I'm trying to derive something and I got something but I wanted to check it.
I was wondering the following:
Suppose X_t is MA(1) with no intercept and MA parameter theta and variance say sigma squared. is there an interesting relation between say X_t+n - X_t ? I showed on scrap that the difference process must be MA(1) with no intercept but does the MA parameter change to p^(n) and the variance change to n sigmasquared. thanks for any help or confirmation.
Dear Mark,
Conceptually you are absolutely right, but the scaling with n needs clarification. Remember that Var[eps_t+..eps_(t-n)]=n times sigma^2.
Econometric Analysis by William H. Greene and references will be a nice starting point.
Hi Peter: I'm not clear on what's wrong with my answer but I appreciate the reference. I have a lot of econometrics texts but not Greene's. Does Green's text talk about functions of ARIMA processes such as nth differences etc ? I have so many texts that I'd only prefer to purchase it, if it does have this sort of thing. Thanks.
See the attachment for the autocovariance function of X(t+n) - X(t). I suggest Chris Chatfield "The Analysis of Time Series: An Introduction" as a good (and slim, good for the backpack) introductory text. Hamilton's "Time Series Analysis" is a very analytical and encyclopedic reference book (bulky, leave in your office).
Bests,
F
Hi Francesco: I'll check out the attachment. It's much appreciated. I do have the books you mentioned. Chatfield is a nice book. IMHO, Hamilton is the best book out there as far as the topics it covers.
I cannot find anything anywhere, be it article or text, that derives the resulting process that result from operations on an ARIMA process. O.D Anderson comes close but the material is contained in detail in his thesis which has so far been unobtainable. Unfortunately, O.D. Anderson passed away in 1995. Thanks for your help.
Hi Francesco: I read what you did and it's very nice and clear. Thank you. I went about it a totally different way and got a totally different answer but maybe they are equivalent. At some point this week, I will make a latex document out of my scrap paper and show you what I did. I'm not sure why I didn't think of doing that earlier. The result is simple so I thought I was on the right track. But maybe not :). Thanks again.
Happy it helped. There are a couple of typos in the file I sent you, but the results are all right. I will have a look at your draft when its ready.
Bests,
F
Francesco: I couldn't find my scrap paper so I started to derive it in latex and eventually realized that you are absolutely correct and I am absolutely wrong. There is no nice process for the nth difference of an MA(1) aside from what you derived. Your help is much appreciated and thank you for your time and generosity.
All right. But the process is not so ugly: it makes sense, because taking differences may induce correlation. If you know the structure - as we do - it's easy to estimate all the parameters. Bests, F
Hi Francesco: Yes, the Yule-Slutzky effect at work !!!!
I think the nth difference of an AR(1) might still be AR(1) but I'm gonna look into that. Thanks again for taking the time and effort to help me.
The n-th difference of an AR(1) is an ARMA. See the attached note for an example and further reference.
Bests,
F
Thanks Francesco. I'll check it out and it's much appreciated. The generosity of some of the people on researchgate is really amazing. I'll let you know my understanding. It's very kind of you to provide it.
Mark
Hi Francesco: Again, I have my result on ugly scrap but this time I'm on the same page as you. :). If you continue your derivation for the nth difference, you end up getting an arma with the original AR(1) parameter ( say phi ) raised to the n and a bunch of differenced epsilon_terms with each one getting multiplied by phi to the eye except for the last one. So, I think the resulting ARMA is still not invertible because of the differenced epsilon terms. But I'll write it out sometime this week to make it more clear and send it you. Thanks a lot. for your great help.
Yes, you get a non invertible arma, as in the example I sent you 5 days ago.
bests,
f
Hi Francesco: Yes, I agree. Thanks so much for your beautiful derivations.
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