Appendix
In this appendix, we elaborate the procedures for estimating asymptotic covariance
matrices of the GMM estimators. All models in the main text, with suitable
interpretations of y, X, Z, e, and Ω, have the form:
(A.1) y = Xβ + e, E( e ) = 0, E( Z 0e ) = 0, E( ee0 ) = Ω,
where y =(y01,...,y0N)0, X =(X01,...,X0N)0, Z =(Z01,...,Z0N)0, and =
(e01 ,...,e0N)0, Zi being the IV matrix of Xi. The two generic GMM estimators
considered are
(A.2) βb =[X0PZX]-1[X0PZy], PZ = Z(Z0Z)-1Z0,
s~∙∙~' -ɪ
(A.3) β = [ X 0P z (Ω) X ] -1 [ X 0P z (Ω) y ],
Let the residual vector calculated from β be be
P Z (Ω) = Z ( Z0 Ω Z )-1Z0.
y - Xβ, and use the notation
SXZ =
XZ
X0Z
SZX =
ZX
Z0X
SZZ =
Z0Z
SeZ =
e0Z
S = 7,
SZe = N ,
SZeeZ =
Z0ee0Z
S
ZebbeZ
Z 0beeb 0 Z
Inserting for y from (A.1) in (A.2) and (A.3), we get
√N ( β - β ) = √N [ X 0P Z X ]-1[ X 0P Z e ] = [ S XZ S-Z S Zχ ]-1 S
XZS
1 Z0e
ZZ √N ,
N ( β - β ) = √N [ X 0P Z (Ω) X ]-1[ X 0P Z (Ω) e ] = [ S χZ S - Ω Z S Zx ]-1
SXZS
1 Z0e
and hence,
N(β - β)(β - β) 0 = [SXZSZ-Z1 SZX]-1[SXZSZ-Z1 SZeeZSZ-Z1 SZX][SXZSZ-Z1 SZX]-1,
N(β - β)(β - β)0 = [SXZS-1ZSzx]-1[SXZS-1ZSZeeZS-1ZSzx][SxzS-1ZSzx]-1
I X-I I X XZ Z Ω Z ZX∖ XZ Z Ω Z ZeeZ Z Ω Z ZXJL XZ Z Ω Z ZXX
The asymptotic covariance matrices of √Nβ and N3e/3 can then, under suitable
regularity conditions, be written as [see Bowden and Turkington (1984, pp. 26, 69)]
. !---------------------- ∙---■ . . . ■---■ . . ∙---■ . . ■---■ ---
aV(√Nβ) = lim E[N(β - β)(β - β)0] = plim[N(β - β)(β - β)0],
aV(√N/3) = lim E[N(β - β)(β - β)0] = plim[N(β - β)(β - β)0].
29
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