The documentation for function "mvregress" states that the return value "beta" is a vector of the regression coefficients. Looking deeper into "Multivariate Normal Regression", we see that matlab uses the same regression coefficients ("beta") for every dimension of the multivariate response variable Y
This is ludicrous. Of course each component of the response variable can have its own set of coefficients. THAT is multivariate linear regression.
Am I missing something? Is this just an inherent shortcoming in matlab's "mvregress" function? If so, what a bizarre design choice...
Is there some way to get real multivariate linear regression, i.e. get a matrix beta of regression coefficients?
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You are missing something. See my answer in this thread for several examples of using design matrices with mvregress():
There are also examples from MathWorks here: http://www.mathworks.com/help/stats/mvregress.html