# Number of dependent variables greater than # IV's in multivariate hypothesis test

I have a multivariate regression model $\mathbf{Y}$ = $\mathbf{XB}$ + $\mathbf{E}$ where $\mathbf{Y}$ is $n \times m$.

In my case $\mathbf{n < m}$, the number of columns in $\mathbf{Y}$ is greater than the number of rows. Is it still possible to do multivariate hypothesis testing with a model like this (compute Wilks Lambda, Pillai's trace etc.)?

I have been getting errors from some programs and I don't see why. I can't find a mention of this in my multivariate analysis book or online anywhere, maybe I'm not googling the right terms?

If computing these statistics are $\mathbf{not}$ a possibility in this circumstance, could someone explain why not?

Statistics like Wilks lambda, Pillai's trace are certain functions of two covariance matrices, see https://en.wikipedia.org/wiki/Wilks%27s_lambda_distribution and https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance. In this way they generalize, in different ways, the F-ratio in univariate anova, which is a function of two sums of squares.

For the case of Wilks lambda, denote the two matrices as $A, B$. Then $$\lambda = \frac{\det A}{\det(A+B)}$$ But in the case $n<m$, the matrices are singular, so the determinants are zero ... But each determinant is a product of eigenvalues, so some of this eigenvalues are zero. The other eigenvalues still contain useful information!

Looking into the details (which I will not do here, now), it is possible to see that everything of interest is described by the nonzero eigenvalues. Specifically, Wilks $\lambda$ is a likelihoodratio test, and going through the details in the $n<m$ case, one will see that the test has the same form, with the determinants above replaced with the product of the nonzero eigenvalues. That could be called a generalized determinant, in the spirit of generalized inverses (I don't know if this is an established name). Probably the distribution theory must be checked for this case, and I do not know of any reference containing this generalizations.

For Pillai's trace much of the same will be true.