High Dimensional Inference in Partially Linear Models

We propose two semiparametric versions of the debiased Lasso procedure for the model $Y_{i}=X_{i}\beta_{0} g_{0}(Z_{i}) \varepsilon_{i}$, where the parameter vector of interest $\beta_{0}$ is high dimensional but sparse (exactly or approximately) and $g_{0}$ is an unknown nuisance function. 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 the vector of covariates $Z_{i}$ is high dimensional provided that the function classes $\mathbb{E}(X_{ij}\vert Z_{i})$s and $\mathbb{E}(Y_{i}\vert Z_{i})$ 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