see, under die null hypothesis die Unite-sample distribution ol / > can differ from its large-sample distribution. Jobson and Korkie (1982) suggest tin adjustment to / < which lias better finite-sample proj>erties. Defining f as the modified statistic, we have

(V - - 2)

./, = --г- h

= -2)[logS 1-loglSl] £ к. (5.3.41)

We will visit the issue of the finite-sample properties of ] > and f in Section 5.4.

Л useful economic interpretation can be made of the lest statistic J\ using results from efficient-set mathematics. Gibbons, Ross, and Shanken (1989) show that

N \+-

where the portfolio denoted by ц represents the ex post tangency portfolio constructed as in (5.2.28) from the N included assets plus die market portfolio. Recall from Section 5.2 that the portfolio with the maximum squared Sharpe ratio of all portfolios is the tangency portfolio. Thus when er. post the market portfolio is the tangency portfolio J\ will be equal to /его, and as the squared Sharpe ratio of the market decreases, J\ will increase, indicating stronger evidence against the efficiency of the market portfolio. In Section 5.7.2 we present an empirical example using J\ after considering the Mack version of the CAPM in the next section.

5. >,2 liltitl; Version

In the absence ol a riskfree asset we consider the Black version of the (АРМ in (5.1.5). Ihe expected return on the zero-beta portfolio l- A ,1 is treated as an uiiobservahle and hence becomes an unknown model parameter. Defining the zero-beta portfolio expected return as у, the Black version is

KR, = i,y +№IR, <] - y)

= U- ft) У +/3KI /? !. (5.3.43)

Willi the Black model, the unconstrained model is the real-return market

model. Define R, as an (iVx 1) vector of real returns for N assets (or portfolios of assets). For these N assets, the real-return market model is

R, = a + ftRmt + e, (5.3.44)

£[£,] = 0 (5.3.45)

E[e,e/] = £ (5.3.46)

Е[Л /] = n,n, 4{R, ,- liaf] = oln (5.3.47)

Cov[R,nl,e,} = 0. (5.3.48)

ft is the (iVx 1) vector of asset betas, R i is the time period t market portfolio return, and a and e, are (Nx 1) vectors of asset return intercepts and disturbances, respectively.

The testable implication of the Black version is apparent from comparing the unconditional expectation of (5.3.44) with (5.3.43). The implication is

a = (t--ft)y. (5.3.49)

This implication is more complicated to test than the zero-intercept restriction of the Sharpe-Lintner version because the parameters ft and у enter in a nonlinear fashion.

Given the IID assumption and the joint normality of returns, the Black version of the CAPM can be estimated and tested using the maximum likelihood approach. The maximum likelihood estimators of the unrestricted model, that is, the real-return market model in (5.3.44), are identical to the estimators of the excess-return market model except that real returns ar e substituted for excess returns. Thus fi, for example, is now the vector of sample mean real returns. For the maximum likelihood estimators of thjp parameters we have

a = (5.3.50)

ft = E-A)№ ,-A, )

Ё = r £(R< ~ W )(R - - (5.3.52)

where

J т ) r

A = fYlKl 3IKl Am = JRml-

Conditional on the real return of the market, /< , R,M.....R r, die distributions are

where

0 ~ {0,-

JE ~ WN(7-2,£),

= - YiRn, - A )2.

(5.3.53)

(5.3.54) (5.3.55)

The covariance of a and (3 is

Cov[6, (3] =

£.

(5.3.56)

j For the constrained model, that is, the Black version of the CAPM, the log-likelihood function is

I NT T

I £(v,/3,E) = -- log(27r)- - logE

-5£(R(->U-/з)-/зlm,)£-,

x (R, - y(L - P) - 0R ,). Differentiating with respect lo y, P, and E, we have ЭС

<mmr

SC y-l

= (*-/9)E- Yj(K,-y<l-p)-pRml) -1=1

У (R, - у( . - /3) - /ЗК, ,) (Я, , - у) г

]Г (R, - yd -/3>- /ЗЛ, ,)

(5.3.57)

(5.3.58) (5.3.59)

x(R,-K(t-/3)-/3RMl)

Е-1. (5.3.60)

Setting (5.3.59), (5.3.59), and (5.3.60) lo /его, we can solve for the maximum likelihood estimators. These are:

p. = (t-p)t~\fi-pftm) (l-PY±*~\t-p)

1 7 /=1

F.qttalions (5.3.61), (5.3.62), and (5.3.03) do not allow us to solve explicitly lor the maximum likelihood estimators. The maximum likelihood estimators can be obtained, given initial consistent estimators of p and E, by iterating over (5.3.61), (5.3.62), and (5.3.63) until convergence. The unconstrained estimators p and Ё can serve as the initial consistent estimators of p and E, respectively.

Given both the constrained and unconstrained maximum likelihood estimators, we can construct an asymptotic likelihood ratio test of the null hypothesis.1 The null and alternative hypotheses are

H : a = (l-P)Y (5.3.64)

Нл: a ф (l-P)y- (5.3.65)

A likelihood ratio lest can be constructed in a manner analogous to the test constructed for the Sharpc-I.inlner version in (5.3.33). Defining J.t as the test statistic, we have

J4 = r[logE*-l gE] ~ xl r (5.3.66)

Notice that the degrees of freedom of the null distribution is jV - 1. Relative to the Sharpc-I.inlner version of the model, the Black version loses one degree оГ freedom because the zero-beta expected return is a free parameter. In addition lo the N(N- l)/2 parameters in the residual covariance matrix, the unconstrained model has 2 N parameters, N parameters comprising the vector a and N comprising the vector /Э. The constrained model has, in addition lo the same number of covariance matrix parameters, N parameters comprising the vector P and die parameter for the expected zero-beta portfolio return y. Thus the unconstrained model has {N - 1) more free parameters than the constrained model.

in the context of tl t- llbi к version of (he CAIM, Gibbons (I.IHJ) liist developed ibis test. Slianken (HNib) provides detailed analysis.

(5.3.01) (5.3.62)

>. / he Capital Asset Tricing Model

We ran also adjnsl /1 lo improve ilie finite-sample properties. Defining /-, as ilie adjusted lest statistic we have

jr, = (г- $ -2)[i .Ks*-4IS] ~ x.v-i- <r .:vr>7)

Ii) finite samples, the null distribution ol Jr, will more closely match the chi-squarc distribution. (See Section 5.! lor a comparison in the context of the Sharpe-l.iiitncr version.)

There are iwo drawbacks to the methods we have just discussed. First, the estimation is somewhat tedious since one must iterate over the first-order conditions. Second, the lest is based on large-sample theory and can have very poor finite-sample properties. We can use the results of Kandel (1981) and Slianken (1980) to overcome these drawbacks. These authors show how to calculate exact maximum likelihood estimators and how to implement an approximate lesi with good linile-sample performance.

For the unconstrained model, consider the market model expressed in terms of returns in excess of die expected zero-beta return y:

R, - у 1. = a -f (){!( , -y)+e,. (5.3.08)

Assume у is known. Then the maximum likelihood estimators for the unconstrained model are

a(y)

I v-

S = y.X]R.-/WWOIIR,-£-/3(K W<J]- (5.3.71) 11

fhe unconstrained estimators of/3 and E do not depend on the value of у but, as indicated, the estimator of a does. The value of the unconstrained log-likelihood function evaluated al die maximum likelihood estimators is

NT T - NT

L ~ - i K(-7t) - - logE - - (5.3.72)

which does not depend on y.

= A -yt- Шт - у).

e/=i(ri - А)(д 1/ -a .)

T.L,ULi - А.-Л2

(5.3.Я9) (5.3.70)

5.3. Statistical Framework for Estimation and Testing

Constraining a to be zero, the constrained estimators are

T.Luin.-Y)2 , 1 7

£* = -(r.-y-Pi-PrJ)

(5.3.73)

(r, - yd-(У) -0*K, ,),

.1.74)

and the value of the constrained likelihood function is

NT T NT

£*(y) = - log(27r) - - logE (y) - - (5.3.75)

Note that the constrained function does depend on y. Forming the logarithm of the likelihood ratio we have

ai(y) = C{y)-C

= -[logE*(y)-IogE]. (5.3.76)

The value of у that minimizes the value of the logarithm of the likelihood ratio will be the value which maximizes the constrained log-likelihood function and thus is the maximum likelihood estimator of y.

Using the same development as for the Sharpe-Lintner version, the log-likelihood ratio can be simplified to

СЩу)

-2lo8

\(Hm-yY +aif J

\(fim-y)z +

x [A-yt-/3(A -y)] + i

(5.3.77)

Minimizing C1Z with respect to у is equivalent to maximizing G where

--l.

--Vt7 [A-K-/3(Am-y)]S l[i-YL-f3{nm-y)].

(5.3[78)

Thus the value of у which maximizes G will be the maximum likelihood estimator. There are two solutions of дС/ду = 0, and these are the ea) rools of the quadratic equation

H(y) = Ayl + By + C,

(5.3.79)