美联储线性因子模型与期望收益估计

Finance and Economics Discussion SeriesFederal Reserve Board, Washington, D.C.ISSN 1936-2854 (Print)ISSN 2767-3898 (Online)Linear Factor Models and the Estimation of Expected ReturnsCisil Sarisoy, Peter de Goeij, and Bas J.M. Werker2024-014Please cite this paper as:Sarisoy, Cisil, Peter de Goeij, and Bas J.M. Werker (2024).“Linear Factor Mod-els and the Estimation of Expected Returns,” Finance and Economics Discussion Se-ries 2024-014.Washington:Board of Governors of the Federal Reserve System,https://doi.org/10.17016/FEDS.2024.014.NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminarymaterials circulated to stimulate discussion and critical comment. The analysis and conclusions set forthare those of the authors and do not indicate concurrence by other members of the research staff or theBoard of Governors. References in publications to the Finance and Economics Discussion Series (other thanacknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Linear Factor Models and the Estimation ofExpected ReturnsCisil Sarisoy∗Federal Reserve BoardPeter de GoeijTilburg UniversityBas J.M. WerkerTilburg UniversityJanuary 2024AbstractThis paper analyzes the properties of expected return estimators on individual assetsimplied by the linear factor models of asset pricing, i.e., the product ofβandλ. Weprovide the asymptotic properties of factor–model–based expected return estimators,which yield the standard errors for risk premium estimators for individual assets. Weshow that using factor-model-based risk premium estimates leads to sizable precisiongains compared to using historical averages. Finally, inference about expected returnsdoes not suffer from a small–beta bias when factors are traded. The more precisefactor–model–based estimates of expected returns translate into sizable improvementsin out–of–sample performance of optimal portfolios.Keywords:Cross Section of Expected Returns, Risk Premium, Smallβ’s.∗We thank Torben G. Andersen, Bertille Antoine, Svetlana Bryzgalova, Frank de Jong, Joost Driessen,Stefano Giglio, Bryan Kelly, Frank Kleibergen, Yinying Li, Paulo Maio, Adam McCloskey, Dino Palazzo,Andrew Patton, Eric Renault, Enrique Sentana, George Tauchen, Viktor Todorov, Brian Weller, DachengXiu, and Guofu Zhou for helpful comments and discussions as well as seminar and conference participantsat BlackRock, Federal Reserve Board, Northwestern University Kellogg School of Management, ErasmusUniversity Rotterdam, Tilburg University, and CIREQ Montreal Econometrics Conference in honor of EricRenault. We also thank Chazz Edington for his excellent research assistance. The views expressed are solelythose of the authors and should not be interpreted as reflecting the views of the Board of Governors of theFederal Reserve System, or of any other person associated with the Federal Reserve System. Correspondingauthor: Cisil Sarisoy, Federal Reserve Board, Washington, D.C. 20551 U.S.A. E-mail: cisil.sarisoy@frb.gov.1 1 IntroductionEstimating expected returns on individual assets or portfolios is perhaps one of the longeststanding challenges in asset pricing. One standard approach at hand is to use historicalaverages. However, it is known that these estimates are generally very noisy. Even usingdaily data does not help much, if at all. There is a long history of papers trying to improveestimates of expected returns by using asset pricing models, in which expected excessreturns on individual assets are linear in their exposures to the risk factors imposed (β).The coefficients in this linear relationship are the prices of risk for the factors (λ). Examplesinclude Sharpe (1964)’s CAPM, Merton (1973)’s ICAPM, Breeden (1979)’s CCAPM, Ross(1976)’ APT and Lettau and Ludvigson (2001)’s conditional CCAPM, among many others.The literature on inference based on factor models mainly concentrates, in a frequentistsetting, on the econometric properties of the prices of risk,λ, and evaluating the ability ofthe models in explaining the cross section of expected returns. In this paper, the focus isdifferent: we analyze the estimation of the expected (excess) returns onindividual assetsorportfoliosbased on linear factor models, i.e., the product of exposuresβand risk pricesλ. In order to have an estimate of the expected (excess) return on an individual asset, bothβandλhave to be estimated, and the dependence between these estimators introduce anontrivial noise structure in the standard errors of the expected (excess) return estimators.Jorion (1991) compares CAPM—based estimators with classical sample averages ofpast returns finding the former outperforming the latter in estimating expected stock re-turns. P ́astor and Stambaugh (1999) investigate, in a Bayesian setting, the impact of prioruncertainty about mispricing in a factor model on the posterior estima