Empirical Asset Pricing Models

The author surveys and discusses linear asset pricing models with the intent to identify some sets of variables or factors with reduced dimensionality to approximate the core or pricing kernel of asset returns. A theoretical foundation may start with discussion on factor pricing models where asset returns are projected onto some lower-dimensional sets of factors that possibly explain the major variations of asset returns. The aim is to identify major determinants for the fluctuations of asset returns where these determinants satisfy some systematic properties that ensure their indispensable roles.

The author discusses the methodologies that are currently applied to empirical asset pricing models on asset returns, including up-to-date coverage on theoretical setting and model specification tests. For instance, factor analysis and (asymptotic) principal component analysis are provided for searching for these pricing cores or kernels of asset returns. Unfortunately, these earlier studies incur the difficulty of observability of these factors and of (economic) interpretation of the principal components. In essence, the application of multi-factor asset pricing models with observed/presumed factors becomes an alternative in the search for the systematic components of asset returns.

Deficiency in the empirical applications of asset pricing models leads research toward some model selection criteria to justify the search. Unfortunately, with criteria that either emphasize the forecastability of models or impose a penalty for the increase of dimensionality (or complexity), the search for empirical asset pricing models tends to ignore the necessary role of the identified variables or factors to portrait the systematic and intrinsic commonality for all asset returns. That is to say, the coherence or strength of this systematic and intrinsic commonality in asset returns should be emphasized—in addition to the dimensionality and complexity. (Asymptotic) non-diversifiability, for instance, is an obvious requirement. In addition, almost all of this field of research into model selection of asset returns assumes that there exists a “true” (or correct) factor-pricing model in the data generating mechanism. Although robustness in asymptotic arguments are all provided in these studies, little discussion is provided for the soundness of such an assumption.

The author develops the alternative methodology in (1) cross-sectional properties of presumed (economic) factors/proxies that are considered essential for asset returns asymptotically, and (2) test statistics that can be applied to test these cross-sectional properties for empirical asset pricing models. Many model specification tests for these models have emphasized the statistical inferences on time-series properties of estimators and test statistics.

This chapter covers the discussions of model selection tests in empirical asset pricing models with the asymptotic properties developed in Chap. 4. In particular, model selection with forward selection for variables in empirical asset pricing models is introduced. The purpose of this chapter is to consider the sequential model search where model selection tests (or criteria) with additional asymptotic properties for common factors of asset returns are used. Differing from the other empirical studies, the emphasis is on the cross-sectional commonality of these presumed variables or factors when the asset returns are projected onto these variables. Given that the underlying intrinsic mechanism of asset returns is unknown, the sequential model search is to pursue the optimality in approximation that the basic requirement for these presumed variables or factors will satisfy the coherence condition where cross-sectional dependence is persistent.