A Complete Introduction To Time Series Analysis (with R):: MA(1)

In the last article, we explored the famous AR(1) process, and we saw that it was a mean-zero stationary process, with an ACF function that presented exponential growth (or decay, depending on the values for phi). This time, we will explore the Moving-average MA(1) process. Without anything else to say, let’s get to it.

Moving-average MA(1)

Dependence graph of the moving MA(1) process

What is this saying? This says that our observations depend on present time noise, and a proportion of past noise, one step back in time. For the MA(1), we have that

This time, it’s easy to see that it is a mean-zero process:

For the ACV, however, we have to consider four cases: h=0,1,-1, |h|>1. First, we have that

Then,

I leave it to you to verify that for the case h=-1, we obtain the exact same result. Is this a coincidence? I think not!

Why is the last case true? This is because

so since abs(h) > 1, these indices will never be the same and therefore the Zt’s will be uncorrelated, therefore, by the laws of expectation, each expectation of…