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Suppose I have a Time Series Model (assume ARIMA) and use it to make one-step ahead prediction.

If I acquire a new data point, (for example I was originally using the first 100 days to fit an Arima model to predict 101st day, now I observe the data of 101st day and want to use 101 days of data to predict the 102nd day), Should I?

  1. Refit everything, including mega parameters $p,d,q$?

  2. Let $p,d,q$ remain what it is, but refit the coefficient of the ARIMA model?

  3. Or if there's other protocol when we face this kind of problem?

I am relatively new to this area so any advice, insights and suggestions will be appreciated. Thanks.

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Think about it this way.

If you believe that the data generating process is in fact an ARIMA model, you have a very clear prescription: if you perform both model selection AND estimation anew, basic asymptotics would show that you will reduce your forecasting error.

Now, what happens if you aren't crazy and you know the model is false, though potentially useful? If you fear structural changes, you would be wise to drop the first observation, add the last one and at least re-estimate model parameters if not also perform model selection anew.

The real constraint here with regards to estimate and model selection is that it is computationally intensive. There is little statistical reason to not take new information into account. Now, if you are backtesting ARIMA models to forecast a handful of variables for a handful of horizons, even a cheap laptop is likely to be able to re-doing everything systematically. But, if you deal with heaps of data and complicated models, you have to impose constraints somewhere.

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Hi: What you describe is a pretty common problem when estimating time-series models and predicting based off of them. I don't think there's oner answer but rather choices like you mentioned and others.

1) use some fixed rolling window of observations to re-restimate the same model and then predict the next value.

2) use all previous observations + new data point to re-estimate the same model and predict the next value.

3) don't re-estimate at all but use the new observation for prediction of the next value.

4) use a kalman filter to re-estimate and predict the next value which is the same as 2) but it's the more "natural" way of doing it since KF is an updating mechanism.

I don't think model choice is generally re-done when a new data point comes in but that's also possible of course. Like I said, there's not one answer.

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