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A computational algorithm for multiple equation models with panel data
Economics Letters North-Holland
143
34 (1990) 143-146
A computational algorithm for multiple equation models with panel data * Terrence Kinal and Kajal Lahiri State University of New York at Albany, Albany, NY 12222, USA Received Accepted
11 December 1989 23 January 1990
By stacking observations by individuals across equations, we propose a simpler computational algorithm to estimate multiple-equations error-components models with or without simultaneity using the Choleski decomposition of the resultant covariance matrix of residuals. This generalizes the Fuller-Batesse transformation to the multiple equation case.
1. Introduction Computation in single equation error components models has been greatly simplified through the use of the data transformation originally suggested by Fuller and Batesse (1973). More recently, Breusch, Mizon and Schmidt (1989) have characterized the optimal instrument set that is required, subsequent ot the Fuller-Batesse transformation, in the context of different types of single equation error-components models with endogeneity. Avery (1977) suggested an appropriate generalized least squares procedure for a multiple equations model with error components, and various full-information procedures to estimate a set of simultaneous equations with error components were suggested and analyzed by Chamberlain and Griliches (1975), Baltagi (1981, 1984), Prucha (1985), Balestra and Varadharajan-Krishnakumar (1987), and Comwell, Schmidt and Wyhowski (1988). The main purpose of the present paper is to show that we can generalize the Fuller-Batesse transformation to the multivariate case. Furthermore, this generalization suggests a much simpler computational algorithm to estimate a multiple equations error components model, with or without simultaneity.
2. The model Consider a typical equation individuals over T periods. Yjil = Xjit
Pj
+
Gji
+
from
#jir
3
a model
j=l
containing
,...,
M,
i=
M equations,
,...,
N,
with
t=l,...,
lxK,
* The authors
acknowledge
01651765/90/$03.50
helpful
comments
from Badi Baltagi,
0 1990 - Elsevier Science Publishers
Alok Bhargava
B.V. (North-Holland)
and Robin
Sickles.
panel
T.
data
on
N
(1)
144
T. Kind,
Now
define
(;ijii,i
..
.
iji)
=
K. Lahiri
/ A computational algorithm
U’ = (uijlujiz _ . . ~,~r), and ns, = Y,: = (J’,iiJ’ji, . . . yj;r), Xii = (XjiiXl;i . . . xJ,r), where i, is a row vector T ones. We make the usual assumptions [cf. Baltagi jlj,iTp
(1984)], i.e., iiji -
N(O,$),
E(iki+ily)
=
i
2”’
9
u,it- N(O,Uij)y E( uj,,ukls) = { ;“” 2
E(ij;jUk,,)= 0,
i=l
,.*-,
if
i=j
N,
otherwise
i=l
j=
i=l
j=l,..., and
l,...,
(2)
M,
t=l,...,
T,
(4)
t=s
(5)
3
otherwise i,k=
M,
’
,--*, N,
if
l,...,
M,
j,l=l,...,
N,
t=l,...,
T.
(6)
The standard arrangement of the data is to stack observations according to individual, i.e., Y’ = x;.= (x;Ix;z.. . xi,), u;.= (u;1u;2.. . U(N), T$= (?&T$*. . . Tj;,>, y$ = <_Y,;y,; . . . y$), . . . . XL.), U’=(ui.u; . . . . uh.),and r’=(ni.n; . . . . TJ~.). Then ( y;.y;. . . . yh.), X’=(x;x; r=xp+
‘k,
(7)
whereB’=(&‘/3;.../3h)and‘k=U+Z’.Now &?=var(*)
=var(U)
+var(!P)
=2,81,,+2,8
[Z,@J,],
(8)
where 2” = II au-U Iland Z, = II uqiq,II. Then a generahzed least squares estimator (GLS) of /? in (7) is given by
&s = ( x’s2-1x)-1(X'K'Y). In most practical applications Q will be far too large to invert by direct numerical methods. Following Nerlove (1971), Avery (1977) and Baltagi (1980) provide expressions for 52-l in terms of its spectral decomposition which permit the calculation of /?oLs. These expressions, however, are rather involved and burdensome to implement since they require the calculation of the characteristic roots of 52. If instead, the rows and columns of D are rearranged by reorganizing the data, a much simpler computational scheme emerges. In this alternative arrangement, standard computer algorithms may be used to obtain GLS estimates. First stack observations by equation and then by individual. Thus put yli = ( y;iy;. . . y,&), xl; = diag(x;,, xii,. . . , XL,), uli = (u;~u&. . . u&~), and nl.i = (T&&. . . ohi). Thus, y’ = (y.‘,y.>. . . y.>), x’ = (xlixl,. . . XI,,,), u’ = (ul,ul,. . . uIN), and n’ = (~l.iql.~. . . ~l.~). Then y=xB+c,
(10)
where /3 is defined above and E = n + u. Now var(e) = var(n) + var( u) = Z,,,8 X = V, where X = II B;j II = II au,, IT + a,,, JT 11, JT = i,i& This alternative setup is more convenient for constructing and interpreting the estimation procedure outlined in section 3.
T. Kind, K. Lahiri / A computational algorithm
3. A simpler computational
145
algorithm
Equation (10) is appropriately estimated by GLS when C, and z,, are known and by feasible GLS otherwise. In this section we first show that the transformation used to perform GLS is a multivariate generalization of Fuller and Batesse’s transformation. Then we outline a computational algorithm for the estimation. It is easy to verify that the positive definite square root of Z is
zP* = 2;”
0
z, -
$ [2’,/’
- (2, +
z-q”*] @
JT
and that
For
a
single
equation
(M = 1)
it
reduces
to
z-‘/*
= (Zr - (h/T)
which is Fuller and Batesse’s transformation
63 JT)/uu,
where
X= 1
except for the factor a,. Thus,
-JW, while the Fuller-Batesse transformation is implicit in Avery (1977) and Baltagi (1980) it becomes explicit here because of the way in which the day have been arranged. We now provide a computationally feasible transformation using the Choleski decomposition which is supplied by most econometric packages. We may write _X= AA’, where A = S, 8 IT = (X/T)(S, - .S,) 60JT and where the lower triangular matrices S, and S, are Choleski decompositions satisfying S,S,’ = 2, and S,Si = z,, + T2,. Therefore, since var(e) = (I,,, @ A)( IN 8 A)‘, put y * = (I,,, 8 A)-‘y, x * = (IN 8 A)-‘x, and c * = (IN 8 A)-le. A convenient diagnostic check is provided by the fact that now var(e *) = ZMNT.The GLS estimator is then obtained by applying OLS to y*
=x*p+c*.
(11)
The variance and covariance components in 1, and I7 alternative techniques [see Baltagi (1981)]. The transformation acts on any variable w.; as follows: w,r = s;lw+
+
(s*-’ - sl-‘) w+= S,‘Kii + s*-‘w.;,
can be estimated
by a number
of
(12)
where w.: = ( wliw2;. . . wMi), i?.‘, = ( WliW2,. . . iQ+), G.; = w.~ - W.i, and Wji= J,w,,/T. Employing the transformations in (12), OLS can be applied directly to (11). Alternatively, if the transformation used is S, A-‘, then the system is estimated by seemingly unrelated regression (SUR). In this case the transformation is completely analogous to Fuller and Batesse. Note that transformation (12) can be applied to the data arranged in the conventional manner. Hence the alternative arrangement discussed in here is helpful for deriuing a computational scheme, but is not necessary for the actual estimation. Also, when a subset of the explanatory variables in (1) is correlated with ?j, (single exogenous) of with ui, and 4, (endogenous), we should apply the instrumental variable (IV) technique using an instrumental variable set appropriate for the mode1 [see Comwell, Schmidt and Wyhowski (1988)].
146
T. Kind,
K. Lohiri / A computational
algorithm
4. Conclusion By arranging the data to simplify construction of transformations for multiple equation error component models, we have shown that the transformations of Avery (1977) and Baltagi (1980, 1981) are multivariate generalizations of the Fuller-Batesse (1973) transformations. This fact allows us to implement a computationally simpler estimation procedure with possibly non-linear restrictions across equations. Not only is this procedure computationally simpler, it also requires less computational power and utilizes the built-in computational algorithms of the standard econometric packages. We have demonstrated these gains with simulated data for the SUR case and with world trade data for the simultaneous equations case [see Kinal and Lahiri (1989)].
References Avery, R., 1977, Error components and seemingly unrelated regressions, Econometrica 45, 199-209. Balestra, P. and J. Varadharajan-Krishnakkumar, 1987, Full information estimation of a system of simultaneous equations with error components structure, Econometric Theory 3, 223-246. Baltagi, B., 1980, On seemingIy unrelated regressions with error components, Econometrica 48, 1547-1551. Baltagi, B., 1981, Simultaneous equations with error components, Journal of Econometrics 17, 189-200. Baltagi, B., 1984, A Monte Carlo study for pooling time series of cross-section data in the simultaneous equations model, International Economic Review 23, 603-624. Breusch, T., G. Mizon and P. Schmidt, 1989, Efficient estimation using panel data, Econometrica 57, 695-700. Chamberlain, G. and Z. Griliches, 1975, Unobservables with variance components structure: Ability, schooling and the economic success of brothers, International Economic Review 16, 422-450. Cornwell, C., P. Schmidt and D. Wyhowski, 1988, Simultaneous equations and panel data, Mimeo., Sept. (Michigan State University, East Lansing, MI). Fuller, W. and G. Batesse, 1973, Transformations for estimation of linear models with nested error structure, Journal of the American Statistical Association 68, 636-642. Kinal, T. and K. Lahiri, 1989, Estimation of simultaneous equations error components models with an application to a model of developing country foreign trade, Discussion paper 89-14 (State University of New York at Albany, New York) Nov. Nerlove, M., 1971, Further evidence on the estimation of dynamic economic relations from a time-series of cross-sections, Econometrica 39,341-358. Prucha, I., 1985, Maximum likelihood and instrumental variable estimation in simultaneous equation systems with error components, International economic Review 26,491-506.
143
34 (1990) 143-146
A computational algorithm for multiple equation models with panel data * Terrence Kinal and Kajal Lahiri State University of New York at Albany, Albany, NY 12222, USA Received Accepted
11 December 1989 23 January 1990
By stacking observations by individuals across equations, we propose a simpler computational algorithm to estimate multiple-equations error-components models with or without simultaneity using the Choleski decomposition of the resultant covariance matrix of residuals. This generalizes the Fuller-Batesse transformation to the multiple equation case.
1. Introduction Computation in single equation error components models has been greatly simplified through the use of the data transformation originally suggested by Fuller and Batesse (1973). More recently, Breusch, Mizon and Schmidt (1989) have characterized the optimal instrument set that is required, subsequent ot the Fuller-Batesse transformation, in the context of different types of single equation error-components models with endogeneity. Avery (1977) suggested an appropriate generalized least squares procedure for a multiple equations model with error components, and various full-information procedures to estimate a set of simultaneous equations with error components were suggested and analyzed by Chamberlain and Griliches (1975), Baltagi (1981, 1984), Prucha (1985), Balestra and Varadharajan-Krishnakumar (1987), and Comwell, Schmidt and Wyhowski (1988). The main purpose of the present paper is to show that we can generalize the Fuller-Batesse transformation to the multivariate case. Furthermore, this generalization suggests a much simpler computational algorithm to estimate a multiple equations error components model, with or without simultaneity.
2. The model Consider a typical equation individuals over T periods. Yjil = Xjit
Pj
+
Gji
+
from
#jir
3
a model
j=l
containing
,...,
M,
i=
M equations,
,...,
N,
with
t=l,...,
lxK,
* The authors
acknowledge
01651765/90/$03.50
helpful
comments
from Badi Baltagi,
0 1990 - Elsevier Science Publishers
Alok Bhargava
B.V. (North-Holland)
and Robin
Sickles.
panel
T.
data
on
N
(1)
144
T. Kind,
Now
define
(;ijii,i
..
.
iji)
=
K. Lahiri
/ A computational algorithm
U’ = (uijlujiz _ . . ~,~r), and ns, = Y,: = (J’,iiJ’ji, . . . yj;r), Xii = (XjiiXl;i . . . xJ,r), where i, is a row vector T ones. We make the usual assumptions [cf. Baltagi jlj,iTp
(1984)], i.e., iiji -
N(O,$),
E(iki+ily)
=
i
2”’
9
u,it- N(O,Uij)y E( uj,,ukls) = { ;“” 2
E(ij;jUk,,)= 0,
i=l
,.*-,
if
i=j
N,
otherwise
i=l
j=
i=l
j=l,..., and
l,...,
(2)
M,
t=l,...,
T,
(4)
t=s
(5)
3
otherwise i,k=
M,
’
,--*, N,
if
l,...,
M,
j,l=l,...,
N,
t=l,...,
T.
(6)
The standard arrangement of the data is to stack observations according to individual, i.e., Y’ = x;.= (x;Ix;z.. . xi,), u;.= (u;1u;2.. . U(N), T$= (?&T$*. . . Tj;,>, y$ = <_Y,;y,; . . . y$), . . . . XL.), U’=(ui.u; . . . . uh.),and r’=(ni.n; . . . . TJ~.). Then ( y;.y;. . . . yh.), X’=(x;x; r=xp+
‘k,
(7)
whereB’=(&‘/3;.../3h)and‘k=U+Z’.Now &?=var(*)
=var(U)
+var(!P)
=2,81,,+2,8
[Z,@J,],
(8)
where 2” = II au-U Iland Z, = II uqiq,II. Then a generahzed least squares estimator (GLS) of /? in (7) is given by
&s = ( x’s2-1x)-1(X'K'Y). In most practical applications Q will be far too large to invert by direct numerical methods. Following Nerlove (1971), Avery (1977) and Baltagi (1980) provide expressions for 52-l in terms of its spectral decomposition which permit the calculation of /?oLs. These expressions, however, are rather involved and burdensome to implement since they require the calculation of the characteristic roots of 52. If instead, the rows and columns of D are rearranged by reorganizing the data, a much simpler computational scheme emerges. In this alternative arrangement, standard computer algorithms may be used to obtain GLS estimates. First stack observations by equation and then by individual. Thus put yli = ( y;iy;. . . y,&), xl; = diag(x;,, xii,. . . , XL,), uli = (u;~u&. . . u&~), and nl.i = (T&&. . . ohi). Thus, y’ = (y.‘,y.>. . . y.>), x’ = (xlixl,. . . XI,,,), u’ = (ul,ul,. . . uIN), and n’ = (~l.iql.~. . . ~l.~). Then y=xB+c,
(10)
where /3 is defined above and E = n + u. Now var(e) = var(n) + var( u) = Z,,,8 X = V, where X = II B;j II = II au,, IT + a,,, JT 11, JT = i,i& This alternative setup is more convenient for constructing and interpreting the estimation procedure outlined in section 3.
T. Kind, K. Lahiri / A computational algorithm
3. A simpler computational
145
algorithm
Equation (10) is appropriately estimated by GLS when C, and z,, are known and by feasible GLS otherwise. In this section we first show that the transformation used to perform GLS is a multivariate generalization of Fuller and Batesse’s transformation. Then we outline a computational algorithm for the estimation. It is easy to verify that the positive definite square root of Z is
zP* = 2;”
0
z, -
$ [2’,/’
- (2, +
z-q”*] @
JT
and that
For
a
single
equation
(M = 1)
it
reduces
to
z-‘/*
= (Zr - (h/T)
which is Fuller and Batesse’s transformation
63 JT)/uu,
where
X= 1
except for the factor a,. Thus,
-JW, while the Fuller-Batesse transformation is implicit in Avery (1977) and Baltagi (1980) it becomes explicit here because of the way in which the day have been arranged. We now provide a computationally feasible transformation using the Choleski decomposition which is supplied by most econometric packages. We may write _X= AA’, where A = S, 8 IT = (X/T)(S, - .S,) 60JT and where the lower triangular matrices S, and S, are Choleski decompositions satisfying S,S,’ = 2, and S,Si = z,, + T2,. Therefore, since var(e) = (I,,, @ A)( IN 8 A)‘, put y * = (I,,, 8 A)-‘y, x * = (IN 8 A)-‘x, and c * = (IN 8 A)-le. A convenient diagnostic check is provided by the fact that now var(e *) = ZMNT.The GLS estimator is then obtained by applying OLS to y*
=x*p+c*.
(11)
The variance and covariance components in 1, and I7 alternative techniques [see Baltagi (1981)]. The transformation acts on any variable w.; as follows: w,r = s;lw+
+
(s*-’ - sl-‘) w+= S,‘Kii + s*-‘w.;,
can be estimated
by a number
of
(12)
where w.: = ( wliw2;. . . wMi), i?.‘, = ( WliW2,. . . iQ+), G.; = w.~ - W.i, and Wji= J,w,,/T. Employing the transformations in (12), OLS can be applied directly to (11). Alternatively, if the transformation used is S, A-‘, then the system is estimated by seemingly unrelated regression (SUR). In this case the transformation is completely analogous to Fuller and Batesse. Note that transformation (12) can be applied to the data arranged in the conventional manner. Hence the alternative arrangement discussed in here is helpful for deriuing a computational scheme, but is not necessary for the actual estimation. Also, when a subset of the explanatory variables in (1) is correlated with ?j, (single exogenous) of with ui, and 4, (endogenous), we should apply the instrumental variable (IV) technique using an instrumental variable set appropriate for the mode1 [see Comwell, Schmidt and Wyhowski (1988)].
146
T. Kind,
K. Lohiri / A computational
algorithm
4. Conclusion By arranging the data to simplify construction of transformations for multiple equation error component models, we have shown that the transformations of Avery (1977) and Baltagi (1980, 1981) are multivariate generalizations of the Fuller-Batesse (1973) transformations. This fact allows us to implement a computationally simpler estimation procedure with possibly non-linear restrictions across equations. Not only is this procedure computationally simpler, it also requires less computational power and utilizes the built-in computational algorithms of the standard econometric packages. We have demonstrated these gains with simulated data for the SUR case and with world trade data for the simultaneous equations case [see Kinal and Lahiri (1989)].
References Avery, R., 1977, Error components and seemingly unrelated regressions, Econometrica 45, 199-209. Balestra, P. and J. Varadharajan-Krishnakkumar, 1987, Full information estimation of a system of simultaneous equations with error components structure, Econometric Theory 3, 223-246. Baltagi, B., 1980, On seemingIy unrelated regressions with error components, Econometrica 48, 1547-1551. Baltagi, B., 1981, Simultaneous equations with error components, Journal of Econometrics 17, 189-200. Baltagi, B., 1984, A Monte Carlo study for pooling time series of cross-section data in the simultaneous equations model, International Economic Review 23, 603-624. Breusch, T., G. Mizon and P. Schmidt, 1989, Efficient estimation using panel data, Econometrica 57, 695-700. Chamberlain, G. and Z. Griliches, 1975, Unobservables with variance components structure: Ability, schooling and the economic success of brothers, International Economic Review 16, 422-450. Cornwell, C., P. Schmidt and D. Wyhowski, 1988, Simultaneous equations and panel data, Mimeo., Sept. (Michigan State University, East Lansing, MI). Fuller, W. and G. Batesse, 1973, Transformations for estimation of linear models with nested error structure, Journal of the American Statistical Association 68, 636-642. Kinal, T. and K. Lahiri, 1989, Estimation of simultaneous equations error components models with an application to a model of developing country foreign trade, Discussion paper 89-14 (State University of New York at Albany, New York) Nov. Nerlove, M., 1971, Further evidence on the estimation of dynamic economic relations from a time-series of cross-sections, Econometrica 39,341-358. Prucha, I., 1985, Maximum likelihood and instrumental variable estimation in simultaneous equation systems with error components, International economic Review 26,491-506.