to impose more structure on the first and second order moments of the uit’s, vit’s,
ξit’s, and αi’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 yit’s and the xit’s, and possibly higher order moments, rather than eliminating
the αi ’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[(∆yipq)it]=β0E[(∆ξipq)0αi]+E[(∆uipq)uit], 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
(32)
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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