Maximum Likelihood Estimation of Stationary Multivariate ARFIMA Processes

This paper considers the maximum likelihood estimation of a class of stationary and invertible vector autoregressive fractionally integrated moving-average (VARFIMA) processes considered in Luceno (1996). The coverage of this class of VARFIMA processes is quite general and includes the model considered in Andersen et al. (2003) for describing the behavior of realized volatility as one of its special case. We suggest an conditional likelihood Durbin-Levsinson (CLDL) algorithm which employs the multivariate Durbin-Levinson algorithm of Whittle (1963) to efficiently evaluate the conditional likelihood function of the VARFIMA processes exactly. The computational cost of implementing this algorithm is much lower than that proposed in Sowell (1989), thus allowing us to conduct a Monte Carlo experiment to investigate the finite sample performance of the CLDL algorithm for 3-dimensional VARFIMA processes under the moderate sample size up to 400. The simulation results are very satisfactory and reveal the great potential of using our method for characterizing the realized volatility in Andersen et al. (2003) and the spatial data studied in Haslett and Raftery (1989).