《金融投资学》教学资源（英文文献）A Mean-Variance Framework for tests of asset pricing models.pdf

Tbe Revlew of Financial Studies /v 2n 2 1989 known that mean-beta linearity is equivalent to mean-variance efficiency of the market portfolio [Fama (1976)Roll (1977),and Ross (1977)].Sim- ilarly,the consumption-beta model implies the mean-variance efficiency of the portfolio having maximal correlation with consumption [Breeden (1979)].In multiple-beta pricing relations,the benchmark portfolio gen- erally cannot be identified specifically but instead is characterized as some combination of a set of reference portfolios.For example,an exact K-factor arbitrage pricing relation is equivalent to the mean-variance efficiency of some portfolio that combines K factor-mimicking portfolios [Grinblatt and Titman (1987)and Huberman,Kandel,and Stambaugh (1987)]. Although the equivalence between linear pricing relations and mean- variance efficiency is well understood at a theoretical level,links between tests of the pricing models and a mean-variance framework are limited to a few special cases.'This article presents a complete framework for the characterization and investigation of likelihood-ratio tests of the pricing restrictions in a mean-variance setting.Our treatment includes tests with either a single beta or multiple betas,with or without a riskless asset,using either a general or a specific alternative hypothesis.We also extend the mean-variance framework to test the pricing relation with a factor that is not a portfolio return.All the tests considered should be viewed as tests of mean-variance efficiency defined in terms of unconditional distributions rather than as tests of conditional mean-variance efficiency. A major virtue of the mean-variance framework presented in this article is that it allows the researcher to represent graphically in two familiar dimensions the outcome of a test of a multidimensional pricing restriction. A pricing model is tested by examining the position of one or more ref- erence portfolios in sample mean-standard-deviation space.In this approach, the likelihood-ratio-test statistic can be viewed not only as the outcome of a numerical procedure but also as a quantity with simple economic and statistical interpretations. One case for which the mean-variance framework has been developed is one in which a pricing model,that includes a riskless asset is tested against a general alternative hypothesis.The likelihood-ratio test in this case can be characterized as comparing the position in sample mean- standard-deviation space of a benchmark portfolio,or a set of reference portfolios,to the position of the sample tangent portfolio [e.g.,Jobson and Korkie (1982)and Gibbons,Ross,and Shanken (1989)].The rejection region in sample mean-standard-deviation space is defined by a pair of lines. We show that,in the absence of a riskless asset,the rejection region for the likelihood-ratio test using a general alternative hypothesis is defined by a hyperbola in sample mean-standard-deviation space.As in the case in which a riskless asset exists,the rejection region depends only on the See Jobson and Kodde(1982,1988),Gibbons,Ross,and Shanken (1989),Kandel (1984,1986),and Roll (1985). 126

known that mean-beta linearity is equivalent to mean-variance efficiency of the market portfolio [Fama (1976) Roll (1977), and Ross (1977)]. Similarly, the consumption-beta model implies the mean-variance efficiency of the portfolio having maximal correlation with consumption [Breeden (1979)]. In multiple-beta pricing relations, the benchmark portfolio generally cannot be identified specifically but instead is characterized as some combination of a set of reference portfolios. For example, an exact K-factor arbitrage pricing relation is equivalent to the mean-variance efficiency of some portfolio that combines K factor-mimicking portfolios [Grinblatt and Titman (1987) and Huberman, Kandel, and Stambaugh (1987)]. Although the equivalence between linear pricing relations and meanvariance efficiency is well understood at a theoretical level, links between tests of the pricing models and a mean-variance framework are limited to a few special cases.’ This article presents a complete framework for the characterization and investigation of likelihood-ratio tests of the pricing restrictions in a mean-variance setting. Our treatment includes tests with either a single beta or multiple betas, with or without a riskless asset, using either a general or a specific alternative hypothesis. We also extend the mean-variance framework to test the pricing relation with a factor that is not a portfolio return. All the tests considered should be viewed as tests of mean-variance efficiency defined in terms of unconditional distributions rather than as tests of conditional ‘mean-variance efficiency. A major virtue of the mean-variance framework presented in this article is that it allows the researcher to represent graphically in two familiar dimensions the outcome of a test of a multidimensional pricing restriction. A pricing model is tested by examining the position of one or more reference portfolios in sample mean-standard-deviation space. In this approach, the likelihood-ratio-test statistic can be viewed not only as the outcome of a numerical procedure but also as a quantity with simple economic and statistical interpretations. One case for which the mean-variance framework has been developed is one in which a pricing model, that includes a riskless asset is tested against a general alternative hypothesis. The likelihood-ratio test in this case can be characterized as comparing the position in sample meanstandard-deviation space of a benchmark portfolio, or a set of reference portfolios, to the position of the sample tangent portfolio [e.g., Jobson and Korkie (1982) and Gibbons, Ross, and Shanken (1989)]. The rejection region in sample mean-standard-deviation space is defined by a pair of lines. We show that, in the absence of a riskless asset, the rejection region for the likelihood-ratio test using a general alternative hypothesis is defined by a hyperbola in sample mean-standard-deviation space. As in the case in which a riskless asset exists, the rejection region depends only on the 1 See Jobson and Kodde (1982, 1988), Gibbons, Ross, and Shanken (1989), Kandel (1984, 1986), and Roll (1985). 126

estimated means and variance-covariance matrix for the observed universe of assets and does not depend on the specified benchmark or reference portfolios.It is not necessary to estimate a zero-beta expected return to conduct the test,With a single benchmark portfolio,the likelihood-ratio test consists of asking whether the position of the benchmark portfolio in sample mean-standard-deviation space lies within the rejection region. With a collection of reference portfolios,the researcher first plots the sample minimum-standard-deviation boundary of all combinations of the reference portfolios.The test then consists of asking whether this entire boundary lies within the rejection region.We illustrate these procedures by testing a zero-beta CAPM and a two-beta pricing model. The mean-variance framework is also used to investigate likelihood-ratio tests of pricing models against specific alternative hypotheses.We consider tests of a K-beta pricing model against a specific K2-beta pricing model. The null hypothesis identifies K,reference portfolios to be used in explain- ing expected returns,and the specific alternative hypothesis identifies an additional set of K2-K reference portfolios.If a riskless asset exists, then a test of a K,-beta model against a K2-beta model is conducted by testing whether the tangent portfolio of the K,portfolios is also the tangent port- folio of the larger set of K,portfolios.The test is identical to the test of a K,-beta model against a general alternative,except that the set of K2 ref- erence portfolios replaces the original universe of n assets.No other infor- mation about the other n-K,assets is used. When a riskless asset is not included,the specific alternative hypothesis is that some combination of the K2 portfolios is efficient with respect to the set of n assets.As in the case with a riskless asset,there is a close correspondence between the mean-variance representations of the tests against the general and specific alternatives,and the critical hyperbolas in sample mean-standard-deviation space are from the same class.Unlike the case with a riskless asset,however,the critical hyperbola in the case without a riskless asset depends on the returns of all n assets,Tests using a specific alternative are illustrated by testing a single-beta model against a two-beta model. We extend the mean-variance framework to tests of a pricing relation with a factor,such as consumption,that is not a portfolio return.In par- ticular,we consider the role of a reference portfolio with weights estimated. within the sample,to approximate those of the portfolio having maximal correlation with the factor.We show that,if a riskless asset exists,then the likelihood-ratio test of a single-beta pricing model,where betas are defined with respect to a factor,is similar to the test of a single-beta model using a reference portfolio with prespecified weights.In both cases,the position of the reference portfolio is compared with the position of the sample tangent portfolio of the observed universe of n assets.With a prespecified reference portfolio,the critical value for this comparison depends only on the sample means and variance-covariance matrix of the n assets.In the test with estimated weights,however,the critical value also depends on 127

estimated means and variance-covariance matrix for the observed universe of assets and does not depend on the specified benchmark or reference portfolios. It is not necessary to estimate a zero-beta expected return to conduct the test, With a single benchmark portfolio, the likelihood-ratio test consists of asking whether the position of the benchmark portfolio in sample mean-standard-deviation space lies within the rejection region. With a collection of reference portfolios, the researcher first plots the sample minimum-standard-deviation boundary of all combinations of the reference portfolios. The test then consists of asking whether this entire boundary lies within the rejection region. We illustrate these procedures by testing a zero-beta CAPM and a two-beta pricing model. The mean-variance framework is also used to investigate likelihood-ratio tests of pricing models against specific alternative hypotheses. We consider tests of a K1 -beta pricing model against a specific K2 -beta pricing model. The null hypothesis identifies K1 reference portfolios to be used in explaining expected returns, and the specific alternative hypothesis identifies an additional set of K2 - K1 reference portfolios. If a riskless asset exists, then a test of a K1 -beta model against a K2 -beta model is conducted by testing whether the tangent portfolio of the K1 portfolios is also the tangent portfolio of the larger set of K2 portfolios. The test is identical to the test of a K1 -beta model against a general alternative, except that the set of K2 reference portfolios replaces the original universe of n assets. No other information about the other n - K2 assets is used. When a riskless asset is not included, the specific alternative hypothesis is that some combination of the K2 portfolios is efficient with respect to the set of n assets. As in the case with a riskless asset, there is a close correspondence between the mean-variance representations of the tests against the general and specific alternatives, and the critical hyperbolas in sample mean-standard-deviation space are from the same class. Unlike the case with a riskless asset, however, the critical hyperbola in the case without a riskless asset depends on the returns of all n assets, Tests using a specific alternative are illustrated by testing a single-beta model against a two-beta model. We extend the mean-variance framework to tests of a pricing relation with a factor, such as consumption, that is not a portfolio return. In particular, we consider the role of a reference portfolio with weights estimated, within the sample, to approximate those of the portfolio having maximal correlation with the factor. We show that, if a riskless asset exists, then the likelihood-ratio test of a single-beta pricing model, where betas are defined with respect to a factor, is similar to the test of a single-beta model using a reference portfolio with prespecified weights. In both cases, the position of the reference portfolio is compared with the position of the sample tangent portfolio of the observed universe of n assets. With a prespecified reference portfolio, the critical value for this comparison depends only on the sample means and variance-covariance matrix of the n assets. In the test with estimated weights, however, the critical value also depends on 127

the sample correlation between the return on the estimated reference portfolio and the factor.We illustrate this procedure by testing the con- sumption-beta model. The mean-variance framework offers directions for future research beyond the scope of this study.For example,the mean-variance framework pre- sented here,coupled with previously developed analysis,allows the researcher to investigate problems associated with measuring accurately the returns on relevant benchmark or reference portfolios.Kandel and Stambaugh (1987)conducted such an investigation for the Sharpe-Lintner form of the CAPM,where a riskless asset is included.They computed the maximum correlation between a given benchmark portfolio and a portfolio that gives a different inference about the model,and they tested the hypoth- esis that the correlation between the benchmark and the ex ante tangent portfolio exceeds a given level.Their analysis combined a mean-variance framework for the likelihood-ratio test with the results of Kandel and Stambaugh (1986),which derived the maximum correlation between a given portfolio and another portfolio with a given location in mean-vari- ance space.Similar analyses can be conducted for other pricing models by combining the mean-variance framework for tests of these models with the results of Kandel and Stambaugh (1986). The article proceeds as follows.Section 1 defines terms and notation used.Section 2 analyzes likelihood-ratio tests using a general alternative hypothesis,and Section 3 presents tests using specific alternative hypoth- eses.As each test is discussed,we include an illustration using weekly returns on stock market indexes and common stock portfolios formed according to firm size.Section 4 extends the framework to models with a factor that is not a portfolio return and provides an illustration using con- sumption data.Section 5 concludes the article. 1.Definitions and Notation We consider a set of n risky assets,which are often portfolios formed from a larger universe of individual assets.An n x matrix G with full column rank contains the weights for portfolios that are combinations of the n assets.A given set of K reference portfolios is represented by the matrix A,a specific choice of G having K columns.A single reference portfolio is denoted by the vector p,a specific choice of G having one column. Let R,denote returns in period t on the K reference portfolios,and let r,denote returns in period t on the remaining n-K assets.If a riskless asset exists,then R,and r,denote excess returns on these assets,that is, returns in excess of the riskless rate reIt is assumed throughout the paper If the riskless rate is changing,the use of excess returns is problematic in terms of defining unconditional mean-variance efficiency.In that our primary goal is to provide a simple framework in which to Interpret previously applied tests,we follow the convention of previous research in our use of excess retums.This issue,along with more general questions about the appropriateness of testing unconditional relations,lie beyond the scope of this study. 128

the sample correlation between the return on the estimated reference portfolio and the factor. We illustrate this procedure by testing the consumption-beta model. The mean-variance framework offers directions for future research beyond the scope of this study. For example, the mean-variance framework presented here, coupled with previously developed analysis, allows the researcher to investigate problems associated with measuring accurately the returns on relevant benchmark or reference portfolios. Kandel and Stambaugh (1987) conducted such an investigation for the Sharpe-Lintner form of the CAPM, where a riskless asset is included. They computed the maximum correlation between a given benchmark portfolio and a portfolio that gives a different inference about the model, and they tested the hypothesis that the correlation between the benchmark and the ex ante tangent portfolio exceeds a given level. Their analysis combined a mean-variance framework for the likelihood-ratio test with the results of Kandel and Stambaugh (1986), which derived the maximum correlation between a given portfolio and another portfolio with a given location in mean-variance space. Similar analyses can be conducted for other pricing models by combining the mean-variance framework for tests of these models with the results of Kandel and Stambaugh (1986). The article proceeds as follows. Section 1 defines terms and notation used. Section 2 analyzes likelihood-ratio tests using a ‘general alternative hypothesis, and Section 3 presents tests using specific alternative hypotheses. As each test is discussed, we include an illustration using weekly returns on stock market indexes and common stock portfolios formed according to firm size. Section 4 extends the framework to models with a factor that is not a portfolio return and provides an illustration using consumption data. Section 5 concludes the article. 1. Definitions and Notation We consider a set of n risky assets, which are often portfolios formed from a larger universe of individual assets. An n × l matrix G with full column rank contains the weights for l portfolios that are combinations of the n assets. A given set of K reference portfolios is represented by the matrix A, a specific choice of G having K columns. A single reference portfolio is denoted by the vector p, a specific choice of G having one column. Let R, denote returns in period t on the K reference portfolios, and let rt denote returns in period t on the remaining n - K assets. If a riskless asset exists, then Rt and rt denote excess returns on these assets, that is, returns in excess of the riskless rate rFt· 2 It is assumed throughout the paper 2 If the riskless rate is changing, the use of excess returns is problematic in terms of defining unconditional mean-variance efficiency. In that our primary goal is to provide a simple framework in which to Interpret previously applied tests, we follow the convention of previous research in our use of excess returns. This issue, along with more general questions about the appropriateness of testing unconditional relations, lie beyond the scope of this study. 128