Estimating Logistic Regressions with Two Stage Least Squares

I develop an algorithm to estimate a flexible binary regression model with endogeneity by repeatedly solving a two-stage least squares problem; the algorithm is numerically stable and guaranteed to converge regardless of starting value. The method is numerically stable even when a successful outcome is rare because it has a uniformly small condition number, unlike Newton methods with maximum likelihood estimation whose condition number is unbounded across potential parameter values. The instrumental variable method does not require choosing a special regressor or making assumptions on the first stage relationship between covariates and instruments other than a rank restriction to ensure the instruments are relevant enough