Seemingly Unrelated Regressions

Abstract

Consider two regression equations corresponding to two different firms 1{\$}{\$}y{\_}i = X{\_}i {\backslash}beta {\_}i + u{\_}{\{}i, {\}} i = 1, 2{\$}{\$}where yi and ui are T{\texttimes}1 and Xi is (T{\texttimes}Ki) with ui∼(0, $σ$iiIT). OLS is BLUE on each equation separately. Zellner's (1962) idea is to combine these Seemingly Unrelated Regressions in one stacked model, i.e., 2{\$}{\$}y{\_}i = X{\_}i {\backslash}beta {\_}i + u{\_}{\{}i, {\}} i = 1, 2{\$}{\$}which can be written as 3{\$}{\$}y = X{\backslash}beta + u{\$}{\$}where y{\textasciiacutex}=(y{\textasciiacutex}1, y{\textasciiacutex}2) and X and u are obtained similarly from (10.2). y and u are 2T{\texttimes}1, X is 2T{\texttimes}(K1+K2) and $β$ is (K1+K2){\texttimes}1. The stacked disturbances have a variance-covariance matrix 4{\$}{\$}{\backslash}sigma {\_}{\{}11{\}} I{\_}T {\$}{\$}where $Σ$=[$σ$ij] for i, j=1, 2; with 5{\$}{\$}y{\_}i = X{\_}i {\backslash}beta {\_}i + u{\_}{\{}i, {\}} i = 1, 2{\$}{\$}measuring the extent of correlation between the two regression equations. The Kronecker product operator ⊗ is defined in the Appendix to Chapter 7. Some important applications of SUR models in economics include the estimation of a system of demand equations or a translog cost function along with its share equations, see Berndt (1991). Briefly, a system of demand equations explains household consumption of several commodities. The correlation among equations could be due to unobservable household specific attributes that influence the consumption of these commodities. Similarly, in estimating a cost equation along with the corresponding input share equations based on firm level data.