Maximum Likelihood Estimation of Stationary Multivariate ARFIMA Processes

This paper considers the maximum likelihood estimation (MLE) of a class of stationary and invertible vector autoregressive fractionally integrated moving-average (VARFIMA) processes considered in (26) of Luceño [1] or Model A of Lobato [2] where each component yi,t is a fractionally integrated process of order di, i = 1, . . . , r. Under the conditions outlined in Assumption 1 of this paper, the conditional likelihood function of this class of VARFIMA models can be efficiently and exactly calculated with a conditional likelihood Durbin-Levinson (CLDL) algorithm proposed herein. This CLDL algorithm is based on the multivariate Durbin-Levinson algorithm of Whittle [3] and the conditional likelihood principle of Box and Jenkins [4]. Furthermore, the conditions in the aforementioned Assumption 1 are general enough to include the model considered in Andersen et al. [5] for describing the behavior of realized volatility and the model studied in Haslett and Raftery [6] for spatial data as its special cases. As the computational cost of implementing the CLDL algorithm is much lower than that of using the algorithms proposed in Sowell [7], we are thus able to conduct a Monte Carlo experiment to investigate the finite sample performance of the CLDL algorithm for the 3-dimensional VARFIMA processes with the sample size of 400. The simulation results are very satisfactory and reveal the great potentials of using the CLDL method for empirical applications.