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

Since SZ_eeZ and SZ_ωZ coincide asymptotically, we get, using bars to denote plims,

Replacing the plims S_xz, S_zx, SZZ and SZ_ωz by their sample counterparts,

Sxz, SZX, SZZ and SZêêZ and dividing by N, we get from (A.4) and (A.5) the
following estimators of the asymptotic covariance matrices of β^b and βe :
(A.6) ^V(β) = N ^[SXZS-ZSZX] 1^[SXZSZZSZ..Z^s-Z^sZX^][sXZ^s-Z^sZX^{] 1}

= [X⁰PZX]^-1[X⁰PZbb⁰PZX][X⁰PZX]^-1,

(A.7) Vdd ) = N S XZ S ^-. . - S ZX ]—¹

= [ X ⁰Z ( Z ⁰bb. ⁰Z )^-1Z ⁰X ]^-1 = [ X ⁰P Z (b b⁰ ) X ]^-1 ∙

These are the generic expressions which we use for estimating variances and co-
variances of the GMM estimators considered.

When calculating β from (A.3) in practice, we replace PZ(Ω) by PZ(bê⁰) =

Z(Z⁰ .b.b⁰Z)^-1Z⁰ [see White (1982, 1984)].

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