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
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Motivation
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Motivation
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Motivation
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Time-series approach
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Time-series approach
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Time-series approach
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Time-series approach
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).
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Time-series approach
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
]′].
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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).
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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′).
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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
.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Cross-sectional approach
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 .
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Comparing approaches
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Comparing approaches
Time-series regressions
Source: Cochrane (2001)Avg excess returns versus betas on CRSP size portfolios, 1926-1998.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Comparing approaches
Cross-sectional regressions
Source: Cochrane (2001)Avg excess returns versus betas on CRSP size portfolios, 1926-1998.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Comparing approaches
SDF regressions
Source: Cochrane (2001)Avg excess returns versus predicted value on CRSP size portfolios, 1926-1998.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Comparing approaches
Estimates and tests
Source: Cochrane (2001)CRSP size portfolios, 1926-1998.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
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 .
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
Characteristic sorts (cont)
Source: Ang and Liui (2004)
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Linear Factor Models Odds and ends
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.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Motivation
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Motivation
What is an event study?
An event study measures the immediate and short-horizondelayed impact of a specific event on the value of a security.
Event studies traditionally answer the questionDoes this event matter? (e.g., index additions)
but are increasingly used to also answer the questionIs the relevant information impounded into prices immediatelyor with delay? (e.g., post earnings announcement drift)
Event studies are popular not only in accounting and finance, butalso in economics and law to examine the role of policy andregulation or determine damages in legal liability cases.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Motivation
Short- versus long-horizon event studiesShort-horizon event studies examine a short time window, rangingfrom hours to weeks, surrounding the event in question.
Since even weekly expected returns are small in magnitude, thisallows us to focus on the information being released and (largely)abstract from modeling discount rates and changes therein.
There is relatively little controversy about the methodology andstatistical properties of short-horizon event studies.
However event study methodology is increasingly being applied tolonger-horizon questions. (e.g., do IPOs under-perform?)
Long-horizon event studies are problematic because the resultsare sensitive to the modeling assumption for expected returns.
The following discussion focuses on short-horizon event studies.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Setup
Basic event study methodology is essentially unchanged sinceBall and Brown (1968) and Fama et al. (1969).
Anatomy of an event study
1. Event Definition and security selection.2. Specification and estimation of the reference model
characterizing "normal" returns (e.g., market model).3. Computation and aggregation of "abnormal" returns.4. Hypothesis testing and interpretation.
#1 and the interpretation of the results in #4 are the real economicmeat of an event study – the rest is fairly mechanical.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Reference models
Common choices of models to characterize "normal" returns
Constant expected returns
ri,t+1 = µi + εi,t+1
Market model
ri,t+1 = αi + βi rm,t+1 + εi,t+1
Linear factor models
ri,t+1 = αi + β′i ft+1 + εi,t+1
(Notice the intercepts.)
Short-horizon event study results tend to be relatively insensitiveto the choice of reference models, so keeping it simple is fine.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Time-line and estimation
A typical event study time-line is
Source: MacKinlay (1997)
The parameters of the reference model are usually estimated overthe estimation window and held fixed over the event window.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Abnormal returns
Using the market model as reference model, define
AR i,τ = ri,τ − αi − βi rm,τ τ = T1 + 1, ...,T 2.
Assuming iid returns
Var[AR i,τ
]= σ2
εi+
1T1 − T0
[1 +
((rm,τ − µm)
σm
)2]
︸ ︷︷ ︸due to estimation error
.
Estimation error also introduces persistence and cross-correlationin the measured abnormal returns, even when true returns are iid,which is an issue particularly for short estimation windows.
Assume for what follows that estimation error can be ignored.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Aggregating abnormal returns
Since the abnormal return on a single stock is extremely noisy,returns are typically aggregated in two dimensions.
1. Average abnormal returns across all firms undergoing thesame event lined up in event time
ARτ =1N∑N
i=1 AR i,τ .
2. Cumulate abnormal returns over the event window
ˆCAR i(T1,T2) =∑T2
τ=T1+1 AR i,τ
or¯CAR(T1,T2) =
∑T2τ=T1+1 ARτ .
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Hypothesis testing
Basic hypothesis testing requires expressions for the variance ofARτ , ˆCAR i(T1,T2), and ¯CAR(T1,T2).
If stock level residuals εi,τ are iid through time
Var[
ˆCAR i(T1,T2)]
= (T2 − T1)σ2εi.
The other two expressions are more problematic because theydepend on the cross-sectional correlation of event returns.
If, in addition, event windows are non-overlapping across stocks
Var[ARτ
]=
1N2
∑Ni=1 σ
2εi
Var[
¯CAR(T1,T2)]
=∑T2
τ=T1+1 Var[ARτ
].
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Basic methodology
Example
Earnings announcements
Source: MacKinlay (1997)
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Bells and whistles
Outline
1 Linear Factor ModelsMotivationTime-series approachCross-sectional approachComparing approachesOdds and ends
2 Event StudiesMotivationBasic methodologyBells and whistles
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Bells and whistles
Dependence
If event windows overlap across stocks, the abnormal returns arecorrelated contemporaneously or at lags.
One solution is Brown and Warner’s (1980) "crude adjustment"
Form a portfolio of firms experiencing an event in a givenmonth or quarter (still lined up in event time, of course).Compute the average abnormal return of this portfolio.Normalize this abnormal return by the standard deviation ofthe portfolio’s abnormal returns over the estimation period.
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Bells and whistles
Heteroskedasticity
Two forms of heteroskedasticity to deal with in event studies
1. Cross-sectional differences in σεi .• Standardize ARi,τ by σεi before aggregating to ARτ .• Jaffe (1974).
2. Event driven changes in σεi ,t .
• Cross-sectional estimate of Var [ARi,τ ] over the event window.• Boehmer et al. (1991).
Michael W. Brandt Methods Lectures: Financial Econometrics
Event Studies Bells and whistles
Regression based approach
An event study can alternatively be run as multivariate regression
r1,t+1 = α1 + β1rm,t+1 +L∑
τ=0
γ1,τ D1,t+1,τ + u1,t+1
r2,t+1 = α2 + β2rm,t+1 +L∑
τ=0
γ2,τ D2,t+1,τ + u2,t+1
...
rn,t+1 = αn + βnrm,t+1 +L∑
τ=0
γn,τ Dn,t+1,τ + un,t+1
where Di,t ,τ = 1 if firm i had an event at date t − τ (= 0 otherwise).
In this case, γi,τ takes the place of the abnormal return ARi,τ .
Michael W. Brandt Methods Lectures: Financial Econometrics