Model misspecification under the partially linear single index model

Abstract

The partially linear single index model has greater flexibility than linear regression models in facilitating the relationship between a continuous response and a set of covariates. This model allows not only linear dependence but also nonlinear dependence of the response variable on the covariates. Such a flexibility is, however, achieved at the price of losing the closed-form estimators of linear regression models. In this paper, we describe an estimation procedure using the spline approach to handle the nonlinear unknown function in the partially linear single index model. To explore the robustness of the partially linear single index model, we establish consistency results for the model parameters in the linear form under certain model misspecification. We identify several important settings with model misspecification where consistent results for the model parameters in the linear form are still retained. Those settings include cases with spurious covariates, covariates omission, covariate measurement error, and misspecfying the distribution of the noise term in the model. Further, we stress the importance of the independence assumption imposed for the noise term and the regressors, the assumption that is often overlooked in the literature. We illustrate, using an example of measurement error models, that the negligence of this independence assumption can yield biases results which would not be the case otherwise. Numerical studies confirm the satisfactory performance of the proposed method under a variety of settings.

Journal of Statistical Research 2021, Vol. 55, No. 1, pp. 59-73