(b) E[(∆xit+1,t-1) 0it] = 0K,1 for t =2,...,T- 1 are K(T -2) essential OC’s for
equations in levels, with IV’s differenced over two periods.
(c) The other OC’s are redundant: among the 1KT(T — 1)(T — 2) conditions in
(33) when T>2, only KT(T- 2) are essential.
For generalizations to the case where eit is a MA(τ) process, see Bi0rn (2000,
section 2.d). These propositions can be (trivially) modified to include also the
essential and redundant OC’s in the y’s or the ∆y’s, given in (33) and (35).
5.c The estimators
We are now in a position to specialize (23) and (24) to define (i) consistent GMM
estimators of β in (5) for one pair of periods (t, θ), utilizing as IV’s for ∆xitθ
all admissible xip ’s, and (ii) consistent GMM estimators of β in (3), i.e., for one
period (t), utilizing as IV’s for xit all admissible ∆xipq ’s. This is a preliminary to
Section 6, in which we combine on the one hand (i) the differenced equations for
all pairs of periods, and on the other hand (ii) the level equations for all periods,
respectively, in one equation system.
We let Ptθ denote the ((T —2) × T) selection matrix obtained by deleting from
IT rows t and θ, and introduce the [(T—2) × T] matrix
d21
Dt =
dt-1,t-2
dt+1,t-1
dt+2,t+1
t=1,...,T,
dT,T-1
which is a one-period differencing matrix, except that dt,t-1 and dt+1,t are re-
placed by their sum, dt+1,t-1, the two-period difference being effective only for
t =2,...,T— 1, and use the notation
yi ( tθ ) = P tθ yi∙, X i ( tθ ) = P tθ X i∙, xi ( tθ ) = vec( X i ( tθ )) 0,
∆ yi ( t ) = Dy,, ∆ X i ( t ) = DtX i∙, ∆ xi ( t ) = vec(∆ X i ( t ))0,
17
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