Asset Pricing with Omitted Factors

Source: Giglio, S. & Xiu, D. (2021) · Journal of Political Economy 129(7), 1947–1990

TL;DR

Standard estimators of risk premia in linear factor models are biased when priced factors are omitted — the usual cure of adding ad hoc controls (e.g. the Fama-French factors) has no theoretical guarantee. Giglio & Xiu propose a three-pass method to estimate the risk premium of any observable factor that is invariant to the rotation of the other factors: as long as the test assets span the true factor space, the premium of the factor of interest is identified even if the model is incomplete and even if the factor is measured with error.

Problem it solves

Cross-sectional estimates of a factor's risk premium (two-pass Fama-MacBeth, mimicking-portfolio projection) flip sign and magnitude depending on which other factors are included, because of omitted-variable bias and measurement error. There was no systematic correction — only case-by-case ad hoc controls.

The method

A three-pass (three-step) estimator combining PCA with cross-sectional regression:

PCA on a large panel of test-asset returns to recover the latent factor space spanned by all priced sources of risk (and a consistent estimator of the number of latent factors).

Cross-sectional regression using only the principal components (excluding the factor of interest g_t) to estimate the latent factors' risk premia.

Project the observed candidate factor g_t onto the latent space; its risk premium is the implied linear combination of the PC premia. This is consistent even when g_t is observed with error or correlated with omitted factors, and it strips out the measurement-error/noise component.

The key identification result: the risk premium of g_t is invariant to the rotation of the control factors, provided the full factor space is recovered. Asymptotics hold as both the number of test assets n and the time dimension T grow.

Assumptions & inputs

A large cross-section of test assets so PCA recovers the true factor space (the approximate-factor / APT structure of Ross 1976; cf. Kelly-Pruitt-Su 2019).

Priced risks are spanned (strong-factor) by the test assets; weak factors not exposed in the cross-section break identification.

Inputs: a panel of test-asset excess returns plus the time series of the candidate factor g_t.

How to use it

A general-purpose tool for evaluating proposed factors. Empirically, the authors find: the market risk premium is positive and significant, close to the time-series average of market excess returns even with an unrestricted zero-beta rate; several macro factors are dominated by noise and carry essentially zero premium once corrected for measurement error and exposure to unobserved factors; but they find empirical support for stockholder consumption growth (Malloy-Moskowitz-Vissing-Jorgensen 2009) and the Pástor-Stambaugh (2003) liquidity factor.

Limitations & pitfalls

Needs a large, well-behaved cross-section; weak/missing factors in the test assets break the spanning assumption.

Identifies the premium of an observable factor, not the deep economic factor itself; latent factors are recovered only up to rotation.

Key references

Giglio, S. & Xiu, D. (2021) — Asset Pricing with Omitted Factors — Journal of Political Economy

Feng, G., Giglio, S. & Xiu, D. (2020) — Taming the Factor Zoo — Journal of Finance

Kelly, B., Pruitt, S. & Su, Y. (2019) — Characteristics Are Covariances — Journal of Financial Economics

Provenance: verified/generated from the paper's full text.