tests of static asset pricing models

8/11/2019 Tests of Static Asset Pricing Models

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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

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