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I seem not to find this in any textbooks. So I post these questions.

  1. Is monthly data better than weekly data for forecasting?
  2. Can there be seasonality in weekly data? Most software/methods don't seem to find seasonality in forecasting data.
  3. Is there a way to aggregate weekly data to monthly data?
  4. How do methods like ARIMA/Exponential Smoothing handle seasonality with weekly data?
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Weekly forecasting is quite inadequate due to the deterministic effect of holidays and other events. Weekly data can be severely skewed by when the holiday occurs and activity before and after the holiday. Daily data analysis can provide not only good daily forecasts but good weekly forecasts. In my opinion most software errs by exclusively using autoprojective schemes when combining deterministic and autoprojective schemes should be used. Care must be taken not to assume the form of the deterministic structure and to allow the data to speak to the identification. The problem with using Fourier procedures is that it enforces a presumed structure which leads to forecasts looking like the fit but not necessarily like the data. This is easily observable by performing diagnostic checks of model residuals from these kinds of assumed procedures. If the error aren't random, the model/parameters should be questioned. If the software you are using doesn't verify the assumption of random errors to ensure that this requirement is in place by both ACF (stochastic effects) and Intervention Detection (deterministic effects) then you might be wary. We have used Fourier approaches and have been stunned by the non-independence of the error terms (residuals)

On the other hand we have found that incorporating day-of-the-week effects , changes in day-of-the-week effects, month-of-the-year effects, long-weekend effects, lead and lag structures around holidays, specific- week-of-the-month effects, specific day-of-the-month effects, changes in either levels or trends over time CAN be very useful in predicting daily, weekly and monthly aggregates.

Seldom are there any repetitious effects for a particular week in the year although we have seen some! If you are insistent on building a weekly model identify which weeks of the year, if any, have a repetitive pattern. When encountering a year with 53 weeks some clients have eliminated one week where the response is low.

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    $\begingroup$ Much excellent advice as always, but this seems to presuppose that data are economic or business data. In practice some physical, chemical, biological and environmental data are collected weekly, if only as a matter of convenience or compromise (daily measurement would be too much work, monthly too coarse). The underlying processes are quite indifferent to whether Homo sapiens somewhere is celebrating a holiday or it is Tuesday. Also, many economic data series are issued once-per-week and do deserve and require analysis. (The question does not stipulate what kind of data.) $\endgroup$ – Nick Cox Aug 14 '13 at 14:20
  • $\begingroup$ @NickCox You are quite correct that the data that you are trying to model can often dictate the "correct granularity". I was trying to point out that pure ARIMA models are often inappropriate while deterministic structure can provide structure. If there are just a few weeks that are important then we should just indicators for those weeks while dealing with Pulses, Level Shifts, Time Trends and any needed short term ARMA. Fourier fitting ( sines/cosines and such ) might be useful for data in the physical sciences but I haven't (often) seen the proof of that. $\endgroup$ – IrishStat Aug 14 '13 at 15:12
  • $\begingroup$ I've often used Fourier ideas for (environmental) time series, but in contexts where neither close modelling of time series nor prediction of future events from the time series alone was a goal. Fourier models can do a good job of catching a climatically-driven seasonal signal (which can vary over time) while leaving a lot of noise. In practice this just needs a decent regression program and is not time series analysis in your style. $\endgroup$ – Nick Cox Aug 14 '13 at 15:20
  • $\begingroup$ @irishstat Thank you for the feedback. I have daily, weekly and monthly data and have not seen any recommendation in any textbooks on which data to use. I have read somewhere that it is better to forecast using monthly data because it has structure to it. Are you suggesting that daily data is better for forecasting ? $\endgroup$ – forecaster Aug 21 '13 at 1:49
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    $\begingroup$ If you are ONLY interested in monthly forecasta , I would probably select monthly data. If you are either interested in daily forecasts OR weekly forecasts OR creating updated monthly forecasts as you come through the month then I would strongly recommend daily data analysis. $\endgroup$ – IrishStat Aug 25 '13 at 1:49
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  1. Better for what? If you want weekly forecasts, use weekly data. If you want monthly forecasts, use monthly data.

  2. Of course. Weekly data is often seasonal. If the software isn't modelling seasonality that is obviously there, then you are either using the wrong model or using the software incorrectly.

  3. No. There isn't an integer number of weeks per month, so there is no way to reliably split the weeks that overlap month boundaries. You could apportion the week according to the number of days which fall within each month, but that doesn't take account of day-of-week effects, and with weekly data there is no way of estimating day-of-week effects.

  4. Badly. See my blog post on this (http://robjhyndman.com/hyndsight/longseasonality/). There are several problems:

    • there are not an integer number of weeks in a year
    • exponential smoothing essentially has one parameter for each week
    • seasonal ARIMA models either difference away a whole year or more of data, or they regress on observations at least a year old.

    A better approach is to handle the seasonality using Fourier terms (as explained in my blog post). That can be done within an exponential smoothing framework (the TBATS model) or within an ARIMA framework (a regression with ARMA errors). I've compared these models in another blog post: http://robjhyndman.com/hyndsight/forecasting-weekly-data/

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