# Testing the validity of a factor model for stock returns

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)?

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You might make it a little more clear using $i$s for the cross-sectional index and $t$s for the time index. –  John Sep 12 '13 at 21:14
Sorry,there was a typo in the interval of $i$. It is fixed now! –  Mariam Sep 13 '13 at 12:27
Can you be a little more clear? What do you mean on a time-series basis? From reading this, one is tempted to think that you want to use lagged variables versus snapshot (cross-sectional). But again, the way the question is asked, it sounds like you are wondering if you should assess the model validity for each stock or do the validity check at portfolio level. I think this confusion is why the question is still hanging. –  mcisse Mar 18 at 1:14

## 1 Answer

This answer depends on the $X^i$

Before jumping on to the solution it should be answered that are $X^i$ traded in the market? i.e. are the returns on these available in the market (Size/Momentum portfolios, ETF returns) or are these economic variables like CPI, Inflation etc.

If it is the former i.e. traded assets then we can do the time series regression to compute the factor loadings i.e. $\beta_i$. however we need to ensure other things while performing the regression, like multi-collinearity check etc. to avoid spurious results. People have used PCA and Clustering techniques to check for these things. Example, Fama-French model, Cahart 4 factor model.

For using economic variables like CPI, inflation, unemployment data we will need to do the cross-sectional regression since we don't know what is the factor risk premium i.e. $$E[R^i - rf]$$

So this will be a two step regression. You can use Fama-Macbeth or a similar procedure. Example, Chen-Ross-Roll model.

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