Lecture 18: Testing CAPM

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Lecture 18 Testing CAPM

• The following topics will be covered
• Time Series Tests
• Sharpe (1964)/Litner (1965) version
• Black (1972) version
• Cross Sectional Tests
• Fama-MacBeth (1973) Approach

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Review of CAPM

• Let there be N risky assets with mean µ and
  variance ?

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Review of CAPM

• This is the case without risk free asset
• We have
• And
• µop is the return of the zero beta portfolio
• This is the Black version of CAPM

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Review of CAPM
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Review of CAPM
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Test of Sharpe-Lintner CAMP
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Time-Series Tests Maximum Likelihood Approach

• There are N assets and hence, N equations.
• For each equation, we can run OLS and obtain
  estimates of ?i and ?i, I 1,,N.
• We could also estimate the equations jointly.
• Is there any advantage to doing this, that is,
  run the seemingly unrelated regression on the
  system?
• As it turns out, joint estimation is useless if
  we only need estimates for ?s and ?s.
• However, for our joint test, its not useless.
  We need the covariance matrix for our joint test.

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The Likelihood Function

• We will assume that the distribution for excess
  returns are jointly normal. This is critical for
  the maximum likelihood approach. However, if we
  use Quasi ML, or GMM, we do not need normality
  assumption.
• Given joint normality of excess returns, the
  likelihood function for the cross-section of
  excess returns in a single period is

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The Likelihood Function

• With T i.i.d. (over time) observations, the
  likelihood function is

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MLE Estimates of Parameters

• Why do it this way? Because if you know the
  distribution, MLEs are
• Consistent
• Asymptotically efficient
• Asymptotically normal
• The log of the joint pdf viewed as a function of
  the unkown parameters, ?, ?, and ?.

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First Order Conditions

• The ML parameter estimates maximize L. To find
  the estimators, set the FOCs to zero
• There are N of these derivataives one for each
  ?i.
• There are N of these as well, one for each ?i.
  Finally,

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Solution

• These are just OLS parameters for ?, and ?.

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Distributions of the Point Estimates

• The distributions of the MLEs conditional on the
  excess return of the market follows from the
  assumed joint normality of the excess returns and
  the i.i.d. assumption.
• The variances and covariances of the estimators
  can be derived using the Fisher Information
  Matrix.
• The information matrix is minus the matrix of
  second partials of the log-likelihood function
  with respect to the parameter vector.
• evaluated at the point estimates.

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Asymptotic Properties of Estimators

• The estimators are consistent and have the
  distributions
• WN(T-2, ?) indicates that the NxN covariance
  matrix T? has a Wishart distribution with T-2
  degrees of freedom, a multivariate generalization
  of the chi-squared distribution.
• Note that is independent of both

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The Test Statistic

• We estimated the unconstrained market model to
  obtain the MLEs.
• Now, we impose the CAPM restrictions.
• If the CAPM is true, under the null
• H0 ? 0
• and under the alternative
• HA ? ? 0
• From your previous econometrics course, you
  probably remember that there are three ways of
  testing this.
• If we only estimate the unconstrained model, we
  can the Wald test.
• We will also consider likelihood ratio and
  Lagrangian multiplier tests.

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The Wald Test

• A straightforward application (see Greene or
  earlier notes).
• which equals
• where weve substituted in for
• Under the null, J0?2(N).
• Note that ? is unknown.
• Substitute a consistent estimate of it into the
  statistic and then under the null the
  distribution is asymptotically chi-squared.
• The MLE of ? is a consistent estimator.

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We Can Do Better

• The Wald test is an asymptotic test.
• We, however, know the finite sample distribution.
• We can use this to do the Gibbons Ross and Shaken
  (1989) test.
• To do so, we will need the following theorem from
  Muirhead (1983).
• Theorem Let the m-vector x be distributed
  N(0,?), let the (mxm) matrix A be distributed
  Wm(n,?) with n?m, and let x and A be independent.
  Then

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GRS Statistic

• Let
• Applying the theorem,
• Under the null, J1 F(N,T-N-1).
• We can construct J1 (and J0) using only the
  estimators from the unconstrained model.

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An Interpretation of J1

• GRS show that
• q is the ex-post tangency portfolio constructed
  from the N assets plus the market portfolio.
• The portfolio with the maximum (squared) Sharpe
  ratio must be the tangency portfolio.
• When the ex-post q is m, J1 0.
• As ms squared SR decreases, J1 increases
  evidence against the efficiency of m.

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The Likelihood Ratio Test

• For the LR test, we must also estimate the
  constrained model, which is the S-L CAPM (?0).
• FOCs

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The Constrained Estimators

• The estimators are consistent and have the
  following distributions (why T-1?)

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The LR Test

• We know from econometrics (CLM p194) that
• This test is based on the fact that 2 times the
  log of the likelihood ratio is asymptotically
  ?2 with d.f. equal to the number of restrictions
  under the null.
• The test statistic is
• CLM (p195) show that there is a monotonic
  relationship between J1 and J2
• Therefore we can derive finite sample
  distribution for J2 based on the finite sample
  distribution of J1

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Jobson and Korkie (1982) Adjustment

• which is also asymptotically distributed as a
• Why do we need different statistics?
• Because although their asymptotic properties are
  similar, they may have different small-sample
  properties.

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Black version of CAMP
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Testable Implication

• This is a nonlinear constraint. It may looks more
  complicated. But if you remember from your
  econometrics course, all three statsistics (Wald,
  Likelihood Ratio, Lagrangian Multiplier) can
  easily test nonlinear restrictions.
• CLM construct test statistics J4, J5, and J6 to
  test the Black CAPM. See CLM p199-203.

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Size and Power

• They also use simulations to compare small sample
  properties of all the statistics (Section 5.4 and
  5.5 )
• Size simulation simulate under the null, and
  compare the rejection rates under simulation with
  the theoretical rejection rates
• Power simulation simulate under the alternative,
  and see if rejection rate is high enough.

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Further Issues

• What if assets returns are not normal?
• One alternative approach is to use quasi-maximum
  likelihood. Under certain regularity conditions
  you can estimate the model as if the returns were
  normally distributed, and the Wald, Likelihood
  ratio, and Lagrangian multiplier tests are still
  valid (after adjusting for the covariance matrix
  for the errors).
• However, small sample properties of QMLE are of
  serious concern.
• Another alternative is to use GMM, which only
  rely on a few momentum conditions.

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Cross-sectional Test

• Consider the cross-sectional model (Security
  Market Line)
• E(Ri) Rf ßi (E(Rm) Rf )
• or, replacing expected returns with average
  returns,
• ave(Ri) Rf ßi (E(Rm) Rf ) ei
• ? ave(Ri) a ? ßi ei
• Sharpe-Lintner CAPM says that in the above
  cross-sectional regression, a should equal Rf and
  ? should equal E(Rm) Rf .
• To perform the above regression, we use ßi as a
  regressor. However, ßi is not directly observed.
  We can estimate ßi using a market model (using
  time series observations) for each stock. But if
  we use the estimated ßi , there is an
  error-in-variable problem for the above
  regression.
• Whats the consequence of error-in-variable
  problem?
• a upward biased and ? downward biased

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Issues with Cross-sectional Tests

• To alleviate the error-in-variable problem, BJS
  and FM group stocks into equally weighted
  portfolios (betas of portfolios are more
  accurate)
• But an arbitrarily formed portfolio tends to have
  beta 1.
• The maximize the power of test, group stocks into
  portfolios based on stocks betas.
• Unsolved problems errors ei are correlated
  across stocks. This causes problems for
  estimating standard deviations of coefficient
  estimates.
• Fama and MacBeth use a procedure that is now
  known as the Fama-MacBeth regression

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Fama and MacBeth (1973)

• Perform the cross-sectional regression in each
  month, to obtain rolling estimates for a and ?.
  Call them at and ?t . Then, calculate the time
  series means and time series t-stats for at and
  ?t .
• Test
• ave(at ) ave(Rf) and ave(?t ) gt0
• t-stat ave(?t)/std(?t)sqrt(T)
• Discussion under what assumptions is this t test
  valid and why?
• They also perform the test using an extended
  model
• Ri ?0 ?1 ßi ?2 ßi2 ?3 si2 ei
• and test ave(?2) ave(?3) 0

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Results from Cross-sectional Tests

• Estimated a seems too high, relative to the
  average riskfree rate.
• Estimated ? too low, relative to the average
  market risk premium.
• Black version of CAPM seems more consistent with
  the data.
• Other variables, such as squared beta and the
  variance of idiosyncratic component of returns,
  do not have marginal power to explain average
  returns.
• In other words, C1 and C2 seem to hold C3 is
  rejected.

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Exercises

• CLM