Calculating Degrees of Freedom in Multivariate Local Polynomial Regression

The matrix that transforms the response variable in a regression to its predicted value is commonly referred to as the hat matrix. The trace of the hat matrix is a standard metric for calculating degrees of freedom. Nonparametric-based hat matrices do not enjoy all properties of their parametric counterpart in part owing to the fact that the former do not always stem directly from a traditional ANOVA decomposition. In the multivariate, local polynomial setup with a mix of continuous and discrete covariates, which include some irrelevant covariates, we formulate asymptotic expressions for the trace of the resultant non-ANOVA and ANOVA-based hat matrix from the estimator of the unknown conditional mean. The asymptotic expression of the trace of the non-ANOVA hat matrix associated with the conditional mean estimator is equal up to a linear combination of kernel-dependent constants to that of the ANOVA-based hat matrix. Additionally, we document that the trace of the ANOVA-based hat matrix converges to 0 in any setting where the bandwidths diverge. This attrition outcome can occur in the presence of irrelevant continuous covariates or it can arise when the underlying data generating process is in fact of polynomial order. Simulated examples demonstrate that our theoretical contributions are valid in finite-sample settings.