An Answer to Roll’s Critique (1977) 45 Years Later

Abstract

We implement a new framework to mitigate the errors-in-variables (EIV) problem in the estimation of asset pricing models. Considering an international data of portfolio stock returns from 1990 to 2021 widely used in empirical studies, we highlight the importance of the estimation method in time-series regressions. We compare the traditional ordinary-least squares (OLS) method to an alternative estimator based on a compact genetic algorithm (CGA) in the case of the CAPM. Based on intercepts, betas, adjusted R2, and the Gibbons et al. (1989) test, we find that the CGA-based method outperforms overall the OLS method. In particular, we obtain less statistically significant intercepts, smoother R2 across different portfolios, and lower GRS test statistics.

Specifically, in line with Roll’s critique (1977) on the unobservability of the market portfolio, we reduce the attenuation bias in market risk premium estimates. Moreover, our results are robust to alternative methods such as instrumental variables estimated with generalized-method of moments (GMM). Our findings have several empirical and managerial implications related to the estimation of asset pricing models as well as their interpretation as a popular tool in terms of corporate financial decision-making.

Some results in this chapter are published in Asset pricing models with measurement error problems: A new framework with Compact Genetic Algorithms, Finance, 2022. https://doi.org/10.3917/fina.432.0001

Appendix: Pseudo-Code of the Algorithm

This appendix provides a plain language description of the EIV-CGA approach developed by Satman and Diyarbakirlioglu (2015). All functions and methods necessary to implement the approach are available with the R package eive (see Satman and Diyarbakirlioglu (2018)).

We conceive the process in two subsequent parts and a penalty function, which establishes the convergence criterion. The first part sets the initial parameter values and generates the search space within which the CGA will search the solution.

Next, we start the iterations until the “best” offspring is obtained among all possible combinations we can extract from the search space. This goes through random sampling of two parent vectors with elements coming from the current state of $P[i]$. Then, we attach the parents, e.g., the samples, a cost function such that the winner to survive the next generation is the one who has the lowest penalty. Next, we update the $P[i]$ using a simple rule in a loop over the entire search space. The algorithm converges when all elements of $P[i]$ are either equal to 0 or 1.

The cost function returns the sum of squared residuals from the regression given in Eq. 9 between two candidate solutions sampled from the search space and declares the “winner” as the one for which this score is lowest. The end result yields the series $X^{CGA}$ that we consider as a filtered version of original observations.