Zongwu Cai, University of North Carolina, Charlotte
Qiwei Yao & Wenyang Zhang, London School of Economics
Abstract
We deal with smoothed estimators for conditional probability functions of discrete-valued time series {Y(t)} under two different settings. When the conditional distribution of Y(t), given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya-Watson estimator stands out as it shares the advantages of both Nadaraya-Watson and local linear regression estimators. The asymptotic normality of the proposed estimators has been established in the manner of sparse asymptotics, which shows that the proposed smoothed methods outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to the above assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994 -- 1995. An ad hoc model selection procedure based on local AIC is proposed to select the significant pollutant indices.