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I noticed that the Holt-Winters function in the "forecast" package in R contains prediction intervals. This was interesting, as it is not intuitively obvious to me how prediction intervals could be calculated for the HW model.

I can/will read through the R code for the forecast package to find out the specific method used in this case, but I just wondered if anyone could explain the basic principle to calculating prediction intervals for the HW model. How would such prediction intervals work? Would they have a confidence interval?

Even just a recommendation of a good paper to read would be appreciated.

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    $\begingroup$ I had the exact same problem last week, and even looking through some really good books I didn't found the answer on then. What I did found was this paper: sciencedirect.com/science/article/pii/016920709090103I $\endgroup$ – Bruna w Nov 21 '17 at 16:05
  • $\begingroup$ You ask "Would they have a confidence interval?" I'm not sure I understand how a PI would "have" a CI. Could you explain what you are asking about? $\endgroup$ – Stephan Kolassa Nov 21 '17 at 17:30
  • $\begingroup$ You are correct - this is clumsy terminology. It would not so much "have" a confidence interval, as be analogous to one. I suppose what I was asking was whether it would be possible to choose a confidence level for the prediction intervals - e.g., a "95% prediction interval", an "80% prediction interval", etc. I would assume from your answer below that this is the case. $\endgroup$ – Statsanalyst Nov 21 '17 at 17:42
  • $\begingroup$ Yes, that is quite possible. I have seen this quantity between 0 and 1 referred to as a "confidence level", but others disagree, see the comments after this post. Both forecast.ets() and forecast.Arima() have so-called level parameters. $\endgroup$ – Stephan Kolassa Nov 21 '17 at 17:53
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You ask about "the HW model and/or more general ARIMA models". Note that ARIMA and smoothing methods are not in general equivalent or subsets of each other.

For state space models like ETS, which generalize exponential smoothing, you can calculate PIs from predictive densities. In simple cases, you can actually calculate this parametrically, but more often, you will simulate. Take a look at Hyndman et al., Forecasting with Exponential Smoothing, Chapter 6.

The article by Yar & Chatfield (1990, IJF) that Bruna w links to (+1) covers the specific Holt-Winters case (and for completeness, also simpler Single and Double Exponential Smoothing).

For ARIMA models, similar approaches are possible: simple models allow theoretical computations, and sometimes you need to simulate. Brockwell & Davis, Time Series: Theory and Methods (§5.4) gives a few hints, but other theoretical books on ARIMA like Box, Jenkins, Reinsel & Ljung Time Series Analysis: Forecasting and Control (which I don't have at hand right now) may go more deeply.

In any case, prediction intervals and predictive densities should ideally account for multiple sources of uncertainty:

  1. Model selection uncertainty
  2. Parameter estimation uncertainty
  3. Future innovation uncertainty

Few if any approaches cover all three. Model selection uncertainty in particular is often ignored, and future innovations are often forecasted by pretending we know the parameters precisely, while parameter estimation uncertainty is sometimes included (especially in decent simulation-based approaches) and sometimes ignored.

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