Computationally efficient methods for two multivariate fractionally integrated models

.  We discuss two distinct multivariate time‐series models that extend the univariate ARFIMA (autoregressive fractionally integrated moving average) model. We discuss the different implications of the two models and describe an extension to fractional cointegration. We describe algorithms for computing the covariances of each model, for computing the quadratic form and approximating the determinant for maximum likelihood estimation and for simulating from each model. We compare the speed and accuracy of each algorithm with existing methods individually. Then, we measure the performance of the maximum likelihood estimator and of existing methods in a Monte Carlo. These algorithms are much more computationally efficient than the existing algorithms and are equally accurate, making it feasible to model multivariate long memory time series and to simulate from these models. We use maximum likelihood to fit models to data on goods and services inflation in the United States.