On the Asymptotic Theory for Least Squares Series: Pointwise and Uniform Results

In this work we consider series estimators for the conditional mean in light of three new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue constant that controls the ratio between the $L^{\infty}$ and $L^{2}$-norms, and (iii) maximal inequalities with data-dependent bounds for processes whose entropy integrals diverge at some rate. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1995), as follows. First, we weaken considerably 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 splines before. Second, under the same weak conditions we derive $L^{2}$ rates and pointwise central limit theorems results when the approximation error vanishes. Under a 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 holds if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the non-parametric function. Finally, we derive uniform rates and inference results for linear functionals of interest of the conditional expectation function such as its partial derivative or conditional average partial derivative.