Unified specification tests in partially linear time series models

Based on a residual marked empirical process, Cramér–von Mises and Kolmogorov–Smirnov tests are proposed for the correct specification of the nonparametric components in partially linear time series models. The tests are unified in the sense that the asymptotic distribution of residual marked empirical process is invariant across different $n^{\nu }$-consistent estimators in calculating residuals (where $\nu >1/4$) under the null. In addition, the residual marked empirical process has the same power property under the sequence of local alternatives regardless of the estimators used. Achieved through a projection method, these features also enable using a computationally convenient multiplier bootstrap to approximate the unified null distributions of the test statistics. Simulations show satisfactory finite-sample performance of the proposed method. The application to validate the parametric form of conditional variance in the ARCH-X model is also highlighted, along with an empirical analysis of the conditional variance of the FTSE 100 index return series.