Realized semicovariances : looking for signs of direction inside the covariance matrix

We propose a new decomposition of the realized covariance matrix into components based on the signs of the underlying high-frequency returns. Under an asymptotic setting in which the sampling interval goes to zero, we derive the asymptotic properties of the resulting realized semicovariance measures. The first-order asymptotic results highlight how the concordant components and the mixed-sign component load differently on economic information concerning stochastic correlation and jumps. The second-order asymptotics, taking the form of a novel non-central limit theorem, further reveals the fine structure underlying the concordant semicovariances, as manifest in the form of co-drifting and dynamic ``leverage'' type effects. In line with this anatomy, we empirically document distinct dynamic dependencies in the different realized semicovariance components based on data for a large cross-section of individual stocks. We further show that the accuracy of portfolio return variance forecasts may be significantly improved by using the realized semicovariance matrices to ``look inside'' the realized covariance matrices for signs of direction