Underspecification of the Empirical Return-Factor Model and a Factor Analytic Augmentation as a Solution to Factor Omission

Abstract

This empirical paper comprehensively sets out the impact of underspecification on a key foundational concept in empirical finance, the linear factor model. It places emphasis on the extensive consequences of factor omission for model estimation and interpretation. Factor omission in time-series models that relate asset returns to pre-specified factor sets is a common problem. A proposed standard and widely-used solution is the inclusion of a residual market factor which is assumed to be a catch-all proxy for omitted factors. This study shows that a specification that incorporates a set of carefully selected macroeconomic factors will be underspecified. The inclusion of residual market factors will alleviate but not eliminate the consequences of underspecification. Although the early use of factor analytically derived factor scores in factor models has been criticized, augmenting a model comprising pre-specified factors with statistical factors derived from the residuals results in an accurately specified model for which the diagonality assumption holds. Consequently, this paper shows that a factor analytic augmentation is an effective and readily implementable solution to the factor omission problem.

Notes

1 Descriptive statistics for the return series are reported in Table A1 and A2 in the Appendix. The Appendix is available from the authors upon request and includes unabridged and supplementary results that are referenced in this study.

2 All orthogonalizing models used to derive residual market factors are estimated by applying the least squares methodology.

3 As there may be ambiguity regarding the distribution of the values for each series, the non-parametric Wilcoxon matched-pairs signed-rank test is applied following each t-test.

4 The RESET test is applied to exclude functional form misspecification. Unreported results indicate that the specifications are free of functional form misspecification. Any reduction is serial correlation can therefore be attributed to reductions in impure serial correlation.

5 To conduct this part of the analysis, a mean is calculated from the means of individual residual series.

6 Fair (1984) states that Theil’s U statistic may also be used to evaluate ex-post forecasts and for comparative purposes across models.

7 A closer examination of the sample reveals that only five sectors have communalities greater than 0,15. The sector with the highest loading on the single extracted factor is the general retailers’ sector, with a loading of - 0,492 and a corresponding communality of 0,242. With this sector excluded, the MAP test fails to identify a single factor implying that the extracted factor may be the result of strong interdependence between a limited number of industrial sectors.