Seemingly unrelated systems of econometric equations

ABSTRACT

A generalization of Zellner's SUR model is derived for sets of seemingly unrelated systems of econometric equations. The resulting structural form – worked out for a set of Cowles Commission-type simultaneous equations systems – is general enough to include any SUR-type or panel-type specification of systems of econometric equations with contemporaneously correlated errors. Maximum estimation efficiency is obtained by treating all the individual subsystems at once rather than in a subsystem-by-subsystem fashion.

KEYWORDS:

• SUR models
• simultaneous equations systems
• Cowles Commission
• estimation efficiency

JEL CLASSIFICATION:

• C30
• C33
• C51

Acknowledgments

Part of this paper is a part of miss Kakarantza's Ph.D. thesis. Miss Kakarantza has benefited from scholarships granted by the George Stavros' Orphanage and the Propondis Foundation for the academic years 2009–2010 and 2010–2012, respectively.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. ${\mathbf{0}}^{\prime }$ stands for a $(1\times n)$ row vector with zero elements.

2. Note that, according to Equation (39(39) ${\mathit{X}}_{\mu }=\left[\right({\delta }_{ij}{\mathit{X}}_i{)}_{i,j=\sum _{{\mu }_{\ast }=1}^{\mu -1}{n}_{{\mu }_{\ast }}+1,\dots ,\sum _{{\mu }_{\ast }=1}^{\mu }{n}_{{\mu }_{\ast }}}]\mathrm{and}\mathit{X}=[\left({\delta }_{ij}{\mathit{X}}_i{)}_{i,j=1,\dots ,n}\right].$(39) ), matrices ${Я}_{\mu }$ and $Я$ can also be written as ${Я}_{\mu }=\left[\right({\delta }_{ij}{Я}_i{)}_{i,j=\sum _{{\mu }_{\ast }=1}^{\mu -1}{n}_{{\mu }_{\ast }}+1,\dots ,\sum _{{\mu }_{\ast }=1}^{\mu }{n}_{{\mu }_{\ast }}}]\mathrm{and}Я=[\left({\delta }_{ij}{Я}_i{)}_{i,j=1,\dots ,n}\right].$

3. Hausman [6] compared the 3SLS, full information maximum likelihood (FIML), and full information instrumental variables (FIIV) estimators. Since the instruments are identical, the FIML estimators can be computed by beginning with a consistent estimate and iterating the FIIV estimators up to convergence. However, this iterative property does not hold for the 3SLS estimators, because they ignore the over-identifying restrictions in the formation of the instruments.

Sargan [8] proved that if (i) a Cowles Commission-type system is fully identified and stable, and (ii) its error variance–covariance matrix is unrestricted, then the 3SLS and FIML estimators have the same asymptotic distribution, and the corresponding estimates differ asymptotically by an order $O\left(1/T\right)$.

Dhrymes [3] showed that, although the 3SLS and FIML estimators have the same asymptotic distribution, they differ in the way they treat the jointly dependent variables and, therefore, the iteration of the 3SLS does not yield the FIML estimator. Furthermore, since the 3SLS estimator may be viewed as a kind of ‘linearized’ FIML, we can compare the 3SLS to the linearized maximum likelihood (LML) and FIML estimators: Under the condition that all the equations of the system are just-identified, the 3SLS and FIML estimators are identical for every sample size.

4. To see that each equation in (51(51) $\begin{array}{rl}\left[\begin{array}{c}{\mathit{Y}}_1\\ ⋮\\ {\mathit{Y}}_M\end{array}\right]& =\left[\begin{array}{c}{\mathit{\Xi }}_1\\ ⋮\\ {\mathit{\Xi }}_M\end{array}\right]Б+\left[\begin{array}{c}{\mathit{E}}_1\\ ⋮\\ {\mathit{E}}_M\end{array}\right]\Rightarrow \\ {\mathit{Y}}_{\mu }& ={\mathit{\Xi }}_{\mu }Б+{\mathit{E}}_{\mu }(\mu =1,\dots ,M),\end{array}$(51) ) is a Cowles Commission-type simultaneous equations system, partition the $\left(\right(n_{\circ }+m_{\circ })\times n_{\circ })$ matrix $Б=\left[\begin{array}{c}{Б}_{\mathit{Y}}\\ \cdot \cdot \\ {Б}_{\mathit{Z}}\end{array}\right]$, where ${Б}_{\mathit{Y}}$ and ${Б}_{\mathit{Z}}$ are the $(n_{\circ }\times n_{\circ })$ and $(m_{\circ }\times n_{\circ })$ matrices of coefficients corresponding to ${\mathit{Y}}_{\mu }$ and ${\mathit{Z}}_{\mu }$, respectively, define the $(n_{\circ }\times n_{\circ })$ matrix ${Б}_{\sim }={\mathit{I}}_{n_{\circ }}-{Б}_{\mathit{Y}}$, and use Equations (8(8) ${\mathit{\Xi }}_{\mu }={\mathit{\Xi }}_{\ast }{\mathit{G}}_{\mu }=[\mathit{Y}{\mathit{G}}_{{\mathit{Y}}_{\mu }}:{\mathit{Z}}_{\ast }{\mathit{G}}_{\ast \mu }]=[{\mathit{Y}}_{\mu }:{\mathit{Z}}_{\mu }].$(8) ), (10(10) ${\mathit{Y}}_{\mu }={\mathit{\Xi }}_{\mu }{Б}_{\mu }+{\mathit{E}}_{\mu }(\mu =1,\dots ,M).$(10) ), (11(11) ${Б}_{\mu }=\left[\begin{array}{c}{Б}_{{\mathit{Y}}_{\mu }}\\ \cdot \cdot \\ {Б}_{{\mathit{Z}}_{\mu }}\end{array}\right],$(11) ), and (12(12) ${Б}_{\sim \mu }={\mathit{I}}_{n_{\mu }}-{Б}_{{\mathit{Y}}_{\mu }}.$(12) ) to express the μth system of equations as ${\mathit{Y}}_{\mu }{Б}_{\sim }+{\mathit{Z}}_{\mu }(-{Б}_{\mathit{Z}})={\mathit{E}}_{\mu }$, which is the structural form of a Cowles Commission-type simultaneous equations system.

5. The matrices $Я$ and ${\mathit{G}}_{\ast }$ in Equation (44(44) ${\mathit{G}}_{\ast }=\left[\right({\delta }_{\mu {\mu }^{\prime }}({\mathit{I}}_{n_{\mu }}\otimes {\mathit{G}}_{\mu }){)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}]\mathrm{and}Я=[\left({\delta }_{\mu {\mu }^{\prime }}{Я}_{\mu }{)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}\right],$(44) ) must be redefined accordingly: Since $n_{\ast }=M{n}_{\circ }(n_{\circ }+m_{\circ })$, $Я=\left[\right({Я}_{\circ }{)}_{\mu =1,\dots ,M}]$ becomes an $(n_{\ast }\times N)$ block-column matrix with submatrices ${Я}_{\circ }$, whilst ${\mathit{G}}_{\ast }=\left[\right({\delta }_{\mu {\mu }^{\prime }}({\mathit{I}}_{n_{\circ }}\otimes {\mathit{G}}_{\mu }){)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}]$ remains an $(n{\nu }_{\ast }\times n_{\ast })$ block-diagonal matrix.

6. One could imagine that the number of exogenous or predetermined explanatory variables would differ between the various individual subsystems of the Cowles Commission-type in Equation (51(51) $\begin{array}{rl}\left[\begin{array}{c}{\mathit{Y}}_1\\ ⋮\\ {\mathit{Y}}_M\end{array}\right]& =\left[\begin{array}{c}{\mathit{\Xi }}_1\\ ⋮\\ {\mathit{\Xi }}_M\end{array}\right]Б+\left[\begin{array}{c}{\mathit{E}}_1\\ ⋮\\ {\mathit{E}}_M\end{array}\right]\Rightarrow \\ {\mathit{Y}}_{\mu }& ={\mathit{\Xi }}_{\mu }Б+{\mathit{E}}_{\mu }(\mu =1,\dots ,M),\end{array}$(51) ). This is not possible, however, since, in such a case, the linear restrictions (52(52) ${\mathit{b}}_i={Я}_i{\mathit{\beta }}_i,$(52) ) would be much more stringent than necessary: Since different numbers of explanatory variables entail different matrices of coefficients, ${Б}_{\mu }(\mu =1,\dots ,M)$ say, for the various subsystems, and since all matrices ${Б}_{\mu }$ must be transformed into a single vector of parameters $\mathit{\beta }$, identical for all subsystems of the form (57(57) ${\mathit{y}}_{\mu }={\mathit{X}}_{\mu }\mathit{\beta }+{\mathit{\epsilon }}_{\mu },$(57) ), the restrictions (52(52) ${\mathit{b}}_i={Я}_i{\mathit{\beta }}_i,$(52) ) are imposed not only on the dimensionality of the corresponding initial parameter vectors, denoted by ${\mathit{b}}_{\mu ,i}$ in this case, but also on their values.

7. In an attempt to avoid the possibility of autocorrelated disturbances, the researcher may have to include the lagged dependent variable as a predetermined explanatory variable. This causes no difficulty since the lagged dependent explanatory variable is uncorrelated with the error terms of the current period.

8. In this case, matrices $Я$ and ${\mathit{G}}_{\ast }$ must retain their original definition given in Equation (44(44) ${\mathit{G}}_{\ast }=\left[\right({\delta }_{\mu {\mu }^{\prime }}({\mathit{I}}_{n_{\mu }}\otimes {\mathit{G}}_{\mu }){)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}]\mathrm{and}Я=[\left({\delta }_{\mu {\mu }^{\prime }}{Я}_{\mu }{)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}\right],$(44) ) with $n_{\ast }=\sum _{\mu =1}^M{n}_{\mu }{m}_{\mu }$.

9. For the possibility of predetermined explanatory variables, see footnote 7.

10. In this case, the matrices $Я$ and ${\mathit{G}}_{\ast }$ in (44(44) ${\mathit{G}}_{\ast }=\left[\right({\delta }_{\mu {\mu }^{\prime }}({\mathit{I}}_{n_{\mu }}\otimes {\mathit{G}}_{\mu }){)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}]\mathrm{and}Я=[\left({\delta }_{\mu {\mu }^{\prime }}{Я}_{\mu }{)}_{\mu ,{\mu }^{\prime }=1,\dots ,M}\right],$(44) ) must be defined as in footnote 5 with $n_{\ast }=M{n}_{\circ }{m}_{\circ }$.

11. For the possibility of predetermined explanatory variables, see footnote 7. Since the lagged dependent variable is uncorrelated with the error terms of the current period, its inclusion as an explanatory variable in the model, does not affect the consistency of the estimators proposed in this subsection.