Keith Vorkink

Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach

We develop new tests of the capital asset pricing model that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is necessary and sufficient for the validity of the CAPM. Our test is based on semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick-tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parameter estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates. Copyright

Efficient Estimation of Conditional Asset-Pricing Models

A semiparametric efficient estimation procedure is developed for the parameters of multivariate generalized autoregressive conditional heteroscedasticity-in-mean models when the disturbances have a conditional distribution assumed to be elliptically symmetric but otherwise unrestricted. Under high-level assumptions, the resulting estimator achieves the asymptotic semiparametric efficiency bound. The elliptical symmetry assumption allows us to avert the curse of dimensionality problem that would otherwise arise in estimating the unknown error distribution. This framework is suitable for the estimation and testing of conditional asset-pricing models, such as the conditional capital asset-pricing model. We apply our procedure in an empirical study of stock prices, with Monte Carlo simulation results also reported.

Multivariate Realized Stock Market Volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of both lagged volatility and returns. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics. We also introduce a new method to track an index using our model of the realized volatility covariance matrix.

Testing the Capital Asset Pricing Model Efficiently Under Elliptical Symmetry: A Semiparametric Approach

We develop new tests of the capital asset pricing model that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is neccessary and sufficient for the validity of the CAPM. Our test is based on semi-parametric efficient estimation procedures for a seemingly unrelated regression model where the multvariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick-tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parametric estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates.