Alternative Asymptotics and the Partially Linear Model with Many Regressors

TL;DR

This paper explores a unified framework for non-standard asymptotics, deriving new distributional results for series estimators in partially linear models with many regressors, enhancing small sample approximations.

Contribution

It introduces a common structure for various non-standard asymptotics and derives a new distribution for series estimators with many terms in partially linear models.

Findings

Unified structure for many asymptotics identified

New asymptotic distribution for series estimators derived

Applicable to models with many regressors or instruments

Abstract

Non-standard distributional approximations have received considerable attention in recent years. They often provide more accurate approximations in small samples, and theoretical improvements in some cases. This paper shows that the seemingly unrelated "many instruments asymptotics" and "small bandwidth asymptotics" share a common structure, where the object determining the limiting distribution is a V-statistic with a remainder that is an asymptotically normal degenerate U-statistic. We illustrate how this general structure can be used to derive new results by obtaining a new asymptotic distribution of a series estimator of the partially linear model when the number of terms in the series approximation possibly grows as fast as the sample size, which we call "many terms asymptotics".