The Role of Factor Strength and Pricing Errors for Estimation and Inference in Asset Pricing Models

44 Pages Posted: 27 Nov 2019

See all articles by M. Hashem Pesaran

M. Hashem Pesaran

University of Southern California - Department of Economics

Ron Smith

Birkbeck College

Date Written: 2019


In this paper we are concerned with the role of factor strength and pricing errors in asset pricing models, and their implications for identification and estimation of risk premia. We establish an explicit relationship between the pricing errors and the presence of weak factors that are correlated with stochastic discount factor. We introduce a measure of factor strength, and distinguish between observed factors and unobserved factors. We show that unobserved factors matter for pricing if they are correlated with the discount factor, and relate the strength of the weak factors to the strength (pervasiveness) of non-zero pricing errors. We then show, that even when the factor loadings are known, the risk premia of a factor can be consistently estimated only if it is strong and if the pricing errors are weak. Similar results hold when factor loadings are estimated, irrespective of whether individual returns or portfolio returns are used. We derive distributional results for two pass estimators of risk premia, allowing for non-zero pricing errors. We show that for inference on risk premia the pricing errors must be sufficiently weak. We consider both when n (the number of securities) is large and T (the number of time periods) is short, and the case of large n and T. Large n is required for consistent estimation of risk premia, whereas the choice of short T is intended to reduce the possibility of time variations in the factor loadings. We provide monthly rolling estimates of the factor strengths for the three Fama-French factors over the period 1989-2018.

Keywords: arbitrage pricing theory, APT, factor strength, identification of risk premia, two-pass regressions, Fama-French factors

JEL Classification: C380, G120

Suggested Citation

Pesaran, M. Hashem and Smith, Ron P., The Role of Factor Strength and Pricing Errors for Estimation and Inference in Asset Pricing Models (2019). CESifo Working Paper No. 7919, Available at SSRN: or

M. Hashem Pesaran (Contact Author)

University of Southern California - Department of Economics ( email )

3620 South Vermont Ave. Kaprielian (KAP) Hall 300
Los Angeles, CA 90089
United States

Ron P. Smith

Birkbeck College ( email )

Malet Street
London WC1E 7HX
United Kingdom
+44 207 631 6413 (Phone)
+44 207 631 6416 (Fax)

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