wlierc

Л s

II =

С э

1 (t - j9) (A - /ЗА ,) - Ц и - /3) if (4 - /3)

fi + 4V-0>E~u-/3)- 4(A-/9A ,)S (А-/ЗА

- (! + 1) c- -/3)s i(A -

S + - Ф- /ЗА ,) £ (A - /ЗА ,).

1ГЛ is greater than zero, the maximum likelihood estimator y is the largest root, and if A is less than zero, then y* is the smallest root. Л will he greater than zero if Am is greater than the mean return on the sample global minimum-variance portfolio; that is, the market portfolio is on the efficient part of the constrained mean-variance frontier. Wc can substitute y\ into (5.3.62) and (5.3.63) to obtain /3 and E without resorting to an itcjrative procedure.

( We can construct an approximate test of the Шаек version using ret tuns in excess of у as in (5.3.68). Uy is known then the same methodology used to {construct the .Sharpe-I.intner version /. -test in (5.3.23) applies to testing the null hypothesis that the zero-beta excess-return market-model intercept is zero. The test statistic is

Mr) =

iT-N- 1) N

(H, ~yV

aiy)£ a(y) ~ /.;,

(5.3.80)

Because у is unknown, the test in (5.3.80) cannot be directly implemented. But an approximate test can be implemented with J(,iy*)- Because у = у minimizes the log-likelihood ratio, it minimizes Jt\iy). Hence Ji,iy) S Ji,(Y<>)> where y0 is the unknown true value of у. Therefore a test using JcAy) will accept too often. If the null hypothesis is rejected using у it will be rejected for any value of y . This testing approach can provide л useful check because the usual asymptotic likelihood ratio test in (5.3.77) has been found to reject loo often.

Finally, wc consider inferences for the expected zero-beta portfolio return. Given the maximum likelihood estimator of/, we require its asymptotic variance to make inferences. Using the Fisher information matrix, the asvmptotic variance of the maximum likelihood of у is

Varin i I ( + (/, fT/)J)lU-/3)S-

(i-/3)]-

(5.3.81)

This estimator can be evaluated al the maximum likelihood estimates, and then inferences concerning the value oiy arc possible given the asymptotic normality of y*.

5.4 Size of Tests

In some econometric models there are no analytical results on the finite-sample properties ofcstiinators. In such cases, it is common lo rely on large-sample statistics lo draw inferences. This reliance opens up the possibility that the size of the lesl will be incorrect if the sample size is not large enough for the asymptotic results to provide a good approximation. Because there is no standard sample size for which large-sample theory can be applied, il is good practice lo investigate the appropriateness of the theory.

The multivariate /-lest we have developed provides an ideal framework for illustrating the problems that can arise if one relies on asymptotic distribution theory for inference. Using die known (mile-sample distribution of the / -lest statistic J\, we can calculate die finite-sample size for the various asymptotic tests. Such calculations arc possible because die asymptotic test statistics are monotonic transformations of j\.

We draw on the relations of f to the large-sample test statistics. Comparing equations (5.3.22) and (5.3.23) for /,> we have

(T- /V - 1)

Recall in (5.3.-10) for }г we have

and for J;i from (5.4.2) and (5.3.41),

Under the null hypothesis, / , J%, and J.\ are all asymptotically distributed chi-squarc with /V degrees of freedom. The exact null distribution of J\ is central / with N degrees of freedom in die numerator and T - N - 1 degrees of freedom in the denominator.

We calculate the exact size of a test based on a given large-sample statistic and its asymptotic 5% critical value. For example, consider a test using f\ with 10 portfolios and 60 months of data. In this case, under the null hypothesis J, is asymptotically distributed as a chi-squarc random varialc with 10 degrees of freedom. Given this distribution, the critical value for a test with an asymptotic size of 5% is 18.31. From (5.4.1) this value of 18.31

У.-Н*]-.). (5.4.2)

211-1

5. V/ic Capital Asset Pricing Mattel

for / corresponds lo a critical value ol 1.495 lor ]\. (liven dial die exact null distribution ol /1 is / wiib 10 degrees of freedom in die numerator and I!) degrees ol freedom in die denominator, a lest using this critical value for /1 lias a size of 17.0%. Thus, the asymptotic 5% lest has a size of 17.0% in a sample of (id months; il rejects the null hypothesis more than three times loo often.

Table 5.1 presents this calculation for / , J- and /1 using 10, 20, and 10 for values of ,V and using 00, 120, 180, 240, and 360 for values of 7. It is apparent that the futile-sample size of the tests is larger than the asymptotic size of 5%. Thus die large-sample tests will reject the null hypothesis too often. This problem is severe for the asymptotic tests based on /0 and Д>. When N = 10 the problem is mostly important for the low values of /. For example, the linile-sample size of a test with an asymptotic size of 5% is 17.0% and 0.0% for / and / ., respectively. As N increases the severiiv of the problem increases. When N = 40 and 7 = 00 the linile-sample size of an asymptotic 5% test is 98.5% for / and 80.5% for /... In these cases, the null hypothesis will be rejected most of the lime even when it is true. With N - 40, the size of a 5% asymptotic test is still overstated considerably even when T = 300.

The asympiolir tesi with a linile-sample adjustment based on /1 performs much belter in finite samples than does its unadjusted counterpart. Only in the case of N = 40 and / = 00 is the exact size significantly overstated. This shows thai linile-sample adjustments of asymptotic test statistics can play an important role.

5.5 Power of Tests

When drawing inferences using a given lest statistic it is important lo consider its power. The power is the probability that the mill hypothesis will be rejected given thai an alternative hypothesis is Hue. Low power against an interesting alternative suggests thai the lest is not useful to discriminate between the alternative and the null hypothesis. On the other hand, if the power is high, then the test can be very informative but il may also reject the null hypothesis against alternatives that are close to the null in economic terms. In this case a rejection may be due to small, economically unimportant deviations from the null.

To document die power of a test, it is necessary to specify the alternative data-generating process and the size of the test. The power for a given size of test is the probability thai the test statistic is greater than the critical value under the null hypothesis, given that the alternative hypothesis is true.

To illustrate the power of tests of the CAPM, we will focus on the tesi of the Shaipe-l.inlner version using /1 from (5.3.23). Ihe power of this lest

5.5. Power of Tests j

The exaci finite-sample size is presented for tests with a size of 5% asymptotically. The finite-sample size uses ihe distribution of J\ and the relation between J\ and the large-sample lest statistics, Ju, }i. and f%. N is the number of dependent portfolios, and T is ihe number of

lime-series observations.

should be representative, and it is convenient to document since the exact finite-sample distribution of J\ is known under both the null and alternative hypotheses. Conditional on the excess return of the market portfolio, for the distribution of J\ as defined in (5.3.23), we have

Ji ~ Fn.t-n-i№, (5.5.1)

where S is the noncentrality parameter of the F distribution and

alTa. (5.5.2)

To specify the distribution of ]\ under both the null and the alternative hypotheses, we need to specify &, N, and 7 .

Under the null hypoihcsis a is zero, so in this case S is zero and we have the previous result that the distribution is central F with N and V - N - 1 degrees of freedom in the numerator and denominator, respectively. Under thj. alternative hypothesis, to specify S wc need to condition on a value of (Aloi and specify the value of a£~a. For the value of ц?т1ащ, given a monthly observation interval, wc choose 0.013 which corresponds to an ex poll annualized mean excess return of 8% and a sample annualized standard deviation of 20%.

For the quadratic term a£~a, rather than separately specifying a and E, we can use the following result of Gibbons, Ross, and Shauk.cn (IDS.)).2 Recalling that q is the tangency portfolio and that m is the market portfolio, wc have

(До.З)

Using this relation, wc need only specify the difference in the squared Sharpe ratio for the tangency portfolio and the market portfolio. The tangency portfolio is for the universe composed of the N included portfolios and the market portfolio. Wc consider four sets of values for the tangency portfolio parameters. For all cases the annualized standard deviation of the tangency portfolio is set to 16%. The annualized expected excess return then lakes on four values, 8.5%, 10.2%, 11.6%, and 13.0%. Using an annualized expected excess return of 8% for the market and an annualized standard deviation of 20% for the markets excess return, these four values correspond to values of 0.01, 0.02, 0.03, and 0.04 for 5/T.

We consider five values for JV: 1, 5, 10, 20, and 40. For 7 wc consider four values-60, 120, 240, and 360-which are chosen to correspond to 5, 10, 20, and 30 years of monthly data. The power is tabulated for a lest with a size of 5%. The results are presented in Table 5.2.

Substantial variation in the power of the test for different experimental designs and alternatives is apparent in Table 5.2. For a fixed value of N, considerable increases in power are possible with larger values of /. For .example, under alternative 2 for N equal to 10, the power increases from 0.082 to 0.380 as T increases from 60 lo 300.

The power gain is substantial when N is reduced for a fixed alternative. For example, under alternative 3, for 7equal lo 120, the power increases from 0.093 to 0.475 as N decreases from 40 to 1. However, such gains would not be feasible in practice. As N is reduced, the Sharpe ratio of the tangency portfolio (and the nonccntrality parameter of the /distribution) will decline unless the portfolios arc combined in proportion to their weightings in that portfolio. The choice of N which maximizes the power will depend on the

We discuss lliis result further in Chapter (i.

Table 5.2. Iinorr of F-test oj Sharpr-I.inliiri CAIM using statistic p.

file alternative hypoihesis is characterized by the value of the rxpei led est ess iciitrn and the value til die standard deviation of the taneiuy ioiilolio. flu- laiigcncy pnitfofio is with respect to the Л included portfolios and the market poi tlolio. tiy is the exper ted excess return of the tangency poi tfolio, and 0, is the annualized standard deviation ol the excess return of the tangency portlolio. Ihe market portfolio is assumed to have an expected excess return of 8.0% and .1 standard deviation of 20%. Under die null hypothesis the market pot tlolio is the tangent у portfolio. Л is ihe number of portfolios included in the lest and 7 is the number of months ol data included.

rate at which the Sharpe ratio of the tangency portfolio declines as assets are grouped together.

While we do not have general results about die optimal design of a multivariate test, we can draw some insights from this power analysis. Increasing the length of the time scries can lead to a significant payoll in terms of power. Further, the power is very sensitive lo the value of N. The analysis suggests that the value of N should be kept small, perhaps no larger than about ten.