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Tests of Static Asset Pricing
Models
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Tests of Static Asset Pricing Models
In general asset pricing models quantify thetradeoff between risk and expected return.
Need to both measure risk and relate it to theexpected return on a risky asset.
The most commonly used models are:
CAPM
APT
FF three factor model
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Testable Implications
These models have testable implications.For the CAPM, for example:
Expected excess return of a risky asset isproportional to the covariance of its return andthat of the market portfolio.
Note, this tells us the measure of risk used and its
relation to expected return.There are other restrictions that depend upon
whether there exists a riskless asset.
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Testable Implications
For the APT,
The expected excess return on a risky asset is
linearly related to the covariance of its returnwith various riskfactors.
These risk factors are left unspecified by the
theory and have been:
Derived from the data (CR (1983), CK)
Exogenously imposed (CRR (1985))
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Plan
Review the basic econometric methodology wewill use to test these models.
Review the CAPM. Test the CAPM.
Traditional tests (FM (1972), BJS (1972), Ferson andHarvey)
ML tests (Gibbons (1982), GRS (1989)) GMM tests
Factor models: APT and FF
Curve fitting vs. ad-hoc theorizing
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Econometric Methodology Review
Maximum Likelihood Estimation
The Wald Test
The F Test
The LM Test
A specialization to linear models and linearrestrictions
A comparison of test statistics
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Review of Maximum Likelihood
Estimation Let {x1, xT} be a sample of T, i.i.d.
random variables.
Call that vector x.
Let xbe continuously distributed with density
f(x|).
Where, is the unknown parameter vector thatdetermines the distribution.
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The Likelihood Function
The joint density for the independent randomvariables is given by:
f(x1|
)f(x2|
)f(x3|
)f(xT|
) This joint density is known as the likelihood
function,L(x|)
L(x|)= f(x1|)f(x2|)f(x3|)f(xT|)
Can you write the joint density andL(x|) thisway when dealing with time-dependentobservations?
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Idea Behind Maximum Likelihood
Estimation Pick the parameter vector estimate, , that
maximizes the likelihood,L(x|), of
observing the particular vector ofrealizations, x.
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MLE Plusses and Minuses
Plusses: Efficient estimation in terms of
picking the estimator with the smallest
covariance matrix.Question: are ML estimators necessarily
unbiased?
Minuses: Strong distributional assumptionsmake robustness a problem.
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MLE Example: Normal Distributions where
OLS assumptions are satisfied
Sample yof size T is normally distributed
with mean xwhere
Xis a T x K matrix of explanatory variables
is a K x 1 vector of parameters
The variance-covariance matrix of the errors
from the true regression is 2
I, whereIis a T x T identity matrix
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The Likelihood Function
The likelihood function for the linear model
with independent normally distributed
errors is:
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The Log-Likelihood Function
With independent draws, it is easier to maximize
the log-likelihood function, because products are
replaced by sums. The log-likelihood is given by:
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The Information MatrixCont
The MLE achieves the Cramer-Rao lower bound, whichmeans that the variance of the estimators equals the inverseof the information matrix:
Now,
note, the off diagonal elements are zero.
).,(
21
I
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The Information MatrixCont
The negative of the expectation is:
The inverse of this is:
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Another way of Writing I(,2)
For a vector, , of parameters, I(), theinformation matrix, can be written in a secondway:
This second form is more convenient forestimation, because it does not require estimatingsecond derivatives.
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Estimation
The Likelihood Ratio Test
Let be a vector of parameters to be estimated.
Let H0be a set of restrictions on theseparameters.
These restrictions could be linear or non-linear.
Let be the MLE of estimated withoutregard to constraints (the unrestricted model).
Let be the constrained MLE.
U
U
R
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The Likelihood Ratio Test Statistic
If and are the likelihoodfunctions evaluated at these two estimates,
the likelihood ratio is given by:
Then, -2ln() = -2(ln( )ln( ) ~2with degrees of freedom equal to thenumber of restrictions imposed.
)( UUL )( RRL
)
(
UUL )
(
RRL
)(
)(
UU
RR
L
L
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Another Look at the LR Test
Concentrated Log-Likelihood: Many problemscan be formulated in terms of partitioning a
parameter vector, into {1, 2} such that thesolution to the optimization problem, can bewritten as a function of , e.g.:
Then, we can concentrate the log-likelihoodfunction as: F*(1, 2) = F(1, t(1)) Fc().
2
1
).( 12 t
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Why Do This?
The unrestricted solution to
then provides the full solution to the
optimization problem, since t is known.
We now use this technique to find estimates
for the classical linear regression model.
)( 11 cFMax
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Ex: Concentrating the Likelihood Function
Inserting this back into the log-likelihood yields:
Because (y- X)(y- X) is just the sum of
squared residuals from the regression (ee) we can
rewrite ln(Lc) as:
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Ex: Concentrating the Likelihood Function
For the restricted model we obtain the restricted concentrated log-likelihood:
So, plugging in these concentrated log-likelihoods into our definitionof the LR test, we obtain:
Or, T times the log of the ratio of the restricted SSR and theunrestricted SSR, a nice intuition.
)(
1ln)2ln(1
2
)ln( ' RRcR ee
T
TL
ee
eeTLR RR
'ln
'
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ExampleCont
The first-order conditions for the estimates and
simply reduce to the OLS normal equations:
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ExampleCont
Solving
Substituting into the FOC for yields:
xy
T
t t
T
t tt
xxyyxx
1
2
1
)()))(((
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ExampleCont
Solve for as before:2
2
1
2 )(1
T
t
tt xyT
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ExampleCont
The restricted model is exactly the same, except that is constrained
to be one, so that the normal equation reduces to:
and
One can then plug in to obtain and form the likelihood ratio, which
is distributed 2(1).
2R
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The Wald Test
The problem with LR test: Need both restricted
and unrestricted model estimates.
One or the other could be hard to compute. The Wald test is an alternative that requires
estimating the unrestricted model only.
Suppose y~ N(X, ), with a sample size of T,
then:21 ~)()'( TXyXy
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The Wald TestCont
Under the null hypothesis that E(y) = X, the
quadratic form above has a 2distribution. If the
hypothesis is false, the quadratic form will belarger, on average, than it would be if the null
were true.
In particular, it will be a non-central 2with the
same degrees of freedom, which looks like acentral 2, but lies to the right.
This is the basis for the test.
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The Restricted Model
Now, step back from the normal and let be the
parameter estimates from the unrestricted model.
Let restrictions be given byH0:f() = 0.
If the restrictions are valid, then should satisfy
them.
If not, should be farther from zero than
would be explained by sampling error alone.
)(f
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Formalism
The Wald statistic is
Under H0in large samples, W ~ 2with d.f. equal to the
number of restrictions. See Greene ch.9 for details.
Lastly, to use the Wald test, we need to compute the
variance term:
)()])([()'( 1 ffVarfW
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Restrictions on Slope Coefficients
If the restrictions are on slope coefficients of a linear regression, then:
where
and K is the number of regressors.
Then, we can write the Wald Statistic:
where J is the number of restrictions.
12 )'(][][ XXsVarVar
22 ' KT
TKTees
][)())'(])'()[(()'( 2112 JfGXXsGfW
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Linear Restrictions
H0: R- q= 0
For example, suppose there were three betas, 1,
2, and 3. Lets look at three tests.(1) 1= 0,
(2) 1= 2,
(3) 1= 0 and 2= 2. Each row of Ris a single linear restriction on the
coefficient vector.
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Writing R
Case 1:
Case 2:
Case3:
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The Wald Statistic
In general, the Wald statistic with J linear
restrictions reduces to:
with J d.f.
We will use these tests extensively in our
discussion of Chapters5 and 6 of CLM.
][]')'([]'[ 112 qRRXXRsqRW
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Why Do We Care?
We care because in a linear model withnormally distributed disturbances under the
null, the test statistic derived above is exact.This will be important later because undernormality, some of our cross-sectional CAPMtests will be of this form and,
A sufficient condition for the (static) CAPM tobe correct is for asset returns to be normallydistributed.
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The LM Test
This is a test that involves computing only
the restricted estimator.
If the hypothesis is valid, at the value of therestricted estimator, the derivative of the log-
likelihood function should be close to zero.
We will next form the LM test with the J
restrictions f() = 0.
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The LM TestCont
This is maximized by choice of and
)(')]()'[(
)2(
1)ln(
2
)2ln(
2
)ln(2
2
FXyXyTT
LLM
.2
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First-order Conditions
and
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The LM TestCont
The test then, is whether the Lagrange
multipliers equal zero. When the
restrictions are linear, the test statisticbecomes (see Greene, chapter 7):
where J is the number of restrictions.
][]')'([]'[ 112 qRRXXRsqRLM R
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W, LR, LM, and F
We compare them forJlinear restrictions in thelinear model with K regressors. It can be shownthat:
and that W > LR > LM.
,FJKT
TW
,1
1ln
FJ
KTTLR
,]))/(1(1)[(FJ
FJKTKTTLM