On-line Supplemental Appendix to Optimal Cross-Sectional Regression

This supplemental appendix contains additional discussion and extra simulation results of the main text Liao and Liu (2021). Section SA provides detailed literature review and highlights the contributions of our paper Liao and Liu (2021) to the literature. Section SB presents additional simulation results which are not reported in the main text. Department of Economics, UCLA, Log Angeles, CA 90095; e-mail: zhipeng.liao@econ.ecla. edu. Krannert Graduate School of Management, Purdue University, West Lafayette, IN 47905; e-mail: liu2746@purdue.edu. SA Literature Review In the context of the classical linear-beta pricing framework, we construct the set of moment restrictions that are related to existing two-pass cross-sectional regression (e.g., the Fama-MacBeth regression) estimators. We then develop a new class of two-pass estimators that dominate existing estimators in terms of estimation efficiency. Furthermore, we establish the asymptotic equivalence between our estimators and the optimally weighted GMM estimators, which are known in the literature to achieve the semiparametric efficiency bound. Therefore, we provide a unified view on existing two-pass regression estimators and identify those that are optimal, i.e., the OCSR estimators, in a well-defined sense. Our research addresses an important gap in the literature on linear-beta pricing models. Under the assumption that idiosyncratic risks are i.i. d. normal and conditionally homoskedastic given the pricing factors, Shanken (1992) proves the equivalence between the GLS two-pass regression estimator and the maximum-likelihood estimator, demonstrating that the GLS estimator is asymptotically efficient in the parametric framework. We improve on Shanken (1992) by characterizing the OCSR estimator among two-pass regression estimators with possibly nonnormal, dependent and conditional heteroskedastic idiosyncratic risks. More importantly, we extend Shanken (1992)’s insight by showing that the optimality of our OCSR goes far beyond the class of two-pass regressors. It actually extends to the realm of GMM estimators. As such, we establish the legitimacy of the usual two-pass regression approach by showing that information loss is not necessary when one goes from the more methodical GMM approach to the simple-to-implement two-pass regressions.1 Our theory unfolds in several steps. First, we introduce the set of moment restrictions that are closely related to the pricing framework in Jagannathan and Wang (1998) and the maximumlikelihood framework in Shanken (1992). These moment restrictions characterize the linear-beta pricing model and naturally map into the usual two-pass regression implementation. Working within the class of two-pass regressors, we derive the asymptotic distribution of a generic crosssectional regression estimator with an arbitrary weighting matrix and characterize the optimal weighting matrix that generates the smallest asymptotic covariance matrix (in a matrix sense). Jagannathan and Wang (1998) establish the asymptotic distribution of a general cross-sectional regression estimator. Our results thus extend Jagannathan and Wang (1998) by: (1) allowing for potential mispricing of certain assets and additional characteristics; and (2) pinning down the optimal weighting matrix for the two-pass estimator. We then examine the efficiency of our estimator and link it to the optimal GMM estimator based on the system of moment restrictions. We prove that our OCSR estimator and the optimal GMM estimator share the same asymptotic variance-covariance matrix and hence are asymptotically Alternative cross-sectional methods have also been proposed in Kim (1995) and Jegadeesh, Noh, Pukthuanthong, Roll, and Wang (2019). Unlike our paper, these methods focus on the N -consistency of the proposed estimators and do not examine the estimation efficiency for the risk premium parameters.