Estimating A Linear Exponential Density When The Weighting Matrix And Mean Parameter Vector Are Functionally Related

Most economic models in essence specify the mean of some explained variables, conditional on a number of explanatory variables. Since the publication of White’s (1982) Econometrica paper, a vast literature has been devoted to the quasi- or pseudo-maximum likelihood estimator (QMLE or PMLE). Among others, it was shown that QMLE of a density from the linear exponential family (LEF) provides a consistent estimate of the true parameters of the conditional mean, despite misspecification of other aspects of the conditional distribution. In this paper, we first show that it is not the case when the weighting matrix of the density and the mean parameter vector are functionally related. A prominent example is an autoregressive moving-average (ARMA) model with generalized autoregressive conditional heteroscedasticity (GARCH) error. As a result, the mean specification test is not readily modified as heteroscedasticity insensitive. However, correct specification of the conditional variance adds conditional moment conditions for estimating the parameters in conditional mean. Based on the recent literature of efficient instrumental variables estimator (IVE) or generalized method of moments (GMM), we propose an estimator which is modified upon the QMLE of a density from the quadratic exponential family (QEF). Moreover, GARCH-M is also allowed. We thus document a detailed comparison between the quadratic exponential QMLE with IVE. The asymptotic variance of this modified QMLE attains the lower bound for minimax risk.