Why is a Moving Average Process not just a noise signal?

Question

I was studying Moving Average Processes, and wanted to ask why adding a bunch of weighted noise terms is not just a noise term. I understand the operations involving mean and variance in a mathematical pov, but I want to address this doubt more intuitively. Could someone help please?

Answer 1

Maybe it would help to think about a very simple moving average process. Imagine I roll a normal six-sided die 20 times, and I get a set of results that looks like this:

4 6 2 4 4 1 2 4 4 3 2 3 4 2 2 2 2 5 1 3

Now let's apply a straightforward 2-term moving average with even weights across these results (i.e. we just add consecutive numbers and divide by 2). That gives us:

5 4 3 4 2.5 1.5 3 4 3.5 2.5 2.5 3.5 3 2 2 2 3.5 3 2

What do you notice? In this case, there are no values of 1 or 6, and in general they won't show up often because they require two consecutive 1s or 6s which is harder than many of the other possible results. Also, if one of the results is high (say, 5 or 6) then the results around it are also likely to be high because it had to have come from a pair of high rolls in the initial sequence. Compare that to if we instead generated a sequence by rolling a fresh pair of dice and averaging them each time, so that high values are just as likely to wind up next to low values.

So to summarise:

• Values tend to be closer to the mean than the un-averaged noise

• There is some level of correlation between consecutive values compared to a set of independent draws with a similar distribution