ESTIMATION OF PARAMETERS
IN DISTRIBUTED LAG ECONOMETRIC MODELS
William S. Cleveland
Department of Statistias
University of North Carolina at Chapel niH
Institute of Statistics Mimeo Series No. 792
December, 1971
ESTIMATION OF PARAMETERS IN DISTRlijUTED LAG ECONor'1ETRIC MODELS
William S. Cleveland
University of North Carolina at ChapeZ Hill
ABSTRACT
This paper presents a method for estimating parameters in distributed lag
econometric models.
A subset of the parameter space is studied which includes
the maximum likelihood estimate (which incorporates no smoothing assumption on
the
parameters)~
an extreme smoothness estimate, and other values which rep-
resent an amount of smoothing ranging between that of the two extremes.
•
•
From
the subset an estimate is chosen which is not judged unlikely by the data but
yet has an amount of smoothness in the coefficients which makes economic sense •
2
1.
One very'
c:c.IOD
INTRODUtTiON
occurrence infittingd:istributed 1al models to econ.oa1c
data 1. that the least square. estimates of the parameters do not make
eeOft~C
sense, sinc. in a_ra1 the exolenous variables and their laged values are so
highly correlated aad the number of observations
80.•mall
that there is
1nforaatton ill the data about the true values of the parameten.
lit~le
OIle re.ecly
is to eUainate- variables froa the model. This 1. . . adequate procedure if
predlctiOD is the only loal. but it does not help the person interested in
_deret_dinl the econOll1c system generating the data or the person who vants
to .... policy decisions.
In this paper t a _thod of e.tiuti"g par... ters in distributed lag mocteu
1s presented.
III Section 2 notation vill be established and one reasOll for the
ledt of preCision ill the least squanta .sU_tea given.
•
In Section 3 a aenera!
approach to estimating paraMten in distributed lag IlOdeb is presented end
two specific i.,lementations are discussed.
pretation of the two 1IIp1eMntations.
Section 4 lives a Bayesian inter-
In Section
.s
the parameters of two dis-
tributed lal equation. fro.'- the St.' Louis Federal Re.erve lank econOMetric
.ade! of the United States econOlly are estiuted US1DI the _thoda auuested
in Section 3.
ttl Section 6 .ewral commel'lts are _de about the methoa and a
COIIpartson .ade vith the Almon 1.. technique.
2.
THE KJDEL AND THE PR08LEM.
To keep notation froa ..tUnc out of had. it will be supposed that there
are CWO exoaenoUlt Y.r1abl..
at
and b t.
The di.CU8SiOll that follow holds
y
for
t · 1, ••••_,
ables with aean
•
t
wbere
0
anel
These
e
i . a sequence of in.peaelellt ooral random vari-
and variance
02
rather than the ueual
tera.
Zt
e -+
(this paraaeteriution of the variaace
will prove convenient later) and y
1s a constant
N equatioM will now be written In matrix fora.
Let
Y~ Z.
be the colwm vectors
Y •
(yl •••• 'yN)'
Z •
(altt ••• ,~) ,
and
For k· 1, ••• ,8+1 and
t · 1, ••• ,N,
k • a+2, ••••a+b+2 and t - 1•••• ,N,
be the
N
lie
(a+b+2)
let
let
d tk • at_(k_l)'
cl tk • bt~(e+2-k).
For
Finally, let
D
matdx
and let 0 be a col... veetor of Nones.
y -
fly
Then the distributed lag 1IOcIe1 is
+ De + z.
One way of reduciol the degree of correlation between the variables in
~.
model is to reparameterize 10 a way that cIoe8 Dot change
of subatracting from eacll
COlU8l
of
D lts _an.
e
and conatet.
-1 \'11
Let d ok • •
Lt-! d tt
4
'thea the· .4istributed la& IIOdel aay J)e writtea as
Y •. Ov + xe + z
!he correlation between 0
anel each colUllln of
The 1IUi1Mll likelihood eaUMte of
e
I
1. zero.
is
Let
SS(8) •
•
(Y-of-X8), (Y-o!-X8) •
Sioce we wat to focus on the prahle. of utiaatina
e
we will stuc17 the . .r-
ataal . likel1hoocl fUlletton of i
L(e)
c
I L(Gtpt.)
d; d+
N-l
c
[5S(9)]-T
!be 1I&I',1"a1 libl:1hood
function (6, p. 256].
L 18 proportional to . . .Itly.riate Student deu1ty
The aut-.- occurs at
cODBtant on eacb e111psoid of the
fo~
a
&. ML1 ael tb. function 1.
• It
(e-.~),
1'b.~
ia,
ne
a
-.st-.
J1~el, value.
li"eUh~ eStimate
..
a fOtat
f.,
is. 00 the bat, of ~he data alone,
3HL1
..
,~ti
6
aUt .. value
~(~) • L<SMtE) 1s not
£cO,q~e
eon.tant.
this 'aail, of ~omoChet1c el1:lps.oid$ 11 the set ,of contour. of L.
e.d1da~e
8004
• "
X'X(e-O
MLE ) •
te of
e
it is in 80_ ,eue tbe -.at
8i.e~
with htgh relative likelihood (1 •.,•• for which
8MLB •
fro- ~) is nearly as ,cod a caodidat, as
qt. very ofte,. bave the two propert1e. that theeolu.l8 of X • •
b1.$h~y ~otr.. lat..
and the val\Jt! of N.
the n_8r of Qbeenations, i . .aall.
Both,~.r~te. ~.cf to pr~d..ee a likeJ,thoo4 funcUOIl which cannotdisttlllU!ah
vttl\ JIlUc:h
pJ;ec;f..iqnM~
".1~, of eWll:tch have very differ.tClt econOflic
<
.'->
~_- "~_.
c""-
.ant-n..That
',,
.: --'
~_-
-
~~'
-
-,
-. -.'- ,'':: ~ -
:t8, they
-
_.c
":
._
.,
to decre.....
of CC)uel_tion
. . . .t
'.~
:.-:
• -_
i
,-'
_:- _, •
_':-"'_"',
-
, ' , .
the
.
ye~
<",::
-
.
110. . .
"
,
"_"
_,_ -
-
~elatlv~
of l
,-c'
ft.....11er
8~!. The iIOre tbe
the "-.ore elongated the
~o be. Thqs.. i
th. ~l't!ct1. . of ~he uj0l' ••t- of th••e contours.
Th~ ~h9
_'
C
JIIOV" Way fr.,.
.~':.'-.'.
81owJ:l"
-
likelihood
1ft .any4irectj,on .,a, from
titRe!
'0'"
differeftt.economic _.aning..
~tweenth. col~
eJ,U"....l eootO\tt'fll of J.
::',:
-
~~IUI to produce a large relf.~ of ,ood ~andidates COIl-
t~
. . 81~r
.. . . '
t ....
,..,
of $ whl4:ta h,"e
~.in~Il,Y.lt18 ..
•
-
lL(e)
3ML1
ill
tends to ctec~as.
Qta ~l.!"ecJo not provide eqoulb infor...Uon
to
enable 9fte
to'lM1t4
•,',," preci••
of
In the lIlOdel •
. ;';:"' . .,;'
- .
. - ,pecific.U.
::'. , .-- ',':,-; .
'.. -- the
. - ,.r-.eter8
.- ,- ,",
..:<
-~.
:~..
A.NEPAL REflfDY fQR THE PROQLOI 4NQ TWO
~"
'Pte one
DOt
,,"
. .
IMPlDEftTATION$
~in.· ~h.t 1, C?+.ar ~rO!! ~el ..t ,et:t1on is that tbe probJ.ea can-
Itfit resolve4
.,_,-. -"'-;
~ECIHC
c.'
by .taU.,tica4101l8.
<: .:
:.::
-.".
':
. _ :.
",
Tllere
.• -:. _ ,.'
~s
I:'O~
:. -.:: ;__ ':::
8\lfftcient
precision 1.0
the
", .-', _','
.~.
6
additional knowledge about the eeonosicsystell aeneratiul",the elata.
•
to .illply
of cIoi". tM.• would be
r:elative l1keUboocl
•
RL(8)
locit over the "gion of
One way
e values vith blgh
and pick out one that ude sellSe with regard to
all the prior bovledae about the economic. systea generatina the data.
~H
thi. procedure generally would not
feasible to carry out aince
RL
But
ta
typicall,. a f.etion of a large nUllber of variables and difficult to UIlder- ,
atand.
thus it would be cl1ffi.clIlt to codify the prior bowledae about the ec-
OGOI'Iic .Y8 tell ill this way.
The solution 18 to settle for something a bit 1.ss.
lnatead of study1lla
over the whole parameter space. the solution is to atudy it over a subset
IL
that Mba 8ena. for the distributed lag model.
ta ._tble to auppoee there is
°0 .....0.
OJ
8 _ 1 or
j
C8rt~ty that
81100tlmeaa in the coefficients
80 .... ,Sb.
and tbe co.fUc1enu
believe that
fJ'Oll
80M
III tbe typical sltuaUOft it
That 1a, there is a t8ftdeney to
does not differ radically from a j _1
8
j
+r
°0
or
Qj+l nor 8j
Now the _xtre_ £01.'11 of this belief would be an absolute
• (11 • ... • ~
a
a
and
~O -
8t:. •.. • $b ." a. ften the.
_del. "'icb will. be named the extreme smoothneas ItOde1. would be
't •
.
8+1
P+G
I
j-t
Xtj
+~
a+b+2
L
j a a+2
Xtj
"
+
Zt·
~l
Let V be the M x 2 matrix wtao8. first eolumn ls Lj-l Xtj for t · 1 ••••••
and vboee s8coacl colUlllR is
likelihood .Itiaatu of
\"a+b+2
Lj a a+2
X
tj
for
t . 1, ... ,N.
Then the aut. .
S would be
G" 'ad
~
•
(y.V)-1 V' Y.
TIlua _cler thts extreme 81IOOtbaes. coadition th. 1I8xbwm likelihood e..tlute of
e
would be • col... vector
...... aext
b
+
1
318M
ele.enta are
whose lint
aiSM•
a
+ I ela.ents are ~!SN and
At the other extreme 1s the 8.Uute
.... whleh 1n"o.1. . absolutel, no 8IIOothinl MSumptioa oa the par_ten.
,,
7
~dy
The general tetea of the
•
will be studieel which
coatalu
.
ts thi..
A subset of the parameter .pace
9ML1 , 8ESM'
and other values
e Wllich
-
repre.umt an . . . .t of sllIOOthina ranging between that of the two extre...
will be stuclied for a number of values in the subset and
aD
estimate
""8
11.- -
chosen
frolll the subset which is not juc1ged unlikely by the data (1.e., for which
RL(I)
i. not too far frOil
1) but yet has an amoUDt of smoothness in the coef-
ficieats whiCh makes econom1c sense.
In the remainder of this section, two possible choices for the subset of
stuely viII be presented •
.
3.1. AOne-Parameter Subset
Let
W be an
(a+b+2»)( (a+b+2)
diagonal_trix.
The fint
a + 1 ele-
_ata on the diagonal have the same value
which 18 the aquareroot of the Bua of aquares about.the mean of the first
s_
exoaenoUi variable
at.
The next
+ 1 ele.ate
b
OIl
the diaaoaal have the
value
N
rt-l1
2
Xt
\
]';
'
a+2
•
which t. the aquare root of the sum of squares about the _an of the seeonel
exogenous variable
The
•
/J.
OIle
8(0) • 'NtE'
•
8(,)
b t.
parameter subset
and al
pt smoother.
P
+ -,
•
8(p)
for
8(p) ... 3
15M
p
•
~
As
For a particular value of
:b defined by
0
P
P.
incr• •es the valuee of
•
9(p)
bas the prop.rtf that
8
RL<8(pj).
~
&
Another way of saying this 18 that over all points
such that
• (6(p)-aESM )' ~2(e(r)-~ESM)' &(p) haa the largest like•
lthoocl. thUll, for a gIven amount 0:; emoothnes6. 8(p)
has the DOst l1kel1-
<a-GSSH)' w%(e-3ESM)
hood.
Theae atatelletats 1I&Y be proved by an argument entirely analogous to that
in (4, p. 59].
e
In practice, an estiaate of
~nd 1L(8(p»
8(p) by calculating and studying e(p)
.et
of
can be chosen from the one-parameter sub-
A particular estimate
p.
for various ·values
@ would be chosen from the subset which has an
8IIount of smoothness that makes econoadc .....e but for wbich
• •11 .. to render
3
unlikely in llsht of the data.
3
timate is a180 aided by noting by how much
s4uares; thus calculating SS(&(p»
3.2.
Let
Gg (1')
For
I'
inflates the residual
and comparing with
be. nUliber in the interval
let
(g+1)
~
(g+1)
SS(i
HLE
)
SUM
of
is useful.
be the
8(r.p)
(a+b+2)
Tbe ftOl'thwest block of Hb ,p)
(b+l»)( (b+l)
t:he value zero.
lC
(a+b+2)
0
p
0
~(r)
(....1»)( (a+l)
4
posttive integer.
•
utnx. the southeast
and the other f#lemenu of the matrix have
a. 1
and
b· 2.
1 rOO 0
•
8
J
LGa(.r)
matTIx,
K(r,p)
and
block diaaonal matrix
_
1
is an
For eX8lllple, if
0 S r < 1
_atrix
H(r,p) •
block is a
The choice of the es-
A Two-Para_teW' Subset
wl1l be the
p" 0
Is not 80
1.(3)
1
p
rl00.0
0 1 r 1'2
0 r 1 r
0 r;~ r 1
o
o
o
then
9
Let
GIg(r)
be the in~erse of Gg (t1.
CI g (r)
is a tri-diagonal matrix
with the f~1lowin8 elements: the fi.rst and last ele,ments of the main diagonal
are U_r 2)-1; aU other dittgonal eletnents arE! (1+r2 ) O_r 2 )-1; all elements
on the two diagonals parallel and adjacent to the main diaronal are
and all other elements are zero.
2 -1 ;
-r(l-r)
For example
c:
GI (r)
2
•
O-r2) -1
GI (r)
3
•
2 -1 [ 1
(1-r ).
-~
and
Let Hl(r,p) be the inverse of H(r,p).
-J
-r
1+r2
-r
-~.
Thus
H(r,p)
For p. 0,
BI(r,p)
is defined to be an
(a+b+2) x (a+b+2)
matrix of zeroe.
The two parameter subset is
O(r,p)
for 0 ~ r
of
e
<
1
and
p ~ O.
c
9(r,p)
lWH(rtp)W + X'X]-l X'y
contains th~ maximum likelihood estimate
sinc.e
•
As p incresses 8(r,p)
moves toward the origin •
What is most interesting 1s the behavior of
•
6(r
t p)
as
r
+
1.
Let
10
trace [4.5) of
•
be an
8
(p)
UM
and
(l
8
(a+b+2»( 1
~ESM(P) and whose next
fint
a + 1 values of
p >. 0
then as
That 1., as
under the extre1lle smoothness concU.tion.
l' ~
r
b
•
..
value. of 8(r,p)
and
When
•
aESM(p).
.
.
e.
13 (r,p)
If
l'
is nearly one all the
Sll111arly as
I,
r
e
set
a(r,p)
a + 1
.
lncreases fro·m
values of
0
to
1
the
BESM(P).
can be chosen from the two-par.-ter sub-
for various values of
and
l'
and .electing a
p
from the subset which has an amount of slIOothness that ukes
A
e
...11 as to render
.
rutdual .ua of aquarea of
.
.01 and .1 will be quite sufficient.
1, e(r.v)
is not so
SS(8(r,p»,
tbe
9(r,p).
Ia many practical situations studying value. of
eX8llple, between
iLe$)
Just. in the one
unlikely in light of the data.
par... ter sub•• t, it will be of interest to also study
ten" to
values.
•
the values of
toward
sense with respect to the economic mechanism, but for which
l'
elements are
b + 1
the next
get closer together, all approachlDa
.et by calculatina 8(r .p)
8
a + 1
0
In practice, an estimate of
particular
whose first
1
increases from
are nearly
~ctor
+ 1 elements are ~ESM(P)' Let ~er.p) be the
6(r,p)
closer (1.e. I 8IlOother).
o(r,p)
colum
Let
p
close to zero, for
In this situation . . .
tends toward a vector which i8 nearly
alSM -
However,
II the colUllln8 of V are hlgbly correlated, larger values of p will often be
neede..
-.te
.
8
lor, in this situation, there will not be much precbion 1n the utl-
f lSM • 3ESK CO)
despite the strong extre.. saoothness ...umption, and a
-
ISM
(P)
eaUaate.
with a larger value of
p
will be needed in oreler to
set a senaible
, 11
4.
A BAYESIAN INTERPRETATION OF THE METHOD
e
Suppose the prior distribution of
e
that
and
...
e
are independent acd
and covariance 1I&trix
and
is
~
given
mul~ivariate
has tbe properties
nontal with _an M
a, which 18 a very
Then the poaterior mean of
c·Co
•
reasonable point estiaate for the para. tel' , is
4.1. Th& On&·Parameter Family
C· P-1 W-2
Suppose
.
and M .. 6~
UM
o
e
Then the poaterior _ _ of
ts
.
This represents for8dng a prior by.
8(p).
"peekina at the data", since
I!
6
ESM
ation about the value of
depends on the values of Y.
The prior
so the data determine the value of tbe posterior:
As
OML£.
e
about the value of
e,
A
__ , which is nearly
p
.ets larger the information in the prior
gets large compared wi tb the tnfonaation in the sapIa.
the prior belief beoo.es stronger that
teada to
9
Dickey (3, p. 1481J puta it,
-1
OJ 18 P-1wI.
If P is Illall, there is little prior illfor-
varianee of
_an of
8S
e
is near
and tbe postarior
'8£SM'
3ESM •
4.2. The Two..Parameter Family
C. W-!u(r,p)w- l
s;',oae
terior mean of
e
is'
ad the prior mean is
prior
_80
i.
For p > 0
and M ia a vector of zeros.
e(r,p).
O.
The. prior Yariance of each
The prior variance of each
8
Then the poa-
1&
j
t.
(lj
wi2p-1
w;2p -1
and the
O.
the ,rior correlation betweem
pdor: correlation between
C)
j
and
~
OJ and 8k
and betveen
Bj
and
i.
S
k
0
-is
and the
r
1j- kl
which depends only on the tiM lag between the variables correspondina to the
par_ten.
As
the tillle laa inereases the correlation deereases which would
" eeem to be a desirable property.
As
r
tends to
1
there is increasingly
,
_re prior infol'll8tiOft that
aj
ak
and
i. _re S*M)thneas in the coefficients
.
- - of
"0' ••• ,a._
.". ,o. ) and heIlee the poaterior
a
.
. .
and
b !tea-r
I'
~.~. . of tbe vanous
8_ ..
oJ
Cl
j
"
or
1,
tend toward the . _
there 1. little prior 10f01'll8t1_
SJ
but a strong prior belief that
are nearly. equal.
the otreM
.
S(r,,).
eleM1lts of o(r.p).
about the value of an individual
nearly the
0
A ai1l11_ 8tat__ t holds for
p 18 near 0
If
0.
,". -.
t . .·.· .•
~
OzsM(P).
".1_.
··.:1
have similar values (i.e~· that 'tbere
...
1111. prior ~..,ttOD ia
8I1OOtlm.S. 888U11ptiOft
and
80
"
tb'.,) ·"ls nearl,
tIS.·
P and
All
I'
tile true value of
i~
_d.'e(r,,)
both get close to ••ro the informat10D ia the prior about
e
set8 ...11 cOIIpared to the information in the • ..,le
aearly
tIeL!.
5.
EXAMPLE
" .
"
The . e of. the ooe-par_ter auba.t
.
.
8Cr,,)
.
&(p)
and the two-par_ter ....at
.
.
will be illustrated by their application to two equations fro- the
econometric
~l
in (2).
The fir.t i . the price equation
..
.
vben 't" t. the chaD•• in the price lewl f!'Oll quarter
1. the .aand pres.ure 1. quarter
t.
..." b t
t -
1
to
at
t.
i. the chanae froa quarter
"
t - 1
to
t
in the _ttcl,atri price lewl.
qurtel' of 1955 &ad I · 60.
col~
of the aatrix X;
t. 1
.
.,:t'"
eone.pona to the fint
DiBplay 1 show the eorrelatiou between the
there 18 very clearly a subetanUel .-natt of
,.-;. -.t ..
correlation.
,
"
.:
"
.~..
•,.-·t
.""
'!"
."
.. Display 2 sbows the results of calculad,ag e(p)
It
'l'ha·i"div.ld~al al_nts of the vector
for 15 values of
p.
are defined by
e(p)
"
3HL1 •
. 'I'be aaxI.. . likelihood (leut squa::-:fW) .atlaate of 0.
&(0).
dou'llOt
.....r to uk_ very 'aucb sensa for the .sti_tee of 02 8ftd CIS are both " . .
•
.
,
~
't '.:
~
.
.
. . . .tft. whi'cb vould imply ,nees iocrease by a smaller amouat durt. .. .li~. tt.-.
4a&rcel" if the deUDd v.. 181''' 3 quarte:-s earlier or 5 quarten ••rii~r.·
~
'<.oil
..
1. quite differeftt
MU)
.91 'ad SS(t
1.
fr~ tML! eve~
9'.7% of' S8(8(.01».
.
..
1L(8(.Ol»
though
is ..
~lah
f~.tun·
'l'b1. is exactly •
, of the lact-of-prec1ai. .···p~ resulting froID hiah correlations ad •
• •11 _ampl• •1" that was eSescribed 1n Section 2.
..
p
incAaa.s. aoiOI froll neaative to positive for a value of p
.01 ad
.05.
All
,
to
.
°•0 (,), ... ,0• 5 (,)
incr...... the values
'l'he choice of·. particular
de""
•
•
02(P)&Ild' 04 (p)
.
8(p)
e
as an estimate of
upoe ODe' _ knowleclp about the econoatc llech_is..
.6 II1lbt uke seue to . .,.
J. • .0 •••••,
ltL(t(,»
POI'
·these .&1.s of
SS(t
) . SS(8(p»
MLE
are not
.0
between
let s~ther.
would. of course.
ira the rase
,
•
p' tbe
.4
~j (p)'" for
1Ilere.... ,asut'
are reacmably saootb and decrease'.. j
and
increu•.
s. .11 as to 41ecl'edtt
i(p)
til
lipt of the elata.
fte s.coact equation ia d\e total apa41n1 equation
. 4 4
Yt
when .. 't' a. ".d b t
"
t
": ·Ciftl,. the Goalnal
.1tun8.
t. 1
~ y +
I
j-O.
Gj
a t _j +
I
'$j bi_j
are the chaeges frOll qual'ter
GNP,
+
Zt'
j-o
t - 1
to
t
of, reapec-
the mae, atock, and hiah-e1llplo,.ent 'ederal expen-
cornspoD. to the fint quarter of 1953 and II· 68.
01.,181 "3 show the c:orrelatiou between tbe col.... of the utn.x I.
I
;
.~
...
,
Di.,l., 4ahaWa the re.ult. of ~iculat1DI &("r)
tit
'
'aDd
The individual ele1llmts >:!if the _etc!:'
r.
squat:.:~)
The aaxt1llUll likeUhoocl (l... t
aro••l,
for yarioua yaluea of
are clefinecl by
6(r,p)
h.
aML1
eatiQAtu
•
cIo not appear
• 8(0,0)
inadequate, but the amal1 valrJe of the estimate of
>
'"
01
ed the nesa-
.
ti_ value of tbe .atimate of
4 miSht be regardecl by SOII8 . . uairealUClc•..
lIi:ght· be a aore reaUstic es:imate for many. Ql(.lS,.6). i.3S'ta
G
'
i(.1.5,.6)
.
lIOn
nearly :in ltne with
.S(.lS,.6) • .06
Cl
.
.
'l(e(~15, .6»
ta
poaiti~.
.
1.53 and a 2 (.15,.6). 1.59, ad
".
SS(8ML1) .
SS(e(.15,.~»
• .23 are not
80
.
• .95 and
.
. .1ipt of the clata.
•....
8("r)
.
ClO(.1~,.6).
low as to c:01llpletely diacredit
.
t(.15, .6)
in
In should be emphaiced that others may find another
_re . . .1b1e, for the choice will (and auat aince there i. not enoup
.
,
,inforutioa in the date to apecify the _del with great predaiOlS) depend U90Il
_ ' a prior bowleclae about the eeonoadc mechenia••
6.
GENERAL COtttENTS
The ,rocecluru deacribed in thi. paper may be applied to the general
·4UtribUtecl la. _del
n. ~
'J t . •
y + It.l
I j-o
I L. Itj ",t-j'+
-cIt
ror chi. lenera1 __1 H(r,,) 1s a block dtaaoaal matrix with n blocks,
_4 tUM and W each have n
-
~'t
'
different value..
lac:h of the
n
veri.ab1e.
.
for- t • 1 .....1' aipt be an exoaenoue variable •• d.-y variable.
_ . a laueel value of the endogenous variable.
or
15
It
More practical experience 18 ne~,~~,d to judae the relative merits of
··'i(~.,) and 8(p). My own tentativE" G?i!llon at the Hnie of writing of this
.....r
. •1 . ·
• ."
.".
.
_ . .
•
1. that ttla easier to use
.
.._
6(p)
_
_
will suffice provided the colu.a of V
. are not too biab!y correlated but that othenrise the
ea-plicated,
i(r,p)
8101'8 del1~te,
and
IIDre
is needed.
A very reasonable c01IIpetitor to the sethod preaented in this paper i8 the
Aaon lag technique (1).
Both employ _ ·e1e.ent of subjectivity; in the aetbod
of this paper a value °f;)f
p
or values of
r
and
p
1IUSt be chosen and il\
the.Al_n lag technique the degree of tbe 81100thtng polynomial lIlUat be chosen.
However, the . .thad here 1e, 1 believe, .impler, more flexible, and provide.
aD
e..ler to understand and IIOre delicate way of codifying prior infontatim about
, the ellOOthne.. of the coefficients.
In addition,
..
SCr,p)
can cope with the
probl. . of hiah correlaUoo between the different excaanous variable., which
. the'~ 1.. technique camot.
The •••ete
•
8(r,p)
and
•
e(p)
while different in detail frOtft thertd..
trace of Boerl _d 'Kennard (4,5) are very much in the . . . epirit.
excel1eot papera
OIl
Thea. two
the ... of the ridS. trace are .ery worthwhile r ••d1na_
........
16
Display 1. Correlations Between ebe ColUft'tB of X for the Price Equation
(values are rounded to 2 places and multiplied by 100).
--------_._-,....,.,--_._."'' ' ' ' '" .....,.--------------.......
Columna
1
6
96
89
81
74
68
7
45
2
3
4
5
3
.2
.,.
4
5
6
.- •. ~.?~-
...'t·":··
96
96
89
81
90
82
74
52
56
,.
96
88
60
96
63
65
-
e
e
Display 2.
1\
p
•
IL(O(p»
SS(t3
MLE
•
)
SS(6(p»
1
0
.01
.05
.1
.2
.3
.4
.5
.6
.7
.8
.91
.65
.55
.46
.41
.37
1
1.00
.99
.98
.91
.97
•
6(p)
. (p)
<l
9
.029
.029
.027
.025
.023
-
for the P1:1 ce Equation
•
a(p~
•
a (p)
a (p)
3
.038
.028
.022
.021
.019
.019
-.026
-.010
.005
.010
.021
.017
.015
.013
.014
.014
.018
.014
.014
2
.014
.014
•
3
(p)
4
.043
.032
.019
.016
.014
.014
•
;5 (p)
BO(p)
-.017
-.009
.002
.005
.869
.864
.008
.009
.010
.837
.011
.826
.011
.012
.824
.822
.852
.845
.97
.021
.020
.34
.96
.020
.018
.014
.014
.014
.014
.32
.96
.96
.019
.018
.018
.015
.014
.014
.015
.014
.014
.015
.015
.015
.015
.015
.014
.015
.014
.014
.012
.Ou
.821
.821
.015
.014
.013
.820
.015
.015
.014
.817
.015
.015
.015
.816
.31
.29
.019
.019
.018
.017
.018
.017
1·
.21
.96
.96
.96
2
.22
.95
.017
.016
5
.19
.94
.016
.016
.9
•
;1 (p)
.28
.017
.831
.828
........
18
Display 3.
CcJrrelatlons Between the Coluims of X for the Total Spending
Equatioo
(values
are rounded to 2_ places
and multiplied by 100).
r· ..
'"'_""""""'_=
"'._ _"="",,=
7
~""'
.w-
>1..~f;~:~·
..j. .
..
.:<,
1
2
3
4
76
33
·'3 .
75
45
4
27
75
47
S
20
30
6
28
33
28
30
10
30
39
39
36
34
39
38
27
4)
33
38
39
37
16
49
31
7
8
e
"Q.., :_
.
_
""""~.::O~-:-.==.
34
40
5
30
31
29
29
34
6
7
8
50
52
38
51
9
52
53
36
30
SO
48
Display 4.
s(r,p)
for the Total Spending Equation
.
•
:=;;:;c;;;a=:::.
.~
,...
r
1L(6(r,p»
SS(9
)
MLs
SS{Q(r,p»
1
II:
_
__~O(r.p) a1(r,p)
.
0.0
e
e
e
.~--~~--
a 2 (T,p)
~3(rtP)
--
------~--~~-
0.2
0.4
.71
0.6
0.8
.58
.35
0.0
0.4
.54
.53
.46
.98
.98
.98
0.6
.35
.97
0.8
.18
.95
0.0
0.2
0.4
0.6
0.8
.33
.35
.31
.23
1.66
1.61
1.56
1.53
1.53
.91.
. 1.06
1.19
1.31
1.41
1.76
1.02
1. 73
1.68
1.04
.11
.97
.97
.97
.96
.94
0.0
0.2
0.4
0.6
0.8
.20
.23
.21
.16
.06
.95
.96
.95
.95
.92
1.58
.97
1.53
1.50
1.11
1.66
1.65
0.2
.99
.99
.99
.98
.97
1.91
1.86
1.to
.66
.77
.89
1.13
1.64
1.05
1.26
1. 76
.82
.96
p • .05
2.09
1.11
2.03
1.13
1.96
1.13
1.85
1.11
1.67
1.03
.81
.78
1.71
1.65
1.60
1.51
1.09
1.24
1.38
8(r,p)
G4 {r,p) 60 (r,p) B1 {r,p) 82 (r,p) B3{r,p) B4 (r,p)
.
.45.
.45
.4S
-.01
.44
.44
.45
.46
.47
1.07
-.19
.43
.42
1.08
-.17
-.12
-.03
.44
.43
.45
.46
.43
.41
.13
.12
.11
.09
.1.5
.46
.36
.06
-.12
-.10
-.04
.43
.39
.12
.43
.44
.40
.11
.10
.06
.45
.93
.24
.43
.2
.99
1.00
-.01
-.04
.42
.02
.12
.43
.43
.41
P • .1
1.90
1.85
1.78
1.69
1.53
.c·~-
1.07
1.03
.96
-.28
.... 27
-.24
-.17
.44
.41
.14
.14
.13
.11
.08
-.54
-.S4
-.53
-.51
-.46
-.43
-.44
-.45
-.46
-.46
-.48
-.49
-.48
-.45
-.39
-.40
-.42
-.43
-.44
-.44
p.. • • 15
1.49
1.50
1.24
1.35
1.41
1.59
1.45
p •
1.60
1.53
1.39
1.02
.97
.99
.95
.91
.31
.42
.08
-.44
-.45
-.44
-.41
.05
-.35
-.41
-.42
-.41
.31
.39
.38
.12
.11
.09
-.40
-.3>
.37
.31
.07
.40
.38
.34
.05
-.41
-.40
-.38
... 31
-.38
-.40
-.38
-.39
-.40
-.39
...
\.Q
REfERENCES
[1]
Almon, Shirley,
Bxpend! tures )"
"The D:latr1.buted Lag Between Capital Appropriations and
Econometri~a
33, 1965, 118-196.
(2) , Andersen, Leona11 C. and Carlson, Keith M., "A Monetadst Model for
Economic Stabilization," HemetJ - Federal Reserrv€ Bank of St. Louis,
April 1970, 7-25.
(3]
Dickey, James M.
1O
"SlIOothing by Cheating,", The Annal.s of Mathematical
Statistics 40, 1969. 1477-1482.
[4]
Hoed, Arthur E. and Kennard, Rebert W.. "Ridge Jeln••ion: Biased
Estilllation for Nonorthogonal Prob li.~n:s," TechnometPios 12. 1970, 55-67.
(5]
Hoerl, Arthur E. and Kennard, Rohert W., "Ridge Regression: Applieation
to Nonorthogonal Problems," T(N!tmometrio8 18, 1970, 69-78.
(6]
Raiffa, Howard and Schlaifer, Robert, Applied StatistiaaZ Decision
Thao'l"lJ, Division of Research, Harvard BudnE'SA School, 1961.