Multill10dality of the likelihood in the bivariate seemingly unrelated regression model
نویسنده
چکیده
Seemingly unrelated regression (SUR) models traditionally appear in econometrics but recently also emerged in likelihood factorizations of Gaussian graphical models. The literature on maximum likelihood estimation in SUR seems not to mention the possibility of a multimodallikelihood. \eVe want to increase the awareness of this phenomenon by a thorough study of a two-equation model illustrated by an example of simulated observations. In this example, the constrained maximization/iterative Zellner algorithm indeed converges to the two different local maxima depending on which of the two usual starting values is used, namely, the identity matrix and the covariance matrix estimate from multivariate analysis of variance. We show that the considered two-equation SUR model has one, three, or five solutions to the likelihood equations for finite sample size but that the probability of multimodality vanishes asymptotically. Monte Carlo simulations suggest that multimodality rarely occurs if the SUR model is the true model but can be more likely if the data come from super-models.
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