Wishart Processes in Statistics and Econometrics: Theory and Applications
Final Report Abstract
In this project we consider numerous theoretical issues related to the Wishart distribution. This type of distribution arises frequently in practice while dealing with sample covariance matrices. The theoretical contributions in the published papers were driven by practical issues encountered in related projects. First, we have studied the product of a Wishart matrix and a Gaussian vector. This combination can be frequently found in Bayesian portfolio theory. We establish the stochastic representation of the product and derive explicit densities in a couple of special cases. The results were extended to the singular Wishart distribution, which is relevant in high-dimensional applications with few observation points. Second, another useful application of the established results can be found in statistical surveillance. If we monitor abrupt changes in the covariance matrix, we would like to use only the most recent observations to calculate a proxy for the covariance matrix. This allows us to reduce the reaction of the procedure, but results in a singular Wishart distribution. The suggested control statistics has to be monitored using multivariate generalizations of EWMA and CUSUM control schemes. Furthermore, we suggest robustified procedures which diminish the sensitivity of control charts for the covariance matrices to changes in the mean vector. Third, we develop several tests for the elements and characteristics Wishart matrices. Important is that we provide the finite distribution of the test statistics which is extremely advantageous compared to the standard likelihood-ratio based asymptotic tests. The possible applications of the obtained results are diverse. We concentrate in the project on the applications to finance, particularly to portfolio theory and forecasting of the volatility on financial markets. First, we assess the statistical equivalence of portfolios on the efficient frontier by using properties of the Wishart distribution. Additionally we determine the risk aversion coefficients as imposed by the Value-at-Risk constrains. Second, we follow the current trend of deploying high-frequency data for asset allocation decision. Particularly, we analyse the time series of realized covariance matrices and use innovative techniques for forecasting. This includes the use of Google search data as a measure of investors interest and the optimal ordering of assets for models based on Cholesky decomposition. We believe that the established results are of fundamental interest in practice and will find a number of applications in further scientific disciplines.
Publications
- (2013). An Exact Test for a Column of the Covariance Matrix based on a Single Observation. Metrika, 76, 847-855
Gupta, A.K. and T. Bodnar
- (2013). Boundaries of the Risk Aversion Coefficient: Should We Invest in the Global Minimum Variance Portfolio? Applied Mathematics and Computation, 219, 5440-5448
Bodnar, T. and Y. Okhrin
- (2013). Elliptically Contoured Models in Statistics and Portfolio Theory. Springer Series in Statistics: Theory and Methods. Springer, New York, 2013
Gupta, A.K., Varga, T. and T. Bodnar
- (2013). On Exact and Approximate Distributions of the Product of the Wishart Matrix and Normal Vector. Journal of Multivariate Analysis, 122, 70-81
Bodnar, T., Mazur, S. and Y. Okhrin
- (2014). An Exact Test about the Covariance Matrix. Journal of Multivariate Analysis, 125, 176-189
Gupta, A.K. and T. Bodnar
(See online at https://doi.org/10.1016/j.jmva.2013.12.007) - (2014). Distribution of the Product of Singular Wishart Matrix and Normal Vector. Theory of Probability and Mathematical Statistics, 91 (2014), p.1-14
Bodnar, T., Mazur, S. and Y. Okhrin
- (2014). Multivariate Elliptically Contoured Autoregressive Process. Statistica, 73, 303-316
Bodnar, T. and A.K. Gupta
(See online at https://doi.org/10.6092/issn.1973-2201/4326) - (2014). Robust Control Charts for the Covariance Matrix Based on a Single Observation. Sankhya A: The Indian Journal of Statistics, 76, 219-256
Bodnar, O., Bodnar, T. and Y. Okhrin
(See online at https://doi.org/10.1007/s13171-013-0044-x) - (2015). Forecasting Volatility with Empirical Similarity and Google Trends? Journal of Economic Behavior & Organisation, Volume 117, September 2015, Pages 62-81
Hamid, A. and M. Heiden
(See online at https://doi.org/10.1016/j.jebo.2015.06.005) - (2017). How Risky is the Optimal Portfolio Which Maximizes the Sharpe Ratio? AStA Advances in Statistical Analysis, January 2017, Volume 101, Issue 1, pp 1–28
Bodnar, T. and T. Zabolotskyy
(See online at https://doi.org/10.1007/s10182-016-0270-3)
