Encompassing Test for Non-nested Linear and Nearest Neighbor Regressions Under Mixing Conditions

Encompassing tests compare two non-nested competing models to determine whether one model contains all the relevant information captured by the other. They were developed based on the assumption of independence, particularly in cases involving parametric and non-parametric regression methods. However, this assumption is too restrictive since econometric dynamic models and time series rarely exhibit perfect independence. One challenge would be extending results for i.i.d. variables to dependent processes. We are interested in the φ-mixing dependence measure because of its properties, such as the fast decay rate. Consequently, the Central Limit Theorem (CLT) was previously obtained under significantly weaker mild conditions than those required for other mixing notions. This leads to better asymptotic behavior propreties for test statistics, despite practical constraints of the φ-mixing condition compared to weaker dependence measures. We examine the encompassing test for linear parametric and nonparametric nearest neighbor regression methods for φ-mixing processes. We establish the asymptotic normality of the encompassing statistics associated with the encompassing hypotheses. Our results are comparable to those obtained using popular methods in the literature, such as kernel regression. We achieve convergence rates of order k−1/2 for the tests related to the nonparametric as rival model, and n−1/2 for the parametric as rival model. These rates are the same as in the i.i.d. case. The asymptotic variances of the tests are well-defined due to the fast decay of the φ-mixing coefficients. Unlike many tests with nonparametric regression methods, ours do not depend on any density.