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This appears to be the same thing, however, in the former case, the benchmark is the FF-Model. This means you assume the model stated in their eq. 9 is correct (as per your regression), and use the vol of the residuals as TE. They go on and explain: The volatility of the residuals in equation [9] is a measure of idiosyncratic (non-systematic) risk.20 ...

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I would like to point out a recent paper by Lewellen, Nagel and Shaken which has changed a little bit the way factor models are tested. The standard procedure was to run time series regression of a factor model on Fama&French 25 size and BE/ME sorted portfolios to obtain factor loadings, and then cross sectional regressions using $R^2$ as a good measure ...

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For calculating the AIC for factor models, I calculate the likelihood based on the multivariate distribution of the factor model. I try to make any assumptions as explicit as possible. Bayesians typically do not use the (so-called) BIC. WAIC (Watanabe-Akaike Information Criteria) is becoming more common among Bayesians. When thinking about the AIC, you ...

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