Self-Tuning Pivotal High-Dimensional Instrumental Variables (STIV) and Sensitivity-Based Confidence Sets

A self-normalized, scale-invariant convex conic methodology for high-dimensional linear instrumental variables regression $Y = X^\top \beta + U$ with $\mathbb{E}[ZU(\beta)] = 0$, $U(b) = Y - X^\top b$, where both the number of covariates $d_X$ and instruments $d_Z$ can exceed the sample size $n$ (introduced by Eric Gautier and Alexandre B. Tsybakov in 2011 and published by Eric Gautier and Christiern Rose in 2021). Using the sample standardization matrices $\mathbf{D}_{\mathbf{W}} = \operatorname{diag}(\mathbb{E}_n[W_k^2]^{-1/2})$ and empirical noise scale $\widehat{\sigma}(b) = \mathbb{E}_n[U(b)^2]^{1/2}$, the self-tuning pivotal moment statistic $\widehat{t}(b) = \frac{|\mathbf{D}_{\mathbf{Z}}\mathbb{E}_n[ZU(b)]|_\infty}{\widehat{\sigma}(b)}$ is controlled at level $1 - \alpha$ by a universal moderate-deviation or multiplier-bootstrap threshold $\widehat{r}$ free of unknown error scale. The STIV estimator $(\widehat{\beta}, \widehat{\sigma})$ (Definition 3.1, Eqs. 6–8) solves the convex conic program $\min_{b \in \widehat{\mathcal{I}}(\widehat{r}, \sigma),\, \sigma \ge 0}\left(|\mathbf{D}_{\mathbf{X}}^{-1} b_{S_Q}|_1 + c\sigma\right)$ over $\widehat{\mathcal{I}}(r, \sigma) = \{b \in \mathcal{B} : |\mathbf{D}_{\mathbf{Z}}\mathbb{E}_n[ZU(b)]|_\infty \le r\sigma,\, \widehat{\sigma}(b) \le \sigma\}$ with $c > 1$, or equivalently $\widehat{\beta} \in \arg\min_{b \in \mathcal{B}}\left(\frac{1}{c}|\mathbf{D}_{\mathbf{X}}^{-1} b_{S_Q}|_1 + \mathcal{O}(b)\right)$ where $\mathcal{O}(b) = \max\left(\widehat{\sigma}(b),\, \frac{1}{\widehat{r}}|\mathbf{D}_{\mathbf{Z}}\mathbb{E}_n[ZU(b)]|_\infty\right)$. By evaluating the empirical cone $\widehat{K}_S$ (Eq. 9), sensitivity characteristics $\widehat{\kappa}_{\ell, S} = \min_{\Delta \in \widehat{K}_S : \ell(\Delta) = 1}|\widehat{\Psi}\Delta|_\infty$ for $\widehat{\Psi} = \mathbf{D}_{\mathbf{Z}}\mathbb{E}_n[ZX^\top]\mathbf{D}_{\mathbf{X}}$ (Eq. 10, Proposition 3.1), and linear-programming lower bounds $\widehat{\kappa}_{\ell_k}(s)$ under an upper bound $s$ on $|S(\beta) \cap S_Q|$ (Table 1, Proposition 3.2), the framework constructs non-asymptotic confidence sets that remain valid even when some or all covariates are weakly identified or unidentified, achieves thresholded sign consistency $\operatorname{sign}(\widehat{\beta}^{\widehat{\omega}}) = \operatorname{sign}(\beta)$ (Theorem 3.2), and enables root-$n$ asymptotically normal inference via bias-corrected STIV (BC-STIV, Eqs. 30–32, Theorem 4.1) and invalid-instrument detection via NV-STIV (Eq. 39, Theorem 5.1).