Anomalies, Roll’s Critique, and Proxy Error

Abstract

Anomalies generate alphas. Roll (1977) notes that these alphas contain biases that result from using the equity market portfolio as a proxy for the unobserved aggregate wealth portfolio of CAPM – the portfolio of all risky assets. While the existence of the bias is well understood, the size of the bias is not. I develop a formal test that uses Ross’ (1976) absence of arbitrage bound to quantify the size of the bias. I find that 12 of 99 documented anomalies have a statistically significant bias, which ranges from .6% to 1.5%, annualized. I show that this bias is a manifestation of an anomaly’s correlation with the omitted variable – the assets that the proxy omits. In fact, when I add an anomaly with a significant bias as a second factor, to make a new proxy, the biases across all the anomalies are removed. Using these anomalies, my test provides a new proxy that may be closer to the aggregate wealth portfolio.