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Methods Lectures: Financial Econometrics

Linear Factor Models and Event Studies

Michael W. Brandt, Duke University and NBER

NBER Summer Institute 2010

Michael W. Brandt Methods Lectures: Financial Econometrics

Outline

1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends

2 Event StudiesMotivationBasic methodologyBells and whistles


Linear Factor Models Motivation

Outline





Linear factor pricing modelsMany asset pricing models can be expressed in linear factor form

E[rt+1 − r0] = β′λ

where β are regression coefficients and λ are factor risk premia.

The most prominent examples are the CAPM

E[ri,t+1 − r0] = βi,mE[rm,t+1 − r0]

and the consumption-based CCAPM

E[ri,t+1 − r0] ' βi,∆ct+1 γVar[∆ct+1]︸︷︷︸λ

Note that in the CAPM the factor is an excess return that itselfmust be priced by the model⇒ an extra testable restriction.



Econometric approaches

Given the popularity of linear factor models, there is a largeliterature on estimating and testing these models.

These techniques can be roughly categorized asTime-series regression based intercept tests for models inwhich the factors are excess returns.Cross-sectional regression based residual tests for models inwhich the factors are not excess returns or for when thealternative being considered includes stock characteristics.

The aim of this first part of the lecture is to survey and put intoperspective these econometric approaches.


Linear Factor Models Time-series approach

Outline





Black, Jensen, and Scholes (1972)

Consider a single factor (everything generalizes to k factors).

When this factor is an excess return that itself must be priced bythe model, the factor risk premium is identified through

λ = ET [ft+1].

With the factor risk premium fixed, the model implies that theintercepts of the following time-series regressions must be zero

ri,t+1 − r0 ≡ rei,t+1 = αi + βi ft+1 + εi,t+1

so that E[rei,t+1] = βiλ.

This restriction can be tested asset-by-asset with standard t-tests.



Joint intercepts testThe model implies, of course, that all intercepts are jointly zero,which we test with standard Wald test

α′ (Var[α])−1 α ∼ χ2N .

To evaluate Var[α] requires further distributions assumptions

With iid residuals εt+1, standard OLS results apply

Var[α

β

]=

1T

[ 1 ET [ft+1]

ET [ft+1] ET [f 2t+1]2

]−1

⊗ Σε

which implies

Var[α] =1T

1 +

(ET [ft+1]

σT [ft+1]

)2 Σε.



Gibbons, Ross, and Shanken (1989)Assuming further that the residuals are iid multivariate normal, theWald test has a finite sample F distribution

(T − N − 1)

N

1 +

(ET [ft+1]

σT [ft+1]

)2−1

α′Σ−1ε α ∼ FN,T−N−1.

In the single-factor case, this test can be further rewritten as

(T − N − 1)

N

(

ET [req,t+1]

σT [req,t+1]

)2

−

(ET [re

m,t+1]

σT [rem,t+1]

)2

1 +

(ET [re

m,t+1]

σT [rem,t+1]

)2

,

where q is the ex-post mean-variance portfolio and m is theex-ante mean variance portfolio (i.e., the market portfolio).



MacKinlay and Richardson (1991)When the residuals are heteroskedastic and autocorrelated, wecan obtain Var[α] by reformulating the set of N regressions as aGMM problem with moments

gT (θ) = ET

[rt+1 − α− βft+1

(rt+1 − α− βft+1)ft+1

]= ET

[εt+1

εt+1ft+1

]= 0

Standard GMM results apply

Var[α

β

]=

1T

[DS−1D′

]−1

with

D =∂gT (θ)

∂θ= −

[1 ET [ft+1]

ET [ft+1] ET [f 2t+1]

]⊗ IN

S =∞∑

j=−∞E

[[εtεt ft

] [εt−j

εt−j ft−j

]′].


Linear Factor Models Cross-sectional approach

Outline





Cross-sectional regressions

When the factor is not an excess return that itself must be pricedby the model, the model does not imply zero time-series intercepts.

Instead one can test cross-sectionally that expected returns areproportional to the time-series betas in two steps

1. Estimate asset-by-asset βi through time-series regressions

ret+1 = a + βi ft+1 + εt+1.

2. Estimate the factor risk premium λ with a cross-sectionalregression of average excess returns on these betas

rei = ET [re

i ] = λβi + αi .

Ignore for now the "generated regressor" problem.



OLS estimatesThe cross-sectional OLS estimates are

λ = (β′β)−1(β rei )

α = rei − λβ.

A Wald test for the residuals is then

αVar[α]−1α ∼ χ2N−1

withVar[α] =

(I − β(β′β)−1β′

) 1T

Σε

(I − β(β′β)−1β′

)Two important notes

Var[α] is singular, so Var[α]−1 is a generalized inverse andthe χ2 distribution has only N − 1 degrees of freedom.Testing the size of the residuals is bizarre, but here we haveadditional information from the time-series regression (in red).



GLS estimates

Since the residuals of the second-stage cross-sectional regressionare likely correlated, it makes sense to instead use GLS.

The relevant GLS expressions are

λGLS = (β′Σ−1ε β)−1(βΣ−1

ε rei )

andVar[αGLS] =

1T

(Σε − β(β′Σ−1

ε β)−1β′).



Shanken (1992)The generated regressor problem, the fact that β is estimated inthe first stage time-series regression, cannot be ignored.

Corrected estimates are given by

Var[αOLS] =(

I − β(β′β)−1β′) 1

TΣε

(I − β(β′β)−1β′

)×

(1 +

λ2

σ2f

)

Var[αGLS] =1T

(Σε − β(β′Σ−1

ε β)−1β′)×

(1 +

λ2

σ2f

)

In the case of the CAPM, for example, the Shanken correction forannual data is roughly

1 +

(ET [rmkt,t+1]

σT [rmkt,t+1]

)2

= 1 +

(0.060.15

)2

= 1.16



Cross-sectional regressions in GMMFormulating cross-sectional regressions as GMM delivers anautomatic "Shanken correction" and allows for non-iid residuals.

The GMM moments are

gT (θ) = ET [rt+1

ret+1 − a− βft+1

(ret+1 − a− βft+1)ft+1

rt+1 − βλ

= 0.

The first two rows are the first-stage time-series regression.

The third row over-identifies λIf those moments are weighted by β′ we get the first orderconditions of the cross-sectional OLS regression.If those moments are instead weighted by β′Σ−1

ε we get thefirst order conditions of the cross-sectional GLS regression.

Everything else is standard GMM.



Linear factor models in SDF formLinear factor models can be rexpressed in SDF form

E[mt+1ret+1] = 0 with mt+1 = 1− bft+1.

To see that a linear SDF implies a linear factor model, substitute in

0 = E[(1− bft+1)ret+1] = E[re

t+1]− b E[ret+1ft+1]︸︷︷︸

Cov[ret+1, ft+1] + E[re

t+1]E[ft+1]

and solve for

E[ret+1] =

b1− bE[ft+1]

Cov[ret+1, ft+1]

=Cov[re

t+1, ft+1]

Var[ft+1]︸︷︷︸β

bVar[ft+1]

1− bE[ft+1]︸︷︷︸λ

.



GMM on SDF pricing errors

This SDF view suggests the following GMM approach

gT (b) = ET[(1 + bft+1)re

t+1]

= ET [ret+1]− bET [re

t+1ft+1]︸︷︷︸pricing errors = π

= 0.

Standard GMM techniques can be used to estimate b and testwhether the pricing errors are jointly zero in population.

GMM estimation of b is equivalent toCross-sectional OLS regression of average returns on thesecond moments of returns with factors (instead of betas)when the GMM weighting matrix is an identity.Cross-sectional GLS regression of average returns on thesecond moments of returns with factors when the GMMweighting matrix is a residual covariance matrix.



Fama and MacBeth (1973)The Fama-MacBeth procedure is one of the original variants ofcross-sectional regressions consisting of three steps

1. Estimate βi from stock or portfolio level rolling or full sampletime-series regressions.

2. Run a cross-sectional regression

ret+1 = λt+1β + αi,t+1

for every time period t resulting in a time-series

{λt , αi,t}Tt=1.

3. Perform statistical tests on the time-series averages

αi =1T∑T

t=1 αi,t and λi = 1T∑T

t=1 λi,t .


Linear Factor Models Comparing approaches

Outline





Time-series regressions

Source: Cochrane (2001)Avg excess returns versus betas on CRSP size portfolios, 1926-1998.



Cross-sectional regressions

Source: Cochrane (2001)Avg excess returns versus betas on CRSP size portfolios, 1926-1998.



SDF regressions

Source: Cochrane (2001)Avg excess returns versus predicted value on CRSP size portfolios, 1926-1998.



Estimates and tests

Source: Cochrane (2001)CRSP size portfolios, 1926-1998.


Linear Factor Models Odds and ends

Outline





Firm characteristicsSo far

H0 : αi = 0 i = 1,2, ...,nHA : αi 6= 0 for some i

We might, typically on the basis of preliminary portfolio sorts, amore specific alternative in mind

H0 : αi = 0 i = 1,2, ...,nHA : αi = γ′xi

for some firm level characteristics xi .We can test against this more specific alternative by including thecharacteristics in the cross-sectional regression

rei == λβi + γ′xi + αi .



Portfolios

Stock level time-series regressions are problematicWay too noisy.Betas may be time-varying.

Fama-MacBeth solve this problem by forming portfolios1. Each portfolio formation period (say annually), obtain a noisy

estimate of firm-level betas through stock-by-stock time seriesregressions over a backward looking estimating period.

2. Sort stocks into "beta portfolios" on the basis of their noisybut unbiased "pre-formation beta".

The resulting portfolios should be fairly homogeneous in theirbeta, should have relatively constant portfolio betas through time,and should have a nice beta spread in the cross-section.



Characteristic sorts

Somewhere along the way sorting on pre-formation factorexposures got replaced by sorting on firm characteristics xi .

Sorting on characteristics is like sorting on ex-ante alphas (theresiduals) insead of betas (the regressor).

It may have unintended consequence1. Characteristic sorted portfolios may not have constant betas.2. Characteristic sorted portfolios could have no cross-sectional

variation in betas and hence no power to identify λ.



Characteristic sorts (cont)

Source: Ang and Liui (2004)



Conditional informationIn a conditional asset pricing model expectations, factorexposures, and factor risk premia all have a time-t subscripts

Et [rt+1 − r0] = β′tλt .

This makes the model fundamentally untestable.

Nevertheless, there are at least two ways to proceed1. Time-series modeling of the moments, such as

βt = β0 − κ(βt−1 − β0) + ηt .

2. Model the moments as (linear) functions of observables

βt = γ′βZt or λt = γ′λZt .

Approach #2 is by far the more popular.



Conditional SDF approachIn the context of SDF regressions

Et [mt+1ret+1] = 0 with mt+1 = 1− bt ft+1.

Assume bt = θ0 + θ1Zt and substitute in

Et[(1− (θ0 + θ1Zt )ft+1) re

t+1]

= 0.

Condition down

E[Et [(1− (θ0 + θ1Zt )ft+1)re

t+1]]

= E[(1− (θ0 + θ1Zt )ft+1)re

t+1]

= 0.

Finally rearrange as unconditional multifactor model

E[(1− θ0f1,t+1 − θ1f2,t+1)re

t+1]

= 0,

where f1,t+1 = ft+1 and f2,t+1 = zt ft+1.


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