L18: CAPM1 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

L18: CAPM2 Review of CAPM Let there be N risky assets with mean µ and variance Ω

L18: CAPM3 Review of CAPM This is the case without risk free asset We have And µ op is the return of the zero beta portfolio This is the Black version of CAPM

L18: CAPM4 Review of CAPM

L18: CAPM5 Review of CAPM

L18: CAPM6 Test of Sharpe-Lintner CAMP

L18: CAPM7 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, it’s not useless. We need the covariance matrix for our joint test.

L18: CAPM8 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:

L18: CAPM9 The Likelihood Function With T i.i.d. (over time) observations, the likelihood function is:

L18: CAPM10 MLE Estimates of Parameters Why do it this way? Because if you know the distribution, MLE’s are –Consistent –Asymptotically efficient –Asymptotically normal The log of the joint pdf viewed as a function of the unkown parameters, , , and .

L18: CAPM11 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,

L18: CAPM12 Solution These are just OLS parameters for , and .

L18: CAPM13 Distributions of the Point Estimates The distributions of the MLE’s 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.

L18: CAPM14 Asymptotic Properties of Estimators The estimators are consistent and have the distributions: W N (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

L18: CAPM15 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: H 0 :  = 0 and under the alternative: H A :   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.

L18: CAPM16 The Wald Test A straightforward application (see Greene or earlier notes). which equals where we’ve substituted in for Under the null, J 0 ~  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.

L18: CAPM17 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 W m (n,  ) with n  m, and let x and A be independent. Then:

L18: CAPM18 GRS Statistic Let Applying the theorem, Under the null, J 1 ~ F(N,T-N-1). We can construct J 1 (and J 0 ) using only the estimators from the unconstrained model.

L18: CAPM19 An Interpretation of J 1 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, J 1 = 0. As m’s squared SR decreases, J 1 increases – evidence against the efficiency of m.

L18: CAPM20 The Likelihood Ratio Test For the LR test, we must also estimate the constrained model, which is the S-L CAPM (  =0). FOCs:

L18: CAPM21 The Constrained Estimators The estimators are consistent and have the following distributions (why T-1?):

L18: CAPM22 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 J 1 and J 2 Therefore we can derive finite sample distribution for J 2 based on the finite sample distribution of J 1

L18: CAPM23 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.

L18: CAPM24 Black version of CAMP

L18: CAPM25 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 J 4, J 5, and J 6 to test the Black CAPM. See CLM p

L18: CAPM26 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.

L18: CAPM27 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.

L18: CAPM28 Cross-sectional Test Consider the cross-sectional model (Security Market Line): E(R i ) = R f + β i (E(R m ) – R f ) or, replacing expected returns with average returns, ave(R i ) = R f + β i (E(R m ) – R f ) + e i  ave(R i ) = α + γ β i + e i Sharpe-Lintner CAPM says that in the above cross-sectional regression, α should equal R f and γ should equal E(R m ) – R f. To perform the above regression, we use β i as a regressor. However, β i is not directly observed. We can estimate β i using a market model (using time series observations) for each stock. But if we use the estimated β i, there is an error-in-variable problem for the above regression. What’s the consequence of error-in-variable problem? –α upward biased and γ downward biased

L18: CAPM29 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 e i 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”

L18: CAPM30 Fama and MacBeth (1973) Perform the cross-sectional regression in each month, to obtain rolling estimates for α and γ. Call them α t and γ t. Then, calculate the time series means and time series t-stats for α t and γ t. Test: ave(α t )= ave(R f ); and ave(γ t ) >0 t-stat: ave(γ t )/std(γ t )*sqrt(T) Discussion: under what assumptions is this t test valid and why? They also perform the test using an extended model: R i = γ 0 + γ 1 β i + γ 2 β i 2 + γ 3 s i 2 + e i and test: ave(γ 2 ) = ave(γ 3 ) = 0

L18: CAPM31 Results from Cross-sectional Tests Estimated α seems too high, relative to the average riskfree rate. Estimated γ 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.

L18: CAPM32 Exercises CLM