| Abstract: |
The motivation of this thesis is to provide a basic framework for treating long-range cross-correlated processes while keeping the methodology and assumptions as general as possible. Starting from the definition of long-range cross-correlated processes as jointly stationary processes with asymptotically power-law decaying cross-correlation function, we show that such definition implies a divergent at origin cross-power spectrum and power-law scaling of covariances of partial sums of the long-range cross-correlated processes. Chapter 2 describes these and other basic definitions and propositions together with necessary proofs. Chapter 3 then introduces several processes which possess long-range cross-correlated series properties. Apart from cases when the memory parameter of the bivariate memory is a simple average of the parameters of the separate processes, we also introduce a new kind of process, which we call the mixed-correlated ARFIMA, which allows to control for both the bivariate and univariate memory parameters. Chapter 4 deals with tests for a presence of long-range cross-correlations. We develop three new tests, and Monte-Carlo-simulation-based statistical power and size of the tests are com- pared. The newly introduced tests strongly surpass the already existing one. In Chapter 5, we cover the estimators of long-range cross-correlation parameter of choice – the bivariate Hurst exponent. The estimators are split into two groups based on their domain of operation – time and frequency. In addition to four already existing estimators, one of which has been introduced by the author of this thesis, we introduce two new estimators. As another novelty, we reconfigure the estimators so that the power law coherency can be estimated as well. Finite sample statistical properties (bias, variance and mean squared error) of the estimators are compared for various specifications. In Chapter 6, we analyze the leverage effect between financial returns and volatility from a perspective of the long-range cross-correlations. We then conclude and hint several challenges for further research. |
