TY - JOUR

T1 - Calculating degrees of freedom in multivariate local polynomial regression

AU - McCloud, Nadine

AU - Parmeter, Christopher F.

N1 - Funding Information:
The authors are grateful for comments from the associate editor and an anonymous referee as well as participants at the Midwest Econometrics Group, The 2nd West Indies Economic Conference, LACEA-LAMES 2018 and the CSDA International Conference on Computational and Financial Econometrics as well as seminars at the University of Miami and Syracuse University. All errors are ours alone.

PY - 2021/1

Y1 - 2021/1

N2 - 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.

AB - 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.

KW - Bandwidth

KW - Effective parameters

KW - Goodness-of-fit

KW - Irrelevant regressors

KW - Trace

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U2 - 10.1016/j.jspi.2020.05.001

DO - 10.1016/j.jspi.2020.05.001

M3 - Article

AN - SCOPUS:85085054930

VL - 210

SP - 141

EP - 160

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

ER -