The Review of Financtal Studies /v 2 n 2 1989 .pe:The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of portfolios represented by the matrix G. 2.Likelihood-Ratio Tests Using a General Alternative Hypothesis Numerous studies have developed and applied tests of asset pricing models against a general (unspecified)alternative hypothesis using the multi- variate regression r=a十BR+ (4) A linear mean-beta pricing relation states that,for some scalar Y, E(r)=Yw-K+EE(R)一r (5) where E()is the expectation operator and denotes an (n-K )-vector of ones.Furthermore,if a riskless asset exists (so that and R,are stated as excess returns),then y=0.The pricing relation in (5)implies the following restriction on the parameters in the multivariate regression in (4): a-Y(n-k-B) (6) which simplifies to the restriction a=0 when a riskless asset exists. The pricing restriction in (5)can be viewed as a set of restrictions on E and V.the true (population)mean vector and variance-covariance matrix of the n risky assets.These restrictions can be written as E=Ytn-k Viz(V22)[E-Yh] (7) Let the parameter vector 0 contain the elements of E and V,and let denote the entire parameter space (wherein E can be any real-valued vector and V can be any symmetric positive-definite matrix).The restrictions in (7)are represented asE(A),where denotes the region of defined by the restrictions.The notation "(A)"is chosen to emphasize the fact that this region depends on the choice of the K reference portfolios.The notation "w(p)"is used with a single reference portfolio p.Let Z denote the sample of T observations ofR)',and let(;Z)denote the likelihood function (given by the multivariate normal distribution). The likelihood ratio for testing a K-beta pricing model with the reference portfolios represented by A against a general alternative is given by max f(0;2) A(A)= max f(0;2) (8) a(万 In the absence of a riskless asset and when K1.a test of the restrictionsands equivalent to a test of "mean-variance spanning."that is,that the mean-variance frontier of the K assets coincides with that of the larger set of n assets.See Huberman and Kandel (1987). 130
The portfolio having the highest absolute value of the sample Sharpe measure of any portfolio constructed from the set of portfolios represented by the matrix G. 2. Likelihood-Ratio Tests Using a General Alternative Hypothesis Numerous studies have developed and applied tests of asset pricing models against a general (unspecified) alternative hypothesis using the multivariate regression (4) A linear mean-beta pricing relation states that, for some scalar (5) where E ( ) is the expectation operator and denotes an ( n - K ) -vector of ones. Furthermore, if a riskless asset exists (so that stated as excess returns), then 0. The pricing relation in (5) implies the following restriction on the parameters in the multivariate regression in (4): which simplifies to the restriction a = 0 when a riskless asset exists.4 The pricing restriction in (5) can be viewed as a set of restrictions on E and V, the true (population) mean vector and variance-covariance matrix of the n risky assets. These restrictions can be written as (7) Let the parameter vector contain the elements of E and V, and let denote the entire parameter space (wherein E can be any real-valued vector and V can be any symmetric positive-definite matrix). The restrictions in (7) are represented as denotes the region of defined by the restrictions. The notation is chosen to emphasize the fact that this region depends on the choice of the K reference portfolios. The notation is used with a single reference portfolio p. Let Z denote the sample of T observations of and let denote the likelihood function (given by the multivariate normal distribution). The likelihood ratio for testing a K -beta pricing model with the reference portfolios represented by A against a general alternative is given by (8) 4 In the absence of a riskless asset and when K> 1, a test of the restrictions a - 0 and is equivalent to a test of “mean-variance spanning,” that is, that the mean-variance frontier of the K assets coincides with that of the larger set of n assets. See Huberman and Kandel (1987). 130
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). 131
When 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 summarize 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 meanvariance 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 samples.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
Tbe Review of Financtal Studtes/v 2 n 2 1989 if the bracketed quantity in (10)is positive,s equals zero otherwise (in which case there is no rejection),F(n-K,T-n)is the critical value for significance level a of the F-distribution with n-K and T-n degrees of freedom,and y=(n-K)(T-n). Proof.See the Appendix. For a given sample of assets and returns,there may exist no specification of the reference portfolio(s)that results in a rejection of the pricing model, This situation,wherein the maximum squared sample Sharpe measure s(p*)2 is less than vF(n-K,T-n)and thus the bracketed quantity in (10)is negative,is more likely to occur as the number of assets (n)grows large relative to the number of time-series observations (T). As the above proposition states,in a test of a single-beta model (K=1) the efficiency of a portfolio can be tested by plotting its position in sample mean-standard-deviation space,where all returns are stated in excess of the riskless rate.The tested portfolio's position is compared to the location of the two critical lines with intercepts of zero and slopes with absolute values equal to s.If the tested portfolio lies between the critical lines, then its efficiency is rejected. Proposition 1 also indicates that in the test of a multiple-beta model (K >1)the portfolio tested is p,the sample tangent portfolio for the set of K assets.The position of portfolio p is compared to the two critical lines in sample mean-standard-deviation space in precisely the same manner as was the single reference portfolio in the case of K=1.(The differences in S between the two cases simply reflect different degrees of freedom.) Note that Is(p)<s,and thus the multiple-beta model is rejected,if and only if the minimum-standard-deviation boundary of the K reference portfolios does not intersect either of the two critical lines. We illustrate here a test of a two-beta pricing model (K =2)with the weekly returns data used by Kandel and Stambaugh (1987)in tests of the Sharpe-Lintner version of the CAPM(K =1).The set of 12 risky assets (n =12)consists of two market proxies-t h e equally weighted and the value- weighted portfolios of stocks on the New York and American Exchanges- and 10 value-weighted portfolios of common stocks formed by ranking all firms on both exchanges by market value at the end of the previous year. The riskless rate is the return on a U.S.Treasury bill with one week to maturity.'A two-beta model is tested using the two market proxies as the two reference portfolios.We choose these proxies simply to illustrate the testing framework rather than to conduct comprehensive new tests of asset pricing models."For the same reason,we use,for Proposition 1 as well as We thank Richard Rogalski for providing the Treasury bill data. 8 The use of the two proxies may be partially motivated by the well-known"size anomaly"of the single- beta CAPM.The value-weighted Index primarily reflects changes In the prices of large firms,whereas the equally weighted index is affected more by the returns on medium-size and small firms. 132
if the bracketed quantity in (10) is positive, equals zero otherwise (in which case there is no rejection), is the critical value for significance level a of the F-distribution with n - K and T - n degrees of freedom, and Proof. See the Appendix. n For a given sample of assets and returns, there may exist no specification of the reference portfolio(s) that results in a rejection of the pricing model, This situation, wherein the maximum squared sample Sharpe measure and thus the bracketed quantity in (10) is negative, is more likely to occur as the number of assets (n) grows large relative to the number of time-series observations (T). As the above proposition states, in a test of a single-beta model ( K = 1) the efficiency of a portfolio can be tested by plotting its position in sample mean-standard-deviation space, where all returns are stated in excess of the riskless rate. The tested portfolio’s position is compared to the location of the two critical lines with intercepts of zero and slopes with absolute values equal to If the tested portfolio lies between the critical lines, then its efficiency is rejected. Proposition 1 also indicates that in the test of a multiple-beta model ( K > 1) the portfolio tested is the sample tangent portfolio for the set of K assets. The position of portfolio is compared to the two critical lines in sample mean-standard-deviation space in precisely the same manner as was the single reference portfolio in the case of K = 1. (The differences in between the two cases simply reflect different degrees of freedom.) Note that and thus the multiple-beta model is rejected, if and only if the minimum-standard-deviation boundary of the K reference portfolios does not intersect either of the two critical lines. We illustrate here a test of a two-beta pricing model ( K = 2) with the weekly returns data used by Kandel and Stambaugh (1987) in tests of the Sharpe-Lintner version of the CAPM ( K = 1). The set of 12 risky assets ( n = 12) consists of two market proxies-the equally weighted and the valueweighted portfolios of stocks on the New York and American Exchangesand 10 value-weighted portfolios of common stocks formed by ranking all firms on both exchanges by market value at the end of the previous year. The riskless rate is the return on a U.S. Treasury bill with one week to maturity.7 A two-beta model is tested using the two market proxies as the two reference portfolios. We choose these proxies simply to illustrate the testing framework rather than to conduct comprehensive new tests of asset pricing models.8 For the same reason, we use, for Proposition 1 as well as 7We thank Richard Rogalski for providing the Treasury bill data. 8 The use of the two proxies may be partially motivated by the well-known “size anomaly" of the singlebeta CAPM. The value-weighted Index primarily reflects changes In the prices of large firms, whereas the equally weighted index is affected more by the returns on medium-size and small firms. 132
Tests of Asset Pricing Models MInimum-standard-devlation 200 boundary Critical ne t50 100 Boundary of the roference portfollos 60 Equally welghted market Q 0 Value-welghted market -50 -100 -150 -200 Crltical ne =250 10 30 60 70 90 110 Standard Devlation (per year) Figurel A likelihood-ratio test of a two-beta pricing model in the presence of a riskless asset The test Is based on weekly retums in excess of a riskless rate.The two reference portfolios 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 lines reflect a 5 percent significance level.The pricing model is not rejected if the boundary of the reference portfolios intersects a critical line. the propositions to follow,only one of the three subperiods examined by Kandel and Stambaugh (1987).The subperiod selected extends from Oct 8,1975,through Dec.23,1981,and includes 324 weekly observations. Figure 1 displays the test at a 5 percent significance level.The hyperbola representing combinations of the two reference portfolios does not Inter- sect either critical line.Thus,the two-beta model is rejected. 2.2 Tests of single-beta models without a riskless asset A likelihood-ratio test of (6)with a single beta,where y is an unknown zero-beta rate,was first proposed by Gibbons(1982).The hypothesis tested is equivalent to the mean-variance efficiency of the benchmark portfolio with respect to the n risky assets.The exact finite-sample distribution of the likelihood-ratio-test statistic has not been obtained for this case,although a lower bound for the distribution is obtained by Shanken (1986).Thus, selection of an appropriate critical value is more difficult than in the case where a riskless asset exists.Once a critical value is specified,however, we show that this test can be conducted in a mean-variance framework. For discussions of finite-sample properties of the likelihood-ratio statistic and other large-sample equiv- alents,see also Stambaugh(1982),Shanken (1985),and Amsler and Schmidt (1985).Shanken (1985) derives an upper bound on the finite-sample distribution of one altemative to the likelihood-ratio statistic. 133
Figure1 A likelihood-ratio test of a two-beta pricing model in the presence of a riskless asset The test Is based on weekly returns in excess of a riskless rate. The two reference portfolios are the valueweighted 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 lines reflect a 5 percent significance level. The pricing model is not rejected if the boundary of the reference portfolios intersects a critical line. the propositions to follow, only one of the three subperiods examined by Kandel and Stambaugh (1987). The subperiod selected extends from Oct. 8, 1975, through Dec. 23, 1981, and includes 324 weekly observations. Figure 1 displays the test at a 5 percent significance level. The hyperbola representing combinations of the two reference portfolios does not Intersect either critical line. Thus, the two-beta model is rejected. 2.2 Tests of single-beta models without a riskless asset A likelihood-ratio test of (6) with a single beta, where g is an unknown zero-beta rate, was first proposed by Gibbons (1982). The hypothesis tested is equivalent to the mean-variance efficiency of the benchmark portfolio with respect to the n risky assets. The exact finite-sample distribution of the likelihood-ratio-test statistic has not been obtained for this case, although a lower bound for the distribution is obtained by Shanken (1986).9 Thus, selection of an appropriate critical value is more difficult than in the case where a riskless asset exists. Once a critical value is specified, however, we show that this test can be conducted in a mean-variance framework. 9 For discussions of finite-sample properties of the likelihood-ratio statistic and other large-sample equivalents, see also Stambaugh (1982), Shanken (1985), and Amsler and Schmidt (1985). Shanken (1985) derives an upper bound on the finite-sample distribution of one alternative to the likelihood-ratio statistic. 133
Tbe Revtew of Financtal Studtes/v 2 n 2 1989 Definition.W(p)=A(p)2/T-1).This is a monotonic transformation of the lkelibood ratio(p)for testing the efficiency of portfolto p witb respect to the set of n assets.T.In[1 W(p)]is asymptotically distributed as x12 witb n-2 degrees of freedom if portfolto p is effictent. Definition.we is tbe critical value for W(p)at tbe chosen significance level.Tbat is,the efficiency of portfolio p is rejected ff w(p)>W. Proposition 2.Tbe likelibood-ratio test rejects tbe effictency ofportfolto p in the absence of a riskless asset,that is w(p)>w,ff and only if 2(p)>i,()+2()·(a(D) (11) ubere the functionsδ，()andi,()are given by 6,(=x+) and 6,(=-Dx+1) (12) Lx-D Lx-D and where L and D are defined in(2). Proof.See the Appendix. Proposition 2 states that the likelihood-ratio test of efficiency can be performed by first constructing a critical parabola in sample mean-variance (2)space given by the equation2=+2).Note that this critical parabola is a linear transformation of the sample minimum-variance bound- ary of the n assets and that neither 6()nor 6()require information about the tested portfolio (other than that the tested portfolio can be constructed from the set of n assets).If the tested portfolio lies inside the convex region defined by this critical parabola,then the efficiency of that portfolio is rejected.The critical parabola becomes a critical hyperbola in sample mean-standard-deviation space,and we use the latter representa- tion in the illustration below. Using the same 12 assets and the same sample period as were used in the previous example,we test the zero-beta CAPM [Black(1972)]with each of the two indexes as the market proxy.Because this formulation of the model does not include a riskless asset,total (not excess)returns are used. The critical value,we,is based on the result by Shanken (1986)that,under the null hypothesis,the lower bound on the distribution of W(p).(T-K -1)is a T-variate with degrees of freedom n-K and T-K-1. Equivalently,the lower bound on the distribution of W(p)(T-n)/n K)is central F with degrees of freedom n-K and T-n.Therefore,for a significance level of 5 percent and for the 324-week sample size the critical value is =1,312)0=0.0641 312 Figure 2 displays the results of this test.Each of the two market proxies lies inside the rejection region defined by the critical hyperbola,and thus 134
Proposition 2 states that the likelihood-ratio test of efficiency can be performed by first constructing a critical parabola in sample mean-variance space given by the equation Note that this critical parabola is a linear transformation of the sample minimum-variance boundary of the n assets and that neither require information about the tested portfolio (other than that the tested portfolio can be constructed from the set of n assets). If the tested portfolio lies inside the convex region defined by this critical parabola, then the efficiency of that portfolio is rejected. The critical parabola becomes a critical hyperbola in sample mean-standard-deviation space, and we use the latter representation in the illustration below. Using the same 12 assets and the same sample period as were used in the previous example, we test the zero-beta CAPM [Black (1972)] with each of the two indexes as the market proxy. Because this formulation of the model does not include a riskless asset, total (not excess) returns are used. The critical value, is based on the result by Shanken (1986) that, under the null hypothesis, the lower bound on the distribution of W(p). (T - K - 1) is a T 2 -variate with degrees of freedom n - K and T - K - 1. Equivalently, the lower bound on the distribution of W(p) · (T - n)/( n - K) is central F with degrees of freedom n - K and T - n. Therefore, for a significance level of 5 percent and for the 324-week sample size the critical value is Figure 2 displays the results of this test. Each of the two market proxies lies inside the rejection region defined by the critical hyperbola, and thus 134