The estimation of the coefficient on the labour inputs and energy are obtained substituting
a third order polynomial approximation in kit and mit to φit and then using OLS. From
this first stage of the estimation routine we obtain βw, ∣.',^b, βe and
φit = yit - βwlwit - βblbit - βeeit ∙
In the second stage, for any candidate value47 of βk* and βm* it is possible to compute ω^it
using ωit = φit - βk kit - βm mit
Subsequently, from the regression
23
ω it = γ0 + γ1ωt-1 + γ 2ωt-1 + γ 3ωt-1 + ηit
we obtain a consistent (non parametric) approximation of E[ωit | ωit-1 ], E[ωit ∣ωit-ɪ] that
is used to compute the sample residual for the production function (as a function of βk*
and βm*)
ε it (βk, βm ) + ξit = yit - βwbwit - βblbit - βeeit - βk kit - βm mit - E
To identify both βk and βm separately we need two moment conditions∙ The first will be
(as in Olley and Pakes) that the capital does not respond to shocks to this period’s
innovation in productivity ξit, providing the population moment:
E[ξit + εit | kit]=0
The second condition, needed to identify βm , uses the fact that last period’s material
choices should be uncorrelated with the innovation in productivity in this period∙
E[ξit + εit | mit-1]=0
Finally, with Zit=( kit ,mit-1 ), βm and βk are obtained minimizing the GMM criterion
function,
Q(β*) = min ∑ ∑ (ξit + εit )Zit
With respect to inference, given the fact that this is a multi stage estimation procedure, the
covariance matrix of the final parameters must account for variance and covariance of
every estimator that enters the routine∙ This problem is solved by bootstrapping standard
errors∙
47 good starting values might be the OLS estimates from the production function∙
36
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