# Why the simulation of a FARIMA process using the autocovariance function should be better?

Let us consider a Fractional Autoregressive Moving Average process:

$$(1 - L)^d y_t = \epsilon_t$$

where $$d \in (-0.5,0.5)$$ and $$\epsilon_t$$ is a white noise sequence. Let $$\gamma(k)$$ be the autocovariance function of the above process at lag $$k$$. It is possible to simulate this process by truncating its MA($$\infty$$) representation.

However, we know from Hosking (1984) that we can simulate this process via the Durbin-Levinson method, by using $$\gamma(k)$$ as an input in the Durbin-Levinson recursion (see Brockwell and Davids (1986)).

Why this last method should be better with respect to the first one? This is true also for a general stationary and invertible ARMA(p,q) process with autocovariance $$\gamma(k)$$?

Reference:

Brockwell, P. J., and Davis, R. A. (1986). Time series: Theory and methods. Berlin, Heidelberg: Springer-Verlag.

Hosking, J. R. M. (1984). Modeling persistence in hydrological time series using fractional differencing. Water Resources Research, 20 (12), 1898-1908. doi: 10.1029/WR020i012p01898

Durbin-Levinson reduces the computational from O($$T^3$$) to O($$T^2$$), since you can translate explicitly known partial autocorrelations to determine the coefficients of a $$MA(\infty)$$ representation of the fractional noise (see HOSKING, J. R. M. (1981). Fractional differencing. Biometrika, 68(1), 165–176. doi:10.1093/biomet/68.1.165 ). The computationally more demanding alternative requires for example a Cholesky decomposition of known autocovariances.
Practially, you can of course simulate ARMA(p,q) processes based on the $$MA(\infty)$$ representation and Durbin-Levinson, however this seems not very practical since this class of processes has a "orderly", finite representation, in contrast to the infinite representation with slowly decaying coefficients in the fractional noise case.
• Ok, therefore the Durbin Levinson method is better because it is computationally faster. What about accuracy? For instance, if I run an experiment by simulating the process 1000 times via both the Durbin-Levinson (DL) method and the $MA(\infty)$ representation, and then I compute the square distance between the log periodogram and the log true spectral density for each simulation, should I expect to find a lower value for the average of the square distance in the DL case with respect to the other one? Oct 11, 2020 at 12:10