Two-Stage Hard Thresholding with Voting and Plurality Rule for Invalid Instruments

A causal-effect estimation and confidence-interval construction methodology (introduced by Zijian Guo, Hyunseung Kang, T. Tony Cai, and Dylan S. Small in 2018) for linear instrumental variables models with potentially invalid instruments ($Y_i = D_i\beta^* + \mathbf{Z}_i^T\boldsymbol{\pi}^* + \mathbf{X}_i^T\boldsymbol{\phi}^* + e_i$, $D_i = \mathbf{Z}_i^T\boldsymbol{\gamma}^* + \mathbf{X}_i^T\boldsymbol{\psi}^* + \epsilon_{i2}$) that replaces the 50% majority-validity condition ($|\mathcal{V}^*| > |\mathcal{S}^*|/2$) with the strictly weaker *plurality rule* $|\mathcal{V}^*| > \max_{c \ne 0}|\{j \in \mathcal{S}^* : \pi_j^*/\gamma_j^* = c\}|$ (Assumption 1, Equation 7). Given reduced-form estimators $\widehat{\boldsymbol{\Gamma}}$ and $\widehat{\boldsymbol{\gamma}}$ (from OLS in low dimensions or Debiased Scaled Lasso in high dimensions $p \gg n$, Equations 5 and 22–26), Stage 1 estimates the set of relevant instruments $\widehat{\mathcal{S}} = \{j : |\widehat{\gamma}_j| \ge \widehat{\sigma}_2\sqrt{\|\widehat{\boldsymbol{\Omega}}_{j\cdot}\|_2^2 / n}\,\sqrt{2.01\log p_z}\}$ via first-stage hard thresholding (Equation 9); Stage 2 computes for each $j \in \widehat{\mathcal{S}}$ the ratio candidate $\widetilde{\beta}^{[j]} = \widehat{\Gamma}_j / \widehat{\gamma}_j$, noise-adjusted violation vector $\widetilde{\boldsymbol{\pi}}^{[j]} = \widehat{\boldsymbol{\Gamma}} - \widetilde{\beta}^{[j]}\widehat{\boldsymbol{\gamma}}$, and standard errors $\widehat{\mathrm{SE}}(\widetilde{\pi}_k^{[j]})$ (Equations 10–11), forms the pairwise symmetric voting matrix $\widehat{V}_{k,j} = \mathbf{1}\left(|\widetilde{\pi}_k^{[j]}| \le \widehat{\mathrm{SE}}(\widetilde{\pi}_k^{[j]})\sqrt{2.01\log|\widehat{\mathcal{S}}|}\,\text{ and }\,|\widetilde{\pi}_j^{[k]}| \le \widehat{\mathrm{SE}}(\widetilde{\pi}_j^{[k]})\sqrt{2.01\log|\widehat{\mathcal{S}}|}\right)$ (Equations 12–14), selects the valid-instrument set $\widehat{\mathcal{V}} = \arg\max_{\mathcal{V}_m}\|\widehat{\mathbf{V}}_{\cdot m}\|_0$ receiving the plurality of votes (Equations 15–16), and constructs the asymptotically normal one-step GMM point estimator $\widehat{\beta}$ and confidence interval $\mathrm{CI}_{\alpha}$ on $\widehat{\mathcal{V}}$ (Equations 17–19, Theorems 1–3).