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My question is regarding the practical definition of Arima (0, 1, 1) posted on this site ARIMA(0,1,1) Forecast.

The random walk part is very clear to me, the current observation is a function of the previous observation. Could anyone explain to me what the MA(1) part practically indicates? for example, if I have population time series data, a random walk shows that the current population or this year's population at time t is equivalent to the previous population t-1. If we add the moving average part (MA(1)), what does really mean in practical terms? thanks

Stackuser
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  • This link shows that effect is transient because the model can be re-written as an infinite AR(1) where the coefficients are exponentially smoothed versions of the original coefficient. https://people.stat.sc.edu/hitchcock/stat520ch7slides.pdf. So, if you think of it from an impulse response plot of a shock, $\theta \epsilon_t$, it eventually goes away. – mlofton Oct 02 '22 at 21:05
  • @thanks Mlofton, so if I have population count data and end up Arima(0,1,1) is my model, how can I interpret it? like I think the random shock is another environmental variable interference (I guess).... – Stackuser Oct 02 '22 at 21:12
  • I didn't answer you question regarding practical applications:The MA(1) was ( and might still be ? ) quite popular in quality control applications because the shock eventually dying out is helpful. – mlofton Oct 02 '22 at 21:12
  • counting is a whole different framework that I'm not familiar with but, as one would hope, someone has. I'll look for the text. It's got "modelling time series counts using box-jenkins methodology" or some variant of that in the title. My version is in storage and I've never looked at it. – mlofton Oct 02 '22 at 21:15
  • I couldn't find the text that I was looking on amazon but the link below is what looks like a good survey paper to the topic. The first page refers to "Integer Valued ARIMA" model which is probably what you want to look at. Definitely the regular ARIMA framework can't handle counts or integers. https://www.econstor.eu/bitstream/10419/21996/1/EWP-2005-08.pdf – mlofton Oct 02 '22 at 21:23
  • @mlofton, thanks a lot! I will look at it and if you come across better resources, please let me know as I am still sturggling to get the practical meaning of it! – Stackuser Oct 02 '22 at 21:26
  • practical interpretation is difficult enough with discretized continuous. Then, with integers and counts, it's even worse !!!!!! – mlofton Oct 02 '22 at 21:33
  • @mlofton, I really appreciate your deep consideration. I think that gives some practical interpretation (by@Stephan Kolassa). I am happy you have understood my intent (practical consideration). – Stackuser Oct 03 '22 at 13:43
  • @mlofton, I got this concept and do you have any idea what does really mean, "the stochastic nature of demographic processes would "generate a random walk for an individual population trajectory," which is what the MA process attempts to describe. – Stackuser Oct 03 '22 at 15:27
  • Stackuser: the MA(process) really isn't a random walk so I would disregard that statement. it sounds like Stephen provided some nice real world examples so thank you Stephan. Think of an MA(1) as reacting to to the previous error term which can be thought of as the previous period's surprise. So, a positive $\theta$ re-enforces that surprise by including it in the next response. A negative$\ theta$ tries to mitigate the positive surprise from the previous period by multiplying it adding the result to the current response. – mlofton Oct 05 '22 at 04:58
  • @Stackuser: Notice that you have two different things to understand here. One is the concept of an MA(1). But the interpretation of the MA(1) in the standard box-jenkins framework is not going to carry over to the counts-integer case. So, I would think of them as two different frameworks. Start off with the standard one and, once you have that down, you can move to the counts. – mlofton Oct 05 '22 at 05:00
  • @mlofton, I really appreciate your effort to help me understand this stuff. I have come across on the post of Richard Hardy on this post "ARIMA(0,1,1) is a random walk with an MA(1) term on top." https://stats.stackexchange.com/questions/175833/arima0-1-1-forecast/175836#175836 but I don't get it properly and if you don't mind, if you tag him or comment on his post, I will appreciate. I can't comment on other posts since my reputation is below 50 on this site. thanks! – Stackuser Oct 06 '22 at 15:01
  • @Stackuser: I don't think it's a legit answer but I'm putting it in an answer because it gives me more room. – mlofton Oct 07 '22 at 22:25

1 Answers1

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@Stackuser: No problem with trying to help. Richard Hardy is correct when he says that an ARIMA(0,1,1) is a random walk with an extra layer (past error) on top of it.

The random walk (ARIMA(0,1,0)) is:

$y_t = y_{t-1} + \epsilon_t $

The ARIMA(0,1,1)

$y_t = y_{t-1} + \theta \epsilon_{t-1} + \epsilon_t $

So, they are pretty close except that the ARIMA(0,1,1) has the coefficient $\theta$ multiplying the previously lagged error term.

So, the forecast in the ARIMA(0,1,1) will only not be constant for one period.

If you understand this, great, but I would think it would be better to get the whole box-jenkins framework down ( because it's a f framework, not just equations ) and THEN, once you understand that, then move on to counts-integers. I've never delved into the latter but I imagine that it's got to be pretty complicated once you are forced to constrain everything to be integer.

As far as books are concerned, I never really liked Box-Jenkins-Reinsel even though it's kind of viewed as the bible. Rob Hyndman has a popular text that I have seen a lot of positive reviews of. Chris Chatfield has a couple of different texts that I like. Abraham and Ledolter is old but still relevant as far as ARIMA goes. Maybe others have other suggestions.

But, let me know if the difference between ARIMA(0,1,0) and ARIMA(0,1,1) is clear ?

mlofton
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  • I really appreciate your explanation. Definitely, I understood what you explained above and I feel that I have a pretty understanding of the mathematical part of the time series, the problem is its practical interpretation. So, can we say the ARIMA(0, 1, 1) is a random walk with autocorrelation error, if so, what does it mean "autocorrelation error"? I know random walk is the current observation is a function of the previous observation! – Stackuser Oct 08 '22 at 15:19
  • Personally, althogh they look somewhat similar, I wouldn't try to connect the random walk with an MA(1). The only reason they look similar is because of the first 1 in ARIMA(0,1,1). Consider the ARMA(0,1) and write that one out and see if that model makes sense to you. I really wouldn't try to relate the ARIMA(0,1,1) to the random walk. That might be personal preference but it's my recommendation. – mlofton Oct 09 '22 at 15:34
  • I got what you meant, so can we interpret Arima (0. 1.1) as the model captures a random fluctuation like temperature or environmental factors from the previous population (the interference of previous random fluctuation on the previous population)? Do you think so? – Stackuser Oct 09 '22 at 15:48
  • Does this help at all ? https://www.youtube.com/watch?v=lUhtcP2SUsg Notre that the process has a lag one autocorrelation and zero correlations after the first lag. There's nothing magical about it which I think is sometimes the problem with newcomers looking at these things. You might be expecting some amazing insight when all it's saying that is that the new respjnse at time $t$, $y_t$, is a function not only of the new error term, $\epsilon_t$ but also of the previous error term, $\epsilon_{t-1}$. – mlofton Oct 09 '22 at 15:48
  • Yes, the ARIMA(0,1,1) is saying: "Let's have 3 components: One that keeps us near the previous value of what the series ( the $y_{t-1}$ ), one that represents the new shock ( the $\epsilon_{t}$ ), and one that represents some kind of linear relation to the previous shock. ( the $\theta \epsilon_{t-1}$ ). This linear function can be positive ( to add to the the effect of the previous impulse response ) or negative ( to lessen the effect of the previous impulse response ). – mlofton Oct 10 '22 at 18:31
  • thanks a lot for this explanation. – Stackuser Oct 11 '22 at 10:23
  • Not sure if it helps that much but the lag 1 autocorrelation results because, if $y_{t}$ is positive and $\theta$ is positive, there will be a positive auto-correlation because $y_{t+1}$ and $y_{t}$. Similarly, if $y_{t}$ is positive and $\theta$ is negative, then there will be a negative auto-correlation between $y_{t+1}$ and $y_{t}$. – mlofton Oct 12 '22 at 12:40
  • it helps verymuch and I will try to connect different concepts with practical interpretation in time series data modelling. – Stackuser Oct 12 '22 at 13:29