High Dimensional Inference in Partially Linear Models

TL;DR

This paper introduces two semiparametric debiased Lasso methods for high-dimensional partially linear models, enabling inference without minimal signal conditions and accommodating high-dimensional covariates with sparsity structures.

Contribution

It develops novel semiparametric debiased Lasso procedures for high-dimensional partially linear models, allowing inference without minimal signal assumptions and handling high-dimensional covariates with sparsity.

Findings

Both methods achieve asymptotic normality.

Inference is valid without minimal signal conditions.

Proposed hypothesis testing handles exponentially many components.

Abstract

We propose two semiparametric versions of the debiased Lasso procedure for the model $Y_i = X_i\beta_0 + g_0(Z_i) + \epsilon_i$, where $\beta_0$ is high dimensional but sparse (exactly or approximately). Both versions are shown to have the same asymptotic normal distribution and do not require the minimal signal condition for statistical inference of any component in $\beta_0$. Our method also works when $Z_i$ is high dimensional provided that the function classes $E(X_{ij} |Z_i)$s and $E(Y_i|Z_i)$ belong to exhibit certain sparsity features, e.g., a sparse additive decomposition structure. We further develop a simultaneous hypothesis testing procedure based on multiplier bootstrap. Our testing method automatically takes into account of the dependence structure within the debiased estimates, and allows the number of tested components to be exponentially high.