(33)
When either (C1) holds and t, θ, p are different,
o or (C2) holds and ∖t — p∖, ∖θ — p∖ > τ, then
, E[Vip,(∆om)] = E[yv(∆yM)] — E[yv(∆Xa)]β = 0.
(34)
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)0ta] = E[(∆X)0Vit] - E[(∆Xipq) Xt]β = 0к,.
(35)
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(∆Xipq) = 01K and E(∆Vipq) = 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(Vit) - c - E(Xit)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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