Efficient Likelihood Ratio Confidence Intervals Using Constrained Optimization

Using constrained optimization, we develop a simple, efficient approach (applicable in both unconstrained and constrained maximum-likelihood estimation problems) to computing profile-likelihood confidence intervals. In contrast to Wald-type or score-based inference, the likelihood ratio confidence intervals use all the information encoded in the likelihood function concerning the parameters, which leads to improved statistical properties. In addition, the method does not suffer from the computational burdens inherent in the bootstrap. Moreover, it allows the computation of confidence intervals for transformations of the parameters—including counter-factual model quantities—in a straightforward fashion. In an application to Rust’s (1987) bus-engine replacement problem, our approach does better than either the Wald or the bootstrap methods, delivering very accurate estimates of the confidence intervals quickly and efficiently. Furthermore, we demonstrate how to compute confidence bands for the model-implied demand curve for engine replacement. An extensive Monte Carlo study reveals that in small samples, only likelihood ratio confidence intervals yield reasonable coverage properties, while at the same time discriminating implausible values.