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For univariate or single independent variable regressions, this formula can be used (details here):

$$R^2_{adjusted} = 1- \dfrac{SSRes}{SSTotal}\dfrac{n-1}{n-p}$$

However, I cannot find a similar formula for multivariate or multiple independent variable regressions (but I did find that the number of degrees of freedom in a multiple regression equals n-k-1, where k is the number of variables).

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    $\begingroup$ How would you define $R^2$ for general multivariate regressions? It wouldn't make much sense without simplifying assumptions, such as spherical covariances. Although you seem to distinguish multivariate regression (multiple response dimensions) from multiple regression (multiple explanatory variables), your last remark casts doubt on that. $\endgroup$
    – whuber
    May 16 at 20:01
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    $\begingroup$ I’m with @whuber that it will be important to define an $R^2$ analogue for the multivariate $y$ setting; then we can discuss an appropriate penalty for including parameters. A post of mine on Data Science might help you in defining such an $R^2$ analogue. // Further, what do you hope to accomplish with the parameter penalty? In regression with a univariate $y$, the particular penalty comes from comparing unbiased estimators of the variances of two models. What do you want in the multivariate setting? $\endgroup$
    – Dave
    Jun 13 at 6:59

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