| The partially linear regression model is a semiparametric extension of the linear regression model. Assume that the covariates are partitioned in two groups, X and T. Then, the regression function is assumed to be the sum of a linear function of X and an arbitrary (possibly non-linear) function of T. Thus, the observations consist of i.i.d. P pairs X1,T1 ,Y1,&ldots;, Xn,Tn,Yn , where each Yi is defined by Yi=bTXi +fTi+W i≡sXi,Ti +Wi, 1 for independent, identically distributed and zero mean error variables W1,&ldots;,Wn , assumed to be independent of the design points. We study this model in two different contexts:;Case A. The number of covariates appearing in the linear part is a priori given, say q.;Case B. We have available q possible regressors for the linear part, but only an unknown (possibly much smaller) subset of them are relevant for Y, hence we would like to select it, and base inference on the chosen model rather than on the full one.;Case A received much attention in the literature, whereas B was not yet studied. In case A, model (1) was first used by Engle, Granger, Rice, and Weiss (1985) to study the relationship between weather and electricity sales. Chen (1988) gives a method of estimation that leads to asymptotically normal estimators of b ; however, the estimation of f is not adaptive.;We have devised a method, based on a model selection strategy, that allows simultaneous selection of b and adaptive estimation of f, in Case B, which specializes to Case A. We have shown that in both cases the estimators of b are asymptotically normal and that f can be estimated adaptively, at the optimal minimax rate. We have studied the performance of our method through a simulation study and illustrated it through an analysis of the ozone level at Chicago O'Hare Airport. |
