The overall GMM estimator corresponding to (34), which we now write as
E[(∆Zi)0i]=0T(T-2)K,1, minimizing [N-10(∆Z)](N-2V∆)-1[N-1(∆Z)0] for
V∆=(∆Z)0(∆Z),can be written as
(46) βbLx = [X0(∆Z)[(∆Z)0(∆Z)]-1(∆Z)0X]-1[X0(∆Z)[(∆Z)0(∆Z)]-1(∆Z)0y]
-1
= [PiXi0(∆Zi)] [Pi(∆Zi)0(∆Zi)]-1 [Pi(∆Zi)0Xi]
× h[Pi Xi0(∆Zi)] [Pi(∆Zi)0(∆Zi)]-1[Pi(∆Zi)0yi]i .
This estimator examplifies (23). The consistency of βLx relies on Assumptions
(A), (B1), (D1), (D2), and (E). If has a non-scalar covariance matrix, a more
efficient estimator is obtained for V ∆ = V (∆Z) = E[(∆Z)00(∆Z)], which gives
βeLx = hX0(∆Z)V(-∆1Z)(∆Z)0Xi-1 hX0(∆Z)V(-∆1Z)(∆Z)0yi .
We can estimate V(δz) JN consistently from the residuals obtained from (46) bi =
yi - XiβLx, by c(∆Z) JN = (1 /N) PN=1(AZi) bibi^Zi). We can here omit the
intercept c; see Section 5.b. The resulting asymptotically optimal GMM estimator,
which examplifies (24), is
-1
(47) βeLx = [PiXi0(∆Zi)] [Pi(∆Zi)0bibi0(∆Zi)]-1 [Pi(∆Zi)0Xi]
× h[PiXi0(∆Zi)] [Pi(∆Zi)0bibi0(∆Zi)]-1[Pi(∆Zi)0yi]i .
The estimators βbLx and βeLx can be modified by extending ∆xi(t) to (∆xi(t) : ∆yi0(t) )
in (45), also exploiting Assumption (C1) and the OC’s in the ∆y’s. This is indicated
be replacing subscript Lx by Ly or Lxy on the estimator symbols. We can also
deduct period means from the level variables in (44) to take account of possible
non-stationarity of these variables and relax (D1) [cf. (36) - (37)].
Tables 24.3 and 24.4 contain the overall GMM estimates obtained from the
complete set of level equations, the first using the untransformed observations and
the second based on observations measured from their year means. The orthog-
onality test statistics (columns 6 and 7) give for materials conclusions similar to
those for the differenced equation in Table 24.2 for Textiles and Chemicals (which
24
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