Auto-covariance of Arima model

The autocovariances of an ARMA model can be computed using several algorithms. JD+ provides two implementations, which are chosen in an automatic way.

Solution 1

See for instance
Brockwell P.J and Davis R.A. [2003], “Time series: Theory and Methods”, second edition, Springer.
(§3.3 Computing the Autocovariance Function of an ARMA(p,q) Process).

The Third Method is implemented in the method defaultComputer of the class demetra.arima.internal.AutoCovarianceComputers

Solution 2

The autocovariance generating function can be factorized as

\(\frac{\Gamma\left(F\right)}{\Phi\left(F\right)} + \frac{\Gamma\left(B\right)}{\Phi\left(B\right)}\)

That is achieved by solving the linear system

\(\Gamma\left(F\right)\Phi\left(B\right) + \Gamma\left(B\right)\Phi\left(F\right) = \Theta\left(F\right)\Theta\left(B\right)\)

See Decomposition of symmetric Filters for a solution of that equation

That method is implemented in defaultSymmetricComputer of the class demetra.arima.internal.AutoCovarianceComputers