Sufficient Conditions for a Linear Estimator to be a Local Polynomial Regression

TL;DR

This paper characterizes when a linear estimator with certain moment conditions can be viewed as a local polynomial regression, linking kernel sign changes to estimator equivalence.

Contribution

It provides necessary and sufficient conditions for linear estimators to be equivalent to local polynomial regression based on kernel sign changes.

Findings

Linear estimators satisfying moment conditions are equivalent to local polynomial regressions if kernel has at most p sign changes.

Symmetric data placement makes linear weights equivalent to quadratic weights.

The kernel's sign change property is key to the estimator's form.

Abstract

It is shown that any linear estimator that satisfies the moment conditions up to order $p$ is equivalent to a local polynomial regression of order $p$ with some non-negative weight function if and only if the kernel has at most $p$ sign changes. If the data points are placed symmetrically about the estimation point, a linear weighting function is equivalent to the standard quadratic weighting function.