A Complete Introduction To Time Series Analysis (with R):: ARMA processes (Part II)

The coefficients of the causal representation of an ARMA(p,q) process are given by the recurrence relation above.

In the last article, we saw that a general ARMA(p,q) process can be written, with the help of the autoregressive and moving-average operators as

. We also discussed three important properties of an ARMA process:

1. Stationarity
2. Causality (current observation-only depends on the past)
3. Invertibility (can correctly represent current noise as a function of the observations from the past)

We illustrated this with the ARMA(1,1) process, and we saw that we had to put some restrictions in order to ensure that the properties above are satisfied. This time, we will generalize these ideas to the ARMA(p,q). For this, we will make use of the basics of complex numbers, so if you need a review on the essentials, you can take a look at this article.

The Unit Disk

Let’s first recall the unit disk defined on complex numbers, given by the set

,that is, the set of all complex numbers with modulo less or equal to 1. Graphically, this is