Market Proxies as Factors in Linear Asset Pricing Models: Still Living with the Roll Critique (Extended Version)

Abstract

A new model misspecification measure for linear asset pricing models is proposed. The origins of this measure are in Shanken (1987) and Kandel and Stambaugh (1985, 1995), where it is argued that the true market return is inherently latent and, as a consequence, only ever partially observed. Tests of asset pricing models that rely on the market return as a risk factor and are based, by necessity, on an observable proxy to this factor are then misspecified. The proposed misspecification measure, which assigns an upper bound to the correlation between the true market return and the observable proxy return used to conduct the test, can be estimated entirely and directly from observable data. This measure is suited both for testing models that include the market return as a pricing factor in a traditional sense (i.e., determining whether the given model does or does not price a collection of risky assets) and ranking those models (i.e., gauging which model performs the best). The measure is used to price portfolios reflecting the size, value, and momentum premiums. While neither the conditional CAPM nor the ICAPM is shown to offer any improvement over the simple CAPM, all three models are shown to perform materially better under the proposed measure, with improvements in model fit of as much as 45%. Also, it is discovered that winner stocks in a momentum portfolio may have higher market betas than loser stocks.