Handling the measurement error problem by means of panel data: Moment methods applied on firm data

or, compactly,

∆yi = (∆Xi)β + ∆i.

The IV matrix (cf. Proposition 1) is the ((2T - 3) × KT(T - 2)) matrix

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[Z_i⁰(∆_i)] = 0_{T (T -2)K,1}, minimizing [N^-1 (∆) ⁰Z](N^-2V )^-1[N^-1Z⁰(∆)] for V =
Z ⁰Z, can be written as

(42)      β^b_Dx = [(∆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 β_Dx 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),
de_i = ∆y_i - (∆Xi)βDx, by cZ(∆,)/N = (1 /N) P= Z'(de_f)(de_f)⁰Z_i. The re-
sulting asymptotically optimal GMM estimator, which examplifies (24), is

0                               -1

(43)     β^eDx = [^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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