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Multimodality of the likelihood in the bivariate seemingly unrelated regressions model

We analyse the simplest two-equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points, and thus there may be up to three local modes. Consequently the estimates obtained via iterative estimation methods may depend on starting values. We further show that the probability of multimodality vanishes asymptotically. Monte Carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true, but can become more frequent if the model is misspecified. The existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature. Copyright Biometrika Trust 2004, Oxford University Press.