Linear beta pricing models: cross-sectional regression tests

1. Linear beta pricing models:cross-sectional regression tests FINA790C Spring 2006 HKUST

2. Motivation • The F test and ML likelihood ratio test are not without drawbacks: • We need T > N To solve this we could form portfolios (but this is not without problems) • When the model is rejected we don’t know why (e.g. do expected returns depend on factor loadings or on characteristics?)

3. Cross-sectional regression • Can we use the information from the whole cross-section of stock returns to test linear beta pricing models? • Fama-MacBeth two-pass cross-sectional regression methodology • Estimate each asset’s beta from time-series regression • Cross-sectional regression of asset returns on constant, betas (and possibly other characteristics) • Run cross-sectional regressions each period, average coefficients over time

4. Linear beta pricing model • At time t the returns on the N securities are Rt = [R1t R2t … RNt]’ with variance matrix R • Let ft = [f1t … fKt]’ be the vector of time-t values taken by the K factors with variance matrix f • The linear beta pricing model is E[Rit] = λ0 + λ’βi for i=1, … ,N or E[Rt] = λ01 + Bλ where B = E[(Rt-E(Rt))(ft-E(ft))’]f

5. Return generating process • From the definition of B the time series for Rt is Rt = E[Rt] + B( ft-E(ft) ) + ut with E[ut] = 0 and E[utft’] = 0NxK • Imposing linear beta pricing model gives Rt = λ01 + B(ft-E(ft)+λ) + ut

6. CSR method:description • Define γ = [λ0λ’]’ ( (K+1)x1 vector ) and X = [ 1 B ] ( N x (K+1) matrix ) • Assume N > K and rank(X) = K+1 • Then E[Rt] = [ 1 B ][λ0λ’]’ = X γ

7. CSR – first pass • In first step we estimate f and B through usual estimators f* = [(1/T)(ft–μf*)(ft–μf*)’] μf* = (1/T) ft B* = [(1/T)(Rt–μR*)(ft–μf*)’]f*-1 μR* = (1/T) Rt • In practice we can use rolling estimation period prior to testing period

8. CSR - second pass • In second step, for each t = 1, … , T we use the estimate B* of the beta matrix and do cross-sectional regression of returns on estimated B γ* = (X*’Q*X*)1X*’Q*Rt (for feasible GLS with weighting matrix Q*) where X* = [1 B* ] • The time-series average is γ** = (1/T)(X*’Q*X*)1X*’Q*Rt = (X*’Q*X*)1X*’Q* μR*

9. Fama-MacBeth OLS • Fama-MacBeth set Q = IN and γOLS* = (X*’X*)1X*’Rt • The time-series average is γOLS** = (X*’X*)1X*’μR* • And the variance of γOLS* is given by (1/T)  (γOLS* - γOLS** )(γOLS* - γOLS** )’

10. Issues in CSR methodology • Don’t observe true beta B, but measured B* with error: what is effect on sampling distribution of estimates? • How is CSR methodology related to maximum likelihood methodology?

11. Sampling distribution of γ** • Let D = (X’QX)-1X’Q, X = [ 1 B ] • Basic Result If (Rt’, ft’)’ is stationary and serially independent then under standard assumptions, as T→∞, √T(γ** -γ) converges in distribution to a multivariate normal with mean zero and covariance V = DRD’ + DΠD’ - D(Γ + Γ’)D’

12. Where does V come from? • Write μR* = X*γ + (μR* - E(Rt)) – (B*-B)λ • So √T( γ** -γ) = (X*’Q*X*)-1X*’Q* √T(μR* - E(Rt)) - (X*’Q*X*)-1X*’Q* √T(B* - B) λ • Error in estimating γcomes from: • Using average rather than expected returns • Using estimated rather than true betas

13. Comparing V to Fama-MacBeth variance estimator • From the definition of γOLS* its asymptotic variance is (X’X) -1X’RX(X’X)-1 = DRD’ • So in general the Fama-MacBeth standard errors are incorrect because of the errors-in-variables problem

14. Special case: conditional homoscedasticity of residuals given factors • Suppose we also assume that conditional on values of the factors ft, the time-series regression residuals ut have zero expectation, constant covariance U and are serially uncorrelated • This will hold if (Rt’, ft’)’ is iid and jointly multivariate normal

15. Asymptotic variance for special case • Recall γ = [λ0λ’]’ ( (K+1)x1 vector ) and define the (K+1)x(K+1) bordered matrix f† = 0 0’K 0K f • Then Basic Result holds with V = f† + (1+ λ’f-1λ)DUD’ Asymptotically valid standard errors are obtained by substituting consistent extimates for the various parameters

16. Example: Sharpe-Lintner-Black CAPM • For k=1, this simplifies to: • The usual Fama-MacBeth variance estimator (ignoring estimation error in betas) understates the correct variance except under the null hypothesis that λ1 (market risk premium) = 0

17. Maximum likelihood and two-pass CSR • MLE estimates B and γ simultaneously and thereby solves the errors-in-variables problem. • Asymptotic covariance matrix of the two-pass cross-sectional regression GLS estimator γ** is the same as that for MLE • I.e. two-pass GLS is consistent and asymptotically efficient as T→∞

18. Two-pass GLS • For given T, as N →∞ however, the two-pass GLS estimator still suffers from an errors-in-variables problem from using B* (i.e. two-pass GLS is not N-consistent) • We can make the two-pass GLS estimator N-consistent as well through a simple modification (see Litzenberger and Ramaswamy (1979), Shanken (1992))

19. Modified two-pass CSR • For example: Sharpe-Lintner-Black CAPM estimated with two-pass OLS • The errors-in-variable problem applies to betas, or the lower right-hand block of X*’X*. Note that E(β*’β*) = β’β + tr(U)/(TσM2*) • So deduct the last term from the lower-right hand block; this adjustment corrects for the EIV problem as N →∞.