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A conditional least squares estimation procedure for a disequilibrium market model with autocorrelated errors
251
Economics Letters 20 (1986) 251-254 North-Holland
A CONDITIONAL LEAST SQUARES ESTIMATION PROCEDURE MARKET MODEL WITH AUTOCORRELATED ERRORS
FOR A DISEQUILIBRIUM
Sunil K. SAPRA State University of New York, Buffalo, NY 14260, USA Received Accepted
12 July 1985 15 October 1985
Maximum likelihood estimation of parameters in a market disequilibrium model with unknown sample separation and autocorrelated errors is computationally intractable [Quandt (1981) and Lee (1984)]. This paper presents a computationally convenient conditional least squares procedure for estimation of parameters in this model.
1. Introduction Disequilibrium market models with known and unknown sample separation have been studied extensively in the literature and are surveyed in Bowden (1978) and Maddala (1983). In almost all of theoretical and applied work on these models the disturbances in demand and supply equations are assumed to be serially uncorrelated. The only exceptions are Laffont and Monfort (1979) for the case of models with known sample separation, and Quandt (1981) and Lee (1984) for the case of models with unknown sample separation. The last two papers studied the maximum likelihood (ML hereafter) estimation in models with unknown sample separation and autocorrelated errors. The difficulties with the ML estimation, however, are that the likelihood function, in addition to being extremely complicated, is unbounded at certain boundary points of the parameter space. It is therefore of interest to explore alternative procedures which are computationally convenient and could be used if ML estimation failed. The purpose of this paper is to propose a computationally convenient conditional least squares algorithm for the estimation of parameters in a disequilibrium market model with unknown sample separation and autocorrelated errors. Our procedure yields consistent estimates of the parameters involved and avoids imposing unrealistic restrictions on error variances which have to be iimposed under ML estimation.
2. The model and the conditional least squares estimation procedure Consider
the following
0, = XLP, +
U1r7
where D, denotes 0165-1765/86/$3.50
4
=
disequilibrium GA
the quantity
+
U2r
3
demanded
model with unknown Q,
=
during
0 1986, Elsevier Science Publishers
min(
0, ,
period
S, >,
sample t=l,2
t; S, denotes
B.V. (North-Holland)
separation: ,...,
T,
the quantity
(W),(3) supplied
during
S. K. Sapra
252
/ Estimation procedure
period t, X,, and X,, are vectors of exogenous actual quantity transacted in the market during In addition to eqs. (l)-(3), we assume that
where E,~ and c2! are i.i.d. bivariate
,3
u12
2
[
u21
u2
normally
for disequilibrium
market model
variables, u,, and uzr are error terms, and Q, is the period t and is observed.
distributed
N(0, .I?) with
1
The probability
(6) t belongs
that observation
to the demand
function
is
v,=Pr(Q,=D,)=Pr(D,
-
GP2
= q/L
where F denotes U ’ = var(
u,, -
u21 < XLP2 -
-
XG4
>
%Pl>
(7)
i32, U2>>
the cumulative
UZr) = a:/(
1 -
distribution p: ) + u;/(
1 -
function p:)
-
of ( ult - uzI) and
2u12/(1
-
PIP2 1.
(8)
Similarly, (9) Furthermore, 0, = P,&,
eqs. (1) and (2) can be rewritten
+ X;rP, - PIXLP,
+ clt>
S, =
in view of (4) and (5) as
p2&1
+ X32
-
P2GrP2
+ f2r.
(10) ,(11)
Then eqs. (3) (10) and (11) yield
E(Q,IQ,_,)=E(Q,~Q,_,, Q,=D,)Pr(Q,=D,>+E(Q,IQ,-,~ = (p,Qr-, +(p,Q,-,
+ X;,P, +
P23
PJ;J%)F(&~
XrP2
-
~2%,P2)(1
-
Qr=%F’r(Qr=&) 0’)
F(bl,
(12)
b21 u*)),
and
E(Q:lQ,-,)=E(Q:IQ,-1, =
[(p,Q,--I
Q,=D,)Pr(Q,=D,>+E(Q:IQ,-,, + XL&
-
P,X;,P,)~
+[(~2Q,~1+x;,82-~2X;,p2)‘+(~22](1-F(P,,
+u:]F(L
Q,=&)Pr(Q,=&> P2,
u’)
P29
0’)).
(13)
S. K. Supra
Treating the initial the following steps: Step
h= i
for drsequrlibrium
as fixed, our proposed
253
marker model
conditional
least squares
with respect to ,B,, &> a2, p, and p2 the criterion
Minimize
I.
conditions
/ Estimation procedure
procedure
involves
function
[Q,-E(Q,IQ,-,)I*
1=2
1=2
=
i
[~,-(p,Q,-,+x;,p,-p,x;,P,)F(p,,
P23a2)
t=2
-(~zQ,p,
+
W32-
~,x;,b,)(~
-F(h
b23
(14)
02)123
where the last equality is obtained by substituting for E(Q, I Q,_ ,) from eq. (12). Note that this step yields estimates of ,8,, &, a’, p,, p2 and y,‘s only, where the estimates are obtained from ?,=@,,
l32,
a2).
of y,‘s
(15)
Step 2. Since the preceding step does not yield estimates of the parameters of, 0: and o,~. we propose as a second step, the minimization of the criterion function CT_, ~7’ with respect to 012, u,’ (after replacing p,, fi2, u2, p, and p2 by their respective estimates), where
6
~~2=t~2[Q:-E(Q~lQ,-,)]2
r=2
-
{ (~2Qr+,
where the last equality
+ GP2
is obtained
-
~2X;rP2)~
by substituting
+.,2)(1
-
F(B,,
l32> 02))]2.
for E(Qf 1Qr_,)
(16)
from eq. (13).
Note the (J,~ cannot be estimated in Steps 1 and 2 since it is not separately identifiable from the can be obtained from eq. (8) once B2, 3:. a$, fi, and criterion functions (14) and (16). However, G,* pr have been obtained in Steps 1 and 2. Thus S,,=$(l
-&fi*)[~:/(1-;:)+8,‘/(1
-$-s*].
Alternatively, the following procedure could be used to estimate and (2) by replacing the unknown parameters by their respective can be classified as (r)
Q,=b,
if
b,<$,
(11)
Q,=$
if
$
The observations
under
(17) ui2: obtain b, and $ from eqs. (1) estimates. Then observations on Q,
(I) and (II) can be used to run the following
regressions:
254
S. K. Saprcl / Estimrrtion procedure for disequilibrium
market model
An estimate of CJ,~can then be obtained as the sample covariance between i,, and i,, obtained from these regressions. The estimates of unknown parameters obtained from these steps are strongly consistent and possess asymptotic normal distribution under conditions stated in Klimko and Nelson (197X), and White and Domowitz (1984). Finally, minimization of the criterion functions in (14) and (16) can be carried out using either iterative or search techniques. Indeed, iterative techniques such as Gauss-Newton or the Davidon-Fletcher-Powell algorithms or a variant of Hildreth-Lu search procedures can be used to this end.
3. Discussion This paper has proposed a computationally convenient conditional least squares procedure for estimation of parameters in a disequilibrium model with unknown sample separation and autocorrelated errors. Our estimation procedure is computationally quite convenient and obviates the need for imposing certain restrictions on error variances which are needed for the proper ML estimates to exist. Although the estimates obtained by our procedure will not be as efficient as the ML estimates, they can be used as initial consistent estimates in the iterative ML procedure to obtain more efficient estimates of the parameters in the model.
References Amemiya, T.. 1983, Nonlinear regression models, in: Z. Griliches and M.D. Intriligator, eds.. Handbook of econometrics, Vol. I. ch. 6 (North-Holland. Amsterdam). Bowden, R.J., 1978, The econometrics of disequilibrium (North-Holland, Amsterdam). Bruesch. T.S. and A.R. Pagan. 1980. The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, Jan., 239-254. Klimko. L.A. and P.I. Nelson. 1978, On conditional least squares estimation for stochastic processes, Annals of Statistics 6. 6299642. Laffont. J.J. and A. Monfort, 1979, Disequilibrium econometrics in dynamic models, Journal of Econometrics 11, 353-363. Lee, L.F.. 1984. The likelihood function and a test for serial correlation in a disequilibrium market model, Economics Letters 14. 195-200. Maddala. G.S., 1983, Limited-dependent and qualitative variables in econometrics (Cambridge University Press, Cambridge). Neyman. J., 1959. Optimal asymptotic tests of composite statistical hypotheses, in: U. Grenander, ed., Probability and statistics (Almquist and Wicksell. Stockholm) 213-234. Quandt. R.E.. 1981. Autocorrelated errors in simple disequilibrium models, Economics Letters 7, 55561. White, H. and I. Domowitz. 1984. Nonlinear regression and dependent observations, Econometrica 52, no. 1, 143-161.
Economics Letters 20 (1986) 251-254 North-Holland
A CONDITIONAL LEAST SQUARES ESTIMATION PROCEDURE MARKET MODEL WITH AUTOCORRELATED ERRORS
FOR A DISEQUILIBRIUM
Sunil K. SAPRA State University of New York, Buffalo, NY 14260, USA Received Accepted
12 July 1985 15 October 1985
Maximum likelihood estimation of parameters in a market disequilibrium model with unknown sample separation and autocorrelated errors is computationally intractable [Quandt (1981) and Lee (1984)]. This paper presents a computationally convenient conditional least squares procedure for estimation of parameters in this model.
1. Introduction Disequilibrium market models with known and unknown sample separation have been studied extensively in the literature and are surveyed in Bowden (1978) and Maddala (1983). In almost all of theoretical and applied work on these models the disturbances in demand and supply equations are assumed to be serially uncorrelated. The only exceptions are Laffont and Monfort (1979) for the case of models with known sample separation, and Quandt (1981) and Lee (1984) for the case of models with unknown sample separation. The last two papers studied the maximum likelihood (ML hereafter) estimation in models with unknown sample separation and autocorrelated errors. The difficulties with the ML estimation, however, are that the likelihood function, in addition to being extremely complicated, is unbounded at certain boundary points of the parameter space. It is therefore of interest to explore alternative procedures which are computationally convenient and could be used if ML estimation failed. The purpose of this paper is to propose a computationally convenient conditional least squares algorithm for the estimation of parameters in a disequilibrium market model with unknown sample separation and autocorrelated errors. Our procedure yields consistent estimates of the parameters involved and avoids imposing unrealistic restrictions on error variances which have to be iimposed under ML estimation.
2. The model and the conditional least squares estimation procedure Consider
the following
0, = XLP, +
U1r7
where D, denotes 0165-1765/86/$3.50
4
=
disequilibrium GA
the quantity
+
U2r
3
demanded
model with unknown Q,
=
during
0 1986, Elsevier Science Publishers
min(
0, ,
period
S, >,
sample t=l,2
t; S, denotes
B.V. (North-Holland)
separation: ,...,
T,
the quantity
(W),(3) supplied
during
S. K. Sapra
252
/ Estimation procedure
period t, X,, and X,, are vectors of exogenous actual quantity transacted in the market during In addition to eqs. (l)-(3), we assume that
where E,~ and c2! are i.i.d. bivariate
,3
u12
2
[
u21
u2
normally
for disequilibrium
market model
variables, u,, and uzr are error terms, and Q, is the period t and is observed.
distributed
N(0, .I?) with
1
The probability
(6) t belongs
that observation
to the demand
function
is
v,=Pr(Q,=D,)=Pr(D,
-
GP2
= q/L
where F denotes U ’ = var(
u,, -
u21 < XLP2 -
-
XG4
>
%Pl>
(7)
i32, U2>>
the cumulative
UZr) = a:/(
1 -
distribution p: ) + u;/(
1 -
function p:)
-
of ( ult - uzI) and
2u12/(1
-
PIP2 1.
(8)
Similarly, (9) Furthermore, 0, = P,&,
eqs. (1) and (2) can be rewritten
+ X;rP, - PIXLP,
+ clt>
S, =
in view of (4) and (5) as
p2&1
+ X32
-
P2GrP2
+ f2r.
(10) ,(11)
Then eqs. (3) (10) and (11) yield
E(Q,IQ,_,)=E(Q,~Q,_,, Q,=D,)Pr(Q,=D,>+E(Q,IQ,-,~ = (p,Qr-, +(p,Q,-,
+ X;,P, +
P23
PJ;J%)F(&~
XrP2
-
~2%,P2)(1
-
Qr=%F’r(Qr=&) 0’)
F(bl,
(12)
b21 u*)),
and
E(Q:lQ,-,)=E(Q:IQ,-1, =
[(p,Q,--I
Q,=D,)Pr(Q,=D,>+E(Q:IQ,-,, + XL&
-
P,X;,P,)~
+[(~2Q,~1+x;,82-~2X;,p2)‘+(~22](1-F(P,,
+u:]F(L
Q,=&)Pr(Q,=&> P2,
u’)
P29
0’)).
(13)
S. K. Supra
Treating the initial the following steps: Step
h= i
for drsequrlibrium
as fixed, our proposed
253
marker model
conditional
least squares
with respect to ,B,, &> a2, p, and p2 the criterion
Minimize
I.
conditions
/ Estimation procedure
procedure
involves
function
[Q,-E(Q,IQ,-,)I*
1=2
1=2
=
i
[~,-(p,Q,-,+x;,p,-p,x;,P,)F(p,,
P23a2)
t=2
-(~zQ,p,
+
W32-
~,x;,b,)(~
-F(h
b23
(14)
02)123
where the last equality is obtained by substituting for E(Q, I Q,_ ,) from eq. (12). Note that this step yields estimates of ,8,, &, a’, p,, p2 and y,‘s only, where the estimates are obtained from ?,=@,,
l32,
a2).
of y,‘s
(15)
Step 2. Since the preceding step does not yield estimates of the parameters of, 0: and o,~. we propose as a second step, the minimization of the criterion function CT_, ~7’ with respect to 012, u,’ (after replacing p,, fi2, u2, p, and p2 by their respective estimates), where
6
~~2=t~2[Q:-E(Q~lQ,-,)]2
r=2
-
{ (~2Qr+,
where the last equality
+ GP2
is obtained
-
~2X;rP2)~
by substituting
+.,2)(1
-
F(B,,
l32> 02))]2.
for E(Qf 1Qr_,)
(16)
from eq. (13).
Note the (J,~ cannot be estimated in Steps 1 and 2 since it is not separately identifiable from the can be obtained from eq. (8) once B2, 3:. a$, fi, and criterion functions (14) and (16). However, G,* pr have been obtained in Steps 1 and 2. Thus S,,=$(l
-&fi*)[~:/(1-;:)+8,‘/(1
-$-s*].
Alternatively, the following procedure could be used to estimate and (2) by replacing the unknown parameters by their respective can be classified as (r)
Q,=b,
if
b,<$,
(11)
Q,=$
if
$
The observations
under
(17) ui2: obtain b, and $ from eqs. (1) estimates. Then observations on Q,
(I) and (II) can be used to run the following
regressions:
254
S. K. Saprcl / Estimrrtion procedure for disequilibrium
market model
An estimate of CJ,~can then be obtained as the sample covariance between i,, and i,, obtained from these regressions. The estimates of unknown parameters obtained from these steps are strongly consistent and possess asymptotic normal distribution under conditions stated in Klimko and Nelson (197X), and White and Domowitz (1984). Finally, minimization of the criterion functions in (14) and (16) can be carried out using either iterative or search techniques. Indeed, iterative techniques such as Gauss-Newton or the Davidon-Fletcher-Powell algorithms or a variant of Hildreth-Lu search procedures can be used to this end.
3. Discussion This paper has proposed a computationally convenient conditional least squares procedure for estimation of parameters in a disequilibrium model with unknown sample separation and autocorrelated errors. Our estimation procedure is computationally quite convenient and obviates the need for imposing certain restrictions on error variances which are needed for the proper ML estimates to exist. Although the estimates obtained by our procedure will not be as efficient as the ML estimates, they can be used as initial consistent estimates in the iterative ML procedure to obtain more efficient estimates of the parameters in the model.
References Amemiya, T.. 1983, Nonlinear regression models, in: Z. Griliches and M.D. Intriligator, eds.. Handbook of econometrics, Vol. I. ch. 6 (North-Holland. Amsterdam). Bowden, R.J., 1978, The econometrics of disequilibrium (North-Holland, Amsterdam). Bruesch. T.S. and A.R. Pagan. 1980. The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, Jan., 239-254. Klimko. L.A. and P.I. Nelson. 1978, On conditional least squares estimation for stochastic processes, Annals of Statistics 6. 6299642. Laffont. J.J. and A. Monfort, 1979, Disequilibrium econometrics in dynamic models, Journal of Econometrics 11, 353-363. Lee, L.F.. 1984. The likelihood function and a test for serial correlation in a disequilibrium market model, Economics Letters 14. 195-200. Maddala. G.S., 1983, Limited-dependent and qualitative variables in econometrics (Cambridge University Press, Cambridge). Neyman. J., 1959. Optimal asymptotic tests of composite statistical hypotheses, in: U. Grenander, ed., Probability and statistics (Almquist and Wicksell. Stockholm) 213-234. Quandt. R.E.. 1981. Autocorrelated errors in simple disequilibrium models, Economics Letters 7, 55561. White, H. and I. Domowitz. 1984. Nonlinear regression and dependent observations, Econometrica 52, no. 1, 143-161.