Simple solutions to the initial conditions problem in dynamic, nonlinear panel data models with unobserved heterogeneity (original) (raw)
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We use a quasi-likelihood function approach to clarify the role of initial values and the relative size of the cross-section dimension N and the time series dimension T in the asymptotic distribution of dynamic panel data models with the presence of individual-speci…c e¤ects. We show that the quasi-maximum likelihood estimator (QMLE) treating initial values as …xed constants is asymptotically biased of order q N T 3 as T goes to in…nity for a time series models and asymptotically biased of order q N T for a model that also contains other covariates that are correlated with the individual-speci…c e¤ects. Using Mundlak-Chamberlain approach to condition the e¤ects on the covariates can reduce the asymptotic bias to the order of q N T 3 , provided the data generating processes for the covariates are homogeneous across cross-sectional units. On the other hand, the QMLE combining the Mundlak-Chamberlain approach with the proper treatment of initial value distribution is asymptotically unbiased if N goes to in…nity whether T is …xed or goes to in…nity. Monte Carlo studies are conducted to demonstrate the importance of properly treating initial values in getting valid statistical inference. The results also suggest that when using the conditional approach to get around the issue of incidental parameters, in …nite sample it is perhaps better to follow Mundlak's (1978) suggestion to simply condition the individual This paper was stimulated by the private communication with Jushan Bai. We would like to thank him for pointing out the terms that cancel the impact of individual-speci…c e¤ects and the impact of correlation between the errors of the equation and the lagged dependent variables. We would also like to thank the editor Oliver Linton, an associate editor, two anonymous referees and Elie Tamer for helpful comments. Partial research support by China NSF #71103004 and #71631004 to the …rst author is also gratefully acknowledged.