The method of sieves has been widely used in estimating semiparametric and nonparametric models. In this paper, we first provide a general theory on the asymptotic normality of plug-in sieve M estimators of possibly irregular functionals of semi/nonparametric time series models. Next, we establish a surprising result that the asymptotic variances of plug-in sieve M estimators of irregular (i.e., slower than root-T estimable) functionals do not depend on temporal dependence. Nevertheless, ignoring the temporal dependence in small samples may not lead to accurate inference. We then propose an easy-to-compute and more accurate inference procedure based on a pre-asymptotic sieve variance estimator that captures temporal dependence. 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 usual Wald test with chi-square critical values.
Weak Dependence Sieve M Estimation Sieve Riesz Representor Irregular Functional Misspecification Pre-asymptotic Variance Orthogonal Series Long Run Variance Estimation F Distribution