A univariate long memory process with memory \(d\) is one in which the spectral density \[ f(\lambda)\sim C| 1-\exp(-i\lambda)| ^{-2d}\;\text{as}\lambda\to0+\;\text{with}\;C>0. \] The process is stationary and invertible in the case where \(0\leq| d| <1/2\). The process is said to have short memory when \(d=0\) and long memory for any \(| d| <1/2\), although some authors reserve the term ‘long memory’ for \(d > 0\) and refer to processes with \(d < 0\) as ‘anti-persistent’ or ‘intermediate memory’; see P.J. Brockwell and R.A. Davis, Time series: theory and methods. 2nd ed., Berlin etc.: Springer (1991; Zbl 0709.62080), for more background. The simplest case of long memory is fractionally integrated white noise, \(\{y_t\}\), defined by the equation \( (1-L)^{d}y_t =\varepsilon_t\), where \(\{\varepsilon_t\}\) is a white noise with variance \(\sigma^2\) and \(L\) is the lag operator, \(Lx_t = x_{t-1}\). The autoregressive fractionally integrated moving average ARFIMA\((p,d,q)\) process \(\{x_t\}\) is defined by the equation \( a(L)(1-L)^{d}x_t =b(L)\varepsilon_t\), where \(a(L)\) and \(b(L)\) are polynomials of degree \(p\) and \(q\), respectively, that have no common roots and all of their roots are outside the unit circle. Together with the assumption that \(| d| <1/2\), these conditions ensure that \(\{x_t\}\) is a stationary and invertible process. We may consider the ARFIMA\((p,d,q)\) process \(\{x_t\}\) as an ARMA\((p,q)\) process driven by fractional white noise that is determined by the equation \(a(L)x_t =b(L)[(1-L)^{-d}]\varepsilon_t\), and we may describe the ARFIMA\((p,d,q)\) process \(\{x_t\}\) as an ordinary ARMA\((p,q)\) which is fractionally integrated and determined by the equation \(x_t =(1-L)^{-d}[b(L)\varepsilon_t/a(L)]\). Since the composition of linear filters is commutative in the univariate case, these two descriptions are identical. The composition of linear filters does not commute in the multivariate case, so there are different multiple possible extensions of univariate ARFIMA processes to VARFIMA processes.
The authors discuss two multivariate generalizations of fractionally integrated autoregressive models. Although FIVAR and VARFI models are similar at the first glance, their implications differ dramatically. Computationally efficient methods for computing the covariances of each model, for computing the quadratic form and approximating the determinant for maximum likelihood estimation and for simulations from each model are proposed. These algorithms are equally accurate, making it feasible to model multivariate long memory time series and to simulate from these models. They are used to fit models to data from goods and services inflations in the United States. For similar results see J. Pai and N. Ravishanker, Stat. Probab. Lett. 79, No. 9, 1282–1289 (2009; Zbl 1160.62348), where the preconditioned conjugate gradient algorithm is applied to the estimation of multivariate long memory time series models.