Tests of Asset Pricing Models When a single-beta pricing model is tested,the matrix 4 is replaced in (8)by p,representing the tested reference portfolio. This section presents a framework in sample mean-variance space for conducting likelihood-ratio tests of the pricing restrictions.We first sum- marize existing results for models with a riskless asset (Section 2.1);we then present new geometrical interpretations for testing models without a riskless asset (Sections 2.2 and 2.3). 2.1 Tests of models with a riskless asset When a riskless asset exists,efficiency is defined with respect to the set of n risky assets plus the riskless asset.If the pricing model contains a single beta,that is,the matrix B in (4)has one column,then a test of the pricing model is equivalent to a test of the mean-variance efficiency of the specified reference portfolio with return R.If the pricing model contains several betas,that is,B has more than one column,then in general one cannot identify a specific benchmark portfolio that is implied by the pricing model to be mean-variance efficient.The linear pricing relation in (5)is equivalent to the statement that some portfolio of the K reference portfolios is mean- variance efficient [Jobson and Korkie (1985),Grinblatt and Titman (1987) and Huberman,Kandel,and Stambaugh (1987)]. The finite-sample distribution of the likelihood-ratio-test statistic for models with a riskless asset is presented by Gibbons,Ross,and Shanken (1989).Following Anderson (1984),they show that a transformation of the likelihood-ratio statistic for testing a =0 in (4)(when r,and R,are stated in excess of the riskless rate)obeys an F-distribution in finite sam- ples.?The following proposition summarizes the sample mean-variance representation of this test provided by Jobson and Korkie (1982)and Gibbons,Ross,and Shanken (1989). Proposition 1.The likelihood-ratio test with significance level a rejects the hypothesis that some portfolio of the K reference portfolios represented by the matrix A is efficient with respect to the set of n assets plus the riskless asset if and only if IS()川<S (9) wbere S S()2-E.(n-KT-n) 1+vF(n-K T-n) (10) s Jobson and Korkie(1985)and MacKinlay(1987)also present the same result for the single-beta CAPM. A similar result is also presented by Jobson and Kodde (1982),except that they characterize what is in fact the finite-sample distribution as being valid only asymptotically.and they misstate the number of degrees of freedom. 6These results are also summarized in a recent paper by Jobson and Korkie(1988). 131When a single-beta pricing model is tested, the matrix A is replaced in (8) by p, representing the tested reference portfolio. This section presents a framework in sample mean-variance space for conducting likelihood-ratio tests of the pricing restrictions. We first sum￾marize existing results for models with a riskless asset (Section 2.1); we then present new geometrical interpretations for testing models without a riskless asset (Sections 2.2 and 2.3). 2.1 Tests of models with a riskless asset When a riskless asset exists, efficiency is defined with respect to the set of n risky assets plus the riskless asset. If the pricing model contains a single beta, that is, the matrix B in (4) has one column, then a test of the pricing model is equivalent to a test of the mean-variance efficiency of the specified reference portfolio with return Rt . If the pricing model contains several betas, that is, B has more than one column, then in general one cannot identify a specific benchmark portfolio that is implied by the pricing model to be mean-variance efficient. The linear pricing relation in (5) is equivalent to the statement that some portfolio of the K reference portfolios is mean￾variance efficient [Jobson and Korkie (1985), Grinblatt and Titman (1987) and Huberman, Kandel, and Stambaugh (1987)]. The finite-sample distribution of the likelihood-ratio-test statistic for models with a riskless asset is presented by Gibbons, Ross, and Shanken (1989). Following Anderson (1984), they show that a transformation of the likelihood-ratio statistic for testing a = 0 in (4) (when rt and Rt are stated in excess of the riskless rate) obeys an F-distribution in finite sam￾ples.5 The following proposition summarizes the sample mean-variance representation of this test provided by Jobson and Korkie (1982) and Gibbons, Ross, and Shanken (1989).6 Proposition 1. The likelihood-ratio test with significance level a rejects the hypothesis that some portfolio of the K reference portfolios represented by the matrix A is efficient with respect to the set of n assets plus the riskless asset if and only if 5 Jobson and Korkie (1985) and MacKinlay (1987) also present the same result for the single-beta CAPM. A similar result is also presented by Jobson and Kodde (1982), except that they characterize what is in fact the finite-sample distribution as being valid only asymptotically. and they misstate the number of degrees of freedom. 6 These results are also summarized in a recent paper by Jobson and Korkie (1988). 131