Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms

Nonparametric series estimation often involves specification search over the different number of series terms due to the unknown smoothness of underlying function. This paper considers pointwise inference in the nonparametric series regression for the conditional mean and introduces test based on the supremum of t-statistics over different series terms. I show that proposed test has correct asymptotic size and it can be used to construct confidence intervals (CI) that have correct asymptotic coverage probability uniform in the number of series terms. With possibly large bias in this setup, I also consider infimum of the t-statistics which is shown to reduce size distortions in such case. Asymptotic distribution of the test statistics, asymptotic size, and the local power results are derived. I investigate the performance of the proposed tests and CIs in various simulation setups as well as an illustrative example, nonparametric estimation of wage elasticity of the expected labor supply from Blomquist and Newey (2002). I also extend our inference methods to the partially linear model setup focusing on finite dimensional parameters given the range of different series terms used for the nonparametric part.