Pitfalls of the Cholesky Decomposition for Forecasting Multivariate Volatility

This paper studies the pitfalls of applying the Cholesky decomposition for forecasting multivariate volatility. We analyze the impact of one of the main issues in empirical application of using the decomposition: The sensitivity of the forecasts to the order of the variables in the covariance matrix. We find that despite being frequently used to guarantee positive semi-definiteness and symmetry of the forecasts, the Cholesky decomposition has to be used with caution, as the ordering of the variables leads to significant differences in forecast performance. A possible solution is provided by studying an alternative, the matrix exponential transformation. We show that in combination with empirical bias correction, forecasting accuracy of both decompositions does not significantly differ. This makes the matrix exponential a valuable option, especially in larger dimensions