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 factor that controls the ratio between the L∞ and L2-norms, and (iii) maximal inequalities 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 (1997), as follows. First, we weaken considerably the condition on the number k of approximating functions used in series estimation from the typical k2/n → 0 to k/n → 0, up to log factors, which was available only for spline and local polynomial partition series before. Second, under the same weak conditions we derive L2 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. 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.