Seemingly unrelated regressions are statistical regression models based on
the Gaussian distribution. They are popular in econometrics but also arise in
graphical modeling of multivariate dependencies. In maximum likelihood
estimation, the parameters of the model are estimated by maximizing the
likelihood function, which maps the parameters to the likelihood of observing
the given data. By transforming this optimization problem into a polynomial
optimization problem, it was recently shown that the likelihood function of a
simple bivariate seemingly unrelated regressions model may have several
stationary points. Thus local maxima may complicate maximum likelihood
estimation. In this paper, we study several more complicated seemingly
unrelated regression models, and show how all stationary points of the
likelihood function can be computed using algebraic geometry.Comment: To appear in the Journal of Symbolic Computation, special issue on
Computational Algebraic Statistics. 11 page