Searching and Sampling Uniform Confidence Intervals for Invalid Instruments

A post-selection-robust causal-inference methodology (introduced by Zijian Guo in 2023) for constructing uniformly valid confidence intervals for a treatment effect $\beta^*$ in linear instrumental-variable models with possibly invalid instruments when locally invalid instruments satisfy $0 < |\pi_j^*/\gamma_j^*| < s_j(n) \asymp \sqrt{\log n / n}$ (Definition 1, Equation 12) and cannot be consistently separated from valid instruments. Given an initial candidate set $\widehat{\mathcal{V}}$ (such as the symmetric-voting plurality set $\widehat{\mathcal{V}}^{\rm TSHT} = \{l \in \widehat{\mathcal{S}} : \max_{k \in \widehat{\mathcal{S}}, j \in \widehat{\mathcal{W}}} \widehat{\Pi}_{j,k}\widehat{\Pi}_{k,l} = 1\}$ satisfying $\mathcal{V} \subset \widehat{\mathcal{V}} \subset \mathcal{V} \cup \mathcal{I}(0, s(n))$ and $|\mathcal{V}| > |\widehat{\mathcal{V}}|/2$ under finite-sample plurality, Equations 30–34), the *Searching Confidence Interval* (Algorithms 1 and 3, Equations 15–20) evaluates for each grid point $\beta \in [L, U]$ the residual $\widehat{\pi}_j(\beta) = \widehat{\Gamma}_j - \beta\widehat{\gamma}_j$ and noise threshold $\widehat{\rho}_j(\beta, \alpha) = \Phi^{-1}(1 - \alpha/(2|\widehat{\mathcal{V}}|))\sqrt{(\widehat{V}_{\Gamma,jj} + \beta^2\widehat{V}_{\gamma,jj} - 2\beta\widehat{C}_{jj})/n}$, counts the implied valid instruments $M(\beta) = \sum_{j \in \widehat{\mathcal{V}}} \mathbf{1}(|\widehat{\pi}_j(\beta)| \le \widehat{\rho}_j(\beta, \alpha))$, and retains $\mathrm{CI}^{\rm search} = [\min_{\beta \in \mathcal{B}: M(\beta) > |\widehat{\mathcal{V}}|/2} \beta,\; \max_{\beta \in \mathcal{B}: M(\beta) > |\widehat{\mathcal{V}}|/2} \beta]$. The *Sampling Confidence Interval* (Algorithms 2 and 3, Equations 21–26) draws $M$ conditionally centered Gaussian resamples $(\widehat{\boldsymbol{\Gamma}}^{[m]}, \widehat{\boldsymbol{\gamma}}^{[m]})_{m=1}^M$ matching the asymptotic covariance $\widehat{\mathbf{Cov}}$, uses a shrinkage threshold $\lambda \asymp (\log n / M)^{1/(2|\widehat{\mathcal{V}}|)}$ (Proposition 1) so the resampled searches $[\beta_{\min}^{[m]}(\lambda), \beta_{\max}^{[m]}(\lambda)]$ using $\lambda\widehat{\rho}_j(\beta, \alpha)$ contain a realization with $\max_{j \in \mathcal{V}}|e_j^{[m]}| \le \lambda\widehat{\rho}_j$, and takes the union $\mathrm{CI}^{\rm sample} = [\min_{m \in \mathcal{M}} \beta_{\min}^{[m]}(\lambda),\; \max_{m \in \mathcal{M}} \beta_{\max}^{[m]}(\lambda)]$, achieving uniform asymptotic coverage $\ge 1 - \alpha$ and parametric $O_p(n^{-1/2})$ length without any separation condition on locally invalid instruments (Theorems 1–3).