Some SUR Model Computations
(Why is it that bGLS is equivalent to equation-by-equation bOLS when the covariates in each equation are the same?)
The seemingly unrelated regression (SUR) model admits the following representation
Yj =Xj j +”j; j=1;:::;M;
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where each of the T 1 vectors Yj = (yj1;:::;yjT)0 is assumed to depend on its own set of covariates Xj
with corresponding regression coe¢ cients j. Stacking the M sets of equations, we can write the model as Y=W +”;
YM 0 0 0 XM M “M
If the errors are contemporaneously correlated, i.e., V (“t) = MM for “t = (“1t; : : : ; “Mt)0, then it follows
0Y11 0X1 0 01 011 0″11 B Y2 C B 0 X2 0 0 C B 2 C B “2 C
Y=B . C; W=B . 0 . C; =B . C; “=B . C: B@ . CA B@ . . 0 CA B@ . CA B@ . CA
that V (“) = IT and the GLS estimator is given by
0 1 1 0 1
bGLS= W( IT) W W( IT) Y:
An important result can be obtained for the case when the covariates in each equation are the same, that is Xj = X for all j. Then, using the Kronecker product notation, the matrix W can be written as W = IM X
and bGLS becomes
bGLS = (IM X) ( IT) (IM X) (IM X) ( IT) Y:
bGLS. In paticular, we have bGLS =
0 1 1 0 1 (IM X) ( IT) (IM X) (IM X) ( IT) Y
0 1 1 0 1
Using the following three properties of Kronecker products, (A B)0 = A0 B0, (A B) (C D) = (AC) (BD) when the matrices are conformable, and (A B) 1 = A 1 B 1, we can simplify the expression for
= (IM X0) 1 IT (IM X) 1 (IM X0) 1 IT Y
= 1 X0 (IM X) 1 1 X0 Y
= 1 X0X 1 1 X0 Y
= (X0X) 1 1 X0 Y
= IM (X0X) 1 X0 Y: Recalling the deÖnition of Y , we have that
(X0X) 1 X0Y1 1 (X0X) 1 X0Y2 C
. CA; (X0X) 1 X0YM
thereby showing that bGLS is simply the collection of OLS estimates for each of the equations done separately. 1
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