Local regression for vector responses

Abstract

We explore a class of vector smoothers based on local polynomial regression for fitting nonparametric regression models which have a vector response. The asymptotic bias and variance for the class of estimators are derived for two different ways of representing the variance matrices within both a seemingly unrelated regression and a vector measurement error framework. We show that the asymptotic behaviour of the estimators is different in these four cases. In addition, the placement of the kernel weights in weighted least squares estimators is very important in the seeming unrelated regressions problem (to ensure that the estimator is asymptotically unbiased) but not in the vector measurement error model. It is shown that the component estimators are asymptotically uncorrelated in the seemingly unrelated regressions model but asymptotically correlated in the vector measurement error model. These new and interesting results extend our understanding of the problem of smoothing dependent data.