Two-dimensional Gauss–Legendre quadrature: Seemingly unrelated dispersion-flexible count regressions

Many contexts in empirical econometrics require nonclosed form two-dimensional (2D) integration for appropriate modeling and estimation design. Applied researchers often avoid such correct but computationally demanding specifications and opt for simpler biased or less efficient modeling designs. The presentation will detail a new Mata implementation of the 2D version of a relatively simple numerical integration technique—Gauss–Legendre quadrature. Although this Mata code is widely applicable, it is mainly designed for estimators that involve 2D integration at the observation level (for example, the likelihood function for a two-equation nonlinear regression system). The user inputs a vector-valued integrand function (for example, a vector of sample log-likelihood integrands) and a matrix of upper and lower limits for each of the two integration dimensions. The code outputs the corresponding vector of integrals (for example, the vector of observation-specific log likelihoods). To illustrate implementation, we estimate a bivariate seemingly unrelated 2D system of dispersion-flexible Conway–Maxwell Poisson regressions for the number of consultations in a two-week period with a 1) doctor and 2) non-doctor health professional, or both. The data were drawn from the 1977–1978 Australian health survey. Results from this model are juxtaposed with those from Conway–Maxwell and simple Poisson specifications in which possible cross-equation correlation is ignored.