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

The overall GMM estimator corresponding to (34), which we now write as
E[(∆Z_i)⁰_i]=0_T(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) β^bLx = [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 β_Lx 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)⁰X^{i-1 h}X⁰(∆Z)V(^-∆¹Z)(∆Z)⁰yⁱ .

We can estimate V(_δz) JN consistently from the residuals obtained from (46) b_i =
yi ^- Xi^βLx^, by ^c(∆Z) JN = ^{(1 /N}) PN=1(AZi) ^bibi^Zi⁾. W^{e can here omit the}
intercept c; see Section 5.b. The resulting asymptotically optimal GMM estimator,
which examplifies (24), is

-1

(47)     β^eLx = [^PiXi⁰(∆Zi)] [^Pi(∆Zi)⁰bibi⁰(∆Zi)]^-1 [^Pi(∆Zi)⁰Xi]

× ^h[^PiXi⁰(∆Zi)] [^Pi(∆Zi)⁰bibi⁰(∆Zi)]^-1[^Pi(∆Zi)⁰yi]ⁱ .

The estimators β^b_Lx and βe_Lx can be modified by extending ∆x_i(t) to (∆x_i(t) : ∆y_i⁰_(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

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