Tests of Asset Pricing Models MInimum-standard-devlation 36 boundary 0 Critical hyperbola 2 ● Equally welghted market 30传 14 12 Value-welghted market 6 12 124 12B 13.2 13.8 14 14.4 14.8 Standard Devlation (per year) Figure 2 Likelihood-ratio tests of the zero-beta CAPM(without a riskless asset) The tests are based on weekly returns,and the market proxies are the value-weighted NYSE-AMEX and the equally weighted NYSE-AMEX.The sample minimum-standard-deviation boundary is constructed using 12 assets:10 size-based portfolios plus the two market proxies.The critical hyperbola reflects a 5 percent significance level.The pricing model is rejected if the market proxy lies to the right of the critical hyperbola. the efficiency of each of the two indexes is rejected at a significance level of no more than 5 percent. 2.3 Tests of multiple-beta models without a riskless asset Gibbons (1982)and Shanken (1985,1986)discuss likelihood-ratio tests of (6)in the absence of a riskless asset for cases where K 1.This restriction [imposed by the pricing equation in (5)]is equivalent to the statement that some combination of the K reference portfolios is efficient with respect to the set of n risky assets [Grinblatt and Titman (1987)and Huberman,Kandel,and Stambaugh (1987)].Proposition 2 states that the critical region for testing the efficiency of a given portfolio,in the absence of a riskless asset,is given by a linear transformation of the sample mini- mum-variance boundary.Proposition 3 establishes a similar result for the test of the efficiency of some combination of the K reference portfolios represented by the matrix A. Definition.W=(A)2/T-1).This is a monotonic transformation of the likelihood ratio for testing the hypothesis that some portfolio of the K assets represented by the matrix A is mean-variance efficient with respect to the set of n assets. 135
Figure 2 Likelihood-ratio tests of the zero-beta CAPM (without a riskless asset) The tests are based on weekly returns, and the market proxies are the value-weighted NYSE-AMEX and the equally weighted NYSE-AMEX. The sample minimum-standard-deviation boundary is constructed using 12 assets: 10 size-based portfolios plus the two market proxies. The critical hyperbola reflects a 5 percent significance level. The pricing model is rejected if the market proxy lies to the right of the critical hyperbola. the efficiency of each of the two indexes is rejected at a significance level of no more than 5 percent. 2.3 Tests of multiple-beta models without a riskless asset Gibbons (1982) and Shanken (1985, 1986) discuss likelihood-ratio tests of (6) in the absence of a riskless asset for cases where K > 1. This restriction [imposed by the pricing equation in (5)] is equivalent to the statement that some combination of the K reference portfolios is efficient with respect to the set of n risky assets [Grinblatt and Titman (1987) and Huberman, Kandel, and Stambaugh (1987)]. Proposition 2 states that the critical region for testing the efficiency of a given portfolio, in the absence of a riskless asset, is given by a linear transformation of the sample minimum-variance boundary. Proposition 3 establishes a similar result for the test of the efficiency of some combination of the K reference portfolios represented by the matrix A. Definition. This is a monotonic transformation of the likelihood ratio for testing the hypothesis that some portfolio of the K assets represented by the matrix A is mean-variance efficient with respect to the set of n assets. 135
Tbe Review of Financtal Studies/v2 n 2 1989 Definition.Wi is the critical value for Wa at the chosen significance level. That is,the null hypothesis is rejected if w>Wi Since an exact small-sample distribution for the likelihood-ratio statistic has not been obtained,choosing the critical value Wi is again more difficult than in the cases where a riskless asset exists.One could use,for example, the lower bound on the distribution obtained by Shanken (1986).Once the critical value is chosen,however,the test can be conducted in sample mean-variance space as shown by the following proposition. Proposition 3.The likelihood-ratio test rejects the hypothesis that some porifolio of the K assets represented by the matrix A is efficient with respect to the n risky assets,that is,W>Wi,tif and only if (m）>,(w)+62(w)(m) for all m (13) ubereδ，'andi,are defined in(12) Proof.See the Appendix. Note the similarity between Propositions 2 and 3.In both cases,the rejection region is defined by a critical parabola that is simply a linear transformation of the sample minimum-variance boundary.(In fact,the definitions of the parabolas are identical except for the possibly different critical values,we and W given portfolio's efficiency is rejected if it lies inside the convex rejection region formed by the critical parabola.The efficiency of any combination of the K assets is rejected if the entire feasible set of portfolios of those K assets lies within that rejection region. We illustrate this test with the same data used to construct the previous two examples.As in the first example,we,test a two-beta model where the value-weighted and equally weighted market proxies are specified as the two reference portfolios.In this case,however,there is no riskless asset. The critical value is computed by the same method used in the previous example,using Shanken's (1986)lower bound,except that K2=2,so 既=民na0,312品-0659% Figure 3 illustrates the results of this test.All combinations of the two reference assets lie within the rejection region (defined by the critical hyperbola,and thus the two-beta model is rejected. 3.Likelihood-Ratio Tests Using Specific Alternative Hypotheses This section examines likelihood-ratio tests of the pricing restriction in (5),where the restriction is tested against a specific alternative.The specific alternative is one in which a multiple-beta model of higher dimension describes expected returns.That is,the relation in (5)with K,reference portfolios,represented by the matrix A,is tested as the mill hypothesis 136
3. Likelihood-Ratio Tests Using Specific Alternative Hypotheses Definition. is the critical value for WA at the chosen significance level. That is, the null hypothesis is rejected Since an exact small-sample distribution for the likelihood-ratio statistic has not been obtained, choosing the critical value is again more difficult than in the cases where a riskless asset exists. One could use, for example, the lower bound on the distribution obtained by Shanken (1986). Once the critical value is chosen, however, the test can be conducted in sample mean-variance space as shown by the following proposition. Proposition 3. The likelihood-ratio test rejects the hypothesis that some portfolio of the K assets represented by the matrix A is efficient with respect to the n risky assets, that is, if and only if Note the similarity between Propositions 2 and 3. In both cases, the rejection region is defined by a critical parabola that is simply a linear transformation of the sample minimum-variance boundary. (In fact, the definitions of the parabolas are identical except for the possibly different critical values, A given portfolio’s efficiency is rejected if it lies inside the convex rejection region formed by the critical parabola. The efficiency of any combination of the K assets is rejected if the entire feasible set of portfolios of those K assets lies within that rejection region. We illustrate this test with the same data used to construct the previous two examples. As in the first example, we, test a two-beta model where the value-weighted and equally weighted market proxies are specified as the two reference portfolios. In this case, however, there is no riskless asset. The critical value is computed by the same method used in the previous example, using Shanken’s (1986) lower bound, except that K2 = 2, so Figure 3 illustrates the results of this test. All combinations of the two reference assets lie within the rejection region (defined by the critical hyperbola, and thus the two-beta model is rejected. This section examines likelihood-ratio tests of the pricing restriction in (5), where the restriction is tested against a specific alternative. The specific alternative is one in which a multiple-beta model of higher dimension describes expected returns. That is, the relation in (5) with K1 reference portfolios, represented by the matrix A1 , is tested as the mill hypothesis 136