Generalised method of moments approach to testing the CAPM Nimesh Mistry Filipp Levin

Introduction Non-normal and non-IID returns Consider tests which accommodate for non-normality, heteroskedasticity and temporal dependence of returns. Interesting for two reasons: - Normality assumption is not necessary to derive the CAPM as a theoretical model. - Departures in monthly security returns from normality have been documented. There is also evidence of heteroskedasticity and temporal dependence in stock returns. Robust tests of the CAPM can be constructed using the GMM framework

GMM approach to testing the CAPM –Within the GMM framework, the distribution of returns conditional on the market return can be both serially dependent and heteroskedastic. –We need only assume that excess returns are stationary and ergodic within finite fourth moments. Comparison to the maximum likelihood method –The parameters of the excess return market model can be estimated using the maximum likelihood, the approach is desirable because given certain regularity conditions, maximum likelihood estimators are consistent, asymptotically efficient and asymptotically normal.

Intuition GMM based on moment restrictions, instead of requiring a complete specification of the data distribution like maximum likelihood. Given the moment restrictions, choose parameter estimate so that corresponding sample moments are equal to zero (orthogonality conditions state that populations moments are equal to zero). Since have non-linear equations (n equations, n unknowns), solve via numerical methods.

References Lecture Notes Chapter 7 p (p in updated) [Part 1, Oliver Linton] Campbell, Lo, & MacKinlay p (Appendix A.2) Note: GMM also covered in Greene (p. 548) and in Hayashi (p. 204)

GMM approach to testing the CAPM Sharpe-Lintner (with riskless asset) version, tests of Black (without a riskless asset - minimum variance zero correlation portfolio to the market portfolio instead) version similar - with cross-equation restrictions on  (Only requires excess returns to be stationary and ergodic with finite fourth moments) Outline of steps 1.Set up the vector of moment conditions (with zero expectation) 2.Choose the GMM estimator (to minimize the quadratic form Q T (  )) 3.Construct the test statistic (J 7 ~ 2 N )

Intuition 1.Errors uncorrelated with anything known prior (and have mean zero) - otherwise non optimal prediction [no predictability  no correlation] E[  t (  )|Z t ] = 0  E[  t (  ) Z 1t ] = 0 … (for every component) E[  t (  ) Z kt ] = 0 2.This moment condition forms the basis for estimation and testing the using the GMM approach. GMM chooses the estimator so that linear combinations of the sample average of this moment condition are zero.

3., = OLS (ordinary least squares) estimates, same as with maximum likelihood 4.Test statistic, J 7, takes advantage of a heteroskedasticity consistent covariance matrix that is also robust to non- normality

1. Set up the vector of moment conditions (with zero expectation) T time series observations and N assets Vector of moment conditions follows from the excess-return market model. The residual vector provides N moment conditions, product of residual vector and excess-return of the market provides another N moment conditions. f t (  ) = h t   t h’ t = [1 Z mt ]2 instruments  t = Z t –  –  Z mt N variables ( residuals )  ’ = [  ’  ’]2N parameters where: Excess return-market model implies the condition: E[f t (  0 )] = 0 (  0 being the true parameter vector)

2. Choose the GMM estimator (to minimize the quadratic form Q T (  )) GMM estimator is such that the linear combinations of the sample average of the moment condition (E[f t (  0 )] = 0) are zero Sample average: GMM estimator ( ) minimizes (quadratic form): W is a positive definite (2N x 2N) weighting matrix

Therefore there are 2N (moment condition) equations, and 2N (unknown) parameters, since the system is exactly identified, can be chosen to set the average of the sample moments g T (  ) equal to zero. Note: is not dependant on W since Q T ( ) will attain the minimum, of zero, for any W. Estimators (equivalent to maximum likelihood estimators) are:

Different from maximum likelihood in that a robust covariance matrix of the estimators can be formed (variances being different than those under maximum likelihood). Covariance matrix of is:V = [D’ 0 S -1 0 D 0 ] -1 where: is asymptotically normal, therefore:

To apply the asymptotically normal result, need consistent estimators of D 0 and S 0, in this case: Assuming that the sum in S 0 reduces to a finite number of terms, S T is a consistent estimator of S 0. Estimators [D 0, S 0 ]

is a (heteroskedasticity) consistent estimator of the covariance matrix of is a robust (to non-normality) estimator of since: where: Estimators [ , Var(  )]

3. Construct the test statistic (J 7 ~ 2 N ) The test statistic is: Under H 0 :  = 0 J 7 ~ 2 N (under alternative converges to infinity) Lookup the critical value in the 2 tables, and if J 7 exceeds, reject the CAPM

References Lecture Notes Lecture Four: The Capital Asset Pricing Model (p. 57) p [Part 2, Gregory Connor] Campbell, Lo, & MacKinlay p (Chapter 5.6) p (Appendix A.3) Note: GMM tests wrt. CAPM also covered in Greene (p. 356) and in Part 1, Oliver Linton’s lecture notes

Comparison to the maximum likelihood method Advantages More flexible than the maximum likelihood approach Doesn’t assume normality Allows heteroskedasticity Allows temporal dependence Generalizes for dynamics (time varying) in  ‘s and  ‘s

Disadvantages More complex, and computationally intensive, than the maximum likelihood approach ML superior if the distribution assumption is true, however GMM can achieve the same accuracy if moment conditions are specified correctly (stating the distribution assumption)

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