The Multivariate Equivalent Of System Essay
Occasionally, multivariate tests can lead to different conclusions than univariate statistics. The multivariate equivalent of system (31) is:
In this system, and are row vectors of N forecasting errors and N pricing errors, respectively, is a row vector of excess returns of the N portfolios used as test assets, is a coefficient matrix used to extract the expected excess returns of the portfolios, is a row vector of N alphas, and is a zero matrix with asset-specific beta coefficients in the main diagonal. For all sets of quintile portfolios, the modelling of the assets betas as a linear function of the lagged default premium introduces perfect collinearities in the system, preventing the Jacobian matrix from being of full rank. To avoid this problem, I assume constant betas in system (32). Given that the pricing equations use expected returns instead of realized returns, system (32) can be seen as the ex-ante analogue to the tests of GRS (1989) and MR (1991). More specifically, the MR test is obtained as a particular case when there is no noise in returns, while the GRS test is nested when returns are further normally distributed.
I consider two multivariate tests of the conditional CAPM. The first test consists in estimating the version of (32) for which all the alphas are restricted to zero and testing the overidentifying restrictions, while the second consists of testing whether the alphas in (32) are jointly zero from the difference of…