Adaptive Lasso and Median-of-Ratios Consistent Invalid Instrument Selection in Linear Instrumental Variables Models

A two-step consistent invalid-instrument selection and structural parameter estimation method (introduced by Frank Windmeijer, Helmut Farbmacher, Neil Davies, and George Davey Smith in 2019) for linear instrumental variables models $y = d\beta + Z\alpha + u$, $d = Z\gamma + v$ with $s < L/2$ invalid instruments ($\alpha_1 \ne 0$, $\alpha_2 = 0$). Standard unweighted $\ell_1$-penalized IV (sisVIVE, Kang et al. 2016) minimizes $\frac{1}{2n}\|P_Z(y - d\beta - Z\alpha)\|_2^2 + \lambda_n\|\alpha\|_1 = \frac{1}{2n}\|\widehat{M}_{\widehat{d}}y - \widehat{M}_{\widehat{d}}Z\alpha\|_2^2 + \lambda_n\|\alpha\|_1$. Proposition 1 proves that the projected instrument design matrix $C = \text{plim}\,n^{-1}Z'\widehat{M}_{\widehat{d}}Z = Q - Q\gamma(\gamma'Q\gamma)^{-1}\gamma'Q$ has rank $L - 1$ with regression block $C_{21}C_{11}^{-1} = (I_{L-s} - \frac{\widetilde{Q}_{22}\gamma_2\gamma_2'}{\gamma_2'\widetilde{Q}_{22}\gamma_2})Q_{21}Q_{11}^{-1} - \frac{\widetilde{Q}_{22}\gamma_2\gamma_1'}{\gamma_2'\widetilde{Q}_{22}\gamma_2}$ where $\widetilde{Q}_{22} = Q_{22} - Q_{21}Q_{11}^{-1}Q_{12}$ (Equation 18). Proposition 2 establishes the exact structural identity $\gamma_2'C_{21}C_{11}^{-1} = -\gamma_1'$, which implies that for orthogonal instruments ($Q = I_L$), the Lasso irrepresentable condition $\|C_{21}C_{11}^{-1}\text{sign}(\alpha_1)\|_\infty < 1$ reduces to $\|\gamma_2\|_\infty |\gamma_1'\text{sign}(\alpha_1)| < \|\gamma_2\|_2^2$ and necessarily fails whenever invalid instruments have relatively large first-stage strengths $\gamma_1$ (Equations 19–22). Theorem 1 proves that under the plurality/majority rule $s < L/2$, the Median-of-Ratios estimator $\widehat{\beta}_m = \text{median}(\{\widehat{\Gamma}_j / \widehat{\gamma}_j\}_{j=1}^L)$ from the OLS reduced-form ratios $\pi_j = \Gamma_j / \gamma_j = \beta + \alpha_j / \gamma_j$ is $\sqrt{n}$-consistent with $\sqrt{n}(\widehat{\beta}_m - \beta) \xrightarrow{d} \mathcal{N}(0, \sigma_{m(l)}^2)$ where $l$ is the median order-statistic index. Plugging the initial consistent invalidness estimate $\widehat{\alpha}_m = \widehat{\Gamma} - \widehat{\gamma}\widehat{\beta}_m$ into the Adaptive Lasso objective $\widehat{\alpha}_{\text{ad}}^n = \arg\min_\alpha \frac{1}{2}\|\widehat{M}_{\widehat{d}}y - \widehat{M}_{\widehat{d}}Z\alpha\|_2^2 + \lambda_n\sum_{j=1}^L \frac{|\alpha_j|}{|\widehat{\alpha}_{m,j}|^\nu}$ (Theorem 2) achieves selection consistency $\mathbb{P}(\text{supp}(\widehat{\alpha}_{\text{ad}}^n) = \text{supp}(\alpha)) \to 1$ and oracle asymptotic normality $\sqrt{n}(\widehat{\beta}_{\text{post}} - \beta) \xrightarrow{d} \mathcal{N}(0, \sigma_u^2 / (\gamma_2'\widetilde{Q}_{22}\gamma_2))$ without any irrepresentable restriction on $C$.