where Xit = lnYt, ξit = lnXit, β = 1 /μ, αi = -(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 yit =
χit + uit and xit = ξit + vit , where uit and vit are measurement errors. This gives
an equation of the form (3). In the more general case where Ft represents a non-
homothetic technology, separability of Gt 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) yit and xit 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 yit∕N, y = Pi Pt yit∕(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,
plim(χ.t - ξ.t) = 0ιк, so that plim[χφe∙t] = 0кι even if plim[(1 /N) Pn=i χl°teit] =
0K1. From (12) and (13) we therefore get
(14) plim[^ sχ ∙t ) ' (δ s y∙t )] = plim[^ sχ ∙t ) ' sχ ∙t )] β,
(15) plim[(χ-t-χ) '( y∙t — У)] = plim[(χ-t- χ) '( χ-t- χ)]β-