A fast likelihood approximation for vector general linear processes with long series: Application to fractional differencing

SUMMARY The quadratic form appearing in the likelihood function of a Gaussian stationary and invertible vector general linear process is considered in this paper. A new expression for this quadratic form is developed, which allows repeated computations to be performed very efficiently by possibly using forecast and backcast values of the series. While the quadratic form itself involves the inverse of the covariance matrix, the new expression uses the autocovariances of the ‘inverse-transpose’ model. The new expression outperforms alternative procedures in terms of their usefulness to compute approximate maximum likelihood estimates for the parameters in the general linear model. Although the methods developed in the paper can be applied to finite order autoregressive-moving average models, they are most useful when the process does not have a representation with finite autoregressive and moving average orders. The case of a vector autoregressive integrated moving average process in which the degree of differencing is allowed to take fractional values is considered in detail.