Computational efficiency of FIML estimation

Analytic derivatives and the computation of GARCH estimates

“…As in Calzolari, Panattoni, and Weihs (1987), for the uni-directional maximization we use a part of Powell's algorithm (Pierre, 1969, pp. 277-280).…”

“…(c) Experiments on other types of models (e.g. Calzolari, Panattoni, and Weihs, 1987) clearly show that the speed of convergence in the first iterations (for instance, starting from OLS initial estimates) is higher using the estimated information matrix rather than the exact Hessian. (d) In practice, almost all applied works base inference on the same information matrix estimate that they use in the maximization algorithm.…”

A Curious Result on Exact FIML and Instrumental Variables

“…It remains to prove that equation (20) is an acceptable choice for the matrix P t . If s Xt .a''' =£ 0 (which, for / > 1, is made possible by the restrictions on E), and we choose w = (/'.,,--Sa'-')/(s ly .a'''), the resulting P, satisfies equation (5) and therefore produces instruments exactly satisfying FIML first-order conditions.…”

“…then building as in Amemiya [1, p. 962] the nT x Sf=i A:, block-diagonal matrices G = diag(Gi,... ,G n ) and G = diag(G],... ,G n ), and stacking the equations (7) for / = 1,2,... ,n, we get for the gradient of the log-likelihood the expression _ _G'(E-' ® / r )vecF. (9) oa The instruments (8) obviously depend on the choice of the matrices P,; however, for any realized value of the matrix F, the gradient (9) is not affected by this choice provided that all P, satisfy equation (5). Thus the high degree of arbitrariness in such a choice (and therefore in the instruments) will not alter the first-order conditions for FIML.…”

“…Iterations stop when G'(E~' ® I T )vec F = 0 but, of course, convergence is not guaranteed. The joint use of the instrumental variable formula and of the line search maximization of the likelihood, analogous to Dagenais [6] or Calzolari, Panattoni, and Weihs [5], is helpful since it improves considerably the probability and the speed of convergence.…”