SUMMARYA random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coe cients. Karhunen-Loeve (K-L) series expansion is based on the eigen-decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non-stationary Gaussian random processes is examined numerically in this paper. The study is based on ÿve common covariance models. The convergence and accuracy of the K-L expansion are investigated by comparing the second-order statistics of the simulated random process with that of the target process. It is shown that the factors a ecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen-solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K-L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more e cient as the K-L expansion method requires substantial computational e ort to solve the integral equation. The main advantage of the K-L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional e ort.
Aiming to investigate the evolution of mean and volatility spillovers between oil and stock marketsin the time and frequency dimensions,we employed WTI crude oil prices, the S&P 500 (USA) index and the MICEX index (Russia) for the period Jan. 2003-Dec. 2014 as sample data. We first applied a wavelet-based GARCH-BEKK method to examine the spillover features in frequency dimension.To consider the evolution of spillover effectsin time dimension at multiple-scales, we then divided the full sample period into three sub-periods, pre-crisis period, crisis period, and post-crisis period. The results indicate thatspillover effects varyacross wavelet scales in terms of strength and direction.By analysis the time-varying linkage, we found the different evolution features of spillover effects between the Oil-US stock market and Oil-Russia stock market. The spillover relationship between oil and US stock market is shifting to short-termwhile the spillover relationship between oil and Russia stock market is changing to all time scales. That result implies that the linkage between oil and US stock market is weakening in the long-term, and the linkage between oil and Russia stock market is getting close in all time scales. This may explain the phenomenon that the US stock index and the Russia stock index showed the opposite trend with the falling of oil price in the post-crisis period. .
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