This study used Monte Carlo simulations to examine the ability of the two-stage least squares (2SLS) estimator and two-stage residual inclusion (2SRI) estimators with varying forms of residuals to estimate the local average and population average treatment effect parameters in models with binary outcome, endogenous binary treatment, and single binary instrument. The rarity of the outcome and the treatment was varied across simulation scenarios. Results showed that 2SLS generated consistent estimates of the local average treatment effects (LATE) and biased estimates of the average treatment effects (ATE) across all scenarios. 2SRI approaches, in general, produced biased estimates of both LATE and ATE under all scenarios. 2SRI using generalized residuals minimized the bias in ATE estimates. Use of 2SLS and 2SRI is illustrated in an empirical application estimating the effects of long-term care insurance on a variety of binary health care utilization outcomes among the near-elderly using the Health and Retirement Study.