HIDDEN ERRORS in REGRESSION-BASED ATTRIBUTION

Quantitative investment managers often use regression-based performance attribution for portfolio monitoring and reporting. It often leaves substantial portions of performance unexplained. Econometrics tells us additionally of potential errors in the regression betas and hence in the explained performance. This paper explores the sizes of these largely ignored errors. It determines exact attributions for three different portfolio construction scenarios. In each case the exact attributions allow us to quantify the errors in regression-based attribution. The overall error is never less than, and can be three or four times greater than, the unexplained performance, and can exceed the portfolio’s full return. Good signal selection is critical for investment success. The errors create incorrect measures of individual signal contributions, and of comparisons across signals. This in turn misleads portfolio managers, clients, consultants, plan sponsors, and asset owners. A commercial solution to these problems is now available.

In active quantitative investing, portfolio managers need to know which inputs are and are not helping their performance. Plan sponsors and consultants in turn look for managers who have that understanding. At times regression is the best, or even the only, way to trace the effects that the inputs of a complex process have on its outputs. When used to attribute the performance of quantitative investment strategies, however, the results often include large residuals, leaving significant portions of performance unexplained. Econometrics additionally warns us of errors in the regression betas, or estimated coefficients (Kennedy 2001, Greene 2000). These create corresponding errors in the attribution to individual performance sources. While we see the unexplained performance, we do not know the size or sign of the errors in the explained portions. Creative variations of the regression process can reduce the unexplained portion, but we do not know whether they increase or decrease the errors. This paper investigates these errors.

As a representative regression-based attribution we use the portfolio-centered approach developed by Grinold (2006) for active quantitative strategies.i It offers the important practical advantage of attributing to the components of the return forecast or “alpha”. For active strategies, these components, called signals or alpha factors, are the primary drivers of performance, and so arguably the most important focus for attribution.

Determining the regression-based attribution errors requires independently determining the correct attribution. We focus here on three strategies for which this is possible: the well-known V- inverse-alpha strategy; a strategy containing a simple version of “vintage alpha”; and a cost-controlled strategy. The paper first compares aggregate attribution error to unexplained performance in general, and then shows results for these quantities for each of the above-mentioned strategies. Appendix 1 derives the exact attribution for the V-inverse-alpha and simple vintage alpha strategies.ii Appendix 2 extends the analysis to include portfolio constraints, and sheds light on attributing to a set of constraints individually.