High-dimensional inference for single-index model with latent factors

TL;DR

This paper introduces a novel factor-augmented single-index model for high-dimensional data with latent factors, providing new testing, estimation, and inference methods that are robust to heavy-tailed errors and outliers.

Contribution

It develops a new model and testing procedure that simplifies implementation and extends inference to heavy-tailed errors, with theoretical guarantees and practical applications.

Findings

Proposed a simpler test statistic that does not require estimating high-dimensional parameters.

Established error bounds for penalized estimators with estimated latent factors.

Constructed confidence intervals using a debiased estimator with asymptotic normality.

Abstract

Models with latent factors recently attract a lot of attention. However, most investigations focus on linear regression models and thus cannot capture nonlinearity. To address this issue, we propose a novel Factor Augmented Single-Index Model. We first address the concern whether it is necessary to consider the augmented part by introducing a score-type test statistic. Compared with previous test statistics, our proposed test statistic does not need to estimate the high-dimensional regression coefficients, nor high-dimensional precision matrix, making it simpler in implementation. We also propose a Gaussian multiplier bootstrap to determine the critical value. The validity of our procedure is theoretically established under suitable conditions. We further investigate the penalized estimation of the regression model. With estimated latent factors, we establish the error bounds of the estimators. Lastly, we introduce debiased estimator and construct confidence interval for individual coefficient based on the asymptotic normality. No moment condition for the error term is imposed for our proposal. Thus our procedures work well when random error follows heavy-tailed distributions or when outliers are present. We demonstrate the finite sample performance of the proposed method through comprehensive numerical studies and its application to an FRED-MD macroeconomics dataset.