Self-Normalized Heteroscedastic LASSO Optimal Instrumental Variables and Weak-Identification Sup-Score Inference

A high-dimensional sparse instrumental variable (IV) estimation and inference framework (introduced by Alexandre Belloni, Daniel Chen, Victor Chernozhukov, and Christian Hansen in 2012) that estimates the conditional expectation $D(z_i) = \mathbb{E}[d_i \mid z_i]$ of endogenous regressors $d_i$ given $p \gg n$ dictionary instruments $f_i = (f_{i1}, \dots, f_{ip})'$ under heteroscedastic and non-Gaussian first-stage disturbances $\zeta_i = d_i - D(z_i)$ via LASSO and Post-LASSO with iterative self-normalized moderate-deviation penalty loadings $\widehat{\gamma}_j = \sqrt{\mathbb{E}_n[f_{ij}^2 \widehat{\zeta}_i^2]}$ and $\lambda = 2c\sqrt{n}\Phi^{-1}(1 - \gamma/(2p))$ ($c > 1$, Algorithm A.1). Grounded in Jing, Shao, and Wang's (2003) self-normalized moderate deviation theorem (Lemma 5), the score-calibrated loadings control the maximal heteroscedastic score $\max_{1 \le j \le p} |\sqrt{n}\mathbb{E}_n[f_{ij}\zeta_i]| / \sqrt{\mathbb{E}_n[f_{ij}^2\zeta_i^2]} \le \Phi^{-1}(1 - \gamma/(2p))$ without Gaussianity or homoscedasticity, delivering near-oracle first-stage rates $\|\widehat{D}_i - D_i\|_{2,n} = O_P(\sqrt{s\log(p \vee n)/n})$ (Theorem 1), semiparametrically efficient structural IV estimation $\sqrt{n}(\widehat{\alpha}^* - \alpha_0) \to_d \mathcal{N}(0, \Omega_0)$ under $s^2\log^2(p \vee n)/n \to 0$ (Theorems 2-4), and weak-identification-robust confidence regions via the Sup-Score test statistic $\widehat{S}(\alpha) = \max_{1 \le j \le p} \frac{|\sqrt{n}\mathbb{E}_n[(y_i - d_i'\alpha)f_{ij}]|}{\sqrt{\mathbb{E}_n[(y_i - d_i'\alpha)^2 f_{ij}^2]}}$ and its exact dual Inverse LASSO null-solution equivalence $\widehat{S}(\alpha) \le \Lambda \iff \widehat{\beta}_{\mathrm{IL}}(\alpha; 2\Lambda\sqrt{n}) = 0$ (Section 3.5, Eqs. 3.17-3.19).