Factor Exposure Analysis: Exploring Residualization

Making the independent variables more independent. What?

June 2021. Reading Time: 10 Minutes. Author: Nicolas Rabener.

SUMMARY

  • Regression analysis is frequently subject to multicollinearity
  • Independent variables can be residualized
  • Using residualized variables in a factor exposure analysis identifies different drivers

DISCLAIMER

“The worth of an econometrics textbook tends to be inversely related to the technical material devoted to multicollinearity” – Williams, R. Economic Record 68, 80-1. (1992) via Kennedy, A Guide to Econometrics (6th edition).

If this quote does not interest you, then the rest of this research note is unlikely to be exciting, so perhaps best to skip this one.

INTRODUCTION

Well, we are glad you made it, so let’s start talking about multicollinearity. The basic concept is that when we have several independent variables in a regression analysis, then these should be independent of each other. If these variables are correlated, then this erodes the statistical significance of the results.

Regression analysis in finance is almost always plagued by multicollinearity. First, many seemingly independent asset classes are more correlated than frequently assumed, e.g. US equities and high yield bonds. Secondly, correlations are not stable and change frequently. Merger arbitrage is uncorrelated to equities when stock markets are calm, but correlations spike when stocks crash (read The Complexity of Factor Exposure Analysis).

Fortunately, there are methodologies to test for and reduce multicollinearity. A variance inflation factor (VIF) can be used to measure it. A Lasso regression can be used to increase the significance of regressions, which typically results in the removal of correlated variables. Another approach is to residualize the variables and make them more independent.

In this research note, we will explore residualizing variables used in a regression analysis.

RESIDUALIZING STOCK MARKET INDICES

Residualizing a variable simply means removing the effects of other variables. Ironically, we are going to use regression to do this as well. As a case study, we use the