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I have a question regarding forecasting with my ARIMA model. Let's say we have found ARIMA(1,1,1) as the best model based on AIC or BIC.

Now, if I want to estimate the value of (t+1) I can substitute in the equation the values of y(0) and e(0). Ok, so I would like to predice now for the horizon t=2. What can I do with the value of e(1) if it is in the future, thus, I don't know the real value and I cannot calculate the error!?

I would say there must be a way to also estimate the error (based on expectancies) or make future errors 0, but then my MA part is completely useless right?

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To get the expected value of $y_{t+2}$, you will need to plug in the expected value of $\epsilon_{t+1}$. Which is zero, since ARIMA assumes normally distributed innovations with mean zero.

You could say that the MA part is "completely useless", yes. Then again, there don't seem to be many other real-life examples of moving average processes, either.

Plus, your third paragraph already hints at an extension: if you are not only interested in point forecasts (where your criticism applies), but also in predictive densities or prediction intervals, then the full density of $\epsilon_{t+1}$ does indeed come in, and depending on your MA parameter estimates may have quite some effect, indeed.

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  • $\begingroup$ Thanks Stephan. Is the interval confidence for the prediction y that distribution of error you mentioned? So Everytime I go ahead on my horizon, that distribution becomes wider? $\endgroup$ – Pablo Ruiz Ruiz Nov 28 '17 at 21:03
  • $\begingroup$ I am referring to the width of the error distribution, i.e., the residual variance. And yes, predictive distributions and predictive intervals should typically get wider as we go farther into the future, at least under ARMA. $\endgroup$ – S. Kolassa - Reinstate Monica Nov 28 '17 at 21:08

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