Confidence intervals for causal effects with invalid instruments by using two‐stage hard thresholding with voting

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The confidence interval method for selecting valid instrumental variables (2019, methodology): “(21)V̂ar � ̂ [j] k � = ̂2j ⎛⎜⎜⎝ � Z�Z �−1 kk −2 � ̂k ̂ j �� Z�Z �−1 kj + � ̂k ̂ j �2 � Z�Z �−1 jj ⎞⎟⎟⎠ , 2We provide detail of the correspondence between the specification in Guo et al. (2018) and our notation in Appendix A.6.”
Statistical inference in two-sample summary-data Mendelian randomization using robust adjusted profile score (2018, methodology): “[23] Guo, Z.”