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

When either (C1) holds and t, θ, p are different,
o or (C2) holds and ∖t — p∖, ∖θ — p∖ > τ, then
, E[Vip,(∆o_m)] = E[y_v(∆y_M)] — E[y_v(∆X_a)]β = 0.

When either (B1), (D1), and (D2) hold and t, p, q are different,
or (B2), (D1), and (D2) hold and ∖t — p∖, ∖t — q∖ >τ, then

. E[(∆Xipq)⁰_ta] = E[(∆X)⁰Vit] - E[(∆Xipq) Xt]β = 0к,.

When either (C1), (D1), and (D2) hold and t, p, q are different,
or (C2), (D1), and (D2) hold and ∖t — p∖, ∖t — q∖ >τ, then

^E^Vipq ) ⁰it^] = ^E^Vipq ) Vit^] - ^E^Vipq ) ^Xit^{] β} = ⁰ ∙

The intercept c needs a comment. When mean stationarity of the latent regres-
sor, (D1), holds, then E(∆X_ipq) = 0_1K and E(∆V_ipq) = 0. If we relax (D1), which
cannot be assumed to hold in many situations due to non-stationarity, we get

^E[^ xipq ) ^βit^] = ^E[^ xipq ) ⁰Vit^] - ^E[^ xipq ) ^{0 ]} c - ^E[^ ^Xipq ) ^Xit^{] β} = ⁰ K1^,

^E[^Vipq)⁰it^] = ^E[^Vipq)Vit^] - ^E[^Vipq^)]c - ^E[^Vipq)xit^]β = ⁰∙

Eliminating c by means of E(_it) = E(V_it) - c - E(X_it)β = 0 leads to the following
modifications of (34) and (35):

When either (B1) and (D2) hold and t, p, q are different,

(36) or (B2) and (D2) hold and ∖t-p∖, ∖t-q∖ >τ, then

E[(∆Xipq) ⁰Oit] = E[(∆Xipq)⁰(Vit- E(Vit))] - E[(∆Xipq)⁰(Xit- E(Xit))]β = 0K,

When either (C1) and (D2) hold and t, p, q are different,

(37) or (C2) and (D2) hold and ∖t-p∖, ∖t-q∖ >τ, then

E[(∆ Vipq ) Oit ] = E[(∆ Vipq )( Vit - E( Vit ))] - E[(∆ Vipq )( Xit - E( Xit ))] β = 0.

The OC’s (32) - (37), corresponding to (19) in the general exposition of the
GMM, will be instrumental in constructing our GMM estimators. Not all of these
OC’s, whose number is substantial even for small T , are, of course, independent.
Let us examine the relationships between the OC’s in (32) - (33) and between
the OC’s in (34) - (35). Some of these conditions are redundant, i.e., linearly

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