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 $\gamma(k)$?


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


1 Answer 1


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.

  • $\begingroup$ 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? $\endgroup$
    – Federico
    Oct 11, 2020 at 12:10
  • $\begingroup$ Provided that your T is not large enough such that matrix inversion will cause numerical imprecisions of relevant magnitude, i see no reason why the coefficients of both methods should not be equal since both methods are exact (provided you use the same exogeneous shocks to simulate if the number of repetitions/periods is small). Only if asymptotic approximations of coefficients are employed for too early lags to be valid, there may be a difference for highly persistent processes. $\endgroup$
    – Henry
    Oct 11, 2020 at 19:16

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