# How can I calculate the parameters of a MA time series model?

I am new to Time Series Analysis and I have problems understanding the MA-model (opposed to the AR model). I read many webpages about it and it is either said that MA is a linear regression with past forecast errors or with white noises. So some label the Epsilons as past forecast errors and others as white noise.

My question is whether there is a difference between those two 'approaches'? Further, I do not understand how we can calculate the forecast errors. As far as I understood MA is used for forecasting itself. So how can I fit a forecasting model that itself relies on an forecast (of past error terms)? So my basic question is how can I calculate the Epsilon-parameters of the MA model?

I'd appreciate every comment.

EDIT: Do you know a website where the MA model is explained in an understandable ways also for people who have just started to learn and use time series? I still do not know how I can calculate the parameters.

• Your question is basically answered by stats.stackexchange.com/questions/26024/…. The difference between "forecast error" and "white noise" is that the forecast errors are $\hat{\epsilon}$, estimates of the driving shocks, while white noise is a common type of shock assumed to drive the system. Oct 12, 2020 at 9:20
• Thanks Henry for your comment. But how can I determine the concrete value of the white noises? I do not understand the answers in your link. At the moment I do not see why I would use an MA model when it requires another forecasting method itself. Oct 12, 2020 at 10:31

My question is whether there is a difference between those two [AR and MA] 'approaches'?

Any stationary $$AR(p)$$ process have an $$MA(\infty)$$ representation, and any invertible $$MA(q)$$ process have an $$AR(\infty)$$ representation.

Further, I do not understand how we can calculate the forecast errors

I give here (Is the MA($\infty$) process with i.i.d. noise strictly stationary?) a formula for the variance of an $$MA(q)$$ process. In is estimated version, if there aren't bias problems, it represent an estimate of mean square forecast error also (MSFE).

So how can I fit a forecasting model that itself relies on an forecast (of past error terms)? So my basic question is how can I calculate the Epsilon-parameters of the MA model?

Actually an $$AR(p)$$ model can be estimated consistently in standard OLS fashion also, while $$MA(q)$$ are not. This happen because the "error series" is not observable. In most software some ML algorithm are implemented; some theoretical points are addressed here: https://www.it.uu.se/research/publications/reports/2006-022/2006-022-nc.pdf

• Thanks markowitz for your answer. Basically I am quite confused about the MA model? Why shall I use it to forecast values, if it itself needs a forecasting method? Of course I can forecast some values e.g. by using a linear regression or artifical neural networks. But if I decide to use them, why shall I want to use an MA-model then for forecasting. So for me it does not make sense to use MA-model (or related models like ARMA, ARIMA). Oct 12, 2020 at 10:28
• MA involve unobserved dependent variables, here is the trick, but it do not need an apart forecast method for working. MA sometimes can work better than AR. ARMA sometimes work better than pure MA or pure AR. Oct 12, 2020 at 12:04
• Thanks markowitz for your answer and effort. I still do not understand how MA works and how I can get the parameters of an MA. You wrote that it involves 'unobserverd dependent variables'. If the variables are unobserved how can I get them without any forecasts as you mentioned? Oct 12, 2020 at 12:48
• AR is obviously quite easy to calculate because you just have to use the observerd old values of the time series. But MA requires to either calculate an error of observations (which I think is impossible without havig a forecasts) or a white noise term. The white noise term can be sampled from a normal distriubtion for example but I do not see how I can use this because at the end when using the white noise term as random variables the output would be also purely random. Oct 12, 2020 at 12:51
• MA process is a combination of white noise one. Estimate MA is not easy and not admit short answers. The link suggested from Henry give a good explanation, you can see there and in ref therein. Also two stage procedure is possible as suggested in the paper that I linked above. Oct 12, 2020 at 13:07