Estimation of dynamic linear models in short panels with ... - Cemmap

Thus:∂ Pr( yit= 1| X∂xit−s, u )ii==∂ρ∂x⎛⎜⎝s−1∏it−sj=0+it−sδ∞∑it−jk = s+1s−1∏j=0δ⎞⎟φ⎠⎛ ρ⎜⎝ δit−j( β'x + u )it−kit−s+it−sk −1∏j=0∞∑k = s+1δit−jρδit−si⎞⎟⎠it−kβk −1∏j=0δit−j∂δ∂x[ φ( α + β'x + u ) − φ( β'x + u )]β(20)The profile of ∂ Pr(y it =1 | X i , u i )/ ∂x it-s is thus considerably more complicated than thegeometric decay implied by the SD model (1).it−sit−sit−siit−si3 Estimation3.1 Initial conditionsIn the SD model, there are two alternative approaches for dealing with the randomeffects u. Heckman (1981b) specifies an approximation to the distribution of y i0 | X i ,u i , and then derives the distribution of y i1 ... y iT | y i0 , X i , u i using sequentialconditioning. The random effects are then integrated out by numerical quadrature.The alternative approach, used by Wooldridge (2000) is to specify instead thedistribution of u i | y i0 , X i . A semi-parametric variant due to Arellano and Carrasco(2003) involves the sequence of conditional means = E( u y ... y ,x ... x )λ ,iti|i0it i0which are estimated as nuisance parameters. The latter approach has many advantagesin models like (1) but is problematic in LAR models, where the lagged dependentvariable is not observable and cannot be conditioned on. Conditioning on itsobservable counterpart complicates matters enormously. For this reason, we use theHeckman treatment of initial conditions, together with an explicit hypothesis testingprocedure to control the bias induced by approximation error in the assumeddistribution of y i0 | X i , u i .Assume that we observe y and x over a period t = 0 … T. The LAR process (2)implies the following distributed lag representation: 3it3 In the case where the Γ r are not observable, we impose the normalisation σ ε = 1 and henceforthand σ u are re-interpreted accordingly.*yit, β- 7 -