Tuesday, 12.30 – 13.50
1
Charles
Charles University
University
Econometrics
Econometrics
Jan
Jan Ámos
Ámos Víšek
Víšek
FSV UK
Institute of Economic Studies
Faculty
of
Faculty
of Social
Social Sciences
Sciences
STAKAN
III
Eighth Lecture
(summer term)
2
Schedule of today talk
● Limited
● Model
Response Variable
with Varying Coefficients
Limited Response Variable
3
The pioneering paper:
James M. Tobin (1958): Estimation of relationship for limited
dependent variables. Econometrica, 26, 24 – 36.
Examples:
● Expenditure of households on durable goods
Purchase is not made until “desire” or “need”
to buy the good exceeds a certain level.
So, either the “desire” is close to this level or is very low,
the zero is in data for the corresponding household.
In other words, the response variable is cut by zero,
since “negative” expenditure corresponding to various
levels of “desire” fall under the treshold.
Limited Response Variable
continued
4
● Wage
of married women
Only the wages of women who are in labor force,
are recorded.
In other words, the wages of women
whose “reservation wage”,
i.e. a prize of work made for family at home,
is larger than their market wage is zero (in data).
It means that all women who work for their families,
without distinction whether her “reservation wage” barely
exceeds their market wage or is much larger than it,
have zero in data (at the column “wage”).
The restriction on response variable matters in the case
when the probability of “cutting the true value of response”
is not negligible, i.e. it is when probability of falling below
the treshold is not very small.
Limited Response Variable
5
continued
Assume that the process is governed by
Yi*  X iT    i* , i  1 ,2 , , n
but we observe Yi* ‘s only if they are positive, i.e. we observe
Yi  Yi*
Yi*  0
Yi  0
Yi*  0 .
and
So, we have
Yi  X iT    i , i  1 ,2 , , n
where Yi‘s are truncated from below by 0 and  i‘s by
 X iT  .
Such model is called censored or Tobit.
Limited Response Variable
6
continued
We may meet with situation when also X i ‘s are truncated
and then we usually speak about truncated model,
Takeshi Amemiya (1984): Tobit models: A survey. J. of Econometrics, 3 – 61.
In what follows – Tobin models, only. We have
P( Y *  0 )  P(  *   X T  ) ,
i
i
(*)
i
so assuming that the disturbances are normally distributed ( with
variance  2 ), we can write

  X iT 

L   ( 1  Z i ) ln 
i 1 
 

n


1
1

2
T
2 
  Z i   ln( 2  ) 
( Yi  X i  )  
2

2
 2



where
Zi  1
Yi  0
and
Zi  0
otherwise.
Limited Response Variable
7
continued
So we put
T



X

2
i 
ˆ
 ,ˆ  arg max  ( 1  Z i ) ln 
 R p ,  0 i  1 
 

n




1
 1
T
2 
 Z i   ln( 2  2 ) 
(
Y

X

)

i
i
2

2
 2

and solve it numerically, e.g. by EM algorithm
(expectation-maximizing algorithm – see e.g.
William H. Greene (2000): Econometric Analysis, Prentice-Hall).
Limited Response Variable
8
continued
Two-Step Estimation - James J. Heckman (1979): Sample selection bias
as a specification error. Econometrica 47, 153 – 161.
Assume that for m from n observation
Yi  Yi* i.e. Yi*  0 .
For them we have
Yi  X iT    i
where Yi‘s and  i‘s have truncated normal distribution ( with
variance  2 ), i.e. we have



   
 Yi Yi*  0  X iT     i Yi*  0
 X iT
i

 i*   X iT 

Limited Response Variable
9
continued


1
  i  i*   X iT   P 1 ( i*   X iT  )
 2

1 ( 

(
1
X iT 

So, we have
with

1
)

2
X 
T
i

)
2



X iT 

t2
 X iT  t exp{ 2 2 }dt

y2
y exp{ 
}dy
2

T

X iT 
y 
1 X i 
}
  (
)  (
)
exp{ 
T
2   Xi 



2



 Yi Yi*  0  X iT      i
i   (
X iT 

)   1 (
X iT 

) .
Limited Response Variable
10
continued
In fact, we have found


  i Yi*  0     i
and so we can write
Yi  X iT      i  ~i
where already
 ~i  0
(1)
.
Were ‘s
available (observable ), we can estimate coeffs in ( 1 )
i
consistently and efficiently.
Nevertheless, putting
Zi  1
Yi*  0
Zi  0
Yi*  0 ,
and
Limited Response Variable
11
continued
we have likelihood
n
( Z 1 , Z 2 , , Z n )   P
Zi
( Zi  1 ) P
( 1 Z i )
( Zi  0 )
i 1
and hence log-likelihood (see ( * ) a few slides back)
n
L   Z i log P ( Z i  1 )  ( 1  Z i ) log P ( Z i  0 )
i 1
n


  Z i log P (  i*   X iT  )  ( 1  Z i ) log P (  i*   X iT  )
i 1

X iT 
X iT 
   Z i log (
)  ( 1  Z i ) log ( 


i 1 
n

)

(2)
But ( 2 ) is of course the log-likelihood of a probit model
and hence we can consistently estimate
X iT 

, i.e. also  i .
Limited Response Variable
continued
12
Then we put
Yi  X iT     ˆ i  ~i ,
and estimate, by means of OLS, consistently
(but not efficiently)  .
However, nowadays there is a way how to improve this estimate
just to add the third step – in fact applying GMM-estimation.
(We shall return to it later – may be in the last lecture of term.)
OLS corrected for asymptotic bias - William H. Greene (1981):
On the asymptotic bias of the ordinary least squares estimator of Tobit model.
Econometrica 49, 505 – 513.
Limited Response Variable
13
continued
Theorem (Greene, 1981, Econometrica, Kmenta)
Assumptions
Assume for the simple regression model that
   Y   Y2  YX  
Y * 

L     N   ,

2
 X 







X


 
YX
X




with
0
2
Y     0  X




and
YX
X .
Assertions
Then
ˆ ( O L S ,n )   X2 YX P ( Y *  0 )   0 P ( Y *  0 )
p


m
*
m

#
i
:
Y

0
Put
, then ˆ 
is the consistent estimator of
i
n
~ n ˆ ( O L S ,n )
0
*
P ( Y  0 ) and hence   
is the consistent estimator of  .
m
Model with Varying Coefficients
14
Systematic coefficients variation
Three examples:
Seasonal variation of coeffs – usually intercept is different for different seasons
Yt  X tT    t , 1   t , t  1,2,, T
with
 t   1 Dt 1   2 Dt 2   3 Dt 3   4 Dt 4
and
D tj  1
for the j-th quarter
D tj  0
otherwise.
j  1 ,2 ,3 ,4 ,
The model is frequently called season-dependent model.
Model with Varying Coefficients
continued
15
Systematic coefficients variation
Time variation of coeffs – usually slopes are dependent on time
Yt  X tT  t   t , t  1 ,2 , ,T
(1)
with
t   0    t .
(2)
The model is frequently called trend-dependent model.
The substitution of ( 2 ) into ( 1 ) gives
Yt  X tT  0  tX tT    t  X tT  0  Z tT    t , t  1 ,2 , ,T
and the test of significance of coeffs  is the test of hypothesis
that the model is not trend-dependent.
Model with Varying Coefficients
continued
16
Systematic coefficients variation
Switching regression – there are several regimes among which the model “moves”
Yt  X tT  ( 1 )   t , t  1 ,2 , , t ( 1 )
and
Yt  X tT  ( 2 )   t , t  t ( 1 )  1 , t ( 1 )  2 , ,T .
The model can be written as
Yt  X tT  ( 1 )  X tT (  ( 2 )   ( 1 ) )Dt   t ,
with
and
Dt  0, t  1,2, , t (1)
Dt  1, t  t (1)  1, t (1)  2, ,T
.
Model with Varying Coefficients
17
Random coefficients variation
Generalizing a bit Time variation of coefficients model
Yt  X tT  t   t , t  1 ,2 , ,T
(1)
t
with
β t  β t  1  v t  β0   v i
i 1
v  i.i.d., independent from ε  ,Evt  0,

t t 1
(2)

t t 1
Σ ν positive definite
( the model is frequently called adaptive regression model)
Substituting ( 2 ) into ( 1 ) gives
Yt  X β0  ε
T
t
*
t
with
ε  εt  X
*
t
T
t
yielding
  t*  0 ,
and
t
v
i 1
i
var  t*   2  tX tT  X t
cov{ε*t , ε*t  s }  (t  s)  X tT Σ ν X s .
,
Model with Varying Coefficients
continued
18
Random coefficients variation
So, the model is heteroscedastic with correlated disturbances.
Nevertheless, from
t -1

Yt  Yt 1  (X t  X t 1 ) β  εt  εt 1  X v t  X t  X t -1
T
0
T
t
 v ,
T
i 1
i
t  2,3, ,T
coefficients can be estimated consistently (of course without bias).
J. Johnston (1984): Econometric Methods. McGraw-Hill, NY.
Convergent parameter model
Yt  X tT  t   t , t  1 ,2 , ,T
with
t

βt  (1  λ)β0  λβt 1  v t   (1  λ)λ i 1 β0  ν i
i 1
vt t 1i.i.d., independent from εt t 1 ,Evt  0,
Substituting again ( 2 ) into ( 1 ) gives

0 λ1
(1)
(2)
Σ ν positive definite
Model with Varying Coefficients
continued
19
Random coefficients variation
t
Yt  X β  ε with β  (1  λ)β
,
0λ
T
t
0
*
t
0
ε  εt  X
*
t
i 1
i 1
It implies
  t*  0 ,
T
t
t
v
i 1
.
i
var  t*   2  tX tT  X t
and
cov{ε*t , ε*t  s }  (t  s)  X tT Σ ν X s .
So, the disturbances are heteroscedastic and correlated. Again

Yt  Yt 1  (X t  X t 1 ) β  εt  εt 1  X v t  X t  λX t -1
T
0
T
t
 v ,
T
t -1
i 1
t  2,3, ,T
Yt  Yt 1  ( X t  X t 1 )T  0  w t
i
,
Model with Varying Coefficients
continued
20
Random coefficients variation
Clearly  w t  0 and
T
2
2
T
var w t  1  λ σ ε  X t ΣX t  t  1X t  λX t - 1  Σ X t  λX t -1 
.
Moreover, Y t  1 is correlated with w t through
 t  1  X t  λX t -1 
T
t -1
v
i 1
i
.
So, we have to employ instrumental variables
( as a “natural” instrument, we can use X t  1 ).
B. Rosenberg (1973): The analysis of a cross section of time series
by stochastically convergent parameter regression.
Ann. of Economic and Social Measurement, 2, 399 - 428.
Model with Varying Coefficients
continued
21
Random coefficients variation
Random coefficient model
C. Hildreth, J. Houck (1968): Some estimator of linear model
with random coefficients.
J. of the American Statistical Association 63, 584 – 595.
Yi  X iT  i   i , i  1 ,2 , , n
with
(1)
L(βi  β 0 )  G(0, Σ β )
where G(.) is a d.f.. So, we may write
 i   0   i , i  1 ,2 , , n
with
L (ξ i )  G(0, Σ β ) .
(2)
Model with Varying Coefficients
continued
22
Random coefficients variation
The substitution of ( 2 ) into ( 1 ) gives
with
and hence
and
Yi  X iT  0   i* , i  1 ,2 , , n
 i*   i  X iT  i
  i*  0 ,
var  i*   2  ( X i  v2 ) 2
cov{  i* , *j }  0 i  j
.
So, the disturbances are heteroscedastic but uncorrelated.
The coefficients can be estimated either by ML or by GMM .
The models are equivalent to ARCH
(autoregressive conditionally heteroscedastic) models
(you will meet with them next year).
What is to be learnt from this lecture for exam ?
• Limited response variable
- Tobit model and idea of its estimation in one step,
- Heckman’s idea of two step estimation,
- Green’s idea of correcting asymptotic bias.
• Model for seasonal variation of coeffs
• Time variation of coefficients model
- adaptive regression model
• Random coefficient model
All what you need is on
http://samba.fsv.cuni.cz/~visek/