Abstract 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. The two prominent theoretical frameworks for studying hat matrices to calculate degrees of freedom in local polynomial regressions – ANOVA and non-ANOVA – abstract from both mixed data and the potential presence of irrelevant covariates, both of which dominate empirical applications. 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 both the non-ANOVA and ANOVA-based hat matrices 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.

中文翻译：

---

计算多元局部多项式回归中的自由度

摘要 将回归中的响应变量转换为其预测值的矩阵通常称为帽子矩阵。帽子矩阵的迹是计算自由度的标准度量。研究帽子矩阵以计算局部多项式回归中的自由度的两个突出的理论框架——方差分析和非方差分析——从混合数据和不相关协变量的潜在存在中抽象出来，这两者都在经验应用中占主导地位。在包含一些不相关协变量的连续和离散协变量混合的多元局部多项式设置中，我们为来自未知条件均值的估计器的非 ANOVA 和基于 ANOVA 的帽子矩阵的迹制定了渐近表达式。与条件均值估计量相关联的非 ANOVA 帽子矩阵的迹的渐近表达式等于内核相关常数的线性组合与基于 ANOVA 的帽子矩阵的线性组合。此外，我们记录了基于方差分析的帽子矩阵的迹线在带宽发散的任何设置中收敛到 0。这种磨损结果可能发生在不相关的连续协变量存在的情况下，或者当基础数据生成过程实际上是多项式顺序时可能会出现。