# 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 $$B$$ as the lag operator, i.e. $$BX_t = X_{t-1}$$, an ARIMA(p, d, q) process is a discrete time stochastic process of the form

$\phi(B) (1 - B)^d X_t = \theta(B)w_t,$

where $$\phi$$ is a polynomial of degree p, and $$\theta$$ 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$$(p, d, q) \times (P, D, Q)_s$$ model is a s.p. of the form

$\Phi(B^s) \phi(B) (1 - B^s)^D (1 - B)^d X_t = \Theta(B^s)\theta(B)w_t,$

where $$\Phi(B)$$ is a polynomial of degree $$P$$, and $$\Theta(B)$$ is a polynomial of degree $$Q$$.

#### Example

Suppose we have the stochastic process

$X_t = \frac 1 2 X_{t-1} + X_{t-4} - \frac 1 2 X_{t-5} + w_t - \frac 1 4 w_{t-4}.$

How can we write this as an ARIMA$$(p, d, q) \times (P, D, Q)_s$$ model? Note that

$(1 - B^4) X_t = \frac 1 2 X_{t-1} - \frac 1 2 X_{t-5} + w_t - \frac 1 4 w_{t-4}.$

We can rewrite this as

$(1 - B^4) X_t - \frac{1}{2} B (1 - B^4)X_t = (1 - \frac 1 4 B^4) w_t,$

or, more concisely,

$(1 - B^4) (1 - \frac 1 2 B) X_t = (1 - \frac 1 4 B^4) w_t.$

Consequently, we can see that $$X_t$$ is an ARIMA$$(1, 0, 0) \times (0, 1, 1)_4$$ process.