Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory

In this paper we derive the nite-sample distribution of the esti- mated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency port- folio where the distribution of the test statistic is obtained under both the null and the alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sam- ple size increase to in nity. The theoretical ndings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index.