Square Root Market Cap Weights in Equity Factor Regressions

Summary

The document asks how square root market capitalization weights in cross-sectional factor regressions interact with a size factor. It describes a two-stage factor construction process: estimate each stock’s factor exposure with time-series regressions, then estimate factor returns or premia with cross-sectional regressions. The cited approach weights observations by the square root of market capitalization to temper the influence of both numerous small companies and a few very large ones.

The central concern is whether this weighting could suppress the size premium or bias estimates toward large stocks. The text gives a rationale for the intermediate weighting scheme: equal weights can let the many small names dominate, while full capitalization weights can make the effective sample too small relative to the number of explanatory variables. It does not provide a resolution, regression results, or details of the cited model, so the interaction between weighting and the size factor remains an open question.

Key ideas

• Factor construction can estimate stock exposures in time series and factor premia in cross section.
• Square root capitalization weights are presented as a compromise between equal weighting and full capitalization weighting.
• Equal weighting may give numerous small-cap stocks disproportionate influence in the regression.
• Full capitalization weighting may reduce the effective number of observations too far.
• The document asks whether these weights affect the estimated size premium but does not answer that question.

Tags

# Size factor - Root Market Cap Weighted


# Size factor - Root Market Cap Weighted












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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.