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

where Xit = lnYt^{, ξ}it = lⁿXit^{, β} = 1 /μ^{, α}i = ^-(¹ /μ)^φi^, Yt is either zero, ^-(¹ /μ)^ψt
or ln hk(wt:), and c is a constant. The observed log-output and log-input are y_it =
χ_it + u_it and x_it = ξ_it + v_it , where u_it and v_it are measurement errors. This gives
an equation of the form (3). In the more general case where F_t represents a non-
homothetic technology, separability of G_t does not hold. Then the input elasticity
β will be different for different inputs and hence cannot be interpreted as an inverse
scale elasticity.

Neither of these model interpretations imposes a specific normalization on (11)
and (3), as observed input and output are both formally endogenous variables. In
the empirical application, two normalizations will be considered: (i) yit and xit
are, respectively, the log of an observed factor input and the log of observed gross
production, both measured as values at constant prices and β corresponds to 1 /μ,
and (ii) y_it and x_it have the reverse interpretation and β corresponds to μ.

3 Estimators based on period means

In this section, we consider various estimators of β constructed from differenced
period means. From (3) we obtain

(12)          ∆_sy._t = ∆sX∙tβ + ∆₈l,_t,      s = 1 ,...,T-1; t = s + 1 ,...,T,

(¹³)            ⁽ y∙t ^- У) = ^(χ ∙t ^{- χ})^{β + (} 4 ^{- ё})^{,          t} =¹ ,^---,^t,

where y∙t = Pi y_it∕N^, y = Pi Pt y_it∕⁽NT)^{, χ}∙t = Pi ^xit^{/N, χ} = Pi Pt ^χit/⁽NT)^,
etc. and ∆s denotes differencing over s periods.

The (weak) law of the large numbers, when (A) is satisfied, implies under weak
conditions [cf. McCabe and Tremayne (1993, section 3.5)],⁴ that plim(e._t) = 0,
pli^m(^χ._t ^- ξ._t) = 0ιк^, so that pli^m[^χφe∙t^] = 0кι even if plⁱm[(¹ /N) Pⁿ=i ^χl°teit^] =
0_K1. From (12) and (13) we therefore get

(¹⁴)                plim[^ s^χ ∙t ) ' (^δ s y∙t )^] = plim[^ s^χ ∙t ) ' s^χ ∙t )^{] β,}

(¹⁵)                pl^im[(^χ-t^-χ) '⁽ y∙t — У)] = pl^im[(^χ-t^{- χ}) '^{( χ}-t^{- χ})]^β-

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