Numerical methods for optimization-based model estimation and inference

Estimators for nonlinear models (such as choice models) are frequently obtained by maximizing or minimizing a statistical criterion function (e.g., maximum likelihood, or nonlinear least squares), requiring solution of nonlinear optimization problems that frequently face a variety of computational challenges. This is particularly true for researchers pursuing more advanced and complex behavioral models that cannot be routinely solved using standard statistical estimation software. For example, such models may have intense computational requirements, and/or have identification properties that make it difficult to know when a valid solution has been produced, thus placing a premium on using fast and efficient optimization algorithms that also provide numerically reliable diagnostics. However, the expertise required to properly implement such methods can be somewhat specialized. Moreover, there is significant overlap between the mathematical underpinnings for optimization and the econometric requirements for computing inference-related quantities (e.g., variance-covariance matrix estimates). This Chapter provides an introduction and overview of the theory and methods for addressing these practical computational issues, including references to more detailed sources in the literature, and an enumeration and evaluation of alternative approaches, to support increased understanding and insight for researchers making methodological choices.