Abstract: We propose an efficient two-step estimation procedure for a parametric modal regression with autoregressive errors. The procedure relies on estimating a parametric transformation of the dependent variable from data using a (penalized) kernel-based objective function. We establish asymptotic normality for the resulting estimator and demonstrate that it possesses oracle properties, as if the true order of autoregressive error structure were known in advance. To numerically estimate modal parameter and determine the order of error structure, two modified (penalized) modal expectation-maximization (MEM) algorithms are developed. Furthermore, we present a modal residual-based autocorrelation test and show that the statistic is asymptotically distributed as a 𝒳² distribution. Monte Carlo simulations and an empirical analysis are conducted to illustrate the finite sample performance of the resultant estimator. We also discuss the extension of the results to a nonparametric modal regression model.

Key words and phrases: Autoregressive error, MEM algorithm, modal regression, oracle property, order selection, residual-based test.