Estimation and inference of high-dimensional partially linear regression models with latent factors

TL;DR

This paper introduces a novel high-dimensional partially linear regression model that combines factor analysis with sparse modeling, providing accurate estimation and inference for complex data with latent factors.

Contribution

It proposes the FAPLM model integrating latent factors with nonparametric effects, along with a penalized estimation method and inference procedures, advancing high-dimensional regression analysis.

Findings

Theoretical error bounds match minimax rates of Lasso.

The proposed method effectively captures dependencies among covariates.

Numerical experiments demonstrate strong finite-sample performance.

Abstract

In this paper, we introduce a novel high-dimensional Factor-Adjusted sparse Partially Linear regression Model (FAPLM), to integrate the linear effects of high-dimensional latent factors with the nonparametric effects of low-dimensional covariates. The proposed FAPLM combines the interpretability of linear models, the flexibility of nonparametric models, with the ability to effectively capture the dependencies among highdimensional covariates. We develop a penalized estimation approach for the model by leveraging B-spline approximations and factor analysis techniques. Theoretical results establish error bounds for the estimators, aligning with the minimax rates of standard Lasso problems. To assess the significance of the linear component, we introduce a factor-adjusted projection debiased procedure and employ the Gaussian multiplier bootstrap method to derive critical values. Theoretical guarantees are provided under regularity conditions. Comprehensive numerical experiments validate the finite-sample performance of the proposed method. Its successful application to a birth weight dataset, the motivating example for this study, highlights both its effectiveness and practical relevance.