Extended Mundlak Device for Heterogeneity in Nonlinear Panel Data Models with Positive Response Variables

Using the models in Wooldridge (1999), We compare three main estimation methods for positive response variable-- FE method for log linear model (LFE), Poisson Quasi-Maximum Likelihood (PQML) and Generalized Method of Moment (GMM) -- by Mont Carlo Simulation and real life data set. It is not surprising that LFE estimator is not consistent when PQML is; however, we do find circumstance where both LFE and PQML estimators are consistent plus LFE is more efficient. With this regard, we introduce GMM to improve the efficiency of PQML estimator as well as keeping the consistency; this way also finds a solution to the problem raised in Wooldridge (1999). From the simulation results, we find that GMM can reduce the standard error of PQML estimator by almost a half. We also apply the GMM to a US domestic airlines data set and the result shows that GMM improves the efficiency by about $10\%$ compared with PQML. On the other hand, to make full use of the PQML method, We propose a semiparametric estimator of average partial effect after consistently estimating the parameters of interest; the result automatically extends the results in Ai and Norton (2008) from cross sectional setting to panel data models. In order to catch the positivity of the unknown conditional expectation function of the unobserved heterogeneity, we borrow the idea of power series approximation of unknown function in Newey(1993,1994) and develop an ``exponential sieves" estimator suggested in Wooldridge(1992a)