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Here is a question Consider an ARMA(1,1) model estimated using a sample of T observations. a) Write the equation of the model assuming zero mean b) Write the expression for a one- and two-step ahead forecasts from the model

As i understand the one step ahead forecast looks like this y(T+1)=py(T)+θuT+uT+1 but how do we know ut?how do we estimate it? and in case of two steps ahead forecast y(T+2)=py(T+1)+θ(uT)+1+u(T+2)

how different is ARMA with zero mean from a normal ARMA model? I do not know how to write the expression Can you please show the difference between a stationary ARMA model expression and ARMA model with a zero mean

Thank you!!

student
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  • Hi: The forecasts for arma models are based on taking expectations of the future error terms which are usually zero. If there are any $X$ that need to be forecasted, then that's a more difficult problem because the $X$ will need to be forecasted. I have never heard of the term "normal ARMA model". – mlofton Dec 24 '22 at 14:14
  • sorry I am just studying this and get confused how ARMA model with zero mean is different from any other ARMA model – student Dec 24 '22 at 21:48
  • Maybe normal here means strictly stationary ARMA (not necessarily with mean 0)? As we know if it has unit root then its mean may drift permanently thus not weakly stationary. – cinch Dec 25 '22 at 01:26
  • Mohottnad yes that is what i meant) – student Dec 27 '22 at 20:17

1 Answers1

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  1. ARMA(1,1) with zero mean. This model is

$$ Y_t = \phi Y_{t-1} + \theta \epsilon_{t-1} + \epsilon_{t}$$

  1. ARMA(1,1) with non-zero mean. This model is

$$ Y_t = \mu + \phi Y_{t-1} + \theta \epsilon_{t-1} + \epsilon_{t}$$

So, the only difference between the two model is that, in the first one, $\mu = 0$.

I'm not sure if this answers your original question but I hope it helps.

User1865345
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mlofton
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  • I am so grateful! You are the best – student Dec 27 '22 at 20:17
  • @student: I'm glad that it helped. If you want to read intro books on time series, keith ord has written a book that's pretty good ( don't remember the title ) for an introduction. Also, chris chatfield has a chapman and hall book that is also quite good. When you go through one of those, then, depending on your background, I can recommend others. – mlofton Dec 29 '22 at 02:05
  • thank you very much! ) I will definitely go trough these books. I guess you meant Time Series: Sir Maurice Kendall and J. Keith Ord. that is what i have found – student Dec 29 '22 at 20:51
  • yes. but that's much older than chatfield. I'm kind of reluctant to recommend books because the person has to spend money and then maybe doesn't like the book. Now that I think about it, I kind of lean towards recommending chatfield ( time series analysis. an introduction using R ) or Rob Hyndman's text. I've never looked at Hyndman's text but it's been recommended quite a bit. Both of these texts are more modern than ord and kendall. But there have been threads where people ask for recommendations so let me find a thread such as that and send it to you. – mlofton Dec 30 '22 at 06:25
  • Here is a link that talks about books but I still lean towards chatfield or hyndman. One has to be very careful when picking time series books because the level can vary greatly. Given our discussions, my best guess is that chatfield or hyndman would work best for you. https://stats.stackexchange.com/questions/20514/books-for-self-studying-time-series-analysis – mlofton Dec 30 '22 at 06:30
  • I cannot thank you enough)) It is so so kind of you! but Keith Ord is very helpful so far, it is easy to read. that is exactly what i was looking for, i tried to read Hamilton, but it was too complicated for me – student Dec 30 '22 at 20:58
  • ok. if you like keith's book, great ( he was my advisor in grad school ). after that, take a look at chatfield or hyndman. I'm glad that my comments helped you. then, if you take aclasses in mathematical statistics and linear algebra, you can move to hamilton. good luck and I'm glad to help. – mlofton Dec 31 '22 at 09:02
  • thank you and happy new year! – student Jan 02 '23 at 17:19