Sieve Inference on Possibly Misspecified Semi-nonparametric Time Series Models∗

This paper first establishes the asymptotic normality of plug-in sieve M estimators of possibly irregular functionals of semi-nonparametric time series models. We show that, even when the sieve score process is not a martingale difference, the asymptotic variances of plug-in sieve M estimators of irregular (i.e., slower than root-T estimable) functionals are the same as those for independent data. Nevertheless, ignoring the temporal dependence leads to inaccurate inference in finite samples. We then propose an easy-to-compute and more accurate inference procedure based on a “pre-asymptotic" sieve variance estimator that captures temporal dependence of unknown forms. We construct a “pre-asymptotic" Wald statistic using an orthonormal series long run variance (OS-LRV) estimator. For sieve M estimators of both regular (i.e., root-T estimable) and irregular functionals, a scaled “pre-asymptotic" Wald statistic is asymptotically F distributed when the series number of terms in the OS-LRV estimator is held fixed. Simulations indicate that our scaled “pre-asymptotic" Wald test with F critical values has more accurate size in finite samples than the conventional Wald test with chi-square critical values. JEL Classification: C12, C14, C32