Consider the following m regression equation system:

$$r^i = X^i \beta^i + \epsilon^i \;\;\; \text{for} \;i=1,2,3,..,n$$

where $r^i$ is a $(T\times 1)$ vector of the T observations of the dependent variable, $X^i$ is a $(T\times k)$ matrix of independent variables, $\beta^i$ is a $(k\times1)$ vector of the regression coefficients and $\epsilon^i$ is the vector of errors for the $T$ observations of the $i^{th}$ regression.

My question is: in order to test the validity of this model for stock returns (i.e. the inclusion of those explanatory variables) using AIC or BIC criterion, should these criterion be computed on a time-series basis (i.e. for each stock), or on a cross-sectional basis (and then averaged over time)?