Econometric Analysis of Cross Section and Panel Data - Free

188
Chapter 8
which has dimension G L, where L ¼ L 1 þ L 2 þþL G . Then, for each i,
Zi 0 u i ¼ðz i1 u i1 ; z i2 u i2 ; ...; z iG u iG Þ 0
and so EðZi 0 u iÞ¼0 reproduces the orthogonality conditions (8.13). Also,
0
Eðzi1 0 x 1
i1Þ 0 0 0
0 Eðz 0
EðZi 0 X i2 x i2Þ 0 0
iÞ¼
. . B
C
@
A
0 0 0 EðziG 0 x iGÞ
ð8:16Þ
ð8:17Þ
where Eðz 0 ig x igÞ is L g K g . Assumption SIV.2 requires that this matrix have full column
rank, where the number of columns is K ¼ K 1 þ K 2 þþK G . A well-known
result from linear algebra says that a block diagonal matrix has full column rank if
and only if each block in the matrix has full column rank. In other words, Assumption
SIV.2 holds in this example if and only if
rank Eðz 0 ig x igÞ ¼K g ; g ¼ 1; 2; ...; G ð8:18Þ
This is exactly the rank condition needed for estimating each equation by 2SLS,
which we know is possible under conditions (8.13) and (8.18). Therefore, identification
of the SUR system is equivalent to identification equation by equation. This
reasoning assumes that the b g are unrestricted across equations. If some prior
restrictions are known, then identification is more complicated, something we cover
explicitly in Chapter 9.
In the important special case where the same instruments, z i , can be used for every
equation, we can write definition (8.15) as Z i ¼ I G n z i .
8.3 Generalized Method of Moments Estimation
8.3.1 A General Weighting Matrix
The orthogonality conditions in Assumption SIV.1 suggest an estimation strategy.
Under Assumptions SIV.1 and SIV.2, b is the unique K 1 vector solving the linear
set population moment conditions
E½Z 0 i ðy i X i bÞŠ ¼ 0 ð8:19Þ
(That b is a solution follows from Assumption SIV.1; that it is unique follows by
Assumption SIV.2.) In other words, if b is any other K 1 vector (so that at least one
element of b is di¤erent from the corresponding element in b), then