# An asset can have a higher unconditional average

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An asset can have a higher unconditional average return than predicted by the unconditional CAPM, if its beta moves with the market risk premium. One way to test a conditional model is to parameterize the variables that shift betas over time. For example, we might write ° imt = ° i 0 + ° i 1 z t : (3.32) The conditional model can then be written as E t R e i;t +1 = ° i 0 E t R e m;t +1 + ° i 1 E t z t R e m;t +1 (3.33) and now we can take unconditional expectations to get E R e i;t +1 = ° i 0 E R e m;t +1 + ° i 1 E z t R e m;t +1 : (3.34) 61
CHAPTER 3. STATIC EQUILIBRIUM ASSET PRICING This is a multifactor model, where the factors are the excess market return and the excess market return scaled by the state variable z t . The scaled ex- cess market return can be interpreted as the return on a dynamic investment strategy that invests more aggressively in the market when z t is high. 3.3 Empirical Evidence 3.3.1 Test methodology In practice the CAPM will never hold exactly. We need a statistical test to tell whether deviations from the model (mean-variance ine¢ ciency of the market portfolio, or equivalently nonzero alphas) are statistically signi±cant. The two leading approaches are time-series and cross-sectional. At a deep level, they are much more similar than they appear to be at ±rst. Time-series approach The time-series approach starts from the regression R e it = ² i + ° im R e mt + " it ; (3.35) where R e it = R it ° R ft and R e mt = R mt ° R ft . The null hypothesis is that ² i = 0 . This is a simple parameter restriction for any one asset; the challenge is to test it jointly for a set of N assets. An asymptotic test is as follows. De±ne ² as the N -vector of inter- cepts ² i , and ´ as the variance-covariance matrix of the regression residuals " it . (Note that this is di∕erent from the matrix ³ , which is the variance- covariance matrix of the raw returns rather than the residuals.) Then as the sample size T increases, asymptotically T 2 4 1 + R e m » ( R e mt ) ! 2 3 5 ° 1 b ² 0 b ´ ° 1 b ² ² À 2 N : (3.36) To see the intuition, suppose there were no market return in the model. Then the vector ² would be a vector of sample mean excess returns, with variance-covariance matrix (1 =T . Thus the quadratic form b ² 0 b ´ ° 1 b ² is a sum of squared intercepts, divided by its variance-covariance matrix, which has a À 2 N distribution. The term in square brackets is a correction for the presence of the market return in the model. Uncertainty about the betas a∕ects the alphas, and more so when the market has a high expected return relative to its variance. 62
CHAPTER 3. STATIC EQUILIBRIUM ASSET PRICING A ±nite-sample test makes a further correction for the fact that the variance-covariance matrix ´ must be estimated. Under the assumption that the " it are serially uncorrelated, homoskedastic, and normal, we have ² T ° N ° 1 N ³ 2 4 1 + R e m » ( R e mt ) !