Finbarr Timbers

ARIMA, ARMA, what's the difference?

I'm working through TSA, and I noticed that some of my classmates are struggling to understand the difference between an ARIMA process, an AR process, and a MA process, not to mention seasonal version of the above.

Using as the lag operator, i.e. , an ARIMA(p, d, q) process is a discrete time stochastic process of the form

where is a polynomial of degree p, and is a polynomial of degree q. An AR(p) process is an ARIMA(p, 0, 0) process, and a MA(q) process is an ARIMA(0, 0, q) process. To make life even more complicated, we introduce the notion of seasonality:

An ARIMA model is a s.p. of the form

where is a polynomial of degree , and is a polynomial of degree .

Example

Suppose we have the stochastic process

How can we write this as an ARIMA model? Note that

We can rewrite this as

or, more concisely,

Consequently, we can see that is an ARIMA process.