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Rick
Posts:
26
Registered:
3/5/10


Re: Parameter estimation in ARIMA
Posted:
Feb 16, 2013 8:59 AM


Milos,
I must admit that I?m now rather confused by your question. To respond to your direct question:
?You say that the parameter estimation has to be done on original time series??
The answer is ?no, that is not what I am saying.? Frankly, the estimation will fit whatever model you specify to whatever data you specify (or at least it will try, but if the model is a really bad description of the data, then it may not converge).
Your first 2 paragraphs
?This data have trend and seasonal fluctuations, so I performed first difference and seasonal diff.
Now, on that adjusted data I determine the order of the model. After that, with obtained model order I have to estimate the coefficients of the model.?
describe what is typically known as model identification. That is, you may want to difference the data, both seasonally and nonseasonally as appropriate, to induce stationarity. Then on this stationary data, you identify and entertain various stationary ARMA models that might explain your data reasonably well. This is the classic Box & Jenkins style of analysis.
You can then use this identification to fit some model to our data. At this point you have 2 options:
(1) You can fit a stationary ARMA model to the differenced data. This is what you just described, and I?ll call it the ?indirect approach?. Here, you are modeling the nonstationarity in your data by adjusting the data, i.e., by data adjustment, as outlined in Box & Jenkins.
However, if you then want to forecast this data, you need to remember that you would then be forecasting the differenced series. After all, the estimation method doesn?t know you differenced your data, and all it will do is fit whatever model you specified to the data you specified.
(2) You can fit a nonstationary ARIMA model directly to the unadjusted data. This is the ?direct approach?, in you are modeling the nonstationarity in your data by allowing the differencing to be performed by the model, i.e., by allowing the underlying lag operator polynomials to do the work. The advantage here is that, for example, if you wanted forecast the data you?d then be forecasting the actual time series of interest, which I assume is what you?d want to do (at least it's easier).
In either case, you will want to perform the same preestimation identification you explained.
Again, your sentence
?This process, according to the literature (not explicitly defined but what I can see) is done also (as the previous phase) on differenced data.?
is describing option (1) above, which is the Box & Jenkins approach. We support both options ?
Beyond this, I?m really not sure what else I could offer.
Rick
"Milos Milenkovic" <m.milenkovic@mathworks.com> wrote in message <kfkukf$nck$1@newscl01ah.mathworks.com>... > Dear Rick, > just once again to be sure. I have a time series. This data have trend and seasonal fluctuations, so I performed first difference and seasonal diff. > Now, on that adjusted data I determine the order of the model. After that, with obtained model order I have to estimate the coefficients of the model. > This process, according to the literature (not explicitly defined but what I can see) is done also (as the previous phase) on differenced data. > You say that the parameter estimation has to be done on original time series? So the differencing is only for model order determining. > > Thanks once again! > Best, > M > > "Rick " <rick.baker@mathworks.com> wrote in message <kfjaj6$ng$1@newscl01ah.mathworks.com>... > > "Milos Milenkovic" <m.milenkovic@mathworks.com> wrote in message <kfidh6$esr$1@newscl01ah.mathworks.com>... > > > Dear, > > > parameter estimation in ARIMA are performed on adjusted time series (first, seasonal differencing) or original nonstationary time series? > > > Best, > > > Milos > > > > Milos, > > > > If I understand you correctly, the parameters of an ARIMA model are estimated using the original, nonstationary series. > > > > To clarify, I mean that the data is not explicitly differenced to remove any seasonal and nonseasonal integration effects, and then that differenced data then fit to an ARMA model. For example, suppose you want to estimate an ARIMA(P,1,Q) model. We do not fit an ARMA(P,Q) model to the first difference of the original, nonstationary data. > > > > In other words, whatever differencing is required is performed by the underlying lag operator polynomials applied directly to the nonstationary data ? the lag operator polynomials do the work. > > > > HTH, > > Rick



