Estimating endogenous ordinal treatment effects with high‐dimensional instruments and covariates

We propose a novel two‐stage approach to estimate the causal effect of an endogenous ordinal treatment variable on an outcome, using high‐dimensional instruments and covariates. In the first stage, we apply a penalized proportional odds reduced‐form model to estimate the ordinal treatment propensity score and optimal instrument. In the second stage, we substitute ordinal data with predicted values and derive the ordinal logistic instrumental variable estimator (OLIVE) for the endogenous causal effect via least squares regression. Our model allows the number of covariates to exceed the sample size; in such scenarios, we use double selection to identify key controls and introduce the double selection plus ordinal logistic instrumental variable estimator (DS‐OLIVE). We prove DS‐OLIVE is root‐n consistent and asymptotically normal. Simulations demonstrate that OLIVE and DS‐OLIVE surpass existing methods in bias and mean squared error. Finally, we apply our method to a case study assessing parental health's impact on children's study performance.