Handling the measurement error problem by means of panel data: Moment methods applied on firm data

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 ,...,e⁰_N)⁰, Z_i being the IV matrix of X_i. The two generic GMM estimators
considered are

(A.2)           β^b =[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

Inserting for y from (A.1) in (A.2) and (A.3), we get

and hence,

N(β - β)(β - β) ⁰ = [S_XZS_Z^-_Z¹ S_ZX]^-1[S_XZS_Z^-_Z¹ S_ZeeZS_Z^-_Z¹ S_ZX][S_XZS_Z^-_Z¹ S_ZX]^-1,

N(β - β)(β - β)⁰ = [S_XZS^-1ZS_zx]^-1[S_XZS^-1ZSZ_eeZS^-1ZS_zx][S_xzS^-1ZS_zx]^-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].

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