On solving bias-corrected non-linear estimation equations with an application to the dynamic linear model

In a seminal paper, Mak, Journal of the Royal Statistical Society B, 55, 1993, 945, derived an efficient algorithm for solving non-linear unbiased estimation equations. In this paper, we show that when Mak's algorithm is applied to biased estimation equations, it results in the estimates that would come from solving a bias-corrected estimation equation, making it a consistent estimator if regularity conditions hold. In addition, the properties that Mak established for his algorithm also apply in the case of biased estimation equations but for estimates from the bias-corrected equations. The marginal likelihood estimator is obtained when the approach is applied to both maximum likelihood and least squares estimation of the covariance matrix parameters in the general linear regression model. The new approach results in two new estimators when applied to the profile and marginal likelihood functions for estimating the lagged dependent variable coefficient in the dynamic linear regression model. Monte Carlo simulation results show the new approach leads to a better estimator when applied to the standard profile likelihood. It is therefore recommended for situations in which standard estimators are known to be biased.