The confidence interval method for selecting valid instrumental variables

We propose a new method, the confidence interval (CI) method, to select valid instruments from a larger set of potential instruments for instrumental variable (IV) estimation of the causal effect of an exposure on an outcome. Invalid instruments are such that they fail the exclusion conditions and enter the model as explanatory variables. The CI method is based on the CIs of the per instrument causal effects estimates and selects the largest group with all CIs overlapping with each other as the set of valid instruments. Under a plurality rule, we show that the resulting standard IV, or two-stage least squares (2SLS) estimator has oracle properties. This result is the same as for the hard thresholding with voting (HT) method of Guo et al. (Journal of the Royal Statistical Society : Series B, 2018, 80, 793-815). Unlike the HT method, the number of instruments selected as valid by the CI method is guaranteed to be monotonically decreasing for decreasing values of the tuning parameter. For the CI method, we can therefore use a downward testing procedure based on the Sargan (Econometrica, 1958, 26, 393-415) test for overidentifying restrictions and a main advantage of the CI downward testing method is that it selects the model with the largest number of instruments selected as valid that passes the Sargan test.