Introduction

Invariance result

Estimator

Empirics

Inference on Risk Premia in the Presence of Omitted Factors Stefano Giglio

Dacheng Xiu

Booth School of Business, University of Chicago

January 6, 2017

Conclusion

Introduction

Invariance result

Estimator

Empirics

Conclusion

Introduction

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Asset pricing models predict that some factors are priced I

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Traded or non-traded

Theoretical models are typically very stylized I

The literal SDF has often poor explanatory power

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Literal models are rejected

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Plausible that the model is missing some factors

What else can we say about the factor predicted by the model, gt , in the presence of omitted factors?

Introduction

Invariance result

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Empirics

Introduction

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Focus on the risk premium of the factor of interest gt I

How much are investors willing to pay to hedge that risk (and just that risk)?

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Easy to do for traded factors: avg excess return of the portfolio

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Nontraded factors: several ways 1. Nonparametric: multidimensional portfolio sorts 2. Parametric: Fama-MacBeth and other two-pass cross-sectional regressions I

Biased when there are omitted factors: Jagannathan and Wang (1998)

Conclusion

Introduction

Invariance result

Estimator

Empirics

Conclusion

This paper

This paper: three-pass methodology to address the omitted factor problem in Fama-MacBeth regressions 1. Rotation invariance result: FM regression will yield the correct risk premium for gt as long as it includes a set of controls that, together with gt , span the entire factor space. I

Risk premia estimate is the same for any rotation of the controls

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No need to know the identities of the omitted factors

2. Recover the factor space via PCA 3. Derive large N, large T asymptotic inference on the risk premium

Introduction

Invariance result

Estimator

Empirics

Conclusion

Model Setup I

True model is linear with p factors vt (zero-mean). rt = ιn γ0 + βγ + βvt + ut

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γ are the factor risk premia

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β are the risk exposures

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Factor of interest gt (tradable or nontradable): gt = ηvt + zt

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zt is measurement error, that we will account for

Risk premium of gt is the expected excess return of a “pure factor” portfolio with beta of 1 with gt (free of measurement error) and 0 with all other risk sources. For gt , it is ηγ.

Introduction

Invariance result

Estimator

Empirics

Conclusion

Rotation invariance result The slope of gt in a Fama-MacBeth regression where gt appears together with any (p − 1) linear combinations of vt is ηγ, the risk premium of gt . 1. Rotate model so gt is the first factor: Hvt where first row of H is η 2. Slopes in rotated model: Hγ, so slope of first factor is ηγ I

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Independent of other rows of H: invariance result

Note: p − 1 control factors can be arbitrary linear combinations of vt I

Only requirement: gt and controls span the entire factor space

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Natural to use PCA to recover the factor space

Introduction

Invariance result

Estimator

Empirics

Conclusion

A three-pass estimator: Invariance result + PCA We propose a three-pass estimator to obtain ηγ. 1. Extract latent factors vt via PC 2. Use cross-sectional regression to estimate latent factor risk premia γˆ 3. Regress gt on the latent factors vt via time-series regression I

This gives us ηˆ

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Tells us how to rotate the model so that gt is the first factor

Risk premium of gt is estimated as ηˆγˆ I

Estimator is consistent and satisfies CLT as T , N → ∞

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Derive asymptotic distribution

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Consistent if a few extra PCs are used

Introduction

Invariance result

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Empirics

A three-pass estimator

Alternative interpretation of the invariance result: I

Risk premium of gt is −Cov (gt , mt ), where m is the SDF I

Steps 1 and 2 recover the SDF (under APT): mt = 1 − γ | Σ−1 v vt

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Step 3 computes risk premia as −Cov (gt , mt )

Invariance holds because SDF is rotation-invariant and risk premia depend on a univariate covariance

Conclusion

Introduction

Invariance result

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Empirics

Empirical Analysis

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Test assets: set of 202 standard equity portfolio from Fama and French

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Consider many tradable and nontradable factors

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We first need to select the number of PCs I

4 PCs have R 2 of 65%

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Robustness to increasing the number of PCs

Conclusion

Introduction

Invariance result

Estimator

Empirics

Conclusion

Results: Tradable factors, Fama-MacBeth vs. 3-pass

FM Model Factors Avg ret FF3

γ

RmRf

0.50

−0.57

SMB

0.23

0.17

HML

0.34

0.23

3-pass, P = 4 stderr

∗∗

∗

γ

stderr

(0.25)

0.37

∗

(0.13)

0.23

∗

(0.13)

0.21

∗

(0.11)

(0.13)

(0.20)

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Robust to using P=4, 5, 6

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These are tradable factors

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If the model is correctly specified, we would expect the risk premia to be equal to the average excess returns

Introduction

Invariance result

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Results: Nontradable factors

Factors IP Liquidity

Avg ret

FM γ stderr

3-pass, P = 4 γ stderr

−0.13∗

(0.07)

−0.00

(0.00)

0.02

(0.97)

0.26∗∗

(0.12)

Conclusion

Introduction

Invariance result

Estimator

Empirics

Conclusion

Conclusion

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Marry PCA with invariance result to correct for omitted variable bias

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Two caveats: 1. PCA might miss latent weak factors I I I

Tradable factor results Robustness to the number of factors Good deal bounds

2. Invariance result does not hold for loading of SDF on gt I

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Alternative methods (Feng, Giglio and Xiu 2017).

Main point: when computing risk premia, we typically use few arbitrary factors as controls. Crucial to to complete the space.