of using ∆X (t) as IV matrix for Xt, we may either, if K = 1, use ∆Y (t), or, for
arbitrary K, (∆X(t) : ∆Y(t)), provided that also (C1) is satisfied. Third, we can
deduct period means from xit and yit and relax the stationarity in mean assumption
of the latent regressor, (D1); cf. (36) - (37).
If we relax Assumptions (B1) or (C1) and replace them by (B2) or (C2), we
must reconstruct the OC’s underlying (38) and (39) to ensure that the variables
in the IV matrix have a lead or lag of at least τ + 1 periods to the regressor, to
‘get clear of’ the τ period memory of the MA(τ) process. The IV sets will then be
reduced.
6 Composite GMM estimators
combining differences and levels
We now take the single equation GMM estimators (38) and (39) and their het-
eroskedasticity robust modifications one step further and construct GMM estima-
tors of the common coefficient vector β when we combine the essential OC’s for
all periods, i.e., for all differences or for all levels. This gives multi-equation, or
overall, GMM estimators for panel data with measurement errors, still belonging
to the general framework described in Section 4. The procedures to be described
in this section, like the single-equation procedures in section 5.c, may be modified
to be applicable to situations with disturbance/error autocorrelation.
Equation in differences, IV’s in levels. Consider the differenced equation (5)
for all θ = t - 1 and θ = t - 2. These (T - 1)+(T - 2) equations stacked for
individual i read
|
∆yi21 . . |
δ xi 21 . . |
δ ei 21 . . | |||
|
(40) |
. δ yi,T,T -1 |
_ |
. δ xi,T,T -1 |
β + |
. δ ei,T,T -1 |
|
δ Vi 31 . . |
δ xi 31 . . |
δ ei 31 . . | |||
|
. ∆yi,T,T -2 |
. ∆xi,T,T -2 |
. ∆i,T,T -2 |
20