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I have an ARIMA(0,2,1) model. How do i estimate the $\hat{e}_t$ component of the model. I have read a whole lot of theories that confuses me the more. Is there any practical way of estimating this $\hat{e}_t$? Does R offer any help to that too?

I know that my ARIMA(0,2,1) model can be written as $Y_{t} = 2Y_{t-1} - Y_{t-2} + e_{t} + \theta e_{t-1}$. I want to forecast 1-time ahead into the future. In that case, my forecast equation is given as $\hat Y_{t}(1) = 2Y_{t} -Y_{t-1} + \theta \hat e_{t}$. I know my value for $ Y_{t} =7.8$ and $ Y_{t-1} =7.8 $. I know my value of \theta as -0.6816, which i obtained from my R output. My problem now is, how do i determine the value for my $\hat e_{t} $ so i could find $ \hat Y_{t}(1)$? I have an R code that gives me all these forecasts though, but i want to know how R generated my first forecast and how it found the estimate for $\hat e_{t} $.

Thanks for looking!

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  • $\begingroup$ Well, it was mostly @COOLSerdash but you're welcome! $\endgroup$ – Gala Jul 11 '13 at 13:53
  • $\begingroup$ @GaëlLaurans My edits were minimal at best :) gregoire_dube did most of it (initially). $\endgroup$ – COOLSerdash Jul 11 '13 at 14:03
  • $\begingroup$ thanks you two. I had followed your edits to present make some nice editing of my own. Thanks. $\endgroup$ – b2amen Jul 11 '13 at 14:07
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I assume you want to find the fitted residual series $\hat{e}_t$ for $t = 1,2,...,T$.

If you are fitting your ARIMA model in R using the arima() command, then you will find the fitted residuals by:

fitted_model <- arima(x) #x is your time series
fitted_residuals <- fitted_model$residuals #the vector of residuals 

Now generate your predictions $n$ timesteps into the future, say $n=100$:

prediction_vector <- predict(fitted_model,n.ahead=100)
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  • $\begingroup$ @gregoire_dude: Thanks for your answer. I have edit my question now. I hope that gives better information of what i really wanted. Could you take a peep at it now? $\endgroup$ – b2amen Jul 11 '13 at 14:02
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    $\begingroup$ OK. I believe you are asking how the model is 'initiated' because, for example, $\hat{Y}_1$ and similarly $\hat{e}_1$ require values from the past which are not observed. It's a good question. Some assumptions are required to generate estimates for these quantities. See the previous question here. $\endgroup$ – gregory_britten Jul 11 '13 at 15:25
  • $\begingroup$ It is a helpful document though, but it doesn't seem to really give me what i wanted $\endgroup$ – b2amen Jul 11 '13 at 15:38
  • $\begingroup$ I'm not 100% on the exact way ARIMA() does this, but it will surely involve a similar approach to the Box-Jenkins methodology which involves 'backcasting'. The way to do this is 1) assume the unobserved data are equal to $\bar{y}$ and the unestimated error equal to 0; 2) estimate the parameters of the ARIMA model by MLE; 3) backcast to estimate the initial values. $\endgroup$ – gregory_britten Jul 11 '13 at 16:33
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    $\begingroup$ I've fixed the answer so the code will now work. It is too hard to do it manually without writing a lot of code. Just use what is already provided. $\endgroup$ – Rob Hyndman Jul 15 '13 at 22:44

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