Orthogonal Score and Weighted Double-Selection Post-Selection Inference in High-Dimensional Quantile Regression

A post-selection inference methodology (introduced by Alexandre Belloni, Victor Chernozhukov, and Kengo Kato in 2019) for a target treatment or regression coefficient $\alpha_\tau$ in high-dimensional approximately sparse conditional quantile regression models $y_i = d_i\alpha_\tau + x_i'\beta_\tau + \epsilon_i$ with $\mathrm{P}(\epsilon_i \le 0 \mid d_i, z_i) = \tau$ and $p \gg n$ control basis functions $x_i = X(z_i)$ without beta-min separation assumptions. To immunize inference against model-selection mistakes in the nuisance vector $\beta_\tau$, the method constructs the Neyman-orthogonal quantile score $\psi_i(\alpha, \beta, v_i) = (\tau - \mathbb{I}\{y_i \le d_i\alpha + x_i'\beta\})v_i$ using the conditional-density-weighted residual instrument $v_i = f_\tau(0 \mid z_i, d_i)(d_i - x_i'\theta_\tau)$, where $f_\tau(0 \mid z_i, d_i)$ is the conditional error density at zero and $\theta_\tau = \arg\min_\theta \mathrm{E}[f_\tau^2(0 \mid z_i, d_i)(d_i - x_i'\theta)^2]$ solves the density-weighted projection of $d_i$ onto $x_i$ so that $\partial_\beta \mathrm{E}[\psi_i(\alpha_\tau, \beta_\tau, v_i)] = 0$ and $\partial_{\theta}\mathrm{E}[\psi_i(\alpha_\tau, \beta_\tau, v_i)] = 0$ (Section 2.1, Equations 2.6–2.9). Given the outcome $\ell_1$-penalized quantile regression support $\widehat{S}_y$ (Equation 2.11), conditional density estimates $\widehat{f}_i$ via difference quotients of local quantile regressions (Equations 2.15–2.16), and the heteroscedastic Lasso auxiliary support $\widehat{S}_d = \operatorname{supp}(\widehat{\theta}_\tau)$ of $\widehat{f}_i d_i$ on $\widehat{f}_i x_i$ (Equation 2.12), the *Orthogonal Score Estimator* $\widehat{\alpha}_\tau$ (Algorithm 2.1, p. 8) refits $\widetilde{\beta}_\tau$ on $\widecheck{T} \supseteq \widehat{S}_y \cup \widehat{S}_d$ and minimizes the self-normalized criterion $L_n(\alpha) = |\mathrm{E}_n[\widehat{\psi}_i(\alpha)]|^2 / \mathrm{E}_n[\widehat{\psi}_i^2(\alpha)]$, while the *Weighted Double-Selection Quantile Estimator* $\widecheck{\alpha}_\tau$ (Algorithm 2.2, p. 9) solves the $\widehat{f}_i$-weighted quantile regression $(\widecheck{\alpha}_\tau, \widecheck{\beta}_\tau) \in \arg\min_{(\alpha, \beta): \operatorname{supp}(\beta) \subseteq \widecheck{T}} \mathrm{E}_n[\widehat{f}_i\rho_\tau(y_i - d_i\alpha - x_i'\beta)]$, with both estimators sharing the first-order Bahadur representation $\sigma_n^{-1}\sqrt{n}(\widehat{\alpha}_\tau - \alpha_\tau) = \sigma_n^{-1}\sqrt{n}(\widecheck{\alpha}_\tau - \alpha_\tau) + o_P(1) \xrightarrow{d} N(0, 1)$ uniformly over approximately sparse models (Theorem 1, p. 14).