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?