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

to impose more structure on the first and second order moments of the u_it’s, v_it’s,
ξ_it’s, and α_i’s - confer the ‘structural approach’ to EIV modelling. In this way we
might obtain more efficient estimators by operating on the full covariance matrix of
the y_it’s and the x_it’s, and possibly higher order moments, rather than eliminating
the α_i ’s by differencing, as elaborated below. Such estimators, however, may be
less robust to specification errors.

5.b Moment equations and orthogonality conditions

A substantial number of moment conditions involving second order moments in yit ,
xit, and it can be derived from Assumptions (A) - (E).

From (1) - (3) and Assumption (A) we obtain the following moment equations
involving observable variables in levels and differences:

(25)                 E[xi⁰p(∆xitθ)]  =  E[ξi⁰p(∆ξitθ)] + E[vi⁰p(∆vitθ)],

(26)                 E[xi⁰p(∆yitθ)]  =  E[ξi⁰p(∆ξitθ)]β,

(27)               E[(∆xipq)⁰yit]  =  E[(∆ξipq)⁰ξit]β+ E[(∆ξipq)⁰(αi+c)]

and involving observable variables and errors/disturbances:

(28)    E[xi⁰p(∆itθ)] = -E[vi⁰p(∆vitθ)]β,

(29)     E[yip(∆itθ)] = E[uip(∆uitθ)],

(30)   E[(∆xipq)⁰it]=E[(∆ξipq)⁰αi]- E[(∆vipq)⁰vit]β,

(31)    E[(∆y_ipq)_it]=β⁰E[(∆ξ_ipq)⁰α_i]+E[(∆u_ipq)u_it],      t,θ,p,q=1,...,T.

The moments on the left hand side of (25) - (27) are structured by Assumptions
(D) and (E). The moments at the left hand side of (28) - (31) are structured by
Assumptions (B) - (D). Depending on which assumptions are valid, some of the
terms on the right hand side of (28) - (31), or all, vanish. Provided that T>2,
(3), (5), and (28) - (31) imply

When either (B1) holds and t, θ, p are different,
o or (B2) holds and ∖t — p∖, ∖θ — p∣ > τ, then

E[ x0p (∆ Citθ )] = E[ x0p (∆ yitθ )] — E[ x0p (∆ Xitθ )] β = 0 к 1.

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