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

0.001) will generally achieve this. Under these circumstances, the remainder terms in(A2)-(A4) are o p (1). Thus we can estimate the cross-partials consistently as follows:Dˆpqn=µ−1M m m mM ∑ m =z vpv2 1 q,p ≠ q(A5)To calculate the remaining second derivatives, we solve the simulated analogue of thesystem of equations (A4), to give:Dˆpp=nM2µNow transform D back to ψ-space:⎡MP−1m m 2∑z⎢vp− ⎜ ⎟∑m= 1 2 + P q=1⎣⎛⎝1⎞⎠vm 2q⎤⎥⎦(A6)Lˆ* −1−1ψψ≈ n K'D K(A7)One advantage of this method is that replications can be made sequentially and theprocedure stopped when the quantities (A5) and (A6) appear to have reachedconvergence. The simulated analogue of expression (A2) can be compared with aconventional numerical derivative to check on the negligibility assumption for theremainder terms. Antithetic variance reduction can be used to improve simulationprecision in (A5) and (A6).- 22 -