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



effect by extending (6) to

(7)                     Xi = eφ Ft ( K. ) = Aeφψ ( Yrl )μ,

where E(φi) = E(ψt) = 0.

The second interpretation is to assume a (neo-classical) technology with one
output and
several inputs, of which two are capital and materials, and output
constrained
cost minimization. Let Xiit denote output, Yiit = (Yiit1 ,...,YiitG) the
vector of
G inputs, and wti = (wti1 ,...,wtiG) the vector of input prices, common to
all firms - all treated as latent variables. We describe the technology by

(8)                                 Xiit = eφiFt(Yiit ),

where Ft is a production function common to all firms, t reflecting that tech-
nological changes are allowed for. We interpret
φi as a constant known to firm
i, but unobserved by the econometrician. The dual cost function can then be
written as
Ciit = Gt(wti,e-φiXiit) [cf., e.g., Jorgenson (1986, section 5)], where
Ciit = PkG=1 wtik Yiit k. Using Shephard’s lemma, we can express firm i’s optimal
input of factor
k in year t as

(9)                                Yiitk =gtk(wti,e-φiXiit),

where gk () = ∂Gt()/∂wi. Assuming that Ft represents a homothetic technology,
so that
Gt can be separated as Gt(wti ,e-φiXiit)=Ht(wti)K(e-φiXiit), where the
functions
Ht and K are monotonically increasing, (9) becomes

Yiitk =htk(wti)K(e-φiXiit),

with hk() = ∂Ht()/∂wi. If, in particular, (8) has a constant scale elasticity μ for
all firms and years, then
K(eiXitt) = e~cφμiμ(X*t)1 /μ for all i,t and hence

(10)                           Y'lk = hk ( w, ) e--∙-'μ ( Xit )1 /μ.

Taking logs, we can then write both (7) and (9) in simplified notation as

(11)                         χit = c + αi+ γt + βξit,



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