Exponential Rarity of Multiple Likelihood Modes in Bivariate Seemingly Unrelated Regression

The likelihood of the crossed bivariate Gaussian seemingly unrelated regression model can have several local modes. Its almost-sure eventual uniqueness is known. We give an explicit finite-sample bound: for fixed noncollinear regressor vectors of correlation r, with d=1-|r| and k=n-2>=2, the probability of more than one stationary point is at most exp[-k d^3/(16(1+|r|))]+2 exp[-k/16]. The bound is uniform over slopes and all positive-definite error covariances. A determinant normal form gives a directly checkable unimodality certificate, which combines with elementary Gaussian tails. Common regressors are allowed by replacing k with n-p-2 after projection. For Gaussian random predictors of population correlation r0, an explicit bound is 11 exp[-(n-2)(1-|r0|)^3/512]. A change-of-measure argument shows that the probability of multiple modes is exp[-Theta(n)] at each fixed nonsingular random-design parameter. Constants are not claimed optimal. A classical algebraic MANOVA certificate distinguishes the two parts of the motivating AIM question.