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

effect by extending (6) to

(7)                     Xi = e^φ Ft ( K. ) = Ae^φ∙ ^ψ ( Y^rl )^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 X_iⁱ_t denote output, Y_iⁱ_t = (Y_iⁱ_t¹ ,...,Y_iⁱ_t^G) the
vector of G inputs, and w_tⁱ = (w_tⁱ¹ ,...,w_t^iG) the vector of input prices, common to
all firms - all treated as latent variables. We describe the technology by

(8)                                 Xiⁱt = eφⁱFt(Yiⁱt ),

where F_t is a production function common to all firms, t reflecting that tech-
nological changes are allowed for. We interpret φ_i as a constant known to firm
i, but unobserved by the econometrician. The dual cost function can then be
written as C_iⁱ_t = G_t(w_tⁱ,e^-φⁱX_iⁱ_t) [cf., e.g., Jorgenson (1986, section 5)], where
C_iⁱ_t = ^P_k^G₌₁ w_t^ik Y_iⁱ_t ^k. Using Shephard’s lemma, we can express firm i’s optimal
input of factor k in year t as

(9)                                Yiⁱt^k =gt^k(wtⁱ,e^-φⁱXiⁱt),

where gk (∙) = ∂G_t(∙)/∂wⁱ. Assuming that F_t represents a homothetic technology,
so that G_t can be separated as G_t(w_tⁱ ,e^-φⁱX_iⁱ_t)=H_t(w_tⁱ)K(e^-φⁱX_iⁱ_t), where the
functions H_t and K are monotonically increasing, (9) becomes

Yiⁱt^k =ht^k(wtⁱ)K(e^-φⁱXiⁱt),

with hk(∙) = ∂H_t(∙)/∂wⁱ. If, in particular, (8) has a constant scale elasticity μ for
all firms and years, then K(e^-φiXit_t) = e~^cφμi∣^μ(X*_t)¹ /^μ for all i,t and hence

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

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

(11)                         χ_it = c + α_i+ γ_t + βξ_it,

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