Underspecification in the macroeconomic Arbitrage Pricing Theory (APT) linear factor model and the role of the residual market factor

Abstract

The linear factor model is a building block of the Arbitrage Pricing Theory (APT). Macroeconomic factors may be used in linear factor models to proxy for the pervasive influences in returns. However, as the true return generating process is unobservable, macroeconomic data is either inaccurate or unavailable and because of the principle of parsimony, the linear factor model is likely to suffer from factor omission and consequent underspecification. Underspecification may adversely affect the interpretation of results, introduce coefficient bias, result in an upward bias in the residual variance and adversely affect predictive ability. The diagonality assumption that underlies the APT linear factor model will also be violated. Consequently, underspecification may pose a challenge to the general validity and interpretation of the linear factor model and the APT model. A widely applied solution to omitted factor bias in APT literature is the Burmeister and Wall (1986) residual market factor, hypothesised to fulfil the role of a wide-ranging proxy for omitted factors. This factor is derived from a broad market aggregate by excluding the influence of other factors that feature in a given linear factor model. This study sets out to determine whether the use of a conventional residual market factor derived from a domestic market aggregate adequately resolves underspecification. This study also considers the impact of underspecification on the linear factor model. The role of a second residual market factor derived from a widely used global market index, the MSCI World Market Index, in resolving factor omission is also considered. A second residual market factor that is orthogonal by contribution to the factor set in the linear factor model should be irrelevant if a conventional residual market factor is an adequate proxy for omitted factors. Consequently, the second residual market factor in this study also fulfils the function of a test of the adequacy of the conventional residual market factor. The approach in this study is comparative; three reduced form models are juxtaposed against a benchmark model and each other. The benchmark model incorporates a macroeconomic factor set, two residual market factors and a factor analytic augmentation as proxies for any remaining unobserved and omitted factors. Each specification is estimated using maximum likelihood (ML) estimation. Conditional variance is modelled as an ARCH(p) or GARCH(p,q) process to permit the structure of conditional variance to enter coefficient estimates and to provide insight into the conditional variance structure of the residuals. It is hypothesised that if factor omission has no impact on representations of the linear factor model and if the residual market factor is an effective and adequate proxy for omitted factors, then a model that comprises macroeconomic factors and a residual market factor should be comparable to the benchmark model in terms of results, general inferences and other aspects. This study finds that a linear factor model incorporating only macroeconomic factors performs poorly. The significance of factors is understated and the model is misidentified. Standard errors and residual variance are inflated, coefficients are biased and predictive and explanatory performance is poor. Significant deviations from the true return generating process are observed and the diagonality assumption is violated. The incorporation of a single residual market factor improves such a specification although there is still evidence of significant omitted factor bias. Violations of the diagonality assumption continue to persist but are not as widespread as for the specification that solely employs macroeconomic factors. The inclusion of a second residual market factor does not significantly alleviate the symptoms of underspecification and this factor is significant in a number of instances suggesting that the residual market factor does not capture all omitted influences by itself. Researchers of the APT and practitioners are encouraged to take note of these findings to avoid misinterpreting the results of macroeconomic linear factor models. The linear factor model is a complex construct and the application of a widely used approach in APT literature to resolve factor omission may not be adequate. This can adversely impact studies focusing on the linear factor model and equilibrium pricing within the APT and studies that apply macroeconomic linear factor models motivated by the APT.