Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

Abstract

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.

Key Words:

• Semiparametric efficiency bounds
• Conditional heteroskedasticity
• Time series

Acknowledgments

For their helpful remarks, I thank the Editor, anonymous referees, Bill Brown, Oliver Linton, Allan Gregory, Vicky Zinde-Walsh and participants in workshops at Princeton, Queen's, Rochester, Texas A&M, Rice, Toronto, UBC, Guelph, and the 1997 Canadian Econometric Study Group. I also acknowledge financial support from the National Science Foundation, Montreal Institute of Mathematical Finance, and the Social Sciences and Humanities Research Council of Canada.