# Using cross-sectional factor model (BARRA type) returns in a time series factor model (Fama-French type)?

This may be seen as a follow up question for the previous discussion on time-series vs cross-sectional factor models: Which approach to estimating fundamental factor models is better, cross-sectional (unobservable) factors or time-series (observable) factors?

Assume that we use a cross-sectional factor model (e.g. BARRA model).

Using cross-sectional regressions, we estimate the pure factor returns for each time period (by regressing stock returns on firm characteristics, such as P/E).

So we obtain time series of pure factor returns.

Then, is it appropriate to estimate a time-series regression where individual stock returns (that are also used in developing the cross sectional model) are regressed on pure factor returns (that are estimated using cross sectional regressions)?

And if yes: 1) What are the econometric implications of such an approach? Since the explanatory variables are also estimates, we may have an errors-in-variables problem. 2) How the betas estimated in time-series regressions compare with the original factor exposures (i.e. firm characteristics)?

Are there any research papers on these issues?

If I were to write down what the model looks like, I think you're talking about something like below $$y_{it}=\alpha_{i}+G_{it}F_{t}\beta_{i}+\varepsilon_{i}$$ where $G$ are firm characteristics, $F$ are the cross-sectional factor estimates and $\beta$ are the time series betas. One thing that stands out is that $F$ and $\beta$ are only identified because you're talking a two-step approach.
It makes more sense to me to do the cross-sectional regressions $$y_{i}=\alpha+G_{i}F+\varepsilon_{i}$$ and get the residuals $$\varepsilon_{i,t}=y_{i,t}-\alpha_{i}-G_{i,t}F_{i}$$ and then do a time series regression of the residuals against the factors to see if there is any lingering exposure $$\varepsilon_{t}=\alpha_{t}+\beta F_{t}+\eta_{t}$$
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