or, compactly,
∆yi = (∆Xi)β + ∆i.
The IV matrix (cf. Proposition 1) is the ((2T - 3) × KT(T - 2)) matrix
xi (21) ’ |
∙∙ 0 . .. |
0∙ . . |
∙∙ 0 .. .. | |
(41) Zi= |
. 0 ∙ |
.. ∙∙ xi(T,T-1) |
. 0∙ |
.. ∙∙ 0 |
0 ∙ |
∙∙ 0 .. |
xi2 ∙ .. |
∙∙ 0 . | |
. . 0∙ |
.. .. ∙∙ 0 |
. . 0∙ |
.. .. ∙∙ xi,T-1 |
We here use different IV’s for the (T - 1)+(T - 2) equations in (40), with β as a
common slope coefficient. Let
∆y= [(∆y1)0,...,(∆yN)0]0, ∆= [(∆1)0,...,(∆N)0]0,
∆X= [(∆X1)0,...,(∆XN)0]0, Z=[Z10,...,ZN0]0.
The overall GMM estimator corresponding to (32), which we now write as
E[Zi0(∆i)] = 0T (T -2)K,1, minimizing [N-1 (∆) 0Z](N-2V )-1[N-1Z0(∆)] for V =
Z 0Z, can be written as
(42) βbDx = [(∆X) 0Z(Z0Z)-1Z0(∆X)]-1 [(∆X) 0Z(Z0Z)-1Z0(∆y)]
-1
= [Pi(∆Xi)0Zi][PiZi0Zi]-1 [PiZi0(∆Xi)]
× h[Pi(∆Xi)0Zi][PiZi0Zi]-1 [Pi Zi0(∆yi)]i .
This estimator examplifies (23). The consistency of βDx relies on Assumptions
(A), (B1), and (E). If ∆ has a non-scalar covariance matrix, a more efficient
estimator is obtained for V = V Z(∆) = E[Z0(∆)(∆)0Z], which gives
-1
βeDx = (∆X)0ZVZ-(1∆)Z0(∆X) (∆X)0ZVZ-(1∆)Z0(∆y) .
We can estimate VZ(δ,) /N consistently from the residuals obtained from (42),
dei = ∆yi - (∆Xi)βDx, by cZ(∆,)/N = (1 /N) P= Z'(def)(def)0Zi. The re-
sulting asymptotically optimal GMM estimator, which examplifies (24), is
0 -1
(43) βeDx = [Pi(∆Xi)0Zi][Pi Zi0∆di∆di0Zi]-1[Pi Zi0(∆Xi)]
0
× [Pi(∆Xi) 0Zi][Pi Zi0∆di∆di0Zi]-1[Pi Zi0(∆yi)] .
21
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