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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.

PeterBe
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  • Your question is basically answered by https://stats.stackexchange.com/questions/26024/moving-average-model-error-terms. 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. – Henry Oct 12 '20 at 09:20
  • 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. – PeterBe Oct 12 '20 at 10:31

1 Answers1

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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

markowitz
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  • 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). – PeterBe Oct 12 '20 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. – markowitz Oct 12 '20 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? – PeterBe Oct 12 '20 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. – PeterBe Oct 12 '20 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. – markowitz Oct 12 '20 at 13:07
  • Thanks for your answer markowitz. Can you tell me one common technique to estimate the error terms of MA? Maybe can you also tell me a very simple method how to do this? – PeterBe Oct 12 '20 at 13:51
  • But did I understand correctly that you use the past data in order to derive the parameters for MA? So this would mean that we indeed need some kind of forecast (e.g. from machine learning), right? Or how could you else derive the parameters? – PeterBe Oct 12 '20 at 13:53
  • Hi markowitz. Thanks for your answers and effort. I upvoted your answer and I'd still like to hear your comments regarding my last 2 comment. I'd really appreciate any furher feedback from you. – PeterBe Oct 13 '20 at 16:40
  • If it helps a MA is some event (a shock) that has impact on future results until it dies out. White Noise is not error as far as I know. Its the absence of serial correlation. When there are no serial patterns (MA, AR, I think non-stationary or seasonality) then you have white noise. ARIMA has methods to estimate the impact of MA (and the level of MA) but I do not know the theoretical basis behind them. – user54285 Oct 13 '20 at 21:51
  • @PeterBe, I pleased to help you but I'm not sure about what add. “But did I understand correctly that you use the past data in order to derive the parameters for MA?” Of couse. “Can you tell me one common technique to estimate the error terms of MA? Maybe can you also tell me a very simple method how to do this?” this explanation is not simple and is not for comments. You looking for MA parameters. Errors can be eventually estimated apart (two stage procedure). I mentioned yet some reference before. I suggest you to look there. – markowitz Oct 14 '20 at 06:41
  • Thanks markowitz for your comment and effort. But when you use the past values to derive the parameters of MA then you use some kind of forecast because the parameters can't be observerd as you said. This confuses me quite heavily as I do not undestand why you use a forecasting method (MA) that itself needs a forecasting method to forecast its paraemter. Why do you not just directly use a forecasting method to forecast the time series instead of doing the indirect way with the MA? – PeterBe Oct 14 '20 at 07:09
  • Thanks markowitz for your comments. Do you have any comment regarding my last comment? I am still confused about MA in general and I'd appreciate further input from you (and/or others). – PeterBe Oct 15 '20 at 08:13
  • MA do not need for a forecasting method in behind, It need an estimation technique. Maybe the two stage procedure put you in the confusion. I suggest you to consider the reply of Al-Ahmadgaid Asaad given here (https://stats.stackexchange.com/questions/26024/moving-average-model-error-terms). Two stage do not appear there. If you consider my answer and suggestions exhaustive you can accept it. – markowitz Oct 15 '20 at 13:18
  • Thanks markowitz for your answers and effort. I accepted your answer (and I have previously upvoted it). Nevertheless, I still have problems with the MA model approach. I read the answer you gave me the link to but I had problems undestanding it, as the answer in the reply where to theoretical and too much linking to some literature for me. I need the estimation to be done with an examples and real numbers. Do you know a website where this is done and explained? I'd really appreciate something like this. – PeterBe Oct 15 '20 at 17:10
  • I don’t know where you can find such kind of explanation. Maybe download this open source software can be useful (http://gretl.sourceforge.net/). It is easy to use, you can use data and learn a lot from them. However let me suggest you to make effort about theoretical points, them are necessary in econometrics and, more in general, in science. – markowitz Oct 16 '20 at 06:31
  • Thanks markowitz for your answer. Basically I also want to learn the theory behind MA but so far all explanations were not good according to my point of view. I think in order to understand something you also need an example where the method is applied and this lacks in all of the explanations about MA that I have read (more than 10). This is kind of weird because on the one hand people say that it is difficult to calculate a MA model and on the other hand no one really explains this in an comprehensive and illustrative way. – PeterBe Oct 16 '20 at 07:02
  • Hi Markowitz, you adviced me to learn the theory about MA models. Do you know a website where this is explained in an understandable ways also for people who have just started to learn and use time series? – PeterBe Oct 19 '20 at 06:58
  • I'd highly appreciate any further comment as I am struggeling on this one. – PeterBe Oct 20 '20 at 08:48
  • I already suggested you some materials, one open source software also. You said that you had been read 10 explanation yet. Give you another book don't seems me a solution. I’m sorry but I do not have more to give you here. This not seems me a good reason for de-accept my answer, however the choice is only yours. I'm busy I can't spend all my time on this site. – markowitz Oct 21 '20 at 08:31
  • Thanks markowitz for your comment. I accepted your answer and I will ask this question in another forum (not StackExchange) because for me it is important to learn about MA. – PeterBe Oct 21 '20 at 10:22