Profile-score Adjustements for Nonlinearfixed-effect Models

Maximum-likelihood estimates of nonlinear panel data models with fixed effects are generally not consistent as the number of units, N, grows large while the number of time periods, T, stays fixed. The inconsistency can be viewed as a consequence of the bias of the score function, where the unit-specific parameters have been profiled out. We investigate ways of adjusting the profile score so as to make it unbiased or approximately unbiased. This leads to estimators, solving an adjusted profile score equation, that are fixed-T consistent or have less asymptotic bias, as T ! 1, than maximum likelihood. One approach to adjusting the profile score is to subtract its bias, evaluated at maximum- likelihood estimates of the fixed effects. When this bias does not depend on the incidental parameters, the adjustment is exact. Otherwise, it does not eliminate the bias entirely but reduces its order (in T), and it can be iterated, reducing the bias order further. We examine a range of nonlinear models with additive fixed effects. In many of these, an exact bias adjustment of the profile score is possible. In others, suitably adjusted profile scores exhibit much less bias than without the adjustment, even for very small T.