I saw in a paper for specifically the Northfield equity risk model that when constructing their factors they use the standard, time series regression to get each stock’s beta to a specific factor and then a cross sectional regression at each time point to get the factor risk premium.
As part of the cross sectional regression however they weight the observations by the square root of the market cap to allow for the fact that there are more small cap stocks in their universe.
The observations in all cross-sectional regressions are weighted by square root of market capitalization, which compensates for the skewness in the distribution of market capitalization. If the observations are equally weighted, the analysis is biased toward small capitalization names that are far more numerous. If the observations are purely capitalization weighted, the effective number of observations gets far too small for the large number of independent variables. This procedure provides essentially the same result as generalized-least squared methods that weight observations by inverse error terms
My question therefore is how would a size factor (which they have) work when the observations are already weighted by market cap? Wouldn’t the weighting to the regression add a significant bias on large cap stocks thus the factor risk premium for small caps would not exist?