n t a l q u a l i t y e x i s t i n g p r i o r to the
i m p l e m e n t a t i o n of a n e w p o l i c y or the o c c u r r e n c e of an
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
activities.
T C M m a y als o be u s e d to
f o r eca st site u s a g e an d
changes
(gains or losses)
in ne t e c o n o m i c v alue due to
se v e r a l
alternative policy directives
Su ch d i r e c t i v e s m a y a f f e c t
a f f e c t i n g site usage.
site acce s s pri c e s
(e.g. , by
i n t r o d u c i n g or c h a n g i n g e n t r a n c e or u s e r fees or by
tax ati on)
or result
of a site
(Ward a n d Loomis,
to p r o j e c t s
the w a t e r
in ch a n g e s to the e n v i r o n m e n t a l
1986).
quality
S uch ch anges m ay be due
suc h as d a m m i n g a r i v e r or p e r m a n e n t l y c h a n g i n g
le vel
of a re servoir.
from an i n v o l u n t a r y
Water Resources
In some cases,
oil or h a z a r d o u s m a t e r i a l
Council
estimating changes
C h a n g e s m a y also result
(1983)
provides
spill.
guidelines
The
for
in site u s a g e a n d e c o n o m i c benefits.®
it m a y be r e a s o n a b l e to use p r e v i o u s l y
estimated relationships
of v a l u a t i o n and n o n - p r i c e v a r i a b l e s
such as si te att r i b u t e s .
i n c i d e n t w h i c h c h a n g e s a sites quality.
The status qu o
w o u l d t h e n m o s t a p p r o p r i a t e l y r e f e r to the b a s e - l i n e
c o n d i t i o n s i m i l a r to the d e f i n i t i o n p r o v i d e d in s e c t i o n
11.14 of T i t l e 43 CFR.
In thi s case it w o u l d be the c r o s s s e c t i o n a l " s n a p - s h o t " of th e site q u a l i t y at the time the
da t a w e r e col l e c t e d .
® W R C r e c o m m e n d s T C M as one of t wo p r e f e r r e d m e t h o d s
to e v a l u a t e c h a n g e s in e c o n o m i c b e n e f i t s an d to fo rec ast
u s a g e w h e n a s s e s s i n g the im p a c t s of w a t e r r e l a t e d pro ject s.
A d d i t i o n a l l y , f e d e r a l r e g u l a t i o n s g o v e r n i n g th e m e t h o d s u s e d
to a s s e s s e c o n o m i c losses due to oil or h a z a r d o u s m a t e r i a l
spi l l s a d o p t the W R C g u i d e l i n e s by r e f e r e n c e (43 CF R S e c t i o n
11.18 (a) (2)) as on e m e a n s for d e t e r m i n i n g e c o n o m i c losses
a s s o c i a t e d w i t h s u c h acci den ts.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11
Statistical Determination
of the F o r m of the F i r s t - S t a g e
Demand Function
B a s e d on the
several
criteria
f o r e g o i n g summary,
s h o u l d be m e t to obta i n a c c u r a t e estim a t e s
of consumer''s surplus.
c o l l e ction,
Other than
demand functions
soun d and c o m ple te data
in gen e r a l
s p e c i f i e d a c c o r d i n g to d e m a n d t h e o r y
O m i s s i o n of key v a r i a b l e s
time
it can be seen that
and substitutes
s h o u l d be fully
(Koutsoyiannis
suc h as the o p p o r t u n i t y cost of
in a T C M d e m a n d f u n c t i o n will g e n e r a l l y
result
in b i a s e d net e c o n o m i c v a l u e e s t i m a t e s
Strong
(1983)).
surplus
form p l a y s
estimate
Howev er,
(see,
S i n c e th e m a g n i t u d e of c o n s u m e r ' s
rel i e s h e a v i l y on p r i c e
functional
elasticities
a key
(see,
of demand,
e.g.,
Ze imer et al.
c orrect
it is a l s o i m p o r t a n t that the m o del be able to
in e s t i m a t i n g c o n s u m e r ' s
sur p l u s
and Duffield
literature
(1988)).
(see,
This
e.g..
is a fac t o r
W a r d a n d Lo omis
secti on reviews the
regarding statistical procedures previously used
s p e c i f y the
f o r m o f th e
A priori
and consumer surplus
McConnell
c a n n o t be a d d i t i v e
i n c o m e varies.
f i r s t — stage d e m a n d function.
s p e c i f i c a t i o n of the form of the d e m a n d
f u n c t i o n has p r o v i d e d
examp le,
surpl us
(1980)).
since p r e d i c t e d trips
to
e.g..
role in the v a l u e of c o n s u m e r ' s
accurately predict trips
(1986)
(1977)).
less t h a n op t i m a l
(see,
(1975)
e.g.,
e s t i m a t i o n of d e m a n d
McConnell
sh ows that the
(1985)).
fun ct io nal
As an
form
s ince p r i c e m u s t be a l l o w e d to v a r y as
This
conclusion
is b a s e d on a n a lysis w h i c h
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
sho ws th e d e m a n d s lope w i t h r e s p e c t t o i n c o m e d epends
c ost s
with
zero.
a nd th at th e
s y m m e t r y of the cross p a r t i a l
r e s p e c t to all o t h e r p r i c e s
T h e refore,
inc orr ect .®
derivatives
an d inc o m e is not eq ual
a linear functional
to
form is t h e o r e t i c a l l y
The l i n e a r form has b e e n u s e d to e s t i m a t e the
first-stage TCM demand function
(1980)).
on
However,
showed that Bowes
Vau gh n,
(see,
Russell,
a n d Loomis'
e.g.
B owes and Loomis
an d H a z i l l a
(1982)
d a t a is m o r e a p p r o p r i a t e l y
d e s c r i b e d by a s e m i - l o g fro m once h e t e r o s c e d a s t i c i t y is
identified and c o r r e c t e d .
W h i l e th e l i n e a r f o r m is ina ppropr iate,
provides
theory
little o t h e r g u i d a n c e on the a p p r o p r i a t e
mathematical
for m of the
f i r s t - s t a g e T C M d e m a n d function.
M u c h w o r k has b e e n d e v o t e d to the d e t e r m i n a t i o n of the
functional
three
for m u s i n g s t a t i s t i c a l methods.
exam ples.
First,
b a s e d on a study of g eneral
in th e D e s o l a t i o n W i l d e r n e s s A r e a
Smith
The f o l l o w i n g are
(1975)
recreation usage
in N o r t h e r n C ali fo rnia ,
s u g g e s t s t h e r e d o e s not a p p e a r to be any
empirical
justification
f a v o r i n g us e of e i t h e r the linear,
semi-log,
or the d o u b l e - l o g
(with log t r a n s f o r m a t i o n s
in the
® Q u i r k (1976) a l s o n otes th at if p r i c e s an d inc o m e
are h o m o g e n e o u s to d e g r e e zero a l i n e a r f u n c t i o n a l f o r m is
no t p o s s i b l e .
A l l s e m i - l o g m o d e l s in this p a p e r r e f e r to a f o r m
in w h i c h th e d e p e n d e n t v a r i a b l e is t r a n s f o r m e d by the
n a t u r a l - l o g a n d the i n d e p e n d e n t v a r i a b l e t a kes its l i n e a r
form.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
13
dependent
a n d i n d e p e n d e n t variables)
models.
a d i s c r i m i n a t i o n test
for n o n - n e s t e d models,
Pesaran's
(Pesaran
neither
N statistic
the
(1974)),
Further,
using
nam e l y
Smit h foun d that
s e m i — log or d o u b l e — log forms repre s e n t the
behavioral patterns
of r e c r e a t i o n usag e d e s c r i b e d by the
data.
In a n o t h e r
Z e i m e r at ai.
double-log
s tudy of w a r m - w a t e r f ishing in Georgia,
(1980)
f o und that the s e m i — log ve rsus the
form fit t h e i r d a t a best
(as d e t e r m i n e d by
a p p l y i n g th e B o x - C o x m e t h o d of m odel d i s c r i m i n a t i o n ) .
Third,
b a s e d on the
f i n d i n g s of Ze ime r et.
(1983)
suggest that
functional
specification
al,.
Stro ng
fo rm is an i mportant
c o n s i d e r a t i o n w h e n a p p l y i n g the T C M approach.
Study Methodology
The goal
mathematical
of th is p a p e r is to d e t e r m i n e the
form of the
u s i n g th e B ox an d Co x
f i r s t - s t a g e TC M d e m a n d fu nction
(1964)
discriminating alternative
is d o n e
functional
forms.
for the b i v a r i a t e d e m a n d function.
d e t e r m i n a t i o n of f u n c t i o n a l
bounds
s t a t i s t i c a l m e t h o d of
As noted,
this
The
fo rm is also l i m i t e d by the
of t h e a p p l i c a b l e e c o n o m i c d e m a n d theory:
“
A l t h o u g h S m i t h (1975) n o tes that the T C M a p p r o a c h
is t y p i c a l l y u s e d for s p e c i f i c r e c r e a t i o n a c t i v i t i e s a n d m a y
n o t be v a l i d for the a p p l i c a t i o n in his 1975 paper, he
a r g u e s t h a t d e m a n d for such d i v e r s e a c t i v i t i e s will be
i n d i r e c t l y r e f l e c t e d in the d e r i v e d d e m a n d for the sites
s erv ices.
S m i t h ' s a n a l y s i s was b a s e d on a si ngle site zonal
a g g r e g a t e t r a v e l cost model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
specifically,
The
outdoor
been
th e i nverse
r e l a t i o n of p r i c e a n d quantity.
s t a n d a r d forms u s e d to e s t i m a t e
recreation demand
generally
functions
in the
l i m i t e d to the linear,
fir s t - s t a g e
lite r a t u r e have
semi-log
(with a
n a t u r a l - l o g t r a n s f o r m a t i o n of the d e p e n d e n t v a r i a b l e ) ,
double-log,
done
and quadratic
forms.
R e l a t i v e l y little has been
in d e t e r m i n i n g w h i c h of t h e s e
in e s t i m a t i n g
Strong
recreational
demand
a nd Zeimer,
Muss er,
(1983);
further investigate
forms
(see,
Montana
functional
fo rms
Duffield's
(1988)
Transformations
(1980))
or to
streams,
for the
(1964)
family of p o w e r
fis hi ng on
thi s p a p e r e x a m i n e s p o s s i b l e
f i r s t - s t a g e d e m a n d functi on u s i n g
findings
as a s t a r t i n g point.
on the d e p e n d e n t a n d p r i c e - p r o x y v a r i a b l e
transformations
given
f a m i l y of p o w e r
in e q u a t i o n
(1):
X*0
z
In
Where
(1975);
c o l l e c t e d for sport
ar e e s t i m a t e d u s i n g the B o x - C o x
(1)
Smith
forms o t h e r t h a n thos e listed.
to the d a t a
cold-water
e.g.
a n d Hill
By a p p l y i n g the B ox a n d Co x
transformations
is most a p p r o priate
z
; 1=0
z is e i t h e r the d e p e n d e n t
or i n d e p e n d e n t v a r i a b l e an d
X is t h e B o x - C o x t r a n s f o r m a t i o n p a r a m e t e r .
The g enera l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
form
15
of th e d e m a n d f u n c t i o n is p r e s e n t e d in e q u a t i o n
(2)
(2);
P. -
Where
is p e r c a p i t a t rips t a k e n from o r i g i n
Dij is a v e r a g e
round-trip distance
from o r i g i n
Po a n d pi ar e e s t i m a t e d pa r a m e t e r s ,
(Xk)
i to site
i to site
e is the e r r o r term,
j,
j,
and
are th e B o x - C o x t r a n s f o r m a t i o n s of t h e i r r e s p e c t i v e
variables.
Much
f l e x i b i l i t y in the
form of the e s t i m a t e d
d e m a n d f u n c t i o n is a f f o r d e d by the B o x - C o x tr a n s f o r m a t i o n .
For
in stan ce,
howev er,
form
if
= 1 the f u n c t i o n is linear.
A>v = 0 an d A.^ = 1 the
( S p itz er
equation
(2)
( 1 9 8 2 a ) T h e
different
th is
stu d i e s
The
on t h e s t r e a m f isheries da ta b a s e u s i n g
results
a n d d i s c u s s e d in m o r e
these
stu dies ,
(log-linear)
on a v e r a g e
and average
zone.
B o x -C o x
of t h e s e
detail
studie s are
(1987)
s u m m a r i z e d here
In the first of
s p e c i f i e d a d o u b l e — log
in w h i c h p e r capi t a trips w e r e
r o u n d - t r i p dis tance,
years
f i r s t — stage T C M d e m a n d
in c h a p t e r 4.
D u f f i e l d et al.
model
in cha p t e r 4.
study is a r e e x a m i n a t i o n of two of
s p e c i f i c a t i o n s of the
function.
s e m i-log
form f l e x i b i l i t y of
is d i s c u s s e d in g r e a t e r detail
As noted,
three prior
form b e c o m e s the
If,
regressed
total t rout c a t c h by site,
of f i s h i n g e x p e r i e n c e of a n g l e r s by o r i g i n
D u e to th is m o d e l ' s
f ailure to p r o d u c e h o m o s c e d a s t i c
K m e n t a (1986) sh ows that as A,^ a p p r o a c h e s zero,
t r a n s f o r m a t i o n b e c o m e s the n a t u r a l - l o g of z.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
16
residuals
a n d its p o o r p r e d i c t i v e power,
examined several
fu nction.
alternative
In this
study,
forms of the
it was
f o r m of th e B o x — Cox t r a n s f o r m a t i o n
(1974)
produced a model which
and prediction
Specifically,
(2),
was
(1988)
first-stage demand
found that
a restricted
s u g g e s t e d b y Zaremb ka
s a t i s f i e d the h o m o s c e d a s t i c i t y
in the D u f f i e l d et al.
th is m o d e l
except
constant
were
concerns
Duffield
(1987)
model.
t o o k a form si m i l a r to e q u a t i o n
replaced with
a d d e d to a v e r a g e
(D^j + C)
where
r o u n d - t r i p distance.
Xy and
set e q u a l to zero to p r o d u c e the d o u b l e - l o g
wa s v a r i e d u n t i l
t r i p p r e d i c t i o n was w i t h i n
o b s e r v e d trips.
This
C is a
form,
.1 p e r c e n t
an d C
of
r e s u l t was a c h i e v e d at C = 90 m i l e s
for a m u l t i v a r i a t e m o d e l .
B a s e d on D u f f i e l d ' s
to lie c l o s e
one.
to
zero a n d
(1988)
findings,
Xy is e x p e c t e d
s h o u l d fall b e t w e e n
zero an d
It is not e x p e c t e d tha t the v a l u e of X^ will be close
to or g r e a t e r t h a n one,
general
polynomial,
g i v e n the p o o r p e r f o r m a n c e of the
s e m i — log,
an d d o u b l e — log forms e s t i m a t e d
" H o m o s c e d a s t i c r e s i d u a l s are t h ose in w h i c h the
v a r i a n c e of e a c h r e s i d u a l is n e a r l y c o n s t a n t ac ro ss all
observations.
W h e n t h e v a r i a n c e of the resi d u a l s are not
c o n s t a n t a c r o s s all o b s e r v a t i o n s , t h e y are said to be
h e t e r o s c e d a s t i c . The c o n s e q u e n c e s of h e t e r o s c e d a s t i c i t y are
inefficient parameter estimates and biased variance
e s t i m a t e s y i e l d i n g i n v a l i d h y p o t h e s i s tests of the
s i g n i f i c a n c e of p a r a m e t e r s (see, e.g., K m e n t a (1986)).
A n o t h e r d i f f e r e n c e b e t w e e n e q u a t i o n (2) an d the
m o d e l e s t i m a t e d by D u f f i e l d (1988) for the e n t i r e sa mpl e was
t h a t th e s e v e r a l o t h e r v a r i a b l e s such as c a t c h a g g r e g a t e d by
site, a s u b s t i t u t e index, a n d c e r t a i n s o c i o - e c o n o m i c an d
d e m o g r a p h i c v a r i a b l e s w e r e i n c l u d e d in the function.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
by D u f f i e l d
(1988).
Since there
d e t e r m i n i n g the
the
is lit tle t h e o r e t i c a l b asis
functional
f o r m m u s t be
form of the T C M d e m a n d equation,
f o u n d by experi m e n t a t i o n .
The mod el
"best" w i l l be d e t e r m i n e d t h r o u g h s t a t i s t i c a l
an aside,
and purely
for
for ill ustration,
judged
inference.
As
the p r e d i c t i v e p o w e r
a n d ow n p r i c e e l a s t i c i t i e s of the m o d e l d e t e r m i n e d best in
this
s tudy are
compared with bivariate
e s t i m a t e d by D u f f i e l d et al.
(1987)
forms of the mod e l s
an d D u f f i e l d
(1988).
S tud y O u t l i n e
Th e b a l a n c e
chapters.
of thi s p a p e r is p r e s e n t e d in four
C h a p t e r two p r o v i d e s
a l i t e r a t u r e r e v i e w of the
use of th e B o x - C o x m e t h o d of e s t i m a t i o n of n o n l i n e a r
regression parameters.
compares
first s ec tion of this c h a p t e r
t wo a p p r o a c h e s u s e d to d e t e r m i n e the
of a model.
This
u s e d to e s t i m a t e
model .
The
This
is
specification
f o l l o w e d by a r e v i e w of the m e t h o d s
the p a r a m e t e r s
of a B o x — Cox r e g r e s s i o n
s e c t i o n clos e s w i t h a su mmary of the r easons
m a x i m u m l i k e l i h o o d e s t i m a t i o n was c h o s e n to e s t i m a t e the
Box-Cox
demand
r e g r e s s i o n of th e
func tio n.
Th e n e x t
first-stage
str e a m fi sheries
se c t i o n sum m a r i z e s the m e t h o d s
u s e d to d i s c r i m i n a t e n e s t e d an d n o n - n e s t e d rival m o d e l s
purposes
of d e t e r m i n i n g m o d e l
inference.
several
The
last
for
s p e c i f i c a t i o n by s t a t i s t i c a l
s e c t i o n of c h a p t e r tw o s u m marizes
d i a g n o s t i c m e a s u r e s u s e d in c o n j u n c t i o n w i t h the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
model
s p e c i f i c a t i o n a p p r o a c h e m p l o y e d in this paper.
Chapter three
a n d the m e t h o d s
statistics
functions
s u m m a r i z e s the dat a u s e d in this
u s e d to g a t h e r this
of the v a r i a b l e s
data.
study
Descriptive
s p e c i f i e d in each of the
e x a m i n e d are a l s o pro vided.
The r e s u l t s of a p p l y i n g th e B o x - C o x m e t h o d of model
d i s c r i m i n a t i o n to th e DFWP
s trea ms
p r o v i d e d in c h a p t e r
I n c l u d e d in this
s u m m a r y of m o d e l s
n o t e d above.
four.
fisheries data is
c hapter is a
e s t i m a t e d in the two p r e v i o u s
Next,
studies
five g e n e r a l B o x — Cox r e g r e s s i o n mo de l s
are e s t i m a t e d s u c h t h a t e ight p l a u s i b l e
attained using equation
(2)
as a basis.
forms c o u l d be
T h es e m o d e l s
are
t h e n d i s c r i m i n a t e d to d e t e r m i n e w h i c h form c o n tains the
parameters
w h i c h ar e m o s t
lik e l y to d e s c r i b e the p o p u l a t i o n
fro m w h i c h the d a t a are drawn.
the b i v a r i a t e m o d e l s
This m o d e l
is c o m p a r e d w i t h
s u m m a r i z e d at the outset of this
chapter.
The s t u d y
in t h i s
analysis
is s u m m a r i z e d in ch a p t e r
five.
Included
c h a p t e r is a s u m m a r y of the l i m i t a t i o n s of the
and suggestions
for fu tur e
research.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CH APTER 2
L I T E R A T U R E R E V I E W OF M O D E L S P E C I F I C A T I O N AND
THE B O X - C O X T R A N S F O R M A T I O N
This
literature
chapter reviews
the e c o n o m e t r i c an d st atis tical
regarding application
f a m i l y of p o w e r t r a n s f o r m a t i o n s
functional
form.
First,
of the ge n e r a l Box -Co x
to m o del
general
s p e c i f i c a t i o n of
e l e ments of m o del
s p e c i f i c a t i o n w i t h i n the c o n t e x t of o r d inary least
(OLS)
are
revi ewed.
specification
Second,
are revi ewed.
tw o a p p r o a c h e s to model
I n c l u d e d in this
s u m m a r y of th e a p p r o a c h u s e d in thi s
tests
u s e d to
Montana
and extended
to estimate
Next,
(BCE)
regression
a s u m m a r y of t h e
econometric
function.
for m of the
Third,
the b a s i c
B o x - C o x t r a n s f o r m a t i o n and me t h o d s used
a Box-Cox
Co x r e g r e s s i o n
demand
re view is a
study an d a list of the
select the m o s t a p p r o p r i a t e
stream fisheries
squares
f u n ction are reviewed.
f o u r m e t h o d s u s e d to esti mat e a Box-
f u n c t i o n a n d a r e v i e w of av ai l a b l e
software
is p r o v i d e d .
This is f o l l o w e d by a
Th e B o x - C o x e x t e n d e d r e g r e s s i o n e q u a t i o n
( a t t r i b u t e d to S a v i n a n d W h i t e (1978) by Seaks and La yso n
(1983)) i n c l u d e s v a r i a t i o n s of e q u a t i o n (2) in whic h B o x - C o x
t r a n s f o r m a t i o n s ar e a p p l i e d t o i n d e p e n d e n t v a r i a b l e s in
a d d i t i o n to th e d e p e n d e n t v a r i a b l e .
(See a l s o Box and Cox
(1964).)
19
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20
s u m m a r y of th e l i k e l i h o o d r a t i o test
nested models
a n d the
j— t e s t
for d i s c r i m i n a t i n g
for d i s c r i m i n a t i n g n o n — n e s t e d
models.
Next,
d i a g n o s t i c t e s t s u s e d in this
reviewed
f o l l o w e d by a s u m m a r y c o n c l u s i o n s
study are
re a c h e d in this
chapter.
General
Elements
of M o d e l
S p e c i f i c a t i o n Fo r OL S E s t i m a t e d
Models
A l t h o u g h this
functional
s t udy
(1986)
f o c u s e d on e s t i m a t i n g the
form of a f i r s t - s t a g e T C M d e m a n d f u n ction u sing
th e B o x - C o x t r a n s f o r m a t i o n ,
th e m o r e
is
the p r o b l e m is c o u c h e d w i t h i n
g e n e r a l p r o b l e m of m o d e l
describes
specification
speci fic atio n.
for mode l s
using
us e of an e s t i m a t i o n t e c h n i q u e w h i c h satisf ies
assumptions.
error term
Application
of OL S esti m a t o r s
(residuals or d i s t u r b a n c e term)
Kme n t a
(OLS)
as the
its g eneral
assume tha t the
is n o r m a l l y an d
i n d e p e n d e n t l y d i s t r i b u t e d w i t h a m e a n of zero and a c o n stant
variance
errors
(e ~ N ( 0 , ) ) ;
is
zero;
th e
c o v a r i a n c e b e t w e e n any two
ea c h of th e e x p l a n a t o r y v a r i a b l e s
measured without
error,
are n o n s t o c h a s t i c ,
are
a n d no "exact"
l i n e a r r e l a t i o n s h i p e x i s t s b e t w e e n any two of these
variables;
a n d the n u m b e r of o b s e r v a t i o n s
of estimable
these
o f th e
coefficients
assumptions
re s u l t s
in th e model.
e xceeds the n u m b e r
S a t i s f a c t i o n of
in b e s t u n b i a s e d e s t i m a t o r s
regression parameters.
ha s m i n i m u m v a r i a n c e a m o n g all
A BUE e s t i m a t o r
(BUE)
is one w h i c h
l i n e a r u n b i a s e d estimat ors .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
A n u n b i a s e d e s t i m a t o r is one in w h i c h t he e x p e c t e d v alue of
the
estimator equals
Kmenta
the t r u e v a l u e
(1986),
narrows
specification by noting that
from failures
to
model,
and correctly
enters
the model.
s p e c i f y th e
complete model
s p e c i f i c a t i o n err ors
as i n d i c a t e d in the literature,
i n c l u d e the above lis ted
of OLS e s t i m ators.
constant error variance
F o r instance,
(Kmenta
(1986)).
form of the
Also,
m a y be t he r e s u l t of a m i s s p e c i f i e d m o d e l
As no ted ,
Du ffield,
a non­
( h e t e r o s c e d a s t i c i t y ) may be the
specified function
equation
form of the
s p e c i f y the w ay in w h i c h the e r r o r term
However,
of an p o o r l y
result
rel evan t to the
"correct" mathematical
specification must
basic assumptions
regression
the d e f i n i t i o n of
to i n c l u d e o n l y the v a r i a b l e s
model,
result
of the parameter.
Loomis,
a n d Broo k s
data ou tli ers
(Kennedy
(1987)
(1992)).
found that the
f i r s t — s t a g e d e m a n d f u n c t i o n e s t i m a t e d for the en tir e
fisheries
this
s a m p l e e x h i b i t e d h e t e r o s c e d a s t i c errors.
a p r i o r i know l e d g e ,
analysis
w o u l d be i n c o m p l e t e w i t h o u t t e s t s
Howe ver ,
th e
for an d p o s s i b l e
focu s of th is
s p e c i f i c a t i o n o f th e m a t h e m a t i c a l
demand
fun ction.
and corrections
T h e refore,
With
of form s p e c i f i c a t i o n
c o r r e c t i o n of a n y of t h e b a s i c a s s u m p t i o n s
estimator.
str e a m
of the OLS
s tudy is l i m i t e d to
fo rm of the
analysis
fir s t - s t a g e
of p o s s i b l e test s of
for h e t e r o s c e d a s t i c e r r o r s w i t h i n the B o x -
Cox transformation
f r a m e w o r k are lef t
for f u r t h e r a n a l y s i s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
on the
stream fisheries data b a s e .
Approaches
to M o d e l
Kennedy
Specification
(1992)”
approaches
to m o d e l
Regression
(AER)",
2)
et a l . (1987)
the M o n t a n a
best,
"Test,
"proceeding
and Duffield's
specific
and 3)
the a n a l y s i s by
(1988)
of the form of
forms are d e t e r m i n e d by,
general model"
residuals
to d e t e r m i n e
' t e s t i n g up'
if the a s s u m p t i o n s
researchers
(1992)).
to a
The A E R
of a p p l y i n g d i a g n o s t i c te sts
of an e s t i m a t e d m o d e l
If the a p r i o r i m o d e l
assumptions,
a nd
(Kennedy
is m a r k e d b y a p r o c e s s
b e e n met.
(TTT )",
c l a s s i f i e d as an A E R approach.
from a simple model
specific more
to th e
Tes t
"Average Economic
s t r e a m f i s h e r i e s d e m a n d f u n c t i o n w o u l d appe a r
approach
approach
Test,
1)
Collectively,
a l t h o u g h not p e r f ectly,
In t h i s
st at e of the art
s p e c i f i c a t i o n as
" F r a g i l i t y Anal y s i s ' ” ®.
Duffield
classifies
k n own to be c orrect
of an OLS e s t i m a t o r have
fails to sa t i s f y the se
u s i n g the A E R a p p r o a c h t u r n first
S e e G a u d r y a n d D a g e n a i s (1979), G r e e n e (1990),
L a h i r i a n d E g y (1981) for e c o n o m e t r i c t r e a t m e n t of
s i m u l t a n e o u s t e s t i n g a n d c o r r e c t i n g for h e t e r o s c e d a s t i c
e r r o r s w i t h i n th e B o x - C o x t r a n s f o r m a t i o n framework.
Also
se e V a u g h n , Ru ssell, a n d H a z i l l a (1983) for an a p p l i c a t i o n
o f th e L a h i r i an d Eg y (1981) m e t h o d to a tra v e l cost model.
Chapter
This
5.
s e c t i o n rel i e s h e a v i l y on K e n n e d y
(1992),
"Fragility Analysis" incorporates a Bayesian method
t o d e t e r m i n e if e s t i m a t e d p a r a m e t e r s of a m o d e l fall w i t h i n
a n a c c e p t a b l e range.
S ince this stu dy u s e d class i c a l
s t a t i s t i c a l me thods, a r e v i e w an d c o m p a r i s o n of this
a p p r o a c h t o t h e A E R a n d TT T a p p r o a c h e s is omitted.
18
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23
to m o r e
errors
OLS
sophisticated estimation techniques
in the residuals.
to r esol ve
F a i l u r e of such m e t h o d s
to sat isf y
a s s u m p t i o n s t h e n leads to t ests of s p e c i f i c a t i o n in
w h i c h h i g h R2
a nd s i g n i f i c a n t t - r a t i o s
are u s e d to select
the b e s t m o d e l .
In contrast,
researchers
begin with a model more general
The g e n e r a l m o d e l
t ests
is t h e n
u s i n g th e TTT a p p r o a c h
than t h a t b e l i e v e d correct.
s u b j e c t e d to s ever al d i a g n o s t i c
r e g a r d i n g th e a s s u m p t i o n s of OLS.
reveal u n s a t i s f a c t o r y
If such tests
a t t a i n m e n t of OL S ass umptions,
that the m o d e l
analyst
concludes
is t h e n
r e s p e c i f i e d a n d the t e s t i n g p r o c e d u r e is repeated.
The o v e r a l l p r o c e s s
considered
That
is,
power,
is m i s s p e c i f i e d .
the
is r e p e a t e d unti l the m o d e l
" c o n g r u e n t w i t h the e v i d e n c e "
The m o del
is
(Kennedy
(1992)).
t h e m o d e l ha s a l o g i c a l l y p l a u s i b l e p r e d i c t i v e
is t h e o r e t i c a l l y an d p a r a m e t r i c a l y consis tent ,
residuals
ar e c o m p l e t e l y
n o i s e ) , and the model
specifica tion s.^ ®
random
The TTT a p p r o a c h is,
TTT approaches
(1992)
they exhibit white
s u c c e s s f u l l y e n c o m p a s s e s all
c o n s i d e r e d a " t e s t i n g down"
Kennedy
(i.e.,
the
rival
therefore,
approach.
n ote s that n e i t h e r th e A E R n o r the
ar e w i t h o u t
criticism.
Some c r i t i c i s m s
p a r a m e t r i c a l y c o n s i s t e n t refe rs to a m o d e l s a b i l i t y
t o p r e d i c t o b s e r v a t i o n s not u s e d to s p e c i f y the model.
Suc h
c o n s i s t e n c y c o u l d be a t t a i n e d u s i n g p o s t - s a m p l e p r e d i c t i o n
t e s t s in w h i c h a p o r t i o n of the data is r e m o v e d fro m the
i n i t i a l s p e c i f i c a t i o n p r o c e s s and u s e d to v a l i d a t e the
s p e c i f i e d model.
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24
d i r e c t e d s p e c i f i c a l l y to the A E R a p p r o a c h are
1)
the use
econometeric analysis
to r e i n f o r c e t h e o r e t i c a l
relationships;
fa i l u r e to con clude the e x i s t e n c e of
2)
t he
s p e c i f i c a t i o n e r r o r s w h e n d i a g n o s t i c te sts
d i s s a t i s f a c t i o n of O L S assumptio ns;
(i.e.,
data m i n i n g ) .
Critics
r e l i a n c e on a d j u s t e d Rz
unique
features
c h a n c e to b e c o m e
(see M a y e r
suggests
a determining
1980)
for instance,
that
sele c t i o n ap p e a l s to the
of t he a n a l y z e d data,
(1975 a n d
this
and 3) m a x i m i z i n g Rz
suggest,
for m odel
sh ow
t h e r e b y a l l o w i n g me re
factor in m o d e l
in K ennedy
(1992)).
specification
Kennedy
l a t t e r c r i t i c i s m supports the n e e d for p o s t ­
s am p l e p r e d i c t i o n t e s t s .
B o t h the A E R a n d TTT app roaches are c r i t i c i z e d for
t h e i r l a c k of a w e l l - d e f i n e d str uctu ral
specification.
Furth er,
a p p r o a c h to model
b o t h ap proaches are c r i t i c i z e d for
t h e e x p e c t e d o c c u r r e n c e of type I errors,
the m u l t i t u d e
a ppr o a c h .
to the
loss
a result due to
of d i a g n o s t i c te sts p e r f o r m e d u n d e r each
U n d e r the TT T approach,
of d e g r e e s
this re sult is also due
of f r e e d o m for g eneral m o d e l
specifications .
Kennedy
model
(1992)
specification:
1)
suggests the foll o w i n g p r i n c i p l e s
for
e c o n o m i c theo r y s h o u l d serve as the
P o s t - s a m p l e p r e d i c t i o n tests were o m i t t e d from this
study.
K e n n e d y (1992) su gges ts r e d ucing the cr itical
r e g i o n or th e p r o b a b i l i t y of a type I e rror to m i t i g a t e
r e s u l t s w h e n the TT T a p p r o a c h is used.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
su ch
25
basis
for m o d e l
residuals
not
specification;
" t e s t i n g up";
4)
etc.)
3)
tests
o m i t t e d v a r iables,
" t e s t i n g down"
functional
form,
functional
misspedfication
(e.g.,
tests
for
homo sced ast ici ty,
of e r r o n e o u s l y selec t i n g one
form);
tests
5)
should encompass
rival m odels;
u s e d to a r r i v e at th e
selected as correct
an d 7)
l i m i t ations of the
r e p o r t e d a l o n g w i t h the m ethods
s e l e c t e d th e model.
compare more
a TT T a p p r o a c h to m o d e l
B e c a u s e these
f a v o r a b l y w i t h the TTT a p p r o a c h t h a n
Kennedy
B a s e d on th is
heteroscedasticity
a large n u m b e r of
6) m o d e l s
s h o u l d be
th e A E R ap pro ach,
(e.g.,
s h o u l d be e m p l o y e d i n c l u d i n g p o s t ­
s a m p l e p r e d i c t i o n tests;
principles
is p r e f e r r e d to
s h o u l d be p e r f o r m e d s i m u l t a n e o u s l y to
misspecification over another
selected model
r e vealing
sh oul d point to
for m i s s p e c i f i c a t i o n
m i t i g a t e th e p o s s i b i l i t y
versus
d i a g n o s t i c tests
s a t i s f y i n g OL S a s s u m p t i o n s
s p e c i f i c a t i o n errors;
outliers,
2)
appears
to lean t o w a r d s u g g e s t i n g
specification.^^
review,
an a p p r o a c h si m i l a r to th e TTT
a p p r o a c h a p p e a r s to h a v e th e g r e a t e s t m e rit in model
s p e c i f i c a t i o n analy sis.
comments
are
sufficiently
to r e c o g n i z e t h a t
model
Kennedy's
lit e r a t u r e review and
c o m p e l l i n g to use a TTT a p p r o a c h
u n s a t i s f i e d OLS a s s u m p t i o n s m a y m e a n the
is m i s s p e c i f i e d .
Furth er,
since this a p p r o a c h b e g i n s
B o t h K e n n e d y (1992) a n d H a r v e y (1990) note th at the
p r o c e s s of m o d e l s e l e c t i o n is c o m p l e x and that no g e n e r a l l y
a c c e p t e d m e t h o d has b e e n a d o p t e d by e c o n o m e t r i c i a n s in
general.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
w i t h th e m o s t ge n e r a l
below,
down
f o r m of th e model,
is the t e s t a b l e h y p o t h e s i s
to a s p e c i f i c
to m o d e l
selection
form,
degree
in w h i c h th e
out liers,
study,
and tests
C o n s i s t e n t with the
for func tion al
h o m o s c e d asti city,
selected model
are r e v i e w e d in thi s
However,
study.
d i a g n o s t i c t ests
p a r a m e t e r consis t e n c y ,
as n ote d
it is the m o s t appl i c a b l e a p p r o a c h
for this
f ocus of th e paper,
of this
which,
encompasses
c h a p t e r for use in this
form,
and the
rival models
study.
due to the n arrow focus on functional form,
simultan eous tests of m i ss p ec i fi c at i on such as
h et e ro s ce d as t ic i ty versus
functional
form are omitted from
the analysis in this s t u d y .
The B o x - C o x F a m i l y of P o w e r T r a n s f o r m a t i o n s
As n o t e d in c h a p t e r one,
sufficient
a priori theoretical
mathematical
Kmenta
becomes
guidance
for compl ete
f o r m s p e c i f i c a t i o n of the d e m a n d function.
(1986)
suggests that
in such c i r c u m s t a n c e s the form
a testable hypothesis
t e s t i n g the
d e m a n d the o r y lacks
a n d lists sever al me t h o d s
for
f o r m s p e c i f i c a t i o n of an e c o n o m e t r i c model.
of t h e m e t h o d s
T wo
s u g g e s t e d by K m e n t a to d e t e r m i n e fu nct io nal
f o r m u s e th e B o x - C o x
f a m i l y of p o w e r t r a n s f o r m a t i o n
d e s c r i b e d in e q u a t i o n s
of the transformation
(1)
and
(2)
in c h a p t e r one.
On e use
is to t e s t a l i n e a r fo rm against an
See the c o m m e n t s r e g a r d i n g l i m i t a t i o n s of the
a n a l y s i s p r e s e n t e d h e r e i n r e g a r d i n g s i m u l t a n e o u s tests
m i s s p e d f l oa t ions in c h a p t e r five,
infra.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
27
alternative
(2),
in w h i c h all Ày a n d Xp,
in t erms of eq uation
are e s t i m a t e d w i t h e q u a l v a l u e
(1986)
a nd Z a r e m b k a
(1968)).
Box-Cox transformed models
l i k e l i h o o d r a tio t e s t
d o u b l e - l o g model,
(see,
In t h i s
e.g.,
case,
ar e d i s c r i m i n a t e d
(described b e l o w ) .
in w h i c h
Km ent a
the line ar and
using a
Similarly,
a
= 0, m a y a l s o be test e d
a g a i n s t a f o r m in w h i c h X^ a n d X^ are e s t i m a t e d with equal
value.
A
s e c o n d us e of the B o x - C o x t r a n s f o r m a t i o n is
s u g g e s t e d w h e n the m o d e l ' s
terms
of e q u a t i o n
The m o d e l ' s
(2)
general
f o r m is questioned.
Xy a n d X^ ar e v a r i e d i nde pe nden tly .
fo rm in th is a p p r o a c h is t e r m e d "fle xibl e" an d
is u s e d to t e s t a p r i o r i
specifications
d e t e r m i n e d by the data.
Use of thi s
n u m b e r of f o r m r e s t r i c t i o n s
hypothesis
a g a i n s t the
specifications
a p p r o a c h allows any
flexible
of th e
f o r m d e t e r m i n e d by the data.
Duffield
Furthermore,
model with a constant
(aver age
(1988)
found that
fo rm of the d e m a n d function
f a i l e d to a c c u r a t e l y d e s c r i b e v a r i a t i o n
for trips.
a g a i n s t a form
on Xv a nd Id as t e s t a b l e
As n o t e d in c h a p t e r one,
several
In
it wa s
in p e r cap ita d e m a n d
f ound that a d o u b l e - l o g
a d d e d to the p r i c e - p r o x y v a r i a b l e
r o u n d - t r i p distan ce)
p r e d i c t i v e p o w e r of th e model.
s i g n i f i c a n t l y e n h a n c e d the
G i v e n the res u l t s of the
D u f f i e l d (1992) i m p r o v e d this a p p r o a c h by r e p l a c i n g
r o u n d - t r i p d i s t a n c e w i t h p r e d i c t e d t otal t r i p co sts b a s e d on
f u n c t i o n s for r e s i d e n t a n d n o n - r e s i d e n t a n g l e r s that were
d e v e l o p e d by Duff iel d, Loomis, a nd B r o o k s (1987) to e s t i m a t e
v a r i a b l e t r a v e l costs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
s p e c i f i c a t i o n s ex amined,
Duffield
(1988)
a f a m i l y of p o w e r t r a n s f o r m a t i o n s
variable.
stream
Thus,
the g e n e r a l
suggested searching
on th e p r i c e - p r o x y
f o r m of th e f i r s t -stage M o n t a n a
f i s h e r i e s d e m a n d f u n c t i o n b e c o m e s the test abl e
hypothesis
in t h i s
McConnell's
(1985)
study.
This
is c o n s i s t e n t wit h
suggestion that
functional
form sh oul d be
t e s t e d w h e n u s i n g the T C M fr amework.
Purpose
Box-Cox
and Assumptions
regression.
initially
The B o x - C o x t r a n s f o r m a t i o n was
i n t e n d e d for us e on th e d e p e n d e n t v a r i a b l e to
a c h i e v e a sim p l e m o d e l
( h o m o s c e d a s t i c i t y ),
(Box a n d Cox
subsequent
str ucture,
an d n o r m a l
(1964)).
c o n s t a n t e rror v a r i a n c e
d i s t r i b u t i o n of the errors
However,
applications
transformation
variables,
of the B a s i c and E x t e n d e d
in the initi al and
it has b e e n
can a l s o be a p p l i e d to the ind e p e n d e n t
resulting
in the e x t e n d e d B o x — Cox model.
e x t e n d e d B o x — C o x r e g r e s s i o n allo w s
determining functional
transformations
the data.
The
s u g g e s t e d that the
The
f l e x i b i l i t y in
form by e s t i m a t i o n of the
of th e v a r i a b l e s
in t he m o d e l a c c o r d i n g to
c o n t i n u o u s n a t u r e of the B o x - C o x
transformation allows
linear and nonlinear
one to e s t i m a t e b o t h i n t r i n s i c a l l y
forms.
E co n om e tr i c models are classified as either
i n t r i n s i c a l l y linear or nonlinear.
An intrinsically linear
f un ction is linear in the estimated parameters but nonlinear
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
in th e v a r iab les.
Functions
s p e c i f i e d w i t h pol yno mial s,
i n t e r a c t i o n terms,
a n d as e i t h e r m u l t i p l i c a t i v e
g e n e r a l l y q u a l i f y as i n t r i n s i c a l l y
or a d d itive
lin e a r forms.
Such forms
are a l s o m a r k e d by the e r r o r t e r m s p e c i f i e d as additive.
Alternatively,
nonlinear
an i n t r i n s i c a l l y n o n l i n e a r funct i o n is
in th e v a r i a b l e s a n d e s t i m a t e d par ameters.
W h ile
su ch f u n c t i o n s m a y a l s o be m u l t i p l i c a t i v e or additive,
they
are m a r k e d by an e r r o r t e r m s p e c i f i e d as m u l t i p l i c a t i v e
the m o d e l .
in
OL S can be u s e d to e s t i m a t e all i n t r i n s i c a l l y
l i n e a r a n d some n o n l i n e a r forms by t r a n s f o r m i n g the
variables
int o l i n e a r forms
(although n o n l i n e a r forms are
generally
estimated using maximum l ikelihood).
However,
w h e n the e r r o r t e r m is s p e c i f i e d as m u l t i p l i c a t i v e
in
n o n l i n e a r models,
leads to
transformations
of the v a r i a b l e s
a n o n l i n e a r d i s t r i b u t i o n of the e r r o r t e r m a n d OLS
e s t i m a t i o n of th e
l eas t
squares
As
Co x
estimation
n o t e d below,
regression
Ho wev er,
f u n c t i o n is a p p r o p r i a t e l y t e r m e d n o n l i n e a r
(see,
e.g.,
such as that s p e c i f i e d in e q u a t i o n
f i x e d in t h i s proces s.
specification,
th e
incrementaly.
However,
a Box-
(2).
are c o n s i d e r e d
a
priori
In an A E R a p p r o a c h to mode l
f o r m of the m o d e l
is d e t e r m i n e d
the B o x - C o x m e t h o d al low s
in d e t e r m i n i n g the
f o r m of the
h o w e a c h v a r i a b l e ente r s the model,
nonlinearly.
(1986)).
OLS c o u l d be u s e d to e s t i m a t e
the B o x - C o x t r a n s f o r m a t i o n s
flexibility
Kmenta
The i n t u i t i v e
appe a l
i.e.,
m o d e l in t e rms of
l i n e a r l y or
of th e B o x - C o x m e t h o d is
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30
that greater
f l e x i b i l i t y is a l l o w e d to d e t e r m i n e the
the m o d e l by a l l o w i n g v a r i a b l e
w i t h th e data an d not
down
f r o m th e m o s t
approach
linearization
fo rm of
in c o n f o r m a t i o n
a p r i o r i e x pectation.
Thus,
testing
g eneral to a s p e c i f i c form u s i n g a TT T
is m a d e p o s s i b l e w i t h the e x t e n d e d B o x — Cox model.
M e t h o d s U s e d to E s t i m a t e a B o x - C o x R e g r e s s i o n .
Spitzer
(1982a a n d 1982b)
lists
estimating a Box-Cox regression
four a p p r o a c h e s to
including maximum likelihood
a n d c o n c e n t r a t e d m a x i m u m likelihood,
squares,
a n d i t e r a t i v e OLS.
This
n o n l i n e a r least
r e v i e w shows that m a x i m u m
l i k e l i h o o d or c o n c e n t r a t e d m a x i m u m l i k e l i h o o d e s t i m a t i o n
provides
th e m o s t
a c c u r a t e a nd le ast c o s t l y e s t i m a t e s
parameters
in a B o x - C o x r e g r e s s i o n e q u a t i o n
hypothesis
testing,
a s s u m i n g the s o f t w a r e
i l l u s t r a t i v e pu rpo ses ,
of e q u a t i o n
summarized
(2).
for p u r p o s e s of
is avail abl e.
E a c h of the abov e l i s t e d m e t h o d s
(MLE)
ca n be d e s c r i b e d
Regression coefficients
e s t i m a t e d u s i n g OLS ar e e s t i m a t e d by m i n i m i z i n g the
squared errors
Further,
determine
th e
of the
assumptions
regression
function
sum of
from t h e i r
to a t t a i n b e s t u n b i a s e d e s t i m a t o r s a n d
confidence
residuals
contr a s t ,
are
in turn.
by c o m p a r i s o n w i t h OL S estima tio n.
mean.
Fo r
thes e m e t h o d s are p r e s e n t e d in t e rms
Maximum likelihood estimation
the
of the
intervals
for th e e s t i m a t e d p a r a m e t e r s ,
are a s s u m e d to s a t i s f y the c l a s s i c a l OL S
s u m m a r i z e d at the o u t s e t of th is
parameters
chapter.
In
e s t i m a t e d u s i n g M L E are t h o s e w h i c h are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
31
m o r e l i k e l y to d e s c r i b e th e p o p u l a t i o n
from w h i c h the da ta
are d r a w n t h e n by an y o t h e r set of p a r a m e t e r s d e s c r i b i n g a
different population
variables
are not
transformations
identical
(1992)).
When the regression
s u b j e c t to t r a n s f o r m a t i o n s
ar e a s s u m e d
fixed,
estimated regression
Kennedy
steps are
{Kennedy
(1992)
or
ML E an d OL S p r o d u c e
co eff icients.
s u m m a r i z e s ML E in four steps.
Thes e
s u m m a r i z e d a s s u m i n g the B o x - C o x r e g r e s s i o n
function given
in e q u a t i o n
(2).
First,
the d i s t r i b u t i o n of
the e r r o r t e r m in th e
r e g r e s s i o n e q u a t i o n is specif ied .
common
a n d that u s e d in this
specification,
the e r r o r t e rms
id e n t i c a l l y ,
Second,
is that
are a s s u m e d to be independe ntly ,
and normally distributed
(e - N ( 0 , ) ) .
th e r e l a t i o n s h i p of th e e r r o r term s are s p e c i f i e d in
t e r m s of th e v a r i a b l e s
g i v e n th e a b o v e
of f(e^)
distribution
log of t h e
in th e
r e g r e s s i o n function.
s p e c i f i c a t i o n of the e r r o r terms,
their independence,
product
study,
On e
the
sa mpl e or the joint p r o b a b i l i t y
f u n c t i o n of the e r r o r terms.
of t he l i k e l i h o o d
namely
l i k e l i h o o d functi on e q u a l s t he
a c r o s s the
likelihood
Third,
S i nce t h e n a t u r a l
f u n c t i o n is a m o n o t o n i e t r a n s f o r m a t i o n
functio n,
the
log-likelihood function
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is
32
s t a t e d as follow s :
(3)
InL =
2
ln{ 2na^)-
20^
N
T
(
- P^-
)
N
+ (A y - 1)
^
In V j
i=l
The
last t e r m in e q u a t i o n
ja c o b i a n d e t e r m i n a n t
accounts
for p o s s i b l e
distributions
(3),
(Xy - 1)
i=i In V^, is the
of the t r a n s f o r m a t i o n .
differences
This t e r m
in the p r o b a b i l i t y
(probability density
functions)
of the e r r o r
t e r m a n d th e d e p e n d e n t v a r i a b l e w h e n the l a t t e r is
transformed.
terms
The l o g — l i k e l i h o o d f u n c t i o n is s p e c i f i e d in
of the unknown,
terms.
However,
variable
a s s u m e d d i s t r i b u t i o n of the e r ror
k n o w n o b s e r v a t i o n s of the d e p e n d e n t
are u s e d to e s t i m a t e
function.
A b s e n t the
j a c o b i a n d e terminant,
likelihood function assumes
terms
the p a r a m e t e r s of the
the d i s t r i b u t i o n of the e r r o r
an d th e d e p e n d e n t v a r i a b l e are the
w h e n th e d e p e n d e n t v a r i a b l e
1) , t h u s th e
(see e q u a t i o n
Differences
d i s t r i b u t i o n of th e e r r o r t e r m s
will
occur
general
in this
This
(i.e.,
is t r u e
Xy =
case w o u l d b e
zero
in the p r o b a b i l i t y
a n d th e d e p e n d e n t v a r i a b l e s
for all t r a n s f o r m a t i o n s
See K e n n e d y (1992),
d e v e l o p m e n t of t h i s
same.
is not t r a n s f o r m e d
jacobian determinant
(3)).
the log-
on th e d e p e n d e n t
c h a p t e r 2 for d e t a i l s of the
function .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
v a r i a b l e s w h e n X^r ^ 1.
i n c l u d e d in the
differences
Thu s
the
jacobian determinant
l o g - l i k e l i h o o d f u n c t i o n to adjust
in th e p r o b a b i l i t y
distributions
d e p e n d e n t v a r i a b l e a n d t he e r r o r t e r m s
and Kmenta
(1986)
Th e
Pi,
X^,
fou r t h
{see K e n n e d y
in e q u a t i o n
(3) .
Estimates
of these
value
of the
(first a n d s e c o n d deriv a t i v e s )
the n e g a t i v e
of the
c o v a r i a n c e matrix.
However ,
comput a t i o n ,
acceptable
A
Box-Cox
res pec tiv ely.
the v a r i a n c e -
du e to th e c o m p l e x i t i e s
n e g a t i v e of th e i n v e r s e of the
produce
of e q u a t i o n
i n v e r s e of the e x p e c t e d
second order conditions yields
i n v o l v e d in th is
(3) w i t h
first an d sec o n d
to be e q ual to zero a n d n e g a t i v e de finite,
Theoretically,
po,
Necessary and sufficient
for a local m a x i m u m r e q u i r e s the
order conditions
(1992)
step is e s t i m a t i o n of th e p a r a m e t e r s
r e s p e c t to e a c h p a r a m e t e r .
(3)
of the
can be f o u n d by m a x i m i z i n g e q u a t i o n
conditions
for the
for f u r t h e r e x p l a n a t i o n s ) .
X-D, a n d
parameters
is
Spitzer
(1982a)
maintains
the
second order conditions
results.
s e c o n d a p p r o a c h to e s t i m a t i n g the p a r a m e t e r s of a
regression
estimate of
f u n c t i o n e x p l o i t s th e
, g i v e n in e q u a t i o n
(4),
fact that the
r educes e q u a t i o n
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(3)
34
to e q u a t i o n
(5)
(see G r e e n e
(1990)
and Spitzer
(1982a) .
N
(4)
e^
^i=l
(5)
InL^ =
(ln(27t)
+ 1) -
2
Th at is,
Inû^ + (1_-1)
2
1^1
<J^ is c o n c e n t r a t e d out of the l o g - l i k e l i h o o d
fu ncti on.
Spitzer
order conditions
identical
(1982a)
shows t h a t the firs t an d s e c o n d
for m a x i m i z a t i o n of e q u a t i o n
to t h ose p r o d u c e d
In a t h i r d a p p r o a c h
by e q u a t i o n
(5) are
(3).
to e s t i m a t i n g e q u a t i o n (2),
i n i t i a l l y a t t r i b u t e d to Z a r e m b k a
by t h e
N
V Iny.
(1968) , d a t a are r e s c a l e d
i n v e r s e of the g e o m e t r i c m e a n of the d e p e n d e n t
variable
r a i s e d to the p o w e r of
that the
resulting concentrated log— likelihood function
reduces
the
Optimal
values
Spitzer
(1982a)
shows
e s t i m a t i o n p r o b l e m to n o n - l i n e a r l e ast squares.
of the e s t i m a t e d r e g r e s s i o n p a r a m e t e r s are
o b t a i n e d b y m i n i m i z i n g t h e s t a n d a r d e r r o r of the e s t i m a t e of
th e
s c a l e d model.
approaches,
condition
However,
the n e g a t i v e of
of the NLS
different
f r o m th e M L E
the i n v e r s e of th e s e c o n d
order
concentrated log-likelihood function
B a s e d on the l i t e r a t u r e r e g a r d i n g s i m u l t a n e o u s
t e s t s o f f u n c t i o n form a nd h e t e r o s c e d a s t i c i t y , the
s p e c i f i c a t i o n of
in e q u a t i o n (4) a s s u m e s th e e r r o r t e rms
a r e h o m o s c e d a s t i c (see, e.g., K m e n t a (1986) a n d G r e e n e
(1990)).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
does no t p r o v i d e the v a r i a n c e - c o v a r i a n c e m a t r i x d i r e c t l y due
to the
r e s c a l i n g of th e data.
Thus,
m a t r i x m u s t be e s t i m a t e d by m e a n s
estimated scaled coefficients
Spitzer
(1982a)
Pi, A.V, kg,
of the
the v a r i a n c e - c o v a r i a n c e
of c o n v e r t i n g the
to t h e i r o r i g i n a l
form
for the a p p r o p r i a t e ad justment) .
an d
are not a d i r e c t
(see
Thus,
p,,.
res u l t of m i n i m i z i n g
s c a l e d model.
E s t i m a t e s of th e r e g r e s s i o n p a r a m e t e r s ma y also be
m a d e u s i n g an i t e r a t i v e O L S / g r i d - s e a r c h method.
to S p i t z e r
Po,
Pi,
(1982a)
a n d 02
a ser ies of r e g r e s s i o n s
u s i n g the
t h i r d a p p r o a c h a b ove
(2),
each
values of
scaled model
are e s t i m a t e d for
d e s c r i b e d for the
is one app roach.
In terms of e q u a t i o n
r e g r e s s i o n w o u l d d i f f e r a c c o r d i n g to e a c h of the
and
a s s u m e d for e a c h estim atio n.
c o m b i n a t i o n of Xy a n d X q w h i c h m i n i m i z e s the
squared errors
for the
g r i d of t h e s e values,
s c a l e d model,
yields
sum of the
f ound by s c a n n i n g a
However,
since the data
s c a l e d a n d Xy a nd X^ are a s s u m e d f i x e d for the
regression
erro rs,
more
The
the same result s as if any of
t h e a b o v e t h r e e m e t h o d s w e r e used.
are
According
f u n c t i o n m i n i m i z i n g th e
th e e s t i m a t e d c o e f f i c i e n t s m u s t be
importantly,
adjusted.
Spitzer
standard errors
downward,
errors
sum of the
s quared
r e s c a l e d and,
the v a r i a n c e - c o v a r i a n c e m a t r i x m u s t be
(1984)
and Kmenta
(1986)
no te that the
of th e e s t i m a t e d c o e f f i c i e n t s will be b i a s e d
yielding inflated t-ratios
in h y p o t h e s i s testin g.
Thus,
r e s u l t i n g in p o s s i b l e
s i m i l a r to th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
scaled-
36
model
a p p r o a c h s u m m a r i z e d above,
the v a r i a n c e - c o v a r i a n c e
m a t r i x m u s t be a d j u s t e d to v a l i d a t e h y p o t h e s i s
Spitzer
(1982a)
testing
(see
for details).
T h e r e ar e
s everal a d v a n t a g e s
to u s i n g the ML E
a p p r o a c h e s to e s t i m a t i n g the B o x - C o x
regression parameters
o v e r th e a l t e r n a t i v e m e t h o d s
l i s t e d here.
there
s o f t w a r e to c o m p u t e a v a l i d
is c u r r e n t l y a v a i l a b l e
variance-covariance matrix directly
conditions
Foremost
from the
is that
second order
of the l o g - l i k e l i h o o d f u n c t i o n s w i t h o u t
adjustment.^’
attractive
Second,
Kennedy
large-sample
(1992)
note s ML E has
a s y m p t o t i c pr o p e r t i e s .
se v e r a l
Namely,
m a x i m u m l i k e l i h o o d e s t i m a t o r s ar e a s y m p t o t i c a l l y un biased,
c ons istent,
an d ef ficient.
These features
are p a r t i c u l a r l y
a t t r a c t i v e w h e n c o m p a r e d to OLS in t h a t e a c h m a y be
c o n s i d e r e d c o m p a r a b l e to u n b i a s e d n e s s ,
respectively,
classical
a s s u m p t i o n s o f OL S
Additionally,
OLS
w h i c h are res u l t s
in t erms
coefficients,
these parameters
transformations
for large
samples.
of the m e c h a n i c s
of e s t i m a t i o n the
of o n l y the c o n s t a n t
whereas MLE produces
in a d d i t i o n to t he v a r i a n c e
(see e.g.,
a p p r o a c h does no t
a n d BUE,
of s a t i s f a c t i o n of the
is l i m i t e d to i n t e r n a l e s t i m a t i o n
and slope
eff ici ency,
Kmenta
(1986)).
require additional
estimates
of
an d B o x - C o x
Finally,
th e M L E
a d j u s t m e n t s to
v a r i a n c e - c o v a r i a n c e m a t r i x as does an NL S or OLS appr oach.
F o r i n s t a n c e S H A Z A M (White (1990)) a n d LIMDEP
(Greene (1990) ar e t w o e c o n o m e t r i c p r o g r a m s f e a t u r i n g
e s t i m a t i o n of B o x - C o x regres s i o n s .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
B a s e d on t h i s anal ysi s,
regression
software
is e s t i m a t e d u s i n g the
in w h i c h a B o x - C o x
full or c o n c e n t r a t e d log-
l i k e l i h o o d a p p r o a c h w o u l d p r o v i d e the best p a r a m e t e r
est im ates,
However,
t h e r e b y m i t i g a t i n g e r r o n e o u s h y p o t h e s i s testing.
s h o u l d an
l O L S / g r i d - s e a r c h a p p r o a c h be necessary,
it w o u l d be d e s i r a b l e
appropriate
n o t e d by
(Greene
these
(1991))
(1982a a n d 1982b).
LIMDEP,
Thus,
version
L I M D E P was cho s e n for the
the f i r s t - s t a g e d e m a n d function.
(1982b)
optimization
the
6.0
u s e s a l g o r i t h m s w h i c h s a t i s f i e s b o t h of
constraints.
Spitzer
s o f tware to i n c o r p o r a t e
a d j u s t m e n t s to the v a r i a n c e - c o v a r i a n c e m a t r i x as
Spitzer
to e s t i m a t e
for the
notes
software
Mo re over ,
th at c o m p u t e r p r o g r a m s u s i n g
a l g o r i t h m s w h i c h are l i m i t e d to us e of th e
first d e r i v a t i v e of the l o g - l i k e l i h o o d f u n c t i o n p r o v i d e
larger variances
o r d e r co n d i t i o n s .
t h a n do p r o g r a m s u s i n g first
Greene
(1991)
fir st a n d s e c o n d o r d e r c o n d i t i o n s
BOXCOX procedure
specifically
a nd s e c o n d
stat es th at
are u s e d for M L E
in the
in LIMDEP.
It a p p e a r s the l O L S / g r i d - s e a r c h m e t h o d p r o v i d e d in
L I M D E P m a k e s the a p p r o p r i a t e a d j u s t m e n t s to the v a r i a n c e c o v a r i a n c e m a t r i x as s u g g e s t e d by S p i t z e r ( 1 9 8 2 a ) . The
m a g n i t u d e s o f the e s t i m a t e d t - r a t i o s r e s u l t i n g from the
L I M D E P lOLS p r o c e d u r e a p p e a r s i m i l a r in m a g n i t u d e to t h o s e
r e s u l t i n g f r o m th e full ML E p r o c e d u r e (compare m o d e l s 3 . a
a n d 5, t a b l e 6, c h a p t e r f o u r ) .
S e v e r a l iss u e s r e g a r d i n g e s t i m a t i o n of a B o x - C o x
r e g r e s s i o n h a v e b e e n i d e n t ified.
F o r instance, the
d i s t r i b u t i o n of th e e r r o r t e r m s wi ll be t r u n c a t e d since the
d a t a for t r a n s f o r m e d v a r i a b l e s m u s t be p o s i t i v e (Smith 1975)
a n d th e e q u a t i o n m u s t c o n t a i n a c o n s t a n t t e r m (Sch l e s s e l m a n
(1971) in S p i t z e r (1982a)).
The last of t h e s e c o n c e r n s is
s a t i s f i e d in this s t u d y si nce the d e m a n d f u n c t i o n is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
Methods of Model discrimination
As
noted,
th e m o d e l
s p e c i f i c a t i o n d e t e r m i n e d be st
u s i n g the T TT a p p r o a c h s h o u l d e n c o m p a s s
specifications.
models
depends
alternative.
on w h e t h e r one m odel
That
(1986)) .
is,
d i s c r i m i n a t i o n t es ts
for n o n - n e s t e d m o d e l s
A model
F o r instance,
such
can be n e s t e d in the
of n e s t e d m o d e l s
(see e.g.,
is n e s t e d in a n o t h e r m o d e l
a r e s t r i c t e d case of the m o r e g eneral m o d e l
nested.
rival
The m e t h o d s u s e d to d i s c r i m i n a t e
m a y not be a p p l i c a b l e
Kmenta
all
if in e q u a t i o n
if it is
in w h i c h it is
(2)
= 1,
this
form w o u l d be c o n s i d e r e d n e s t e d in a form in w h i c h Xy
w e r e a l l o w e d to v a r y
summarizes
independently
.
section briefly
two a p p r o a c h e s u s e d to d i s c r i m i n a t e n e s t e d an d
non-nested models
in this
study:
and D a v i d s o n a n d M a c K i n n o n ' s
the l i k e l i h o o d r atio test
J— Test.
The Likelihood Ratio T e s t .
test
This
is u s e d to d i s c r i m i n a t e genera l
The l i k e l i h o o d rat io
(unrestricted)
r e g r e s s i o n s w i t h t h e i r r e s t r i c t e d cou nterparts.
of the t e s t
function
t he
The c o n c e p t
is t h a t th e v a l u e of the m a x i m i z e d l i k e l i h o o d
for th e r e s t r i c t e d a n d u n r e s t r i c t e d m o d e l s w i l l be
s i m i l a r if the r e s t r i c t e d m o d e l
Thus,
Box-Cox
is c orrect
(Kmenta
r a tio of the l i k e l i h o o d f u n c t i o n
for the
(1986)).
r e s t r i c t e d a n d u n r e s t r i c t e d m o d e l s w o u l d c o n v e r g e to a v a l u e
of one.
More
specifically,
the rati o of th e e s t i m a t e d
s p e c i f i e d w i t h a c o n s t a n t term.
not c o n s i d e r e d in t h i s study.
The r e m a i n i n g c o n c e r n s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are
39
regression variances
by K m e n t a
test
model
(1986),
is t h a t the
are
the null
restrictions
are not
for the
similar.
This r e s u l t s
is that the
If th e null hyp o t h e s i s
is
restricted and unrestricted models
are
in a l o g - 1 i k e l i h o o d r a t i o g iven in
(6):
LR=-2[Lik^)
In e q u a t i o n
likelihood
(6)
-LiX^)]~xl
L (Xp) a n d L(?i„)
e qu al
n u m b e r of p a r a m e t e r
respectively,
restrictions.
for example,
L(X,r)
a n d L (X^) w o u l d be
of th e m a x i m i z e d l o g - 1 i k e l i h o o d
a n d w h e n X^ a n d
independently,
were
(%2)
form of e q u a t i o n
r e p l a c e d w i t h val u e s
f u n c t i o n whe n
= ^ 0 = 1
As
n o t e d in e q u a t i o n
d i s t r i b u t i o n w i t h d e g r e e s of freedom,
T w o riva l m o d e l s
(6),
m,
the
equal
in th e r e s t r i c t e d model.
can be d i s c r i m i n a t e d u s i n g a log-
r atio t e s t p r o v i d i n g t h e r e s t r i c t e d m o d e l
special
(2),
is a s y m p t o t i c a l l y d i s t r i b u t e d as a c h i -
to t h e n u m b e r of r e s t r i c t i o n s
n e s t e d as
an d m equa l s the
a l l o w e d to v a r y e i t h e r t o g e t h e r or
respectively.
log-1ikelihood ratio
are
In terms of a test
b e t w e e n th e l i n e a r a n d m o s t g e n e r a l
likelihood
the v a l u e s of the log-
f u n c t i o n w h e n the B o x — Cox t r a n s f o r m a t i o n s
r e s t r i c t e d a nd u n r e s t r i c t e d ,
squared
in a l i k e l i h o o d ratio
of the m a x i m i z e d l o g - 1 i k e l i h o o d
functions
(6 )
as n o t e d
i m p o s e d on the u n r e s t r i c t e d
correct.
t h e n the v a l u e s
equation
hypothesis
Thus,
c o r r e c t a n d the a l t e r n a t e h y p o t h e s i s
restrictions
true,
w o u l d be n e a r l y equal.
case s of th e u n r e s t r i c t e d o r less
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
can be
40
restricted model
r a t i o test
(Greene
(1990)).
However,
the
ca n not be u s e d to d i s c r i m i n a t e two
w h i c h n e i t h e r can be n e s t e d in th e other.
likelihood
func t i o n s
in
Fo r instance,
a
s e m i — log can not be d i s c r i m i n a t e d f r o m a d o u b l e — log m odel
u s i n g a l i k e l i h o o d r a tio test.
t est s
such as the J-te s t
In su ch i n s t a n c e s
are appl icable ,
o the r
w h i c h is sum m arizes
below.
Test s
for N o n - n e s t e d M o d e l s .
rival m o d e l s are non-nested,
discrimination
test)
Kennedy
(1992)
traditional methods
such as an a n a l y s i s
or a d j u s t e d - R 2
lists
Generally,
of v a r i a n c e
are not a p p l i c a b l e
se v e r a l m e t h o d s
(Kmenta
if two
of m o d e l
(nested F(1986)).^°
for s t a t i s t i c a l l y
d i s c r i m i n a t i n g rival n o n - n e s t e d m o d e l s w h i c h are c l a s s i f i e d
as v a r i a n t s of e i t h e r C o x's te st or D a v i d s o n an d M a c K i n n o n ' s
J— t e s t .
The J— test c o n s i s t s of m o d e l d i s c r i m i n a t i o n
b a s e d on w h e t h e r the p r e d i c t i v e p o w e r of one m o d e l is
e n h a n c e d by the p r e d i c t i v e p o w e r of a rival m o d e l .
c o m p l e t e test e x a m i n e s
h o w e a c h p a i r w i s e c o m b i n a t i o n of
riva l m o d e l s e n c o m p a s s e s
Details
The
e a c h o t h e r ' s p r e d i c t i v e power.
of a p p l i c a t i o n of the
J-t e s t are s u m m a r i z e d in
a p p e n d i x B.
Model d is c rimination using
is applicable when
the dependent variables are equally defined and transformed.
For the readers information, variants of Cox's test
are u s e d to discriminate models ba se d on their variances.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
D i a g n o s t i c T ests U s e f u l
for M o d e l
Specification
On e a s p e c t of th e T T T a p p r o a c h to m o d e l
specification
is to s ubjec t an e s t i m a t e d m o d e l to a series
of d i a g n o s t i c tests.
The d i a g n o s t i c tests u s e d in this
s t u d y s u m m a r i z e d b e l o w in c l u d e test s
homoscedasticity assumptions
and measurement
et al.
plots
as one m e a n s
(1989)
sugge s t s
residuals
the e x p e c t e d v a l u e s
of the
line
(or s t a n d a r d i z e d residuals)
of the
residuals
from a straight,
L a rge
f o rty — five
of the
c o n f o r m i n g p r e c i s e l y to n o r m a l i t y s uggest
fr om no rmality.
homoscedasticity
c o n s i s t of v i s u a l
a n d formal s t a t i s t i c a l
T e st s
for
e x a m i n a t i o n of r e s i d u a l
tests.
s u g g e s t s e x a m i n a t i o n of r e s i d u a l s
r e s iduals)
of
against
r e s i d u a l s u n d e r normal ity .
D e t e c t i o n of H o m o s c e d a s t i c i t y .
plots
resid u a l s are
A no r m a l p r o b a b i l i t y p l o t c o n s i s t s
r e p r e s e n t i n g the e x p e c t e d v a l u e s
residuals when
deviations
the use of n o r m a l p r o b a b i l i t y
of d e t e r m i n i n g w h e t h e r th e
normally distributed.
degree
and d e t e c t i o n
for T e s t i n g the N o r m a l i t y of R e s i d u a l s .
Neter
deviations
of the r e s i d u a l s
of the i n f l u e n c e of d a t a o u t l i e r s .
Methods
p l o t t i n g the
for n o r m a l i t y and
N e t e r et ai.
(1989)
(or s t a n d a r d i z e d
p l o t t e d a g a i n s t the p r e d i c t e d or fitted v a l u e s
of
the d e p e n d e n t v a r i a b l e to d e t e r m i n e w h e t h e r the v a r i a n c e of
the
residuals
are constant.
If the
s c a t t e r of the p l o t t e d
residuals
t e n d s to
fl air e i t h e r to th e
r ight or left,
residuals
d i s p l a y a m o n o t o n i e n o n - c o n s t a n t var ianc e.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
42
Bulges,
e i t h e r at the
left a nd right of the plot or a r o u n d
the c e n t e r of the p l o t m a y m e a n th at the resi d u a l s are n o n ­
constant
and non-monotonic.
If the p l o t t e d r es iduals
e v e n l y d i s t r i b u t e d a r o u n d a v a l u e of zero a l o n g the
axis,
heteroscedasticity
regression model
important
is b e y o n d the scope of this
to no te w h e t h e r the B o x - C o x m o d e l
residuals.
it is
satisfies
the
for h e t e r o s c e d a s t i c i t y are u s e d in
study.
means
W h i t e ' s test
(see K m e n t a
(1986))
provides
a
o f d e t e c t i n g h e t e r o s c e d a s t i c i t y w i t h o u t k n o w l e d g e of
f o r m of the n o n - c o n s t a n t e r r o r v a r i a n c e
by th e G l e j s e r test.
More ove r,
of r e g r e s s i n g the s q u a r e d erro r s
t e s t e d on the v a r i a b l e s
th e
such as
White's
test
of th e m o d e l b e i n g
in t he m o d e l in t h e i r m o d e l e d
form
s q u a r e of e a c h v a r i a b l e a n d i n t e r a c t i o n v a r i a b l e s
for e a c h p a i r e d c o m b i n a t i o n of the v a r i a b l e s
Th e n u l l
required
the test does not r e q u i r e
r e s i d u a l s to be n o r m a l l y dis tributed.
consists
plus
study,
i n c l u d i n g the h o m o s c e d a s t i c i t y of the
Two t e s t s
First,
the
for
in c o n j u n c t i o n w i t h the B o x - C o x
a s s u m p t i o n s of OLS,
the
r e s idual
the v a r i a n c e s are c o n s i d e r e d h o m o s c e d a s t i c .
A l t h o u g h d e t e c t i o n of a nd c o r r e c t i o n s
this
appear
hypothesis
of W h i t e ' s test is that the v a r i a n c e s
t h e r e s i d u a l s ar e cons tant.
hypothesis
also
in the m o d e l .
of
F a i l u r e to reje c t this
im p l i e s th at any h e t e r o s c e d a s t i c i t y
caused by
s a m p l i n g error.
The a sym pt otic ,
statistic
is c o m p u t e d by N(R^„)
is
large-sample
w h i c h is d i s t r i b u t e d as a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
with p degrees
regression
of freedom.
function used
A c o m p l e t e d e s c r i p t i o n of the
in W h i t e ' s
tes t is p r o v i d e d in
Appendix C .
Additionally,
the G l e j s e r test
the d e g r e e of c o l i n e a r i t y p r o h i b i t s
The G l e j s e r t e s t
significant
the
consists
is u s e d in cases wh en
use of W h i t e ' s test.
of d e t e r m i n i n g w h e t h e r a
c o r r e l a t i o n e x i s t s b e t w e e n the a b s o l u t e v a l u e of
residuals
a n d the v a r i a b l e
a s s u m e d to be the c au se of
h e t e r o s c e d a s t i c i t y u s i n g a r e g r e s s i o n model.
This
d e t e r m i n a t i o n is m a d e by c h o o s i n g from an a r r a y of p r e s u m e d
forms of h e t e r o s c e d a s t i c i t y
regression equations
Methods
Outliers.
as r e f l e c t e d in d i f f e r e n t
(see K o u t s o y i a n n i s
(1977)).
for D e t e c t i n g and M e a s u r i n g th e
On e a p p r o a c h to d e t e c t i n g da ta o u t l i e r s
d o n e by v i s u a l l y e x a m i n i n g s c a t t e r plots.
(1988)
are
I n f l u e n c e of
identify
several
s u m m a r i z e d here.
us ef u l
First,
is
N e t e r et al.
s c a t t e r plots,
tw o of w h i c h
p l o t t i n g o b s e r v a t i o n s of the
d e p e n d e n t w i t h an i n d e p e n d e n t v a r i a b l e can reveal
observation
l y i n g o u t w a r d of the c l u s t e r of o b s e r v a t i o n s
a r o u n d the e s t i m a t e d
(1980)).
Second,
r e g r e s s i o n line
in a p l o t
(see a l s o W e i s b e r g
of the s t a n d a r d i z e d r e s i d u a l s
a g a i n s t an i n d e p e n d e n t v a r i a b l e or p r e d i c t e d valu e s
dependent variable
observations
of the
appearing isolated from
r e m a i n i n g r e s i d u a l s m i g h t be c o n s i d e r e d outl iers.
This section is based largely on Neter et al
(1989).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
N e t e r et ai.
(1989)
D F B E T A S m e a s u r e to d e t e c t
sugges t use of DFF I T S
influential
w h i c h m a k e use of l e v e r a g e values.
influence
and
obser vat ions ,
b o t h of
D F F I T S m e a s u r e s the
a p a r t i c u l a r case has on the fit of a r e g r e s s i o n
eq uation.
DFBETAS measures
the i n f l u e n c e a p a r t i c u l a r
o b s e r v a t i o n ha s on e a c h of the
la rge da t a
sets,
r e g r e s s i o n c o e f fici ents.
In
th e a b s o l u t e v a l u e of D F F I T S va lue s
e x c e e d i n g the 2 (k/n)
statistic,
w h e r e k an d n are the
n u m b e r of e s t i m a t e d c o e f f i c i e n t s an d n equ a ls the
sa mpl e
size,
fit of a
indicates
model.
an i n f l u e n t i a l
case a f f e c t i n g the
A b s o l u t e v a l u e s of D F B E T A S v a l u e s e x c e e d i n g 2/(n)'"
indicates
cases w h i c h a f f e c t the e s t i m a t e d const a n t or
c o e f f i c i e n t of th e p a r t i c u l a r v a r i a b l e u n d e r eval uation.
Details
measures
of the e q u a t i o n s m a k i n g up the D F F I T S an d D F B E T A S
are
s u m m a r i z e d in A p p e n d i x C.
O n c e da ta o u t l i e r s are identified,
the
function with dummy variables
collective
can be u s e d to asse s s the
impact a g r o u p of o u t l i e r s has on c e r t a i n
coefficients
h o w thi s
s p e c i f i c a t i o n of
i n c l u d i n g the int ercept.
Kmenta
can be d o n e u s i n g a d u m m y v a r i a b l e
(1986)
shows
s p e c i f i e d in th e
f u n c t i o n as v a r i a b l e a f f e c t i n g th e i n t e r c e p t a n d as an
i n t e r a c t i o n t e r m w i t h any of th e v a r i a b l e s
t h e r e b y a f f e c t i n g th e
in the f u n c t i o n
slop e of t h ese vari able s.
s u c h an a p p l i c a t i o n are
s u m m a r i z e d in A p p e n d i x C.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Details
of
45
Conclusions
and Summary
This
chapter
a p p r o a c h to m o d e l
mitigating type
s h o w e d t h a t a TTT,
vers u s
an A E R
s p e c i f i c a t i o n is p r e f e r r e d for p u r p o s e s
I errors
in the p r o c e s s
of s e l e c t i n g an
a p p r o p r i a t e s p e c i f i c a t i o n of an e c o n o m e t r i c model.
TTT a p p r o a c h to s t a t i s t i c a l
form of the
study.
first-stage
However,
of
d e t e r m i n a t i o n of the
Thus,
a
functional
T C M d e m a n d f u n c t i o n is u s e d in this
due to th e i n f o r m a t i o n p r e s e n t e d in
previous
analyses
of th e D F W P
(namely,
D u f f i e l d et ai.
approach resulting
s t r e a m fishe r i e s
(1987)
fr om t h i s
and Duffield
data b a s e
(1988)
the
stud y a n d p r i o r s tudies w o u l d
m o r e a c c u r a t e l y be l a b e l e d as an A E R / T T T h y b r i d app roa ch.
Second,
use of a full M L E m e t h o d was
found
p r e f e r a b l e to the NL S or l O L S / g r i d - s e a r c h a pproaches.
is due l a r g e l y to th e b i a s
in th e us e of t h e s e
l a t t e r tw o met hod s.
likelihood ratio test
regression models
in v a r i a n c e e s t i m a t e s
resulting
log-
fo r d i s c r i m i n a t i n g r e s t r i c t e d B o x — Cox
from more general
r e s t r i c t e d forms was
Third,
This
rev iew ed.
D a v i d s o n an d M a c K i n n o n
J-Test
spe cif ications,
non—
A l s o r e v i e w e d was the
for d i s c r i m i n a t i n g n o n - n e s t e d
models.
Final ly,
model
in a c c o r d a n c e w i t h the TTT a p p r o a c h to
discrimination,
se v e r a l
d i a g n o s t i c tests us eful
determining functional
f o r m by s t a t i s t i c a l
rev iewed.
included,
T h e s e t ests
w h e t h e r th e r e s i d u a l s
1) m e t h o d s
for
inference were
to d e t e r m i n e
s a t i s f y the OL S a s s u m p t i o n of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
normality,
including visual
p r o b a b i l i t y plots,
residuals
e x a m i n a t i o n of normal
2) m e t h o d s to d e t e r m i n e w h e t h e r the
s a t i s f y the OL S
a s s u m p t i o n of homoscedasticity,
i n c l u d i n g W h i t e ' s a n d G l e j s e r ' s tests,
identifying outliers
residuals
and 3) m e t h o d s
for
suc h as p l o t s of the data and of
an d a n a l y t i c a l m e t h o d s
such as D F F I T S and D F B E T A S
measures.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
DATA SOURCES AND DESCRIPTION
A s n o t e d in c h a p t e r one,
r e e x a m i n a t i o n of th e o r i g i n a l
by D u f f i e l d et al.
(1987)
this
s t u d y is a
TCM demand analysis performed
an d D u f f i e l d
(1988)
on M o n t a n a
c o l d w a t e r s t r e a m fishe rie s.
This
d e s c r i p t i o n of this
(or th e c o l d - w a t e r stre a m
fi s h e r i e s
data base
data b a s e ) .
i n c l u d e d in the st udy
d e s c r i p t i o n of the
th e m e t h o d s
a description
is p rovided.
so urc es
the c o l d - w a t e r s t r e a m s
Last,
First,
Second,
u s e d to gather,
a
of the sites
a list a nd
of d a t a u s e d in this
fisheries
all p r i m a r y data i n c l u d e d
chapter provides
study
from
data b a s e is pr ovided.
organize,
and aggregate
in the data b a s e a n d u s e d in this
s t udy ar e summarized.
D e s c r i p t i o n of the Sites
C o m p r i s i n g the M o n t a n a
Stream Fishery
The data set u s e d in this p a p e r c o n t a i n s TC M data
for 28 t r i b u t a r i e s
in M o n t a n a
a n d / o r w a t e r s h e d s an d 20
"unique w a t e r s "
i d e n t i f i e d by th e M o n t a n a D e p a r t m e n t of Fish,
The data u s e d in this s tudy w e r e d e v e l o p e d for
e s t i m a t i o n of d e m a n d a n d net e c o n o m i c v a l u e s for cold w a t e r
f i s h i n g b y D u f f i e l d et ai. (1987).
No a l t e r a t i o n s w e r e m a d e
t o t h e u n t r a n s f o r m e d da ta u s e d in t h a t study.
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
Wildlife,
and Parks
unique waters
watersheds
These
(Duffie ld et ai.
include tributaries
(1987)).
The 28 n o n -
to u n i q u e waters
or
c o m p r i s e d of a r i v e r a n d its tr ibu tar ies .
streams and tributaries
d e f i n e the r e g i o n a l
f i s h e r y u s e d in this p a p e r . T a b l e s
non-unique waters
str e a m
1 a n d 2 list the 28
a n d the 20 u n i q u e waters,
respec tiv ely.
A r e g i o n a l T C M is g e n e r a l l y u s e d to a n a l y z e d e m a n d
fo r a c o l l e c t i o n of r e c r e a t i o n sites for a s p e c i f i c
r e c r e a t i o n a c t i v i t y such as h u n t i n g or fishing.
T h u s , site
s p e c i f i c m o d e l s p e r t a i n o n l y to one site.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CD
■D
O
Q.
8
Q.
§
TABLE 1. —
Montana Stream Fisheries: Non-Unique Waters.
1— H
T3
CD
3
(/)'
o'
River
Code
11
12
8
14
15
16
17
CD
21
22
3
3"
23
CD
24
CD
T3
25
31
32
33
34
35
36
37
41
42
43
O
Q.
C
a
o
3
T3
O
CD
Q.
44
T3
CD
(/)
(/)
52
56
61
62
71
Source:
Definition
River
Entire drainage
Entire drainage
Entire drainage
River and tributaries below confluence with south fork and
excluding river 89
Lower Clark Fork
River and tributaries below confluence with Flathead
(Paradise)
Excludes mainstem river 91
Kootenai Tributaries
Upper Clark Fork Tributaries
Above confluence with blackfoot (Milltown) and excluding
mainstem river 86
Excludes mainstem river 83
Blackfoot Tributaries
Excludes mainstem river 94
Rock Creek Tributaries
Excludes mainstem river 82
Bitterroot Tributaries
Paradise to Milltown excluding mainstem river 87
Middle Clark Fork Tributaries
Springdale to Gardener excluding mainstem river 98
Upper Yellowstone Tributaries
Excludes mainstem river 90 and 88
Gallatin Tributaries
River and tributaries from Threeforks to Canyon Ferry
Upper Missouri (Region 3)
Excludes mainstem river 92
Madison Tributaries
Entire drainage
Jefferson
Excludes mainstem river 80
Beaverhead Tributaries
Excludes mainstem river 81
Big Hole Tributaries
River and Tributaries below Marias River and above Fort Peck
Middle Missouri
Excludes mainstem river 95
Smith Tributaries
Canyon Ferry to Marias River excluding mainstem river 93
Upper Missouri (Region 4)
Entire drainage
Marias
Springdale to confluence with Bighorn excluding rivers 99,
Middle Yellowstone Tributaries
84, 85, 96, 55, and 56
Excludes mainstem river 96
Stillwater Tributaries
Excludes mainstem river 84
Boulder Tributaries
River and tributaries from upper end of Fort Peck Reservoir
Lower Missouri
to North Dakota Border
Entire drainage
Mi Ik
River and tributaries below confluence with Big Horn
Lower Yellowstone
Flathead, South Fork
Flathead, Middle Fork
Flathead, North Fork
Flathead
Duffield et. al.
(1987,
table 1.)
CO
CD
■D
O
Q.
g
Q.
■CDD
c/)
c/)
TABLE 2
River
Code
80
8
81
82
83
ci'
85
84
86
87
88
89
90
3
3
CD
91
92
CD
93
"
■D
O
Q.
C
a
O
3
"O
O
94
95
96
97
98
99
Montana Stream Fisheries: Unique Waters
River
Definition
Beaverhead
Bighole
Bitterroot
Blackfoot
Boulder
Bighorn
Upper Clark Fork
Middle Clark Fork
East Gallatin
Upper Flathead
Gallatin
Kootenai
Madison
Missouri
Rock Creek
Smith
Stillwater
Swan
Upper Yellowstone
Middle Yellowstone
Mainstem
Mainstem
Mainstem to confluence with East and West Forks
Mainstem
Mainstem
Mainstem
Mainstem above Milltown
Mainstem Milltown to Paradise
Mainstem
Mainstem above Flathead Lake to confluence of South Fork
Mainstem
Mainstem
Mainstem
Mainstem, Bolter to Cascade
Mainstem near Missoula
Mainstem
Mainstem near Absarokee
Mainstem
Mainstem Springdale to Gardener
Mainstem Springdale to confluence with Bighorn
CD
Q.
Source :
Duffield et. al.
(1987,
table 1.)
■CDD
C/)
C/)
o
51
Data Sources
I n c l u d e d in th e C o l d - W a t e r S t r e a m F i s h e r i e s
D a t a Ba se
Th e c o l d - w a t e r
is c o m p r i s e d of t h r e e
are t he
survey data
Fish Wildlife
(DFWP)
Survey
henceforth
supplemental telephone
origins,
sources were
were
for the 1985 l i c e n s e y e a r
fisheries
survey)
regarding
sites
and travel
harvest
rates,
and time costs.
T h e s e two
reported distance traveled were
(Duf f i e l d
Mexico
T h e s e dat a
fished,
(1988)),
Distances
in t h e s e
c o m p u t e d f r o m th e R a n d M c N a l l y R o a d Atla s:
Canada,
a nd a
supplemented with estimated map distances
in c a s e s w h e r e th e
be in e r r o r
of
t h r o u g h the M o n t a n a S t a t e w i d e
d i s t a n c e to the site,
s o c i o - e c o n o m i c data,
da ta
sources
sur v e y c o n d u c t e d in 1985.
so u r c e s p r o v i d e d i n f o r m a t i o n
anglers'
C h i e f a m o n g thes e
study
c o l l e c t e d by the M o n t a n a D e p a r t m e n t
and Parks
1989,
data b a s e u s e d in this
sources.
Angling Pressure Mail
(McFarland,
st r e a m s
f o u n d to
cases
U.S.,.
(1977).
Data Preparation
for F i r s t — S t a g e T C M D e m a n d F u n c t i o n
Estimation
C o l l e c t i o n of S u r v e v D a t a .
fisheries
through
DFWP c o n d u c t e d the
s u r v e y d u r i n g e a c h of the l icense y e ars of 1982
1985 b e g i n n i n g
spring
1982
(see,
Mc Farland ,
Q u e s t i o n n a i r e s w e r e m a i l e d m o n t h l y to r e s i d e n t
p.
2).
and n o n ­
resident
l i c e n s e d a n g l e r s w i t h i n one m o n t h of the a n g l e r ' s
purchase
of his
or he r license.
Approximately
1, 500 a n d 100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
sur v e y s w e r e
month,
sent to r e s i d e n t
respectively,
w i t h th e
1984
1982 t h r o u g h 1984.
l i c e n s e y e a r the
d i s t r i b u t i o n was
for th e m o n t h s
during
a n d n o n - r e s i d e n t an g l e r s pe r
generally
freq u e n c y of survey
i n c r e a s e d to b i w e e k l y m a i l i n g s
of M a r c h t h r o u g h October,
h i g h f i s h i n g p r e s s u r e m onths.
d i s t r i b u t i o n rate
for th e
This
corresponding with
in crease in the
same n u m b e r of m o n t h l y s urveys was
i m p l e m e n t e d to m i t i g a t e m e m o r y bias.
w ho p u r c h a s e d 2— day
Beginning
N o n - r e s i d e n t an g l e r s
l i c e n s e s w e r e s u r v e y e d on an annual
basis.
A
r a n d o m d r a w i n g of ang l e r s was e n s u r e d by u s i n g a
stratified sampling procedure
a n d 144 -145).
Questionnaires
detailed information
particular
Th is
(see,
(1989),
pp.
3
sent d u r i n g 1985 sou ght
for e a c h fi s h i n g tri p t a k e n d u r i n g th e
sampling period
information
McFarland
for w h i c h the sur v e y was made.
can be n o t e d
fro m a copy of the s urveys
sent p r o v i d e d in A p p e n d i x D.
Th e n u m b e r o f m o n t h l y
i n c r e a s e d to a b o u t
This
th e d a t a n e e d s
Study.
increase
O f the 3 6 , 0 0 0
The
overall
al.
(1987)).
li c e n s e holders,
surveys
a n d 8% w e r e
response
for e a c h
respectively,
in f r e q u e n c y was m a d e to a c c o m m o d a t e
fo r th e p u r p o s e s
sent to r e s i d e n t
s u rveys was
3 , 0 0 0 a n d 250 m o n t h l y m a i l i n g s
of r e s i d e n t a n d n o n - r e s i d e n t
in 1985.
fisher ies
ra te wa s
of the M o n t a n a B i o e c o n o m i c s
sent duri n g 1985,
92% w e r e
sent to n o n r e s i d e n t anglers.
54% or 19,271.
(Duffie ld et
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
F o r th e p u r p o s e s
c o l l e c t e d t h r o u g h the
follows.
First,
of T C M an alysis the dat a sa mple
fisheries
respondents
survey were
r e d u c e d as
th at ha d not fis h e d d u r i n g the
m o n t h or t w o - w e e k t i m e p e r i o d s p e c i f i e d on the su rvey were
exclu d ed.
Additionally,
b e e n on m u l t i - p u r p o s e
t h o s e wh o were d e t e r m i n e d to have
a n d / o r m u l t i - s i t e trips and tho se
m a k i n g o v e r - n i g h t t r i p s w e r e a l s o p u r g e d from the data set.
F i s h i n g site s w e r e t h e n
designations.
Thus,
data
includes those who had
in t a b l e s
c o d e d a c c o r d i n g to DFWP m a n a g e m e n t
for the c o l d - w a t e r stream a n a l y s i s
f i s h e d in the trout
str e a m d e s i g n a t e d
1 a n d 2.
The s e c o n d m a j o r d a t a source used in this
taken
from a supplemental
survey.
The d a t a
s u r v e y of 2, 000
s u r v e y was
of
1985.
resident
non-resident
fis heries
a n d n o n r e s i d e n t angler.
d u r i n g S e p t e m b e r and O c t o b e r
anglers
a n d p r o d u c e d a respon se rate of 80%
for n o n r e s i d e n t s .
analysis.
T hree t y p e s
survey:
socio-economic
income,
behavior,
fishing,
as,
data,
2)
su c h as f i s h i n g exper ien ce,
data c h a r a c t e r i z i n g
s p ent at the site,
a n d e q u i p m e n t used;
expenditures,
for T C M
of d a t a w e r e c o l l e c t e d in this
and education;
such as t i m e
for
The same c r i t e r i a n o t e d
a b o v e was u s e d for s e l e c t i n g q u a l i f i e d respondents
age,
This
sample c o n s i s t e d of 1,600 resident a nd 400
a n d 52%
1)
is
set w e r e g a t h e r e d by a t e l e p h o n e
a d m i n i s t e r by DFWP
Thi s
residents
in t h i s
s u r v e y to the 1985
s tudy
3)
t r a v e l time,
time
site use
spent
an d tr av el cos t d a t a s u c h
a n d distance.
Socio-economic
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
data
f r o m the
fisheries
supplemental
s u r v e y w e r e a p p e n d e d to the
sur vey by origin.
a p p e n d e d by site.
Tra v e l
c osts
o r i g i n / d e s t i n â t ion pairs.
McFarland
Site c h a r a c t e r i s t i c data wer e
da ta w ere a p p e n d e d by
(See D u f f i e l d et ai.
(1987)
(1985a a n d b ) ).
A g g r e a a t i o n of the S u p p l e m e n t e d Fish e r i e s
Data.
As noted,
th e d e m a n d
stud y are b a s e d on zonal
given
region.
Thus,
Responses
a g g r e g a t i o n s of fi s h i n g t rip s
individual
are a g g r e g a t e d b y
u s i n g the
Survev
func t i o n s e s t i m a t e d in this
su rvey r e s p o n s e s
t a k e n to a g i v e n site w e r e a g g r e g a t e d by ori g i n
origins,
and
following
for a
for tr ips
zones.
zones of near l y equal d i s t a n t
rules :
1.
A n o r i g i n zone c o n s i s t s of a single c o u n t y if
the c o u n t y c o n t a i n s the d e s t i n a t i o n site or
is c o n t i g u o u s to the county or counties
c o n t a i n i n g th e site.
2.
S everal c o u n t i e s w e r e l u m p e d t o g e t h e r to
d e f i n e a zone of i n t e r m e d i a t e d i s t a n c e
observations.
This was done to p revent
c o n c e n t r i c gaps w h i c h w o u l d o t h e r w i s e re sul t
fr om zero o b s e r v a t i o n s from countie s of
i n t e r m e d i a t e distanc es.
Therefore, o b s e r v e d
trip s f r o m t h e s e " s u p e r - c o u n t i e s " r e p r e s e n t s
the p o p u l a t i o n of coun t i e s w h e r e no t rips
we re r e c o r d e d w i t h i n the sample.
3.
C o n t i g u o u s c o u n t i e s of states b o r d e r i n g
M o n t a n a w e r e t r e a t e d as ind i v i d u a l zones.
4.
F i v e n o n r e s i d e n t m a r k e t regio ns were d e f i n e d
to a l l o w n o n r e s i d e n t trips to repre s e n t the
p o p u l a t i o n of t h e i r h o m e state an d tha t of
all st at es t h e y t r a v e l e d t h r o u g h to get to
th e site, p r o v i d i n g no trips from these
st ates w e r e i n c l u d e d in the d a t a base.
These
m a r k e t r e g i o n s w e r e c o n s t r u c t e d as sp ok es
e x t e n d i n g o u t w a r d from M o n t a n a t h r o u g h o u t th e
c o n t i n e n t a l U n i t e d States.
Again, this
m e t h o d of a g g r e g a t i o n wa s u s e d to a v o i d any
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
c o n c e n t r i c g a p s in the s u r r o u n d i n g m a r k e t
zones b e t w e e n the sit e a n d the n o n - r e s i d e n t ' s
origin.
Independent variables were
aggregation process
s u m m e d t o g e t h e r an d the
r e s u l t e d in 839 o r i g i n - d e s t i n a t i o n pair s
for t he
cold-water stream fisheries
methods
of o r i g i n - d e s t i n a t i o n data b y
problems with
d a t a base.
Aggregation
site p r e s e n t e d
r e g a r d to the a c c u r a c y of d i s t a n c e traveled.
C a ses w e r e
f o u n d in w h i c h a n g l e r s w h o t r a v e l e d to a
particular
site
f r o m the
same o r i g i n
d i s p a r a t e d i s tance s.
In fact,
were
for 50% of the o r i g i n - d e s t i n a t i o n
found unreliable
pairs.
In t h e s e cases,
reported distances
were
recomputed
an d p o p u l a t i o n w e i g h t e d avera ge d i s t a n c e s
size
1988) .
for th e e s t i m a t e d d e m a n d f u n c t i o n s
p r e s e n t e d in c h a p t e r fou r is
This
th e
siz e
with previous
Th e
11,
23,
complete
four tr i b u t a r i e s
5 6 , a n d 61)
w e r e e x c l u d e d fro m
regio ns
{see t a ble
i n f o r m a t i o n on s o c i o - e c o n o m i c a n d
m o d e l i n g effort s.
statistics
data set
to remain c o n s i s t e n t
f u r t h e r r e d u c e d to o r i g i n - d e s t i n a t i o n
d e m o g r a p h i c v a r i a b l e s as wa s
previous
First,
set of DFWP a d m i n i s t r a t i v e
s a m p l e was
pairs with
data base.
s t u d i e s on t h i s
(river codes
th e c o m p l e t e
1).
741 o r i g i n - d e s t i n a t i o n pairs.
is th e c o m b i n e d r e s u l t of t w o adj u s t m e n t s m a d e on
stream fisheries
regions
s u r v e y da ta on d i s t a n c e
m a p d i s t a n c e s w e r e u s e d in lieu of
(Duffield,
Th e s a m p l e
the
zone m a y have t r a v e l e d
for the v a r i a b l e s
th e ca se
Table
for samples u s e d for
3 provides
descriptive
s p e c i f i e d in the models
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
p r e s e n t e d in C h a p t e r 4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CD
■D
O
Q.
C
g
a.
Table 3.—
Descriptive Statistics for the Variables Specified in the Models Presented
in Chapter 4.
Variable
Mean
Standard
Deviation
8
TRIPS
6.79
16.89
(O'
POP
4.324 E6
TRIPCAP
2.71 E-4
LTRIPCAP
-11.71
3.04
-17.96
-4.19
CAVEDIST
1119.55
1196.89
4.00
4937.50
LCVEDIST
6.21
1.54
1.39
8.50
Natural Log of Average Round-Trip Distance
LKVEDIST
6.49
1.18
4.22
8.52
Natural Log of Average Round-Trip Distance plus 64
-11.59
2.98
-17.70
-4.18
Box-Cox of Per Capita Trips at
■CDD
C/)
W
o'
3
0
3
CD
3"
1
3
9.966 E6
1.02 E-3
Min
1.0
1.0 E3
1.58 E-8
Max
153.00
7.809 E7
Label
Zonal Aggregated Trips
Zonal Aggregated Population
1.51 E-2 Per Capita Trips
CD
Natural Log of Per Capita Trips
"n
c
3
.
3
"
Average Round-Trip Distance
CD
CD
■D
O
Q.
C
a
o
3
TRIP3A
.0016
■D
O
CDIST3A
11.54
4.16
1. 56
18.99
Box-Cox of Average Round-Trip Distance at
CD
TRIP3B
-9.34
1.97
-13.02
-3.87
Box-Cox of Per Capita Trips at .038
7.06
1. 91
1.42
10.04
Box-Cox of Average Round-Trip Distance at
TRIP4A
-13.05
3.70
-21.00
-4 .34
Box-Cox of Per Capita Trips at -.017
TRIP4B
-9.78
2.16
-13.88
-3.94
Box-Cox of Per Capita Trips at
CDIST5
11.70
4.25
1.57
19.33
Box-Cox of Average Round-Trip Distance at .172
Q.
CDIST3B
■CDD
. 169
.038
.03
C/)
C/)
The sample size for all of the ■variables listed in this table is 741.
vn
CHAPTER 4
E S T I M A T I O N A N D A N A L Y S I S OF DE M A N D F U N C T I O N S
This
chapter provides
a s ummary of the m a t h e m a t i c a l
f o r m a n a l y s i s p e r f o r m e d on the b i v a r i a t e
demand function
for the
As n o t e d in C h a p t e r 2,
specification
approaches.
s t r e a m f isheries data.
this
study ta kes a m o d e l
a p p r o a c h w h i c h c o m b i n e s the A E R and TTT
The A E R a s p e c t of this a n a l y s i s
use of t w o p r i o r
d a t a base.
consists
of the
st u d i e s p e r f o r m e d on the s t r e a m f i s h e r i e s
Th e TTT a s p e c t
t e s t s on the b i v a r i a t e
c onsist of se veral d i a g n o s t i c
forms of th e two p r e v i o u s l y e s t i m a t e d
d e m a n d f u n c t i o n s u m m a r i z e d below,
t e s t i n g of the B o x - C o x
comparison
first-stage TCM
estimation and diagnostic
forms of th e T C M d e m a n d function,
of thi s m o d e l w i t h the b i v a r i a t e
p r e v i o u s l y e s t i m a t e d mod els.
This
and
forms of the
c h a p t e r is o r g a n i z e d as
follows.
Th e
first
D u f f i e l d et al.
different
s e c t i o n rev i e w s the p r i n c i p l e s a p p l i e d by
(1987)
functional
and Duffield
(1988)
to e s t i m a t e two
forms of th e T C M d e m a n d function.
B i v a r i a t e m o d e l s b a s e d on t h e s e p r i n c i p l e s w e r e e s t i m a t e d
a n d are p r e s e n t e d in this
is
section.
The fit of these m o d e l s
s u m m a r i z e d a l o n g w i t h an a n a l y s i s of a d h e r e n c e to c e r t a i n
58
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59
a s s u m p t i o n s of OLS.
for the a n a l y s i s
This
s e c t i o n serves
of the regional,
as a s t a r t i n g p o i n t
zonal a g g r e g a t e T C M d e m a n d
f u n c t i o n s p r e s e n t e d in this paper.
B a s e d on the
f o u n d a t i o n set in sec t i o n one,
d e t e r m i n a t i o n of the m o s t
functional
form(s)
statistically
magnitudes
This
theoretical
section includes a synopsis
expectations
of the p a r a m e t e r s
specifications
of t he signs
an d
estimated using various
of the o r i g i n a l B o x - C o x a n d e x t e n d e d B o x - C o x
r e g r e s s i o n a p p r o aches.
This
is f o l l o w e d by a s u m m a r y of
five m o d e l s e s t i m a t e d u s i n g v a r i o u s
Cox t r a n s f o r m a t i o n s
v ari a b l e s .
s ound an d p r a c t i c a l
of the b i v a r i a t e T C M d e m a n d f u n ction is
p r e s e n t e d in s e c t i o n two.
of the a p r i o r i
a
c o m b i n a t i o n s of the B o x -
on the d e p e n d e n t
These models
and i n d e p e n d e n t
are t h e n d i s c r i m i n a t e d to d e t e r m i n e
the form which best
satis f i e s the m a x i m u m l i k e l i h o o d
principle
field.
fr o m this
An a n a l y s i s of thi s m o d e l ' s
a d h e r e n c e to c e r t a i n OL S a s s u m p t i o n s
Finally,
estimated model
c o m p a r i s o n s b e t w e e n th e o p t i m a l B o x - C o x
a n d the b i v a r i a t e
previously estimated models
c o n t i n u e d in the
is al so summarized.
forms of the two
r e v i e w in s e c t i o n one are
last section.
This
of th e p r e d i c t i v e p o w e r of e a c h m o d e l
includes
a comparison
and a conclusion
r e g a r d i n g w h i c h m o d e l d e s c r i b e s p e r c a p i t a t r i p d e m a n d best.
As
an aside,
th e e l a s t i c i t i e s
p r o v i d e d in t h i s
resulting
fr om e a c h m o d e l
section.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are
60
Bivariate Estimates
Tw o P r e v i o u s l y E s t i m a t e d D e m a n d
Functions
As
n o t e d in c h a p t e r
u s e d to e s t i m a t e th e
previous
et al.
studies.
(1987)
linear)
1, two functional
f i r s t - s t a g e d e m a n d fu nc tion in two
In t he
fi rst of thes e studies,
s p e c i f i e d a m u l t ivariate,
form.
This
forms were
f o r m wa s
Duffield
double-log
(log-
s p e c i f i e d p r i m a r i l y to s ati sfy
th e c o n s t r a i n t
of e c o n o m i c t h e o r y that there be d i m i n i s h i n g
marginal
value
for e a c h a d d i t i o n a l
wa s a l s o
s e l e c t e d to a v o i d the p o s s i b i l i t y of p r e d i c t i n g
negative per capita trips
fish caught.
This
form
from dista nt ori gin zones a n d to
m i n i m i z e h e t e r o s c e d a s t i c i t y of the res iduals e x p e c t e d to
occur when population varies
across ori gin zones.
B o t h of
t h e s e l a t t e r t w o p r o b l e m s w e r e e x p e c t e d to o c c u r w i t h the
l i n e a r form.
However,
D u f f i e l d et al.
(1987)
as noted,
the mo del e s t i m a t e d by
f a i l e d to pr o d u c e h o m o s c e d a s t i c
residuals
a n d o v e r p r e d i c t e d o b s e r v e d trips by about
p e r cent.
The o v e r p r e d i c t i o n was
i n f l u e n c e d by t r i p s t a k e n
site.
The m o d e l
distances
60
fou nd to be l argely
f r o m origin s wit h i n 20 m i l e s
a l s o u n d e r p r e d i c t e d trips
a n d o v e r p r e d i c t e d t r ips
of a
fr om i n t e r m e d i a t e
form long er distances.
D r i v e n p r i m a r i l y by the concer ns of
h e t e r o s c e d a s t i c i t y a n d p r e d i ction,
three general
polynomial,
polynomial
alternative
s e m i — log,
Duffield
functional
and double-log
specifications,
forms
forms.
(1988)
examined
including
A l t h o u g h the
including additions
of d i s t a n c e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
squared and cubed terms
and a hybrid double-log-polynomial
form enhanced trip prediction,
for d i s t a n c e s
greater than
d e m a n d was p o s i t i v e l y
3,000 miles.
The
s p e c i f i c a t i o n a l s o e n h a n c e d t r i p pre diction.
c o n v e r s e to the d o u b l e - l o g model,
o v e r p r e d i c t e d t r ips
u n d e r p r e d i c t e d t r ips
sh ort
semi-log
However,
the s e m i - l o g model
at i n t e r m e d i a t e d i s t a n c e s
at
sloped
and
an d long distances.
Further,
the c o e f f i c i e n t of d e t e r m i n a t i o n
for the s e m i - l o g m odel was
less t h a n that of th e d o u b l e - l o g
funct i o n
log v e r s u s
.78 for the d o u b l e - l o g
polynomial
semi-log
(.22 for the
form).
A hybrid
f u n c t i o n was also estimated,
bu t d i d not
c o r r e c t th e h e t e r o s c e d a s t i c i t y p r o b l e m s n o t e d above.
model
also s h o w e d a p o s i t i v e
unacceptable theoretical
As
Duf field
a res ult
(1988)
in w h i c h a v e r a g e
of
semi­
This
slop e at about 3,000 miles,
an
result.
of th e
found that
above
s u m m a r i z e d analysis,
a m u l t ivariate,
d o u b l e — log form
r o u n d - t r i p d i s t a n c e was s h i f t e d by c o n s t a n t
90 e n h a n c e d t r i p p r e d i c t i o n to w i t h i n
.1 p e r c e n t of
o b s e r v e d t rips an d s a t i s f i e d th e a s s u m p t i o n of h o m o s c e d a s t i c
residuals .
Table 4 summarizes
on t h e
form s
and Duffield
estimated bivariate models based
s p e c i f i e d by D u f f i e l d
(1988)
(model
2).
et al.
However,
(1987)
(model
different
1)
fr om
A c o n s t a n t v a r i a n c e of the r e s i d u a l s in this m o d e l
w a s c o n c l u d e d b a s e d on v i s u a l e x a m i n a t i o n of a plot of the
r e s i d u a l s v e r s u s the p r e d i c t e d v a l u e s of the m o d e l (Duffield
(1988)).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
Duffield"s
average
(1988)
an al ysis ,
the
r o u n d - t r i p d i s t a n c e wa s
shift p a r a m e t e r a dded to
f o u n d to be
64 for the
b i v a r i a t e model.
T he o p t i m a l shift p a r a m e t e r of 64 m i l e s was found
b y s e a r c h i n g a r ange of p a r a m e t e r s u n t i l a d j u s ted-R^ was
locally maximized.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■o
I
I
■o
CD
Table 4.—
(/>
Bivariate Specifications of Models Specified by Duffield et ai
(1987) (Model 1) and Duffield (1988) (Model 2).
o'
3
O
Model
Log-Likelihood
Po
Pi
R2
F
7,331.34
-1.03
(-4.53)
(<.001)
-1.71
(-48.32)
{<.001)
.759
2,335.68
3.09
(10.68)
(.000)
-2.28
(-51.91)
(<.001)
.784
8
(S'
3
CD
C
p.
1.
(t-ratio)
(p-value)
0(R)
2.
0(R)
0(R)
0(R)
7,372.34
(<.001)
2,695.48
(<.001)
CD
■o
O
Q.
C
a
o
3
■o
o
Pi is the estimated parameter for average round-trip distance and average
round-trip distance plus 64 in each of models 1 and 2 respectively.
The
critical values for t and F at a 5 percent probability of a type I error
1.6471 and 3.854, respectively.
The sample size for both models is 741.
CD
Q.
O
C
■o
CD
C/i
C/i
LU
64
Diagnostic
summarizes
the
T e s t s on M o d e l s
fit a n d s t a t i s t i c a l
estimated parameters
assumptions
values
1 a n d 2.
following
A l s o OLS
d i s t r i b u t i o n of the r e s i d u a l s
from models
and
Further potential
1 and 2 are summarized.
can be n o t e d in t a b l e
do u b l e - l o g ,
The
s i g n i f i c a n c e of the
c o n s i s t e n c y are ex amined.
resulting
As
in m o d e l s
of a n o r m a l
their variance
outliers
1 and 2 .
4,
b o t h the d o u b l e - l o g and
s h i f t e d d i s t a n c e m o d e l s have h i g h ad jus ted-R 2
s u g g e s t i n g the m a j o r i t y
respectively)
(75.9 an d 78.4 percent,
of the v a r i a t i o n in the n a t u r a l - l o g of pe r
capita trips
are d e s c r i b e d by the i n d e p e n d e n t v a r i a b l e s
e a c h mo de l.
Furthe r,
degrees
w i t h a cr itical v a l u e of 3.854,
of f r e e d o m e q u a l to 1 and 739,
in
with
the h y p o t h e s i s that
is z e r o is r e j e c t e d at a 5 p e r c e n t p r o b a b i l i t y of a type
I error.
Thus,
each model
a l s o n o t e d th at th e
signs
c o n s t i t u t e s a g o o d fit.
It is
of Pi are a c c e p t e d as c orrect at a
5 p e r c e n t p r o b a b i l i t y of a t y p e
I error.
E x a m i n a t i o n of th e no rmal p r o b a b i l i t y p lots of the
residuals
resulting
respectively)
from models
1 an d 2
(figures
1 a n d 2,
suggest these models yield nearly normally
d i s t r i b u t e d residuals.
greater normality
in th e
M oreover,
model
resi duals.
1 ap p e a r s to y i e l d
Fig u r e s
s t a n d a r d i z e d s c a t t e r p l o t s of the r e s i d u a l s
3 and 4 are
and predicted
n a t u r a l - l o g of p e r c a p i t a t r i p s p r o d u c e d by e a c h of M o d e l s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
65
and 2
(table 4).^^
Four general
from t h e s e plots.
log,
First,
the
shifted distance model
a r o u n d th e
south-west
t h e do u b l e - l o g ,
of the d o u b l e ­
a p p e a r mo re e v e n l y d i s t r i b u t e d
(even t h o u g h th ere a p p e a r mo re p o i n t s
q u a d r a n t then any o t h e r q u a d r a n t ) .
shifted distance model
heteroscedasticity
model.
resi d u a l s
can be m a d e
c e n t e r of the p l o t t h a n the r e s i d u a l s p r o d u c e d by
the d o u b l e - l o g m o d e l
in th e
conclusions
Howeve r,
in the
th e
Thus,
a p p e a r s to y i e l d less
r e s i d u a l s th an th e d o u b l e - l o g
re sul ts
of W h i t e ' s test
for
h o m o s c e d a s t i c i t y w i t h a null h y p o t h e s i s of c o n s t a n t v a r i a n c e
and alternate hypothesis
residuals
of n o n - c o n s t a n t v a r i a n c e a c r o s s the
s u g g e s t s that at a 5 p e r c e n t p r o b a b i l i t y of a typ e
I e r r o r th e r e s i d u a l s
in b o t h m o d e l s
1 a nd 2 are not
constant.
Secondly,
th e i n v e r t e d — U shape of the
residual/prediction plot
functional
form.
do es not a c h i e v e
displays
in fi gur e 3 su ggests a m i s s p e c i f i e d
Th at is,
complete
the d o u b l e - l o g f u n c t i o n a l
linea rit y.
In contrast,
form
fig u r e 4
less of an i n v e r t e d - U shape s u g g e s t i n g the d o u b l e -
Th e p l o t s in all o f the fig ure s in this c h a p t e r
were pro d u c e d using SPSSX software which were uniformly
r e s i z e d in W O R D P E R F E C T 5.1.
This a n a l y s i s
a n d K u t n e r (1989).
re lies h e a v i l y on Neter,
nR2 fo r e a c h of m o d e l s 1 an d 2
respectively.
The p - v a l u e s for e a c h of
d i s t r i b u t i o n s w i t h 2 d e g r e e s of f r e e d o m
,22498 E-9, r e s p e c t i v e l y .
Further, the
is 5.9915.
Wasse rma n,
are 4 2.66 a n d 44.43,
t hese
are .54513 E-9 an d
95% c r i t i c a l v a l u e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
log,
shifted distance model
Third,
model tends
as n o t e d by D u f f i e l d
ca n be
and underpredicts
readily
s c a t t e r p l o t of the
residuals
model
at i n t e r m e d i a t e distances.
2 mitigates
r e s u l t i n g from mod el
r o u n d - t r i p distance."”
shape of a s i m i l a r p l o t
th a t m o d e l
(1988) , the d o u b l e - l o g
seen in figure 5 w h i c h is a s t a n d a r d i z e d
n a t u r a l - l o g of a v e r a g e
the
g r e a t e r li n e a r i t y / ' ’
to o v e r p r e d i c t p e r c a p i t a tr ips at short and
lo n g d i s t a n c e s
This
achieves
for m o d e l
In contrast,
2 in figure
6 s u g gests
t h e s e p r e d i c t i o n err o r s a l t h o u g h this
c o n t i n u e s to o v e r p r e d i c t at short d i s t a n c e s
underpredict
1 an d the
and
at i n t e r m e d i a t e distances.
Th e s t a n d a r d for l i n e a r i t y u s e d in this a n a l y s i s is
t h a t a c o m p l e t e l y r a n d o m d i s t r i b u t i o n of the po ints on the
s t a n d a r d i z e d r e s i d u a l s / p r e d i c t i o n p l o t m e a n s th e f u n c t i o n is
linear.
P o i n t s on figu r e 5 a n d 6 f a l l i n g b e l o w z e r o - v a l u e d
r e s i d u a l s not c o u n t e r w e i g h t e d by p o i n t s a b o v e z e r o - v a l u e d
r e s i d u a l s s u g g e s t n et o v e r - p r e d i c t i o n .
The o p p o s i t e is t r u e
for u n d e r - p r e d i c t i o n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
1.0
+-
** I
**
I
I
I
.75 +
o
b
s
e
r
I
I
I
.5 +
I
V
e
d
I
I
I
.25 +
+
I
I
I ★-k
+*--
--+ -
—
.25
+.
.75
F i g u r e 1. —
Normal Probability
R e s i d u a l s for M o d e l 1.
1.0
I
I
—
+ Expected
1.0
(P— P ) Plot of S t a n d a r d i z e d
+-
.75 +
I
O
1
b
s
e
r
I
I
.5 +
I
V
e
d
I
I
I
.25 +
i
.**
I ***
+ *
—
+-
.25
.5
75
Figure 2.—
Normal Probability
R e s i d u a l s for M o d e l 2.
+ Expected
1.0
(P-P) Plot of S t a n d a r d i z e d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
Across
Out ++'
3 +
*PRED
Down - *RESID
+ - . —+ ’
-+
+
I
I
2 +
I
I
+
**
*
if
:*
I
I
1
Max N
+
1 +
I
I
0 +
—
Symbols :
I
I
if if ■
*
+
.
I
*
it
4 .0
8.0
17 .0
if if
+
I
I
-2
+
I
I
—3 +
Out ++-3
+
-+ +
- + -
-2
0
2
3 Out
Figure 3.—
S c a t t e r p l o t of the R e s i d u a l s a nd P r e d i c t e d
N a t u r a l — Lo g of P e r C a p i t a T r i p s P r o d u c e d by M o d e l 1.
Across
Out ++3 +
*PRED
+ .
Down - *RESID
-+ +
+
Symbols :
1
Max N
I
2 +
4 .0
8 .0
16.0
1 +
0
+
if •
-1
+
-2
+
-3 +
Out ++-3
*
* .*
+
-+-
-2
+.
-1
-++
-+■
0
1
3 Out
Figure 4.—
S c a t t e r p l o t of t h e R e s i d u a l s a n d P r e d i c t e d
N a t u r a l — Log of P e r C a p i t a T r i p s P r o d u c e d by M o del 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
Across
Out + + 3 +
L CVEDIST
Down - *RESID
—+ .
h-
I
'++
+
Symbols :
I
I
+
2 +
Max N
4 .0
1
1 +
+
*■*
*
I
I
0
8 .0
I
17 .0
«** *
+
*•
I
I
*
*★
-1 +
~3 +
Out ++-3
+
-—
4- -
-+ +
-1
2
3 Out
Figure 5.—
S c a t t e r p l o t of th e R e s i d u a l s P r o d u c e d by M o del
a n d th e N a t u r a l - l o g of A v e r a g e R o u n d - t r i p Distance.
Across - LKVEDIST
Out + + ■
3 +
1
Down - *RESID
— H
.
+-
-+ +
+
Symbols :
I
Max N
I
2 +
4 .0
1 +
:
8.0
*
16.0
0 +
-1
+
—2
4I
I
-3 +
Out ++-3
+
- + -
-2
-—
4- -
-1
-+ +
3 Out
Figure 6.—
S c a t t e r p l o t of t h e R e s i d u a l s P r o d u c e d by M o d e l 2
a n d t h e N a t u r a l - l o g of S h i f t e d A v e r a g e R o u n d - t r i p Distance.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
Finally,
be
several
in e a c h of fi g u r e s
observations
c l u s t e r of p o i n t s
o u t l i e r s was
models
observations
distance
most
in e a c h plot.
a f f e c t i n g the
of th e o b s e r v a t i o n s
from the general
Thus,
The D F F I T S m e a s u r e
(2 to 30 m i l e s
parameters
lying outward
conducted using DFFITS
1 a n d 2.
3 an d 4 t here a p p e a r to
a formal an alysis of
an d D F B E T A S m e a s u r e s on
r e v e a l e d several
fit of the m o d e l s
fr om v a r i o u s
for s horter
sites).
Similarly,
i n f l u e n c i n g the cons t a n t a n d slope
in t h e s e models,
as m e a s u r e d by DFBETAS,
were
o b s e r v a t i o n s w i t h o n e - w a y d i s t a n c e s of 2 to 50 miles.
im p o r t a n t l y ,
however,
D F B E T A S measures)
observations
parameters
represent
all t h r e e m e a s u r e s
i d e n t i f y t rips t a k e n
i n f l u e n c i n g the
(DFFITS an d the two
from A l a s k a as
fit a n d c o n s t a n t an d sl ope
of th e t wo models.
all
Most
of the t rips m a d e
T h ese
20 o b s e r v a t i o n s
f r o m A l a s k a in the 741
o b s e r v a t i o n a g g r e g a t e d sa mple a n d are the only o b s e r v a t i o n s
which were aggregated using a method different
s u m m a r i z e d in C h a p t e r 3.
from that
The e x c e p t i o n is due to a p h y s i c a l
g a p in th e d e f i n e d m a r k e t b e t w e e n M o n t a n a and Alaska,
Can ada.
Recall that when trips
with Montana
fr om states not c o n t i g u o u s
in w h i c h t h e r e w e r e no trip s m a d e to a site
f r o m s t a t e s b e t w e e n th e
th e p o p u l a t i o n of th e
Canada was
state of origin,
i n t e r v e n i n g states.
t h e s e trips c a r r i e d
Thus,
since
e x c l u d e d f r o m the n o n - r e s i d e n t i a l market,
f r o m A l a s k a c a r r y o n l y the p o p u l a t i o n of Alask a,
trips
namely
from origins
whereas
of s i m i l a r d i s t a n c e s to the east
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
t rips
coast
71
carry greater populations.
trips
to
standout
recreationists
to m o d e l
results
rea s o n
to this
anomaly
c o r r e c t e d to a u n i f o r m
in a p p l i c a t i o n of the
ru les p r e s e n t e d in c h a p t e r 3,
trips
an ef f ort was m a d e
f r o m A l a s k a w i t h a d u m m y va ria ble.
of this
effort
a n d 2,
interactive
dummy variable
(Append ix C ) .
Thus,
aggregation process
in m o d e l s
in m o d e l s
1 an d 2 and la a n d 2a
in the
for t rips o r i g i n a t i n g in A l a s k a has not
There for e,
the a d j u s t m e n t to the
for tri ps o r i g i n a t i n g in A l a s k a is o m i t t e d
f r o m the p r i m a r y p o r t i o n of this
study.
readers
of m o d e l i n g trips
Alaska
information,
as n o t e d a b o v e
Box-Cox models
1
is c o n s i s t e n t w i t h the ge n e r a l
c o r r e c t i o n for the a n o m a l y
f u l l y expl ore d.
bivariate models
from A l a s k a
k n o w n w i t h c e r t a i n t y tha t a d d i n g the
heteroscedasticity prevalent
been
trip s
t he fit a n d s a t i s f a c t i o n of n o r m a l i t y
it is not
The
are s u m m a r i z e d in A p p e n d i x C.
A l t h o u g h th e m e t h o d s u s e d to m o d e l
improves
fo r the A l a s k a
is that d i s t a n c e s t r a v e l e d by
f r o m A l a s k a w e r e all
distance.Due
aggregation
Another
the
results
However,
for the
from
in t he m o d e l p r o v i n g best among the
estimated
(model
5)
is p r e s e n t e d in A p p e n d i x
C.
D i s t a n c e s t r a v e l e d f r o m A l a s k a to s it es in M o n t a n a
w e r e a m o n g t h o s e f o u n d to be in e r r o r as n o t e d by D u f f i e l d
( 1 9 8 8 a ) . T h e s e d i s t a n c e s w e r e c o r r e c t e d u s i n g the d i s t a n c e
f r o m A n c h o r a g e , A l a s k a to G r e a t Falls, Montana, via the
A L C A N h i g hway.
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72
General Model
Specifications
In the nex t
functional
step of the analysis,
forms w e r e
general
8 alternative
s p e c i f i e d b a s e d on the gen er al
s p e c i f i c a t i o n of e q u a t i o n
plausible
for the B o x - C o x R e g r e s s i o n s
(2).
Table
5 s u m m a r i z e s these
forms an d the d i f f e r e n t
r e s t r i c t i o n s on
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73
Table
5.—
P l a u s i b l e F o r m s of E q u a t i o n (2) O t h e r than the
D o u b l e - l o g , S e m i - l o g a n d L i n e a r f o r m s ."
Plausible
(a)
A.y
Forms
Restrictions :
^ =Po~Pi---; ^
Aq
_2iv_
__
N/R
N/R
0
N/R
X
ib)
lnl/=Po -
^ +e
vlj -1
(c)— \
=Pq - p ^ l n D j j + e
m
/R
Ay
(d)
y^j =pQ-p^:^{.- + e
1
(e)
- j .^ =po-PiD^^ + e
N/R
Ay
(f)
In
(g)
In ^ j = P o
(h)
N/R
=Po-p^lnP^j+e
y^j=Po-p,ZPjj + e
+e
0
1
The e r r o r t e r m in all p l a u s i b l e forms e s t i m a t e d was
a s s u m e d to e n t e r a d d i c t i v e l y .
Thus, th e func t i o n a l f o r m is
l i m i t e d to i n t r i n s i c a l l y l i n e a r forms.
R e s t r i c t i o n s of 0 a n d 1 r e f e r to a n a t u r a l - l o g
t r a n s f o r m a t i o n a n d a l i n e a r re striction, resp e c t i v e l y .
N/R
r e f e r s to an e s t i m a t e d B o x - C o x t r a n s f o r m a t i o n p a r ameter.
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74
a n d X q r e q u i r e d to o b t a i n t h e s e
1,
th e p r i m a r y t h e o r e t i c a l
that th e
signs a n d sizes
a n d À.D r e s u l t
are
inversely
traveled
thi s
if
is
Po,
in a f u n c t i o n in w h i c h p e r capita trips
r o u n d - t r i p di stance
from origin
constraint
l i s t e d in t a b l e
positive
(2)
of th e e s t i m a t e d p a r a m e t e r s
(the p r i c e proxy)
exception occurs
As n o t e d in ch a p t e r
c o n s t r a i n t on e q u a t i o n
r e l a t e d to a v e r a g e
one e x c eption,
forms
forms.
is less t h a n
in p l a u s i b l e
With
is s a t i s f i e d in all of the
5 when
is negati ve.
i to site j.
form
(b)
zero.
in w h i c h
This
m u s t be
An additional constraint
for
the m o d e l s to be c o n s i d e r e d d e m a n d func t i o n s is that the
estimated parameters
in e q u a t i o n
i n t e r c e p t or c o n s t a n t term.
not a l w a y s be
P q du e to th e
transformation.
(table 5)
equation
As
(2) must y i e l d a p o s i t i v e
one m ay note this t e r m will
s p e c i f i c a t i o n of the B o x - C o x
F o r instance,
is e x p r e s s e d in its
c o n s i d e r p l a u s i b l e form
(a)
r e d u c e d form as shown in
(7):
(7)
V=X,
If t h e a b s o l u t e v a l u e of P i/^d > Po + 1 a n d Xy and po are
positive,
the
i n t e r c e p t t e r m in p l a u s i b l e
negative.
The
Po^
a nd Ip c o n t r i b u t i n g to the
Pi,
term
Xy,
combinations
in e a c h of p l a u s i b l e
form
(a) w i l l be
of p o s s i b l e v a l u e s and signs of
forms
sign of the i n t e r c e p t
(a) t h r o u g h
(e)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(table 5)
75
are t o o n u m e r o u s
to a n a l y z e a n d list here.
intercept
terras in e a c h of the e s t i m a t e d
below are
independently assessed
Thus,
the
forms p r e s e n t e d
for t h e o r e t i c a l
acceptability.
Form Analysis
of th e B i v a r i a t e F i r s t - S t a g e
T CM D e m a n d
Function
Th is
s e c t i o n p r e s e n t s the e s t i m a t i o n a n d m odel
d i s c r i m i n a t i o n p e r f o r m e d on the
first-stage TCM demand
functions
e s t i m a t e d u s i n g th e p l a u s i b l e
table
First,
5.
e a c h of th e m o d e l s
s e c t i o n are g e n e r a l l y reviewed.
models^
fit of models,
s u m m a r i z e d in
e s t i m a t e d in this
This
a d h e r e n c e to the t h e o r e t i c a l
o v e r v i e w o f th e
forms
includes
a s u m m a r y of
constrain ts,
an
an d r e v i e w of the
s i g n i f i c a n c e of th e B o x — Cox t r a n s f o r m a t i o n pa ram eters.
Next,
the
r esults of d i s c r i m i n a t i n g t h e s e m o d e l s u s i n g the
log-likelihood ratio test
is su mm ariz ed.
b y a r e v i e w of the r e s u l t s
tests to model
s ever al
is f o l l o w e d
diagnostic
5.
Estimated M o d e l s .
estimating
of a p p l y i n g
This
five
forms
Table
of th e b i v a r i a t e
demand
f u n c t i o n l i s t e d in t a b l e
(1988)
findings
linear
forms
(f)
6 s u m m a r i z e s the r e s u l t s of
5.
specific estimates
and
Collectively,
m o d e l s in t a b l e 6 was
f o r m s l i s t e d in t a b l e
(h)
(table 5),
f i r s t - s t a g e T CM
B a s e d on Duf field's
of the
s e m i - l o g and
res pec tively,
were
t he a p p r o a c h u s e d to e s t i m a t e the
d e s i g n e d to a l l o w all of the p l a u s i b l e
5 to result.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
omitted
f r o m th is
At
the o u t s e t only m o d e l s
were estimated.
3 . a to be
an al ysis .
However,
3.b,
4 . a,
and 4.b
due to the fa i l u r e of A,v in mode l
significantly different
p r o b a b i l i t y of a t y p e
3 . a,
I error,
fro m zero at a 5 p e r c e n t
model
5,
in w h i c h Xy is
r e s t r i c t e d to a v a l u e of zero,
was
fr om the
the h y p o t h e s e s that the
insignificance
of Xy,
als o esti mated.
remaining
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Aside
CD
■D
O
Q.
C
g
Q.
Table 6.—
Estimated Box-Cox Bivariate First-Stage TCM Demand Functions
■CDD
Maximum Likelihood Estimates
C/)
C/)
Model
OLS
4
^0
Log-Likelihood
Po
Pi
R2
F
CD
8
3. BCE*
a,
/ X-ju
.0016
(.172)
.169
(6.78)
7,361.84
-4.30
(-11.57)
-.629
(-7.01)
.778
2, 601
b.
X.0
.038
(5.45)
.038
(5.45)
7,345.91
-2.92
(-11.32)
-.907
(-8.56)
.775
2,550
-.017
(-6.68)
1 (R***)
7,129.75
-13.59
(-16.20)
-.0045
(-7.35)
.592
1,076
.030
(3.35)
0(R)
7,337.43
-2.15
(-7.50)
-1.22
(10.25)
.768
2,451
0(R)
.172
(6.87)
7,361.82
-4.33
(11.49)
-.630
(-6.99)
.778
2,599
CD
3
.
3
"
CD
CD
■D
O
Q.
C
a
O
3
"O
O
CD
Q.
■CDD
C/)
C/)
4. BC**
a • X-y
b.
5. Xa (BCE)
BCE stands for use of the Box-Cox extended estimation method.
BC stands for use of the classical Box-Cox estimation method.
HRt. signifies a restriction of the Box-Cox transformation parameter at the indicated value.
Pi are the estimated coefficients for per capita trips transformed by the respective value of h .
The p-value for
in model 3.a is ,4317.
All other p-values are less than .001.
The critical values
for t and F at a 5 percent probability of a type I error are 1.6471 and 3.854, respectively. T-ratios
were computed from the variance-covariance matrix resulting from the maximum likelihood estimation of
each of models 3.a, 3.b, 4.a, and 4,b.
Model 5 was estimated using an OLS/grid-search method.
Adjusted-R2 and F-statistics were computed using OLS regression with the appropriate transformations on
the dependent and independent variables as listed in columns 2 and 3.
78
estimated coefficients
in t a b l e
6,
equal
at a 5 p e r c e n t p r o b a b i l i t y of a t y p e
hypothesis
p e r c e n t p r o b a b i l i t y of a type
in th e
Th e
model
3 . a,
the
signs
correct.
W i t h the exception of
of all e s t i m a t e d an d
in e a c h model.
significantly different
in M o del
3 . a is
from zero at the 95 p e r c e n t
i n t e r v a l t h e r e b y y i e l d i n g th e i ntercept t e r m in
3 . a e qual to zero.
Model Discrimi nati on.
p r e s e n t e d in t a b l e
several
D i s c r i m i n a t i o n of the models
6 was p e r f o r m e d u s i n g a series of
l i k e l i h o o d r a t i o tests,
are
listed in
in th e m o d e l s p r e s e n t e d result in
intercept terms
confidence
I e r r o r for all of the mod els
an d m a g n i t u d e s
restricted parameters
model
the
ar e r e j e c t e d at a 5
signs on pp in e a c h of th e m o d e l s
6 are t h e o r e t i c a l l y
not
Further,
same table.
table
positive
I error.
that no r e l a t i o n s h i p e x i s t s b e t w e e n the
independent and dependent variables
presented
zero is rejec t e d
as
combinations
s u m m a r i z e d in C h a p t e r 2.
of m o d e l s
in table
T here
6 that sat isf y
t h e r e s t r i c t i o n t h a t o n l y m o d e l s t h a t can be n e s t e d in t h e i r
n o n — r e s t r i c t e d c o u n t e r p a r t s m a y be d i s c r i m i n a t e d w i t h this
c o u n t e r p a r t u s i n g a l i k e l i h o o d r a tio test.
However,
on t he
u s e d to e s t i m a t e
likelihood principle
the models
in t a b l e
6
a nd the m e t h o d s
(see t a b l e
combinations
of models
combinations
c o n s i s t of m o d e l s
6 an d preamble),
n e e d to be tested.
restricted models of model
3.b,
3.a .
4 . a,
based
o n l y four
These
4.b,
and 5 as
L i k e l i h o o d - r a t i o test s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
79
these
combinations
hypothesis
is s u m m a r i z e d in t able
in this t a b l e
s i m i l a r to the ge n e r a l m o del
The a l t e r n a t i v e h y p o t h e s i s
r e s t r i c t e d and g e n e r a l m o d e l s
Table
The null
is t h a t t he r e s t r i c t e d models
l i s t e d in c o l u m n one are
in c o l u m n two.
7.
list e d
is that the
are dissi mil ar.
7.—
L i k e l i h o o d R a t i o T ests of B i v a r i a t e Mode l s with
M o d e l 3 . a. as th e General, U n r e s t r i c t e d Case
Restricted
Model
Ge n e r a l
Model
D e g r e e s of
Freedom
5.
3 .a .
1
.04
3 .b.
3 .a.
1
31. 86
<.00 01
4 .b .
3. a .
1
48. 82
<.0001
4 .a .
3. a .
1
464.18
1.
3. a .
2
61 .00
Likelihood
R a t i o (% 2 )
P - Val u e
.841
.000
<.0001
It can be c o n c l u d e d f r o m th is a n a l y s i s that only
model
5,
of the m o d e l s p r e s e n t e d in t able
significantly different
6,
is not
f r o m the m o s t g e n e r a l model
p e r c e n t p r o b a b i l i t y of a ty pe
I error
at a 5
(the critical % ^ v a lue
w i t h o n e d e g r e e of f r e e d o m is 3.8415).
Thus,
t h e m a x i m u m l i k e l i h o o d pr i n c i p l e ,
5 c o n s i s t s of t h o s e
p a r a m e t e r s which,
remaining models
model
a m o n g th e p a r a m e t e r s
in t a b l e
a c c o r d i n g to
e s t i m a t e d for the
7, b e s t d e s c r i b e the p o p u l a t i o n of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
p e r c a p i t a d e m a n d for f i s h i n g t rips d u r i n g 1985.
f u r t h e r c o n c l u d e d t hat m o d e l
the r e m a i n i n g m o d e l s
5 is s t a t i s t i c a l l y
in t able
p e r f o r m e d w i t h th e
for the
hypothesis
a l i k e l i h o o d ratio test was
s u m m a r i z e d in t a ble
s u m m a r i z e d in t able
(the c r i t i c a l
7,
7.
The results
s ugg est that the null
v a l u e w i t h tw o d e g r e e s
of
5.9915).
Diagnostic Tests
probability
4,
c o u l d is r e j e c t e d at a 5 p e r c e n t p r o b a b i l i t y of a
I error
f r e e d o m is
p r e s e n t e d in table
same null h y p o t h e s i s as that e s t a b l i s h e d
remaining tests
of th is test,
type
3.a,
super i o r to
7.
S i n c e t he d o u b l e — log model,
can be n e s t e d in m o d e l
It can be
(P-P)
plot
produces
fairly normal
1 a n d 2,
model
on M o d e l
for m o d e l
5.
5.
res iduals.
5 appears
to p r o d u c e
Figure
7 is the n orm a l
As can be seen,
model
5
In c o m p a r i s o n w i t h m o d e l s
slightly more normally
d i s t r i b u t e d resi dual s,
a l t h o u g h t h e y are s l i g h t l y
skewed
left.
s t a n d a r d i z e d r e s idual p l o t
for m odel
5.
Figure
8 is the
In c o m p a r i s o n w i t h the s i m i l a r pl ots
(figures
3 a n d 4),
model
d i s t r i b u t i o n of t h e
t h a n do m o d e l s
1 and 2
5 ap p e a r s to re sul t in a g r e a t e r
residuals
1 a n d 2,
for m o d e l s
a r o u n d the c e n t e r of th e plot
thereby
s u g g e s t i n g less
h e t e r o s c e d a s t i c i t y of the r e s i d u a l s p r o d u c e d by m o d e l
is n o t e d ho wever,
homoscedasticity
t h a t the
r esults of W h i t e ' s test
s u g g e s t s th e r e s i d u a l s
for
in m o d e l 5 are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.
It
81
heteroscedastic.
1.0
+-
** I
ic-k-k
I
75 +
O
b
s
e
r
+
I
I
I
I
V
e
d
Î5 +
+
I
I
I
I ***
+*---
—
■ - + -
.25
.5
+
—
-
.75
I
+ Expected
1 .0
Figure 7.—
N o r m a l P r o b a b i l i t y (P— P ) Plot of the
S t a n d a r d i z e d R e s i d u a l s for M o d e l 5, Tab le 6.
Across - *PRED
Out + + -- ------------ + .
3 +
*
1
" ~ H ------------ ---- — - +
1
+
1
★
.
-
1
+
+
*
1
1
+
•
•
★
k
■k k
1
k k
1
-1
k
+
k
* k
k k k
• • “
•
•
•
• k k
★
» k
•
k
•
k
‘ *
•
•
+
•
1
..
1
k
+
1
1
-3
Out
i
+
+
1
!
1
1
+
+
+
6.0
13.0
1
• - k
•
3 .0
1
• • •
• k
.
k
.
k
1
“2
Max N
+
1
0
Symbols :
1
1
1
+
+
*•
1
2
Down - *RESID
—
----------- \
+
-3
-- -- _|_--- -- + - ---- + —
- 2
- 1
0
1
2
-+ +
3 Out
F i g u r e 8.
S t a n d a r d i z e d S c a t t e r p l o t of p r e d i c t e d Per C a p i t a
T r i p s a n d R e s i d u a l s for M o d e l 5, T a ble 6.
nR2 for m o d e l 5 is 52.74 w i t h a p - v a l u e of this
d i s t r i b u t i o n w i t h 2 d e g r e e s of f r e e d o m less t h a n .0001.
Fu rther, th e 95% c r i t i c a l v a l u e is 5.9915.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
Figure
9 is a s t a n d a r d i z e d sc a t t e r p l o t of the
r e s i d u a l s p r o d u c e d by m o d e l
5 and a v e r a g e roun d - t r i p
d i s t a n c e t r a n s f o r m e d by the B o x - C o x t r a n s f o r m a t i o n p a r a m e t e r
for the
same mode l.
in s i m i l a r p l o t s
model
A l t h o u g h the i n v e r t e d - U
for m o d e l s
5 t h a n it do es
su g g e s t the
specified.'*^
for m o d e l
functional
of figures
1.
d i s p e r s e m e n t of p o i n t s
performance
1,
the
for
shape cont i n u e s to
6 and 9 al so su ggests
of th e o v e r / u n d e r t r i p p r e d i c t i o n p r o b l e m
e n c o u n t e r e d in m o d e l
the d i s p e r s e m e n t
and 2 a ppears less e vident
f o r m m a y not be c o r r e c t l y
Comparison
further resolution
1
shape pre s e n t
This is e v i d e n t by
around
of p o i n t s
of m o d e l s
1,
2,
the w i d e r
the cent e r of figure
in figure
a n d 5 are
6.
9 verses
The p r e d i c t i v e
furthe r d i s c u s s e d
below.
As n o t e d in c h a p t e r 2, the t r a d e - o f f b e t w e e n
h e t e r o s c e d a s t i c i t y a n d f u n c t i o n a l f o r m is o m i t t e d fr o m thi s
study.
It m a y be th e c a s e that the i n v e r t e d - U shape in
f i g u r e 9 is p a r t i a l l y du e to the lack of c o r r e c t i o n for
heteroscedasticity.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
Across
Out + + 3 +
DIST5
Down - *RESID
-+ +
+
S y m b o l s
I
I
:
Max N
2 +
I
I
3 .0
6.0
1 +
13.0
0 +
-1
-2
+
+
I
I
+
+
I
I
+
I
I
+
-3 +
Out + + -
-+ +
-2
-1
2
3 Out
F i g u r e 9.—
S t a n d a r d i z e d S c a t t e r p l o t of R o u n d - T r i p D i s t a n c e
T r a n s f o r m e d b y th e B o x - C o x P a r a m e t e r l i s t e d for M o d e l 5,
T a b l e 6, w i t h t he r e s i d u a l s from the same model.
Lastly,
presence
e x a m i n a t i o n of figure s
of s e v e r a l outliers.
c o n d u c t e d u s i n g th e
The
results
which trips
with average
Thus,
8 an d 9 sugge st the
o u t l i e r a n a l y s i s was
same m e a s u r e s u s e d for m o d e l s
of this a n a l y s i s
for m o d e l
1 an d 2.
5 a nd a m o d e l
in
f r o m A l a s k a w e r e m o d e l e d as an i n t e r a c t i o n t e r m
r o u n d - t r i p d i s t a n c e are p r e s e n t e d in A p p e n d i x
C.
Discrimination of Models
The last
2 and 5.
s t e p in the a n a l y s i s
discr i m i n a t i n g between model
2 a n d 5,
consists
of
the t w o remaining.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
models
c o n s i d e r e d in thi s paper.
no n - n ested,
a J-te s t wa s u s e d
Since these two mo d e l s
for d iscri min ati on.
r esu lts of the h y p o t h e s i z e d m o d e l s
are
The
are su mmarized in table
8.
T abl e
8
Results
Model
(Hypothesis)
of J - T e s t for D i s c r i m i n a t i o n B e t w e e n
M o d e l s 2 a n d 5.
{io
Pi*
P 2**
4 .724
-3 . 5 0 8
-.541
(t-ratio)
(4.793)
(-4.932)
(-1.728)
(p-value)
{<.0001)
(<.0001)
( .042)
2 . (Null)
5.
(Alt)
2 .308
.341
1. 523
(1.703)
(1,728)
( 4.932)
( -044)
(.042)
(<.0001)
R2
F
.785
1, 352
.785
1, 352
Pi are the e s t i m a t e d c o e f f i c i e n t s of ave r a g e r o u n d - t r i p
d i s t a n c e pl us 64 a n d a v e r a g e r o u n d - t r i p d i s tance s u b j e c t e d
to th e B o x — Cox t r a n s f o r m a t i o n w i t h X,= .172.
P 2 ar e the e s t i m a t e d c o e f f i c i e n t s of the p r e d i c t e d
v a l u e s r e s u l t i n g f r o m e s t i m a t i n g mode l s 5 a n d 2 w i t h OLS,
respectively.
The s a m p l e size is 741 a n d t*=1.647 and F =3.007 at the 5
p e r c e n t p r o b a b i l i t y of a ty pe I error.
In t e rms of the m e t h o d
models
s u m m a r i z e d in A p p e n d i x B,
2 an d 5 ar e c o n s i d e r e d th e null and a l t e r n a t e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
hypothesis models
(for g e n e r a l
(B-1)
respectively.
and
( B-2)),
and 5 parallel
and
(B-3).
th e g e n e r a l
The h y p o t h e s e s
of p 2 in b o t h m o d e l s
specifications
see eq uat ions
The e s t i m a t e s
specifications
of mod els
of equations
2
(B-4)
that the e s t i m a t e d coeffi cie nts
2 and 5
(table
8)
are diff ere nt
from
zero are r e j e c t e d for b o t h h y p o t h e s i z e d models at the 5
p e r c e n t p r o b a b i l i t y of a type
criteria
s u m m a r i z e d in t a b l e
I error.
10
The decisi on
(Appendix B)
suggests that
neither model
2 or 5 is a c c e p t a b l e or s u f f i c i e n t l y ex plains
the v a r i a t i o n
in the n a t u r a l - l o g of p e r capi t a trips when
c o m p a r e d to t h e other.
provided inconclusive
Unfortunately,
results.
Th at is,
however,
this test
t h e i r is
i n s u f f i c i e n t e v i d e n c e at a 5 p e r c e n t p r o b a b i l i t y of a type I
e r r o r to r e j e c t the null h y p o t h e s e s
predicted values
to th e
provide
of t h e n a t u r a l - l o g of p e r capita t rip s adds
fit of the rival model.
drawbacks
that ea ch models'
of the J-test.
This
resu lt
Speci f i c a l l y ,
is one of the
the J-test does not
c o n c l u s i v e e v i d e n c e t h a t e i t h e r m o d e l 2 or 5 is
statistically
insufficient
conditional
two models
s u p e r i o r o v e r the other.
This
is cau s e d by
i n f o r m a t i o n r e g a r d i n g the c o m p a r i s o n of the
distributions
(see M a d d a l a
of th e d e p e n d e n t v ariables
in the
(1992)) .
M a d a l l a (1992) s u g g e s t s s u p p l e m e n t i n g the J-test
w i t h an a n a l y s i s of v a r i a n c e in w h i c h the mode l s are
c o m b i n e d a nd t he c o e f f i c i e n t s on the d i f f e r i n g va riables are
t e s t e d for s i g n i f i c a n c e .
However, due to strong c o l i n e a r i t y
b e t w e e n t h e i n d e p e n d e n t v a r i a b l e s in m o d e l s 2 and 5, such
a n a l y s i s was u n s u c c e s s f u l .
A d d i t i o n a l l y , M a d d a l a cites
M i z o n a n d R i c h a r d (1986) for a m e a n s to o v e r c o m e the lack of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
An alternative means
of d i s c r i m i n a t i n g mo dels
5 is to c o m p a r e t h e i r a d j u s t e d - R ^ s .
and
.778
for m o d e l s
d i f f e r e n c e of
explains
2 an d 5,
T h e s e values are
respectively.
.006 b e t w e e n t h ese v a l u e s
less t h a n
dependent variable
2 and
.784
The slight
(i.e.,
mo del
2
1 p e r c e n t m o r e of th e v a r i a t i o n in the
than m o d e l
5)
s u g g e s t s the two m o d e l s may
be c o n s i d e r e d s t a t i s t i c a l l y c omparable.
B a s e d on the a n a l y s i s
models
in this
chapter,
although
2 a nd 5 m a y be c o n s i d e r e d s t a t i s t i c a l l y c o m p a r a b l e in
t e rms of s l i g h t d i f f e r e n c e s
c o n c l u d e d th at m o d e l
in a d j u s t e d —
5 provides
, it m a y also be
an a l t e r n a t i v e m e a n s of
a d j u s t i n g for p r e d i c t i o n erro rs p r i m a r i l y due to m e a s u r e m e n t
e r r o r at
shorter distances
c o m p a r i s o n of fi g u r e s
s c a t t e r p l o t s below)
the
(see D u f f i e l d
11 an d 10
(1988)).
(see s u b s e c t i o n d i s c u s s i n g
shows that b o t h m o d e l s
2 an d 5 en h a n c e
l i n e a r p r i c e / q u a n t i t y r e l a t i o n s h i p of m o d e l
unlike model
2, m o d e l
5 achieves
(1988).
n o n l i n e a r i t y not
Furthe r m o r e ,
model
s p e c i f i e d in m o d e l
to n o t e t h a t b o t h m o d e l s
multivariate
5 s u g ge sts
2.
r e l a t i o n s h i p as s u g g e s t e d
s p e c i f i c a t i o n of m o d e l
a d e g r e e of
It is also i m p o r t a n t
s uggest tha t th e
i n f o r m a t i o n on th e c o n d i t i o n a l
a l t e r n a t e h y p o t h e s i s models.
However,
right as n o t e d by
r e l a t i o n s h i p of
p e r — c a p i t a t r i p d e m a n d for v a r y i n g d i s t a n c e s
n ot a o n e - t o - o n e
1.
thi s e n h a n c e m e n t w i t h o u t
s h i f t i n g th e e n t i r e d e m a n d f u n c t i o n to the
Duffield
A
from a site is
for a
2 by D u f f i e l d
distributions
(1988).
in the null an d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
It is a l s o n o t e d t h a t th e t h e o r e t i c a l ba sis
multivariate
specification of model
has g r e a t e r m e r i t
model
5.
measure
of the
th e c o n s t a n t
for m o d e l
of m a r g i n a l
at least,
characteristics
model
5 m i g h t be that
d i m i n i s h i n g r eturn s
as d i s t a n c e i n c r e a s e s m o r e
However,
2 m ay be a
models
A
it descr i b e s the
for p e r - c a p i t a trips
a c c u r a t e l y t h a n m odel
1.
2 a n d 5 a p p e a r to c apture
of p e r — c a p i t a trip s d e m a n d not
c a p t u r e d in
1.
O v e r v i e w of M o d e l s
1,
2,
and 5
The b a l a n c e of t h i s
comparison
5 in t e r m s
models
chapter provides
of th e b e n c h - m a r k m o d e l s
I n c l u d e d in this
This
in m o d e l
for
f i x e d costs a s s o c i a t e d w i t h m a k i n g a trip.
theoretical basis
de g r e e
2 s u g g e s t e d by D u f f i e l d
t h a n a p o s s i b l e t h e o r e t i c a l ba sis
Specifically,
for a
section
(1 and 2)
an o v e r v i e w
an d m o d e l
is a c o m p a r i s o n of m o d e l s
of s c a t t e r p l o t s
and line-plots
1,
5.
2,
and
of the t hree
a n d t h e i r p r e d i c t i v e p e r f o r m a n c e across the sample.
section
is c l o s e d w i t h a s ummary of the e l a s t i c i t i e s
of
each m o d e l .
Scatter and Line P l o t s .
scatter plots
of th e n a t u r a l
Figures
10,
11,
a n d 12 are
log of p e r - c a p i t a trips v e r s u s
The m o d e l e s t i m a t i o n a n d d i s c r i m i n a t i o n analy s i s
p r e s e n t e d a b o v e is c o n s i d e r e d s u f f i c i e n t for a c o n c l u s i o n
t h a t m o d e l s 2 a n d 5 are s t a t i s t i c a l l y s i m i l a r in d e s c r i b i n g
t h e v a r i a t i o n in th e n a t u r a l - l o g of p e r - c a p i t a t r i p demand.
The pl ots, p r e d i c t i v e p e r f o r m a n c e a n d e l a s t i c i t i e s of each
m o d e l ar e p r o v i d e d for i l l u s t r a t i v e p u r p o s e s only.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
the t r a n s f o r m a t i o n s
in e a c h of m o d e l s
1,
of a v e r a g e
2,
a n d 5,
r o u n d - t r i p d i s tanc e a p p l i e d
re spectively. 50
W h e n c o m p a r i n g f i gures 10, 11, an d 12 it s h o u l d be
n o t e d t h a t t h e h o r i z o n t a l sca le in figure 12 is
more
c o m p r e s s e d t h e n t h a t of figur es 10 an d 11.
This was done to
m a x i m i z e th e s c a t t e r of o b s e r v a t i o n s across the plo t a n d to
i n c l u d e th e m e a n of the i n d e p e n d e n t v a r i a b l e in fi gu re 12.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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I
+
-20 +
+ +
0
+
+-
1.538462
h — — — — + -
3.076923
+---- + 4 .615385
-+ +
6.153846
7.692308
9.230769
N atural Log of Average Round-Trip Distance
Figure 10.—
S c a t t e r p l o t of the n a t u r a l - l o g of p e r - c a p i t a trips
w i t h the n a t u r a l - l o g of a v e r a g e r o u n d - t r i p distance.
Poi n t s w i t h
n u m e r i c v a l u e s (1-9) r e p r e s e n t the same n u m b e r of obser v a t i o n s .
A l p h a b e t i c letter s, A t h r o u g h Z, repre s e n t clu sters of 10 t h r o u g h
35 points, r e s p e c t i v e l y .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N
a
t
u
r
a
1
511
35 1
132
2421
323\ 21
12334
3111
111257
7111
1163
112 341521\31
^223 4415
111 21434
3142
12 222 5344
1 3144
12 145573
1144543
23294
3256768
2254A9
343334
13B45E3
4134272
1115B6322
286333
26221
25
L
0
g
0
f
p
e
r
C
a
P
-10
-
1 2 .5+
1
t
a
T
r
i
P
s
-15
-17 .5 +
1
+
-20 +
++ -
0
■-+
+ ------- +
.538462
+ -------+
3.076923
+ -
4.615385
■- +
+ -------+
^153846
+ ------ +
7.692308
+ ------ + +
9.230769
N atural Log of Average Round-Trip Distance plus 64
Figure 11.—
S c a t t e r p l o t of the n a t u r a l - l o g of p e r - c a p i t a trips
w i t h t he n a t u r a l - l o g of a v e r a g e r o u n d - t r i p dist anc e plus 64.
P o i n t s w i t h n u m e r i c v a l u e s (1-9) r epr esent the same n u m b e r of
A t h r o u g h Z, r e p r e s e n t
observations.
A l p h a b e t i c l etters
c l u s t e r s of 10 t h r o u g h 35 points, r e s p e c t i v e l y
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
N
a
t
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a
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-15 +
-17.5+
121
1
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Nil
11 123111
2^N2 23
111
1
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11
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11
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11
lir442111251 11
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2212
+
-20 +
+
- + ---------------- + -
076923
-4-———-4-153846
- +
- + -
2 :0769
12.30769
15.38462
-+ +
+ -
). 46154
A v e r a g e Round-Trip Distance Subject to a Box-Cox Transformation of
.172
Figure 12.—
S c a t t e r p l o t o f t he n a t u r a l — log of p e r - c a p i t a
t r i p s w i t h a v e r a g e r o u n d - t r i p d i s t a n c e r a i s e d to the .172
p o w e r . P o i n t s w i t h n u m e r i c v a l u e s (1-9) rep res ent the same
n u m b e r of o b s e r v a t i o n s . A l p h a b e t i c letters, A thr o u g h Z,
r e p r e s e n t c l u s t e r s of 10 t h r o u g h 35 points. r e s p e c t i v e l y .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
A s one can see in t h e s e plots,
in t h r e e
tighter
improvements over model
c l u s t e r of o b s e r v a t i o n s
fewer outlying observations
1.
mo dels 2 and 5 result
These include,
a
a r o u n d the regress ion line,
at s horter distance s
(this is
c o n f i r m e d by c o m p a r i s o n of th e D F F I T S a nd DFBETA S for these
tw o model s),
an d g r e a t e r l i n e a r i t y
relationship.
in the p r i c e — quan tit y
The first tw o of these
c o n s i d e r e d as i m p r o v e m e n t s
d o u b l e - l o g model.
improvem ent s
of the i n t e n d e d use of the
Specif ica lly,
s e l e c t e d by D u f f i e l d et al.
the d o u b l e - l o g mod el was
(1987),
in part,
m i t i g a t i n g h e t e r o s c e d a s t i c i t y w h i c h res ul ts
the
can be
as a m eans of
from c o m p r e s s i n g
r a n g e of o b s e r v a t i o n s by m e a n s of the d o u b l e — log form
(see,
e.g.
Gujarati
(1988)).
If on e w e r e to pl ot m o d e l s
1,
w i t h p e r - c a p i t a t rips on the y-axis,
data points
models
plots
ac ross the
2 and 5 on one graph
u s ing a c o n t i n u u m of
range of a verage r o u nd-trip distance,
2 and 5 would appear most
similar.
for all t h r e e m o d e l s are non-linear,
l eas t a m o u n t of n o n l i n e a r i t y
Fu rther,
at
round-trip distances
th e p l o t s of m o d e l s
These plots
an d model
model
2 shows the
1 the m o s t .
less than 48 and 64 miles,
2 an d 5 are b e l o w m o del
r e m a i n above m o d e l
A l t h o u g h the
1,
respectively.
1 up to di sta nce s of about
This pl ot was o m i t t e d from the text due to
illegibility.
The f u n c t i o n s u s e d for t h ese p lots were roughl y
1.71
M o d e l 1 V = .357 D
V = 21 ,977 (D + 64)
Model 2
M o d e l 5 V = ex p (-.667 - 3.663 D-^"^)
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
1,20 0 a n d 1,500 m i l e s
respectively.
model
1
greater
Thus,
for each of m o d e l s
models
2 and 5,
2 a n d 5 d a m p e n the c urvature of
(p arti c ularly at s h o r t e r distances)
linearity
in the p r i c e / q u a n t i t y
t r a n s f o r m e d form of t h e s e models.
This
provi d i n g
r e l a t i o n s h i p in the
illus tra tes models'
2 a n d 5 m i t i g a t i o n of th e o v e r / u n d e r t r i p p r e d i c t i o n pr o b l e m
with model
1.
Predictive P o w e r .
models
1,
2,
The r e s u l t i n g p r e d i c t i v e p ower of
and 5 provide additional
the p e r f o r m a n c e of t h e se models.
i n f o r m a t i o n re gar ding
As expected,
o v e r p r e d i c t s th e 5, 029 a g g r e g a t e d trips
2 ,82 0
or 56.1 percent.
by 287 or 5.7 p e r c e n t
However,
and model
model
model
1
in the sample by
2 u n d e r p r e d i c t s trips
5 o v e r p r e d i c t s trips by 72
or 1.4 p e r c e n t .
Elasticities.
i l l u s t r a t i v e pur poses,
As an a side and p u r e l y for
t able
9 s u m m a r i z e s the e l asticities
e v a l u a t e d at the m e a n of a v e r a g e
(1,119.5)
for e a c h of m o d e l s
1,
r o u n d - t r i p distanc e
2,
a n d 5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
Table
9. —
Estimated Own-Price Elasticities
for M o d e l s 1, 2, a n d 5
Model
of De ma nd
E s t i m a t e d Own- Price
E l a s t i c i t y of De man d
1.
-1.718
2.
-2 .281
5.
-2.1 15
The own p r i c e e l a s t i c i t y
Pi (D^)
for m o d e l
5 is c o m p u t e d by
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 5
CONCLUSIONS
This
limitations
chapter provides
and suggestions
a s ummary of this
study,
for fu r t h e r research.
its
E a c h are
a d d r e s s e d in turn.
Summary
Th e p u r p o s e of this
fo r m of th e
study has b e e n to e xamine the
f i r s t - s t a g e d e m a n d fun ction for the c o l d - w a t e r
stream fisheries
in M o n t a n a d u r i n g the 1985 season.
one p r o v i d e d an o v e r v i e w of the tr av el
This
cost met hodology.
i n c l u d e d a s u m m a r y of the p e r t i n e n t e c o n o m i c t h e o r y
u n d e r l y i n g the T C M m e t hodology,
definitions
s i m p l i f y i n g as sumptions,
of the p r i c e a n d q u a ntity variables,
requirements,
a n d e x a m p l e s of T C M app lic ati ons.
Additionally,
th e l i t e r a t u r e
functional
data
regar d i n g d e t e r m i n a t i o n of the
fo rm of the T C M d e m a n d functio n s u p p o r t i n g the
use of s t a t i s t i c a l
was
Chapter
i n f e r e n c e to deter m i n e the corre ct
form
rev iewed.
C h a p t e r t wo o u t l i n e d the a p p r o a c h to model
s p e c i f i c a t i o n u s e d in this paper.
best
a p p r o a c h wa s to test down
functional
the
form.
However,
It was c o n c l u d e d th at the
from a g e n e r a l
to a speci f i c
due to the i n f o r m a t i o n r e g a r d i n g
f o r m of th e f i r s t - s t a g e T C M d e m a n d f u n c t i o n as
95
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96
i n v e s t i g a t e d by D u f f i e l d et al.
(1987)
an h y b r i d of a t e s t i n g down an d up
an d D u f f i e l d
(AER/TTT)
c o n c l u d e d as the b e s t a p p r o a c h to use in this
is,
the
s t udy r e l i e d on the
(1988),
app r o a c h e s
study.
was
That
findings of the two p r ior
stu dies n o t e d to limi t the f i e l d of p l a u s i b l e
forms of the
d e m a n d function.
C h a p t e r tw o a l s o p r o v i d e d an o v e r v i e w of the Bo x - C o x
methodology.
approaches
w h i c h th e
I n c l u d e d in this
rev i e w we r e four p o s s i b l e
to e s t i m a t i n g a B o x - C o x r e g r e s s i o n function,
of
full m a x i m u m l i k e l i h o o d m e t h o d was d e e m e d b e s t due
to the a b i l i t y to e s t i m a t e the v a r i a n c e — c o v a r i a n c e m a t r i x
e n c o m p a s s i n g all e s t i m a t e d param ete rs.
model
d i s c r i m i n a t i o n w e r e reviewed,
r ati o test,
Box—Cox
Next,
five p l a u s i b l e
a nd the d o u b l e - l o g model
p r o p o s e d for the d a t a b a s e by D u f f i e l d et al.
Davidson
an d M a c k i n n o n ' s
n e s t e d m o d e l s was a l s o
J- test
ini ti a l l y
(1987).
for d i s c r i m i n a t i n g n o n ­
reviewed.
w i t h th e TTT a p p r o a c h to m o d e l
of
i n c l u d i n g the l i k e l i h o o d
w h i c h wa s u s e d to d i s c r i m i n a t e
regression models
two m e t h o d s
Finally,
in a c c o r d a n c e
specification,
several
d i a g n o s t i c t ests on th e a s s u m p t i o n s of OLS u s e d in this
s tud y w e r e
reviewed.
The r e s u l t s of the e c o n o m e t r i c a n a lysis p e r f o r m e d in
th is
s tudy w e r e r e p o r t e d in c h a p t e r four.
for the a n a l y s i s
in th is
study,
As a b e n c h - m a r k
two b i v a r i a t e models were
e s t i m a t e d a n d s u b j e c t e d to seve ral d i a g n o s t i c tests
regarding
c e r t a i n a s s u m p t i o n s of OLS.
These m o d e l s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
followed
97
th e p r i n c i p l e s
model)
u s e d by D u f f i e l d et al.
and Duffield
d i s t a n c e model)
(1988)
{1987)
(a d o u b l e — log,
(a d o u b l e - l o g
with shif ted
to e s t i m a t e m u l t i v a r i a t e c o unter parts to
t h e s e models.
B a s e d on t he
resu l ts of p r i o r studies,
five Bo x - C o x
r e g r e s s i o n m o d e l s w e r e e s t i m a t e d such that eight p l a u s i b l e
forms
c o u l d result.
T h e s e m o d e l s were tes ted for t h eir
a d h e r e n c e to g e n e r a l i t i e s
the
si gns a n d m a g n i t u d e s
coefficients
r e g a r d i n g a pri o r i e x p e c t a t i o n s
of the e s t i m a t e d r e g ression
a n d B o x - C o x t r a n s f o r m a t i o n par ameters.
the m o s t g e n e r a l
a form c o n s i s t i n g of the nat u r a l
r e g r e s s e d on a v e r a g e
(model
the most
form
i n v e s t i g a t e d in t h i s
log of p e r - c a p i t a trips
r o u n d - t r i p d i s t a n c e rai sed to the
(model 3 . a)
study.
c o m p a r i n g th e m o s t g e n e r a l
findings
it was foun d that
5) w a s th e only for m s t a t i s t i c a l l y
general
of D u f f i e l d
95 p e r c e n t
of the TC M d e m a n d f u n c t i o n
A l i k e l i h o o d ratio test
form w i t h Model
et ai.
from the
confidence
.172
si m i l a r to
(1987),
1, b a s e d on the
was also conducted.
Th is m o d e l was a l s o d i s s i m i l a r to the most general
th e
Only
f o r m f a i l e d to m e e t thes e requ irements.
U s i n g a series of l i k e l i h o o d ratio tests,
power
of
interval,
for m at
thus e l i m i n a t i n g m o d e l
1
f i e l d of s t a t i s t i c a l l y a c c e p t a b l e models.
A n a t t e m p t to d i s c r i m i n a t e M o d e l 5 and the b i v a r i a t e
s p e c i f i c a t i o n of t h e double-log,
a s hift
f a c t o r of 64
inconclusive
results.
(model 2)
Thus,
s h i f t e d di st an ce m o d e l w i t h
u s i n g a J-test p r o v i d e d
t h e s e t wo mo d e l s we re c o m p a r e d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
b a s e d on t h e i r a d j u s t e d —
of t h e s e
statistics were
statistics.
B e c a u s e the va lues
f ound to be cl ose
for m o d e l s
2 a n d 5,
two m o d e l s
are s t a t i s t i c a l l y comp ara ble.
was
Moreover,
1985 than the does the d o u b l e - l o g model.
5 r e v e a l e d th at m o d e l s
2,
and
2 an d 5 re sult in five imp rov e m e n t s
re la tionship,
a r o u n d the
a t i g h t e r clu s t e r of dat a
regression
heteroscedasticity
of o u t l i e r s
1,
T h e s e in c l u d e g r e a t e r l i n e a r i t y in the
price/quantity
points
5
i n f o r m a t i o n r e g a r d i n g the
C o m p a r i s o n s of the s c a t t e r p l o ts of models
1.
mod el
of the p e r - c a p i t a d e m a n d for stream fis hi ng
in M o n t a n a d u r i n g
over model
.778
r e s p e c t i v e l y ) , it was c o n c l u d e d that the
f o u n d to p r o v i d e m o r e
characteristics
(.784 and
lines,
(compare f igures
for s h o r t e r d i s t a n c e s
furth er m i t i g a t i o n of
3,
4,
and 8),
mitigation
(possibly re l a t e d to the
m i t i g a t i o n of h e t e r o s c e d a s t i c i t y ) , and e n h a n c e d p r e d i c t i v e
power.
As
re g a r d s th e
latt e r of thes e i mpro v e m e n t s mode l s
2
a n d 5 p r e d i c t e d t r i p s w i t h i n 5.7 an d 1.4 pe r c e n t of o b s e r v e d
trips,
whereas model
1 o v e r p r e d i c t e d trips by 56.1 percent.
L i m i t a t i o n s an d S u g g e s t i o n s
This
study,
these
for F u r t h e r R e s e a r c h
section highlights
t h r e e of w h i c h m a y m e r i t
issues
fu r t h e r research.
E a c h of
i n c l u d e h e t e r o s c e d a s t i c i t y of the residuals,
t r u n c a t e d d i s t r i b u t i o n of the
in m e a s u r e m e n t
bias.
four l i m i t ations to this
residuals
of r o u n d - t r i p di stance,
An additional
includes estimation
suggestion
in m o d e l 5,
a
err ors
a n d om i t t e d v a r i a b l e
for f u rther re se arch
of the d e m a n d f u n c t i o n to accou nt
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for
99
possible
segments
E a c h of t h e s e
in the m a r k e t by use of a spline
iss u e s
are s u m m a r i z e d in turn.
Heteroscedasticit v .
As p r e v i o u s l y noted,
a t t e m p t s w e r e m a d e to m e a s u r e
study.
Moreover,
2 a n d 5 a p p e a r to m i t i g a t e h e t e r o s c e d a s t i c i t y ,
as is e v i d e n t
in the p l o t s
of the models"
t h e i r p r e d i c t e d values.
W h i t e ' s test
are no t h o m o s c e d a s t i c .
Thus,
residuals
the e s t i m a t e d c o efficients in
or l i n e a r estimates,
t h e y be c o n s i d e r e d e f f i c i e n t
(see K m e n t a
s o l u t i o n to this
again st
sugges ts the res iduals
t h e s e m o d e l are n e i t h e r b e s t
possible
no
or c orrect h e t e r o s c e d a s t i c i t y
in any of the m o d e l s p r e s e n t e d in this
while models
function.
no r can
(1986)).
A
l i m i t a t i o n m a y be d e t e r m i n e d by
r e e s t i m a t i n g the m o d e l s p r e s e n t e d in this p a p e r by
specifying likelihood functions
ca p a b l e of sim u l t a n e o u s l y
e s t i m a t i n g the B o x — Cox p a r a m e t e r s
s h ift p a r a m e t e r
(model 2),
heteroscedast icity
a n d the
(1975)
n otes
resi d u a l s .
parameters
function
As
the
form of
of the Resid u a l s
in M o del
5.
th at us e of th e B o x - C o x t r a n s f o r m a t i o n on
the d e p e n d e n t v a r i a b l e
th i s v a r i a b l e
form,
.
Truncated Distribution
Smith
for fu nctiona l
does
not a l l o w for negat ive values
in
r e s u l t i n g in a n o n - n o r m a l d i s t r i b u t i o n of the
a consequence,
he n o t e s that esti mates of the
in a B o x — Co x r e g r e s s i o n m o d e l u s i n g a l i k e l i h o o d
specified with a normal
d i s t r i b u t i o n of the
S ee f o o t n o t e 19, C h a p t e r 2 for p e r t i n e n t cita t i o n s
o f l i t e r a t u r e a d d r e s s i n g thi s issue.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
100
residuals
Th e
r esults
in a p p r o x i m a t i o n s
likelihood function
Co x m o d e l s p r e s e n t e d
s p e c i f i e d for e s t i m a t e s of the Box-
in t a b l e
n o r m a l l y distri b u t e d .
of these parameters.
6 a s s u m e d the residuals were
There for e,
models
3 . a,
3.b,
4 . a,
4.b,
a n d 5 are c o n s i d e r e d o n l y a p p r o x i m a t i o n s of the mo dels that
w o u l d be e s t i m a t e d g i v e n th e
la titudes)
(or
c a p s u l a t e d in e a c h of t h ese m o d e l s when the
truncated distribution
anal ysi s.
same r e s t r i c t i o n s
issue
is i n c o r p o r a t e d in the
No f u r t h e r a n a l y s i s on this issue was conducted.
E rro r s
in M e a s u r e m e n t of the D i s t a n c e V a r i a b l e .
n o t e d in c h a p t e r 3,
se v e r a l of the r e p o r t e d d i s t a n c e s were
f o u n d to be in e r r o r a n d w e r e
m a p dista nce s.
As
r e p l a c e d in the data b a s e w i t h
T h e s e d i s t a n c e s we re m e a s u r e d by a s s u m i n g
tr ip s o r i g i n a t e d from th e p o p u l a t i o n cent ers of each county
or state.
small
Duffield
sample
size
(1988)
n otes this is a result of the
in the data
set.
He also notes that
a g g r e g a t i o n of t rips w i t h d i s p a r a t e d i s t a n c e s wit hin or igin
zones
to a common
site a d d to the eff ect of the d i s tance
measurement error occurring
dist a n c e s .
He s u g g e s t s
that
l a r g e l y in the range of sho r t e r
s h i f t i n g d i s t a n c e by a constant
a p p e a r s to m i t i g a t e th e e f f e c t s
s h o r t e r d i s t a n c e s by
these
of the m e a s u r e m e n t e r r o r at
s h i f t i n g the d e m a n d fu nc ti on away from
s h o r t e r distance.^'*
D u f f i e l d (1992) f u r t h e r r e f i n e d th e sh if te d
dist a n c e , d o u b l e - l o g m o d e l by t r a n s f o r m i n g a v e r a g e - r o u n d
t r i p d i s t a n c e for r e s i d e n t a n d n o n - r e s i d e n t angl ers by the
t o t a l cost f u n c t i o n s e s t i m a t e d for e a c h clas s in D u f f i e l d et
al (1987).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 0 1
Con ver sely,
c a p t u r e d by m o d e l
figures
model
1 , which
10 an d 12.
can be
However,
error-in-variables model
errors
5 adjusts
for the n o n l i n e a r i t y not
seen by comp a r i s o n of
an e x p l i c i t e x a m i n a t i o n of an
or o t h e r m e a n s
of cor rection
for
in m e a s u r e m e n t w i t h i n the f r a m e w o r k of the Bo x - C o x
r e g r e s s i o n m o d e l s w e r e not e x a m i n e d in this study.
further analysis
of thi s
Thus,
sort m a y p r o v i d e a furth er
e n h a n c e m e n t to th e analysis.
Omitted Variable B i a s .
s t u d y was
Since the scope of this
l i m i t e d to d e t e r m i n i n g the f u n ctional
b i v a r i a t e T C M d e m a n d function,
the mod e l s e s t i m a t e d in this
s t udy ar e p r o n e to o m i t t e d v a r i a b l e bias.
e x a m i n a t i o n of the
functional
attributes,
T hese v a r i a b l e s m i g h t
in the m o d e l
p arameter estimates
include
site
an d several
/ demographic variables
a p p r o x i m a t e tas t e s a n d p r e f e r e n c e s .
these variables
fur ther
include the a d d i t i o n of
a m e a s u r e of subs tit ute s,
socio-economic
Thus,
form of the d e m a n d funct i o n
for s t r e a m f i s h i n g in M o n t a n a m i g h t
n o n - p r i c e v a riables.
form of the
d e s i g n e d to
F a i l u r e to include
w o u l d re sult
in b i a s e d
a n d t h e r e b y b i a s e d estim a t e s of c o n s u m e r
A ge n e r a l B o x - C o x m o d e l s i m i l a r to m odel 3 .a was
e s t i m a t e d for m o d e l 2.
However, it was f ound that the BoxC o x t r a n s f o r m a t i o n on t he d e p e n d e n t an d i n d e p endent
v a r i a b l e s b o t h c o n v e r g e d to zero, y i e l d i n g the initial
d o u b l e - l o g form as that w h i c h m a x i m i z e d the l i k e l i h o o d
function.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
surplus
(see,
e.g.,
D w y e r et al.
(1977)
and W alsh
( 1 9 8 6 ) ) . T h e se n o n - p r i c e v a r i a b l e s are in tended to
explain potential
sh ift s
in demand.
u s e d to e v a l u a t e the e ffects
benefits
site
on d e m a n d and net econo mic
c h a n g i n g the a t t r i b u t e s of one or more
or c h a n g i n g a site's access
In thi s
Duffield
regard,
(1988)
or l i m i t a t i o n s
costs.
the r e a d e r is a d v i s e d to e xamine
in w h i c h he lists three a d d ition al
to the
the m e a s u r e m e n t
stre a m f isheries data set.
e r r o r issue,
the se issues
economic/demographic variables
of th e a n g l e r ' s of an o r i g i n
Market
Segmentation.
research regarding functional
(1987)
o v e r p r e d i c t e d trips
underpredicted trips
analysis
plots
use of s o c i o ­
zone.
A n o t h e r s u g g e s t i o n for further
fo rm w o u l d be to e s t i m a t e p e r fu nction
(Greene
(1990)).
fou nd th at t h e i r d o u b l e - l o g m o d e l
for sho rt an d longer dist ances an d
for m e d i u m range distances.
of the r e s i d u a l s v e r s u s
in f igures
O t h e r tha n
l i m i t e d to the r e p r e s e n t a t i o n
c a p i t a t r i p d e m a n d u s i n g a spl ine
et al.
concerns
include
m i s s p e c i f i c a t i o n of the s u b s t i t u t e variable,
Duffield
can be
of a d d i n g a n e w site to or d e l e t i n g an e x i sti ng
f r o m a region,
sites,
These variab les
6 and
average
9 for m o d e l s
Also,
round-trip distance
2 and 5,
respectively.
S u c h bi as c o n s u m e r su r p l u s is e x p e c t e d to o c c u r if
t h e b i v a r i a t e d e m a n d f u n c t i o n p r e s e n t e d in this p a p e r is
u s e d to c o m p u t e c o n s u m e r s urplus values.
R eade rs are
r e m i n d e d th at the p u r p o s e of this p a p e r is to exa min e the
f u n c t i o n a l f o r m of the b i v a r i a t e model.
Thus, the n o n — p rice
v a r i a b l e s l i s t e d in thi s s e c t i o n are i n t e n t i o n a l l y o m i t t e d
f r o m t h e s p e c i f i c a t i o n p r e s e n t e d in c h a p t e r four.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
reveals
t h e t h i s p o o r p r e d i c t i o n p r o b l e m wa s not en tirely
addressed.
to s e g m e n t
Simply
stated,
a spline
the m a r k e t by i n c r e m e n t s
testing whether different distance
different market
variables
segm ents.
delimiting market
a log— likelihood
function,
which also accounts
f u n c t i o n could be used
in distance,
intervals
there by
represent
In a b r o a d e r scope,
the dummy
s e g ments c o u l d be e s t i m a t e d w i t h
as s u g g e s t e d by Greene
for fun c t i o n a l
form an d
h e t e r o s c e d a s t i s t i c effects.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(1990),
104
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109
APPENDIX A
T E C H N I C A L D E S C R I P T I O N OF THE T R A V E L - C O S T M O D E L
As n o t e d in c h a p t e r one,
of statistically
recreation
terms
A l t h o u g h the T C M has been
o f h o u s e h o l d pro duc tion,
(1985)
zonal TCM consis ts
e s t i m a t i n g a d e m a n d funct i o n for use of a
site.
justification
the
the
justified in
following provides a
in t erms of u t i l i t y m a x i m ization.
provides
the f o l l o w i n g g eneral
d e v e l o p m e n t of the
f i r s t — s tage T C M d e m a n d f u n c t i o n b a s e d on the
quasi-concave
time
utility
McConnell
following
f u n c t i o n s ubject to income an d
constraints:
(A-1)
u(x, z) s. C. y = c x + p z , T = h ^ xi
+ t:^)
Where
X
= N u m b e r of t r i p s to ag i v e n site
z
= Hicksian composite
commodity
h
= Time spe nt w o r k i n g
tj= t r a v e l time p e r t r i p
tz = t i m e s p ent on site p e r t r i p
T
= t otal a v a i l a b l e tim e
y
= e x o g e n o u s inc o m e
c
= o u t - o f - p o c k e t costs p e r trip
p
= P r i c e of the c o m p o s i t e c o m m o d i t y
y
= yo + wh
w
= the w a g e rate
This m o d e l
as s u m e s
recreationists
can tr ade b e t w e e n
H e n d e r s o n a n d Q u a n d t (1980) sh ow the u t i l i t y
f u n c t i o n m u s t be s t r i c t l y q u a s i - c o n c a v e in t e r m s of the
u t i l i t y a n d p r i c e space.
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1 1 0
w o r k a n d l e i s u r e at a c o n s t a n t
h are
time
all m e a s u r e d in the
constraints
(A- 2 )
are
rate w h i c h m e ans
same units.
t2 f and
W h e n the income and
c o m b i n e d u t i l i t y m a x i m i z a t i o n becomes;
u(x, z) + [y* - c* x - pz]
Where
(A— 3)
(A-4)
With
y * = y + wT
c* = w
+ t2 ) + c
f i rst o r d e r
which results
(A- 6 )
in d e m a n d g i ven by:
XT = f i c \ p, y*)
which reduces
{A-1 )
c o n d i t i o n of:
X
to:
= f {c*, y*)
w h e n p is a s s u m e d c o n s t a n t in th e cross section,
the time
f r a m e t y p i c a l l y u s e d fo r T C M d e m a n d f u n c t i o n estimation.
Thus,
Full
d e m a n d is a f u n c t i o n of full
costs
opportunity
generally
inco me an d full costs.
c o n s t i t u t e o u t - o f - p o c k e t costs and the
cost of t r a v e l time.
A c c o r d i n g to gen era l
d e m a n d t h e o r y t h e s e v a r i a b l e w o u l d be i n v e r s e l y r e l a t e d to
t h e q u a n t i t y of r e c r e a t i o n t rips demanded.
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I l l
APPENDIX B
O V E R V I E W OF THE A P P L I C A T I O N OF
THE D A V I D S O N A N D M A C K I N N O N J-TEST^®
The D a v i d s o n a n d M a c K i n n o n J— t e s t
th e p r e d i c t i v e p o w e r of tw o rival,
n o n - n e s t e d models.
following
s u m m a r i z e s the m e t h o d o l o g y
which may
i n c l u d e B o x — Cox t r a n s f o r m a t i o n s
variables.
is a joint test of
The
for a general case
in the re gression
The null a n d a l t e r n a t e h y p o t h e s i s models are
s p e c i f i e d as in e q u a t i o n s
(B-1)
and
(B-2)
in whic h the e r ror
t e r m of e a c h is a s s u m e d n o r m a l l y and i n d e p e n d e n t l y
distributed with
zero m e a n a n d c o n s t a n t variance:
K
(B-l)
k=l
K
iB-2)
y = Y o + E 1ft
k=l
The test
is b a s e d on an a r t i f i c i a l n e s t i n g of the two models
a n d is p e r f o r m e d in t w o stages.
nu l l
hypothesis model
(equation
In th e
(B-l))
first stage,
is a r t i f i c i a l l y
n e s t e d in the a l t e r n a t e h y p o t h e s i s m o d e l
58
This
the
(equation
s u m m a r y rel i e s h e a v i l y on M a d d a l a
(B-2)).
(1992).
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112
This
is do ne by
(B-l)
s p e c i f y i n g the p r e d i c t e d valu e s of e q u a t i o n
resulting
variable
f r o m OL S e s t i m a t i o n as an add itional
in e q u a t i o n
(B -2) to form e q u a tion
(B-3).
K
y=Yo+E
{B-3)
ic=l
Vo r e p r e s e n t s th e p r e d i c t e d val u e s
of the null hy p o t h e s i s
f u n c t i o n a n d ct is t h e e s t i m a t e d parameter.
hypothesis
Ha : (X
test
0.
function
is t h e n e s t a b l i s h e d in w h i c h Hg: a = 0 an d
If t h i s t e s t
ac c e p table,
A secondary
shows the null hypo t h e s i s
is
it is c o n c l u d e d th at the null h y p o t h e s i s
(eq uation
( B -l)) does not add any furt her
d e s c r i p t i o n of th e v a r i a t i o n in Y th en that p r o v i d e d by the
independent variables
information
(B-2).
However,
this
is i n s u f f i c i e n t to deter m i n e w h e t h e r the null
hypothesis model
model.
in e q u a t i o n
Thus,
e n c o m p a s s e s the a l t e r n a t i v e hyp o t h e s i s
the
s e c o n d stage of the test cons ist s of
a r t i f i c i a l l y n e s t i n g the a l t e r n a t i v e h y p o t h e s i s mode l
(e quat ion
1)).
(B-2))
in th e null h y p o t h e s i s m o d e
S i m i l a r to t h e
f irs t
(equation
st age of the test,
by s p e c i f y i n g th e p r e d i c t e d v a l u e s of e q u a t i o n
resulting
equation
this is done
(B-2)
fr om O L S e s t i m a t i o n as an a d d i t i o n a l v a r i a b l e
(B-l)
to f o r m e q u a t i o n
(B-4).
K
(5-4)
(B-
+
k=l
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in
113
Ya r e p r e s e n t s
hypothesis
test
test is then H,, : 5 = 0 and H,:
shows th e null h y p o t h e s i s
c o n c l u d e d t h a t the a l t e r n a t e h y p o t h e s i s
(B-2))
of the a l t e r n a t e
f u n c t i o n a n d ô is the e s t i m a t e d parameter.
secondary hypothesis
If thi s
the p r e d i c t e d va lue s
does no t a d d any
8
is acceptable,
fun ction
# 0.
it is
{equation
fur t h e r d e s c r i p t i o n of the v a r i a t i o n
in Y t h e n t h a t p r o v i d e d by the i n d e p endent v a r i a b l e s
equation
The
in
(B-l).
Maddala
the J - t e s t
(1992)
in a t a b l e
summarizes
the p o s s i b l e o u t c o m e s
of
s i m i l a r to that p r o v i d e d in t able
10
below.
Table
10.—
P o s s i b l e O u t c o m e s of The J-test.
Hypot hes is:
Hypothesis :
8 = 0
a = 0
Not R e j e c t e d
Rejected
Not Rejected
B o t h Ho : an d H^ :
a re a c c e p t a b l e
Hg : is a c c e p t a b l e
Ho : is not a c c e p t a b l e
Rejected
Hq : is a c c e p t a b l e
H 3 : is not a c c e p t a b l e
N e i t h e r H q : nor
Hg : are a c c e p t a b l e
Source:
Maddala
(1992)
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114
APPENDIX C
TECHNICAL DESCRIPTION
OF D I A G N O S T I C TESTS
AND MODELING
This
several
appendix provides
t echn i c a l d e scriptions of
of the d i a g n o s t i c t est s
Technical
DFFITS
OF OUTLIERS
descriptions
sum m a r i z e d in c hap ter 2 .
of W h i t e ' s test for h o m o s c e d a s t i c i t y ,
and DFBETAS measures
for d e t e c t i n g outliers,
and a
m e t h o d of m o d e l i n g the e f f e c t s of outli ers usin g dumm y
variables
are
s u m m a r i z e d in turn.
This is fo llowed by a
s u m m a r y of p r e l i m i n a r y m o d e l i n g eff ort s d e s i g n e d to a ddress
th e a n o m a l y
trips
in the d a t a a g g r e g a t i o n proc ess p r e s e n t e d by
o r i g i n a t i n g in A l a s k a n o t e d
specifications
of
models
1,
in Chapter 4.A l t e r n a t i v e
2 (table 4),
and
5(table 6 ) are
p r e s e n t e d in t u r n .
White's
Test
for H o m o s c e d a s t i c i t y
White's test
r e g r e s s i n g the
for h o m o s c e d a s t i c i t y consists of
squared errors
on the variab les
in the
r e g r e s s i o n m o d e l b e i n g t e s t e d u s i n g the general for m in
equation
(C-1).
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115
(C- 1 )
e| = 5 o + E
p=l
In t h i s
equation
regression
are the
coefficients,
residuals,
Z^, and p is d e t e r m i n e d by the
number of independent variables
b e i n g t e s t e d u s i n g th e
(1986) .
and
When there
= Xij a n d
variables,
, and
in the r e g r e s s i o n
Z^2 ~
Zj5 = X^^^.
■
(X) , p = 2
W h e n t here are tw o i n d e p e n d e n t
= X^j,
Regressors
likewise
function
f o l l o w i n g e x a m p l e t a k e n from Kmenta
is one i n d e p e n d e n t v a r i a b l e
p = 3 and
determined
Ô are e s t i m a t e d
Z^^ = X^^,
Zj^ = X^jX^^,
in e q u a t i o n
(C-1)
for r e g r e s s i o n s w i t h m o r e
Z^^ =
are
independent
variables.
T he nu ll h y p o t h e s i s
variances
this
of th e r e s i d u a l s
hypothesis
also
of W h i t e ' s te st is that the
are constant.
Fa i l u r e to reject
says t h a t any h e t e r o s c e d a s t i c i t y
c a u s e d by
s a m p l i n g error.
The asymptotic,
statistic
is c o m p u t e d by n (R^J
is
large-sample
w h i c h is d i s t r i b u t e d as a
w i t h p d e g r e e s of freedom.
DFFITS
and DFBETAS Measures
Th e g e n e r a l
i m p a c t of o u t l i e r s
study
a p p r o a c h to d e t e c t i n g and m e a s u r i n g the
on a r e g r e s s i o n
f u n c t i o n u s e d in this
is b a s e d on th e l e v e r a g e a single o b s e r v a t i o n or g r o u p
of observations
has on the m e a n a n d v a r i a n c e of the
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116
r e m a i n i n g obse r v a t i o n s .
those
These methods
in w h i c h o b s e r v a t i o n s
variables
can be d i v i d e d into
on the d e p e n d e n t and i n d e p endent
are e x a m i n e d as o u t l i e r s
i n f l u e n c i n g the
the
regression
and t h o s e w h i c h i d e n t i f y cases
th e
fit of the r e g r e s s i o n e q u a t i o n or the val u e s of
e s t i m a t e d coef f i c i e n t s .
values
influencing
le verag e
(h^^) w h i c h c o n s i s t of the d i a g o n a l el eme nts of the
hat m a t r i x
variables
Greene
Both methods utilize
fit of
s t e m m i n g fr om the data
(see,
(1990)
e.g.
Neter
et al.
for the i n d e p e n d e n t
(1989),
Kmenta
(1986)
for d e v e l o p m e n t of the hat m a t r i x ) .
e x p l a n a t i o n of the
or
An
s e c o n d of t h e s e g e n e r a l m e t h o d s
c l a s s i f i e d he re an d use of l e v erage val u e s
in each are
s u m m a r i z e d below.
N e t e r et al.
(1989)
sugges t use of DFF I T S and
D F B E T A S m e a s u r e to d e t e c t i n f l u e n t i a l
w h i c h m a k e use of l e v e r a g e values.
observ ati ons ,
b o t h of
D F F I T S m e a s u r e s the
i n f l u e n c e a p a r t i c u l a r case has on the fit of a r e g r e s s i o n
equation.
equation
(C-2)
In t h i s
(C— 2):
D FFITS, = J i Z l i U L
equation
variable)
fo r t h e
This m e a s u r e m a y be c o m p u t e d for ea ch case by
is the
for th e ith case,
f i t t e d v a l u e of Y
Y^(,, is the p r e d i c t e d v a l u e of Y
ith case w h e n it is o m i t t e d
r e g r e s s i o n equation,
(the d e p e n d e n t
from the e s t i m a t e d
MSE^ is the m e a n
s q u a r e d e r r o r w h e n the
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117
i th c a s e
i th
is omitted,
case.
Thus,
deviation
of the
w h e n the
and
is th e
DFFITS^ is a s t a n d a r d i z e d m e a s u r e of the
regression
it h case
fit w h e n all data are us e d and
is removed.
In larg e dat a sets,
a b s o l u t e v a l u e of D F F I T S v a l u e s
indicate
an i n f l u e n t i a l
estimated coefficients
sample
size,
l e v e r a g e v alue of the
the
e x c e e d i n g the 2 (k/n )'
case.
k a n d n are the number of
in the
r e g r e s s i o n e q u a t i o n and the
respe c t i v e l y .
D F B E T A S m e a s u r e s th e i n f l u e n c e a case has on the
s l o p e p a r a m e t e r of a p a r t i c u l a r v a r i a b l e or on the co nstant
term.
DFBETAS values
(C-3)
are c o m p u t e d u s i n g e q u a t i o n
(C-3):
DFBETAS^^
yjMSE~c^
In t h i s
cases
equation
are
[3^ is the e s t i m a t e d c o e f f i c i e n t when all N
i n c l u d e d for
regression
estimated
coefficient
the ith case,
is
w h e n the ith case
is o mi t t e d from the
r e g r e s s i o n equat ion,
MSE^ is the m e a n
e r r o r w h e n the ith case is omitted,
diagonal
e l e m e n t o f th e
the e s t i m a t e d
and
(X'X)~^ matri x.
squa red
is the kth
DFBETASis
a
s t a n d a r d i z e d m e a s u r e o f the d i f f e r e n c e b e t w e e n the e s t i m a t e d
regression
when
coefficient
it is omitte d.
w h e n th e ith case
In l a r g e d a t a sets N e t e r et al.
r e c o m m e n d th a t a b s o l u t e v a l u e s
2 / (n)^ s h o u l d
in i n c l u d e d and
(1989)
of D F B E T A S v a l u e s e x c e e d i n g
be c o n s i d e r e d i n f luential.
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118
M o d e l i n g the E f f e c t s of O u t l i e r s
O n c e dat a o u t l i e r s
th e
are i den ti fied ,
function with dummy variables
collective
s p e c i f i c a t i o n of
can be u s e d to assess the
imp act a g r o u p of o u t l i e r s
coefficients
how this
Using Dummy Variables
has on certain
i n c l u d i n g the intercept.
Kmenta
can be done u s i n g a d u m m y v a r i a b l e
(1986)
shows
s p e c i f i e d in the
f u n c t i o n as v a r i a b l e a f f e c t i n g the i n t e r c e p t an d as an
i n t e r a c t i o n t e r m w i t h any of the v a r i a b l e s
thereby
a f f e c t i n g the
instanc e,
slope of t h ese variabl es.
p r e l i m i n a r y a n a l y s i s of the
f u n c t i o n e s t i m a t e d in this
per capita visits
in the
functi on
Fo r
f i r s t — stage d e m a n d
study shows that a g g r e g a t i o n of
f r o m A l a s k a do not
foll o w the rules u s e d
to a g g r e g a t e the b a l a n c e of the o r i g i n — d e s t i n a t i o n p a i r s
the
s t r e a m f i s h e r i e s d a t a set
account
(see C h a p t e r F o u r ) .
for th i s d i f f e r e n c e e q u a t i o n
in
To
(2 ) c o u l d be s p e c i f i e d
as
(C-4)
Po - PiD^j + ^ 2 A K +
w h e r e A K is a d u m m y v a r i a b l e
observations
all t r i p s
allows
of trips
term
Specifically,
equation
in w h i c h A K = 1 for
o r i g i n a t i n g in A l a s k a a nd A K = 0 for
fr om all o t h e r origins.
analysis
constant
p,AK(D,j) + e
(C-5)
of ho w tr ips
(p^)
This
s p e c i f i c a t i o n then
from A l a s k a
and price coefficient
if A K = I then
inf luence the
(pj .
equation (C-4) would reduce to
in which the impact of
trips taken from
A l a s k a w o u l d influence the constant and slope coefficients
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119
of e q u a t i o n
(C-5)
(2 ) in its l i n e a r form.
V^j= (po + Pa) + (p3 - Pi)£>ij + e
Howeve r,
if A K = 0,
equation
(C— 4) w o u l d reduce to e q u ation
(C— 6 ) in w h i c h t h e r e w o u l d be no im pac t on either the
constant
(C-6)
or slope c o e f f i c i e n t s
due to trips
from Alaska.
y , - Po - PiD^j + e
The i m p a c t of t r i p s
coefficients
statistical
Expansion
from A l a s k a on the c o n st ant and slope
can be d e t e r m i n e d fr om h y p o t h e s i s tests of the
s i g n i f i c a n c e of
a n d P^,
respectively.
of the a n a l y s i s to i n c o r p o r a t e the Box-Co x
t r a n s f o r m a t i o n on
would result
and
in e q u a t i o n
as s p e c i f i e d in equa t i o n
(C-7)
(2 )
w h e r e the Box-Co x
t r a n s f o r m a t i o n on ave r a g e r o u n d - t r i p d i s t a n c e is the same
fo r b o t h o c c u r r e n c e s
Thus,
of this v a r i a b l e
(C-7)
(C-7).
i n c l u s i o n of A K in this wa y al lows ad justmen t of the
Box-Cox transformation
trips
in equ ati on
a c c o r d i n g to the influen ce of
f r o m Alaska.
+ PzAAT + p 3 A f C ( D y ° ’) - e
ReprocJucecJ with permission of the copyright owner. Further reproctuction prohibitect without permission.
1 2 0
Analysis
o f M o d e l i n g Trips
in M o d e l s
from A l a s k a as a Dummy V a r i a b l e
and 2
1
To u n d e r s t a n d the affe ct of the A l a s k a trips on each
of m o d e l s
1
an d 2,
for t r i p s
from A l a s k a a f f e c t i n g the model b o t h as a shift
a n d s lope p a r a m e t e r
A l a s k a t rips
equation
an a t t e m p t to inc l u d e a dummy v a r iable
{i.e.,
as an i n t e r a c t i o n t e r m b e t w e e n
a n d the log of the d i s t a n c e variables)
for e a c h m o d e l
fa iled due to h i g h c o l i n e a r i t y
b e t w e e n t h e s e tw o va riables.
including a dummy variable
intercept
Thus,
two models,
an d a n o t h e r the slope p a r a m e t e r s were e s t i m a t e d
h o l d i n g t he
constant.
f u nctional
variables
form an d s h i f t e d d i s t a n c e
counterparts
(table 4)
show e d the a d d i t i o n a l
to be s i g n i f i c a n t additions.
Further,
adjusted-R2
for e a c h m o d e l was enhanced.
significant
re su lt of thes e n e w m o d e l s
different
dummy variable
specifications
a n d i n t e r c e p t p a r a m e t e r s equally.
represent
distances)
variable.
the
The most
is that the two
i m p a c t e d the slope
However,
r e l a t i v e l y ex t r e m e d i s t a n c e s
(versus a m o r e u n i f o r m d i s p e r s e m e n t
variable
sh i f t e d d i s t a n c e
The a n a l y s i s of v a r i a n c e of t h ese n e w m o d e l s w i t h
their original
trips
one
for A l a s k a trips a f f e c t i n g the
for e a c h of th e d o u b l e - l o g and double-log,
mo d e l s ,
in one
since the A l a s k a
in the data set
acr oss the range of
it was d e c i d e d to m o d e l A l a s k a trips as a d um my
a f f e c t i n g th e
The
slope p a r a m e t e r u s i n g an i n t e r a c t i o n
r e s u l t i n g e q u a t i o n s are p r e s e n t e d in t a ble
11 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■o
I
I
Table 11.—
Estimates of Models 1 and 2 with Alaska Modeled as an
Interaction Parameter Dummy Variable.
Model
Log-Likelihood
Po
Pi
P2
R2
F
c5'
1 .a .
7,386.29
.418
(1 0 .8 6 )
(<.0 0 1 )
1,411
(t-ratios)
(p-values)
-1.805
(-53.10)
(<.0 0 1 )
.792
3
-.587
(-2.72)
(.003)
4 .01
(15.24)
(<.0 0 1 )
-2.44
(-60.64)
(.0 0 0 )
.508
(14.59)
(<.0 0 1 )
.832
C/)
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3
8
CD
C
p
.
3
"
CD
2 .a.
7,446.27
(<.001)
1,840
(<.001)
CD
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3
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CD
Û.
Po, Pi, and P2 are the estimated parameters for the intercept, average round-trip
distance (plus 64 in model 2.a) and the dummy variable for Alaska trips times the
natural log of distance (plus 64 in model 2.a), respectively.
The dummy variable
for Alaska trips was set equal to one for trips made from Alaska and zero
otherwise.
The critical values for t and F for both models are 1.6471 and 3.0079,
respectively, at a 5 percent probability of a type I error.
Finally, the sample
size for both models is 741.
O
c
■CDo
%
o
3
ro
122
A l t h o u g h not p r o v i d e d in this paper,
p r o bability plots
skewness
for m o d e l s
than models
1
and 2
b o w e d to the left th an t h ose
Furthermore,
models
the normal
la an d 2 a show less negative
{i.e.
o b s e r v e d poin t s
in f igures
1
an d 2 ).
the r e s i d u a l v e r s u s p r e d i c t e d v a l u e plot
la a n d 2 a sh ow g r e a t e r d i s p e r s e m e n t of points
the c e n t e r of e a c h g r a p h a nd m o d e l
2 a appears
h e t e r o s c e d a s t i c in the re siduals.
However,
with a linear
form
c o r r e l a t i o n of the
2a
are less
ar ound
less
G l e j s e r tests
reveal a n e g a t i v e
an d sig n i f i c a n t
d i s t a n c e ter ms
ea ch of mode l s
s u g g e s t i n g the r e s i d u a l s
in t h e s e m o d e l s .
for
in
continue
Finally, m o d e l
la
la an d
to be h e t e r o s c e d a s t i c
cont i n u e s to show the
sam e o v e r a n d u n d e r p r e d i c t i o n p a t t e r n o b s e r v e d for m odel
Yet,
the o v e r p r e d i c t i o n
for l o n g e r dist ances
in model
1 .
2a
a p p e a r s m i t i g a t e d w h e n c o m p a r e d to m odel 2 .
Analysis
of M o d e l i n g Trips
in M o d e l
5
from A l a s k a as a Dummy V a r i a b l e
O u t l i e r a n a l y s i s wa s c o n d u c t e d for m o d e l 5 u s i n g the
D F F I T S a n d D F B E T A S measures.
models
1 a n d 2,
measures
C o l l e c t i v e l y a n d s imi lar to
the D F F I T S m e a s u r e an d the two DFBET AS
on th e c o n s t a n t an d slope p a r a m e t e r s
influential observations
for o n e - w a y d i s t a n c e s
revea l e d
of 2 to 50
The G l e j s e r test for h e t e r o s c e d a s t i c i t y was chosen
fo r m o d e l s la a n d 2 a due to a hi gh de gree c o l i n e a r i t y a m ong
t he s q u a r e d a n d i n t e r a c t i o n t e r m s i n c l u d i n g the A l a s k a d u m my
v a r i a b l e s r e q u i r e d for W h i t e ' s test.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
mil e s .
Additionally,
all t h r e e m e a s u r e s
trips
f r o m A l a s k a as i n f l u e n t i a l
5 wa s
reestimated with a dummy variable
Alaska
a g a i n re vealed
observations.
Thus,
for trips
mo del
from
s p e c i f i e d as an i n t e r a c t i o n t e r m w i t h average ro und -
t r i p d i s t a n c e t r a n s f o r m e d by the B o x - C o x t r a n s f o r m a t i o n as
s how n in t a b l e
This m o d e l
6.
Thi s m o d e l
is p r e s e n t e d in t a b l e
variance between models
th e
is d e s i g n a t e d as model
12.
5 . a.
The a nal ys is of
5 and 5 . a shows th at
i nclusion of
i n t e r a c t i o n A l a s k a — trip / B o x — Cox t r a n s f o r m e d aver age
r o u n d - t r i p d i s t a n c e v a r i a b l e e n h a n c e d the
fit of m odel
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.
■D
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Q.
C
g
Q.
■CDD
Table 12.—
w.
CD
Model
Estimation of Model 5 with Alaska Trips as an Interaction Dummy
Variable with Round-Trip Distance.
Log-Likelihood
Po
Pi
P2
R2
F
7, 540.69
-3.900
(-28.29)
(<.0 0 1 )
-.677
(-60.1)
(<.0 0 1 )
.235
(15,02)
(<.0 0 1 )
.830
1,807
8
5.a.
(t-ratios)
(p-values)
§1
^
-g
§.
a
o
"O
o
(<.0 0 1 )
^ 0/ Pi/ and p 2 are the estimated parameters for the intercept, average
round-trip distance transformed by the Box-Cox parameter estimated for
model 5 (table 6 ) and the interactive dummy variable for Alaska trips.
The
dummy variable for Alaska trips was set equal to one for trips made from
Alaska and zero otherwise.
The critical values for t and F in this model
are 1.6471 and 3.854, respectively, at a 5 percent probability of a type I
error.
The sample size is 741.
CD
Q.
■CDD
C/)
C/)
N3
4^*
125
APPENDIX D
SU R V E Y FOR MS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■D
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Q.
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FISHERIES SURVEY
C/)
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DID YOU FISH IN MONTANA DURING THE MONTH OF MAY ?
n YES
Q
IF YES, HOW MANY DAYS DID YOU FISH IN MAY 7
8
(O '
a
3
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"CDO
O
Q.
n
IF YOU ARE A PIONEER («2 OR OLDER) OR A YOUTH <12 T 0 14). DO YOU PLAN TO USE YOUR CONSERVATION LICENSE TO Q
FISH 7
^
^
□
BOTH?
PLEASE REFER TO THE MAPS TO HELP US IDENTIFY THE WATERS YOU FISHED. IF YOU NEED MORE SPACE. PLEASE USE A SEPARATE
PIECE OF PAPER AND RETURN IT WITH THIS SURVEY.
THANK YOU FOR YOUR TIME AND COOPERATION.
TOTAL NUMBER OF
TOTAL NlIMBEROf WAS THE o o Y a ROOM)
SECTION
TOTAL
NEAREST
TOWN
AND/
FBH
CAUGHT
PER
DAY
r PER DAY PURPOSE STAY
FKHIŒPl
DATE
NAME OF LAKE
TRIP
OF YOUR OVER­
NUKCERF OR POINT OF access HOURS
FISHED
OR STREAM
DBTANCE
OTHER
OTHER TROUT
TROUT
INDICATED OR LANDMARK
TRIP TO
FISHED
NIGHT? TRAVELED
FISHED
IN
SPORT
SPORT AND
AND
ON MAP
FISH?
PER DAY
(YorN
MAY
FISH*
FISH' SALMON
SALMON
(YorN)
C
a
O
3
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CD
ENTER EACH DAY AND EACH WATER FISHED ON A SEPARATE UNE. LIST ALL FISHING IN MONTANA. NOT JUST WATERS WDICATED ON MAPS.
5/
/e<
SI
m
St
m
Q.
SI 166
SI
■CDD
St
I6(
SI
161
C/)
C/)
SI
SI
tSf
m
St
mt
SI
I6i
SI
mt
• SUCH AS: THE NUMBER OF WHITEFtSH, PERCH, BASS, ECT.
" IF YOU STAYED OVERNIGHT, PLEASE MAKE A SEPARATE ENTRY FOR
EACH FISHING TRIP.
THIS INFORMATION WILL BE HELD IN STRICT CONFIDENCE AND USED FOR MANAGEMENT PURPOSES ONLY.
K)
m
(D
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Interview d«ter_
ln te iT l« v « l
PLACE LABEL HERE
I)
UÏH
TOU f I S H
IN
MOMTAKA T H l J i
13/p*) :/P6>
SFASON?
w o-2
“V
I
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nA P M T i.
The nett few qucatlona a*# about aowe
of the trip detail# and etpeftdlturea.
1
(/)
2%
o'
r n
MOST
3
M F f
TO
A f< f» fT
AS#
M 'P .n P lC A U .Ï,
tlS R ItH .
3'M l#
WAS
THF
IIS H IM T .
T hK
THE
T R IP ,
ONF
sou r
riS H IH G
V irfT lO N S
T U IP
a n d
|.A # e ,
ON
fO R
O Nf.T
WHEPF
R IV F R ,
AB IM rr
MONTANA
T R IP
l a s t
M A IN
JN
YO U
OR
_?
10) HOW DIO TOD TRAVEL PROM TOUR NOME TO
Y O fl P
-
N H IT H
REASON
WER E
MODE:
1Ï
YOU
(Llat all tranapoFtatIon «odea)
P R IM A R IL Y
I_ .*.
.
71_____
STREAM.
CO
H |ni«ivl«w I# ended heie heceuee
fI ehImq wAB NOT ma In reason or 1 1 Ip
wea multiple destination, wrIte
down reason and name# of multiple
deatInst Ions.
8
ë '
NAME OF l.AHf./STRFAM:
CmSF
3
CD
?
1 __
)
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é
7
II)
Special Regulation Sires»
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5Y
5f
vehicle
WERE YOU IN?
TOWN-
414W Drive
SIRecteatlonal
RCASONr
11A1
^
day
>
UJ
>
3) WHEN DID YOU RARE TRIS TRIP TO
EC
41 WHAT TYPE OP f l S R I H C E O U Î P M F H T DI D YOO
_7 D A T E , | _ |
3
liRalt
7>P)les
lILurea
IlCoabI n
a t
9
10
OSE?
I
II
17
n
12)
_I E O Ü Ï P M B H T
z
o
8) HOW MUrn
AT
.
f T 'jta l tim e spent at
a mo un t o f t i » c t h e y
I
_ I . I
17
s i t e , not t h e
a c tu a lly fis h e d i
I_
About how many hours a day did you fl»h7
UJ
_J
IF r.rss
O
t h a n oni
IP
I
mODRS
I DAVf
. .« •.. 1 ROdrs
71
J2
day;
2)
PEN
DAY
9)
Money apent on auch thing# aa lirea
should not be Included beceuae they
a r e n ’t apeclflcally for the trip.
CODE:
1 ».
I?
1»
3»
t
>0
31
I_'
_13 I
3Î
*--C- -3S- - I--» I__C
« I___
3«
3«
17
3f
39
l._.t
10
I.41
— 'I__I
41
41
l._ . I
47
14
I_
I
4S
4«
TOO CONSIDER WEAR AND TE AR ON
THE VEHICLE DRIVEN AS PART OP THE
transportation cost?
17) WHA T WOUL D TOD ESTIMATE AS THE
COST PER MILE Of WEAR AND TEAR
ON THE VEHICLE DRIVEN?
66
I
M il F
AMOUKY SPFWT
L71. I73
73
74
■
(_
77 ?
79
«
« »_
PI
f3
t GAS ONLY
I
PI
lal
• CAUGHT
'— I— 'I
I
6) #4
I69 *_
70
I
18) HOW MUCH OP THIS COST WAS FOR GAS?
SPECIES:
-
<7
tC cat 0 ( other vehicle wa# uaed, AfR 119 - 131.
If NOT, ahfp to #33.
APOIlT HOW MANY PISH (OP FACR TTPtl DID YOU
CAtCR AMD HOW MANY DID YOU REEP?
73 _
14) WOULD YOU e s t i m a t e THE AMOUWT OF
MONEY YOU SPENT O H TRANSPORTATION
TO AND PROM __ 7
7
6
WHAT WAS THE PRIMARY TYPE OP
FI SB YOU WERE TRYING TO CATCH?
L 1ST ALL
13) COULD YOD ESTIMATE THE DISTANCE YOU
TRAVELED PROM YOUR HOME T O
?
(One-way diatance, eu* of all mode#
Including atop# along the way)
» . 1 . _ I HttHR.S
About how many hour a did you 11 ah?
$)
I
«9
I
35
z
B O A T /S H O R E
39 3h
7| IP MORP TRAN OWE D A Y ,
K
» 1_
•) S7
^
1S
]
X
<
tjmf did you spend
I
h o w LONG DID IT TARE TO TRAVEL
PROM YOUR HOME T O _ _ _ _ _ ?
about
(0
IThla ahould atart when they left home
and end when they arrived at the flahlng
attc. Including atcpa along the way.)
Ion
8^ DID YOU SPEND MOST OP YOUR TIME PISRINC
f r o m a BOAT OR PROM SRORC7
llRoat 71 Shore
I
M O f Ivlmg 7)RldlA9 ))«oth
I
( 0
111
CODE
WERE YOU DRIVING, RlOlNC IN THE
VEHICLE, OR HOTBT
«
1/1__ l _ t / 1
p
a
&
WHAT TYPE OP
11 C o m p a c t (4 Cylindera)
23 lnter«ed1atel( Cyltndera)
3 3 P u U - S I % e f # Cylinder#)
DESTlNATIONS;
C
p.
S<
If car waa driven, AS# 911 and #1 la.
If MOT, 90 to #17.
1..1 _ I
S
5S
I
U
I
ts
YES*I
N^»3
».
F7
•'II__
ep
#9
I PER Mlir
90
CODE:
1#) O N WHAT TYPE OP ROAD DID YOU SPEND MOST
or YOUR time TRAVELING?
JtnlqTiway
3 3Rut, M p « v « d )
43 Rural funp.v.d)
ro
128
o
«
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.