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 127estimated 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 explain￾ing 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 port￾folio 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 ref￾erence portfolios replaces the original universe of n assets. No other infor￾mation 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 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