Some New Asymptotic Theory for Least Squares Series: Pointwise and Uniform Results

TL;DR

This paper develops new asymptotic theory for least squares series estimators, relaxing conditions on the number of basis functions and providing pointwise and uniform inference results, applicable even with non-vanishing approximation errors.

Contribution

It extends the asymptotic theory for series estimators by weakening growth conditions on basis functions and deriving uniform and pointwise inference results, including under model misspecification.

Findings

Weaker condition on the number of basis functions: $k/n o 0$.

Derived $L_2$ rates and pointwise CLTs for vanishing approximation errors.

Established uniform rates and functional CLTs regardless of approximation error.

Abstract

In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis functions, where $k$ is allowed to grow with the sample size $n$. We consider series estimators for the conditional mean in light of: (i) sharp LLNs for matrices derived from the noncommutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the $L^\infty$ and $L_2$-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge, and (iv) strong approximations to series-type processes. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken the condition on the number $k$ of approximating functions used in series estimation from the typical $k^2/n \to 0$ to $k/n \to 0$, up to log factors, which was available only for spline series before. Second, we derive $L_2$ rates and pointwise central limit theorems results when the approximation error vanishes. Under an incorrectly specified model, i.e. when the approximation error does not vanish, analogous results are also shown. Third, under stronger conditions we derive uniform rates and functional central limit theorems that hold if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the nonparametric function. We derive uniform rates, Gaussian approximations, and uniform confidence bands for a wide collection of linear functionals of the conditional expectation function.