Local linear regression with nonparametrically generated covariates for weakly dependent data

Abstract

This paper studies the behaviour of the local linear (LL) estimator for regression with nonparametrically generated regressors under weak dependence conditions, effectively extending the main result of Mammen et al. (2012) to the case with strongly mixing (orα-mixing) data. Specifically, we establish that the convergence rate in their Theorem 1 carries over to the case with geometrically α-mixing data. In contrast, this convergence rate does not necessarily remain the same for polynomially α-mixing data. We then apply the obtained uniform stochastic expansion of the second-step LL estimator to derive normal approximations for the LL estimator of a nonparametric censored autoregressive model and the three-step nonparametric estimator of the risk–return regression in finance. A rule for bandwidth selection in nonparametric regressions with estimated covariates to obtain valid inference is also suggested. We then provide a simulation study and a real data application to illustrate the practical relevance of the proposed approach. Lastly, technically speaking, to establish a uniform bound for the upper tail probability of a summation involving polynomially strong-mixing empirical processes, we propose a new exponential inequality, which plays a pivotal role in the proof of the main theorem and is of independent interest.