# ARIMA model, daily data, weekly external regressor

I have a data set of daily values that I would like to fit an ARIMA model. I also have an additional data set of weekly values that I believe would serve well as an external regressor in the ARIMA model. From my understanding, the regressor must have the same length as the primary time series. I am wondering how to handle a situation like this. Would it make more sense to aggregate the daily data to weekly, somehow expand the weekly data to the daily level (e.g. repeat values), or perhaps a different approach entirely? For what it may be worth, I am generally using R and the forecast package.

If we extend the question a little further, let's say there are two external regressors, one at the daily level and another at the weekly. Would the approach differ here as well?

• It depends on the nature of the data. It's not hard to imagine cases where you could use the weekly values as-is because it makes sense to create a model in which their effect on each day of that week should be the same. (For instance, this could be a financial time series and the weekly values could be related information released by an agency to the public before the beginning of each week.) I would therefore like to suggest that no general answer to your question can be objectively supported, because we don't know enough about your data. – whuber Aug 12 '17 at 20:12

If external values are sum of days in week, then having same values at day level would will be incorrect. If weekly values are like avg temp etc then these can be kept same for all days of week ( Not completely correct).

It's better if you aggregate daily external variables and make them weekly.

Most importantly it depends on your requirement. If weekly forecasting prediction is acceptable instead of weekly go with 2nd approach.

• Aggregating the daily values to weeks would likely lose a lot of power. That seems like a poor recommendation. – whuber Aug 12 '17 at 20:08
• Amen .. If you have daily data then use them ! – IrishStat Aug 15 '17 at 23:06
• @whuber i do not understand. You mean to say even if weekly forecast is required.( like total sale). Can forecasting at daily level and then aggregating it, would give better results 'always'? I feel if initial day forecast is incorrect, chances are we will land up with high error which could have been avoided if we would have built model itself at weekly level. Can you share some literature on this if you have? – Arpit Sisodia Aug 16 '17 at 7:26
• Arpit, it is easy to prove mathematically that aggregating the data cannot be an improvement. The demonstration is trivial: anyone working with the daily data can always do the equivalent of constructing the weekly aggregate from them and using that. Therefore--if they are able to select and apply an optimal procedure for the daily data--their results cannot possibly be any worse (no matter how "worse" might be measured). – whuber Aug 16 '17 at 14:07

Since daily data is by it's nature more reproducible i.e. we repeat activity based upon day of the week and to a much lesser degree week of the year while also including holiday effects I would suggest predicting daily values and then (if needed ) obtain aggregate forecasts. weekly values are distorted by holiday effects which don't a;way fall in the same week of the year . Using the "same value" across 7 days for a predictor is not necessarily bad just think of a design matrix of 6 daily dummies and 51 weekly dummies. You might look at http://www.autobox.com/cms/index.php/afs-university/intro-to-forecasting/doc_download/53-capabilities-presentation .. slide 49-68 .

Aggregated forecasts and their uncertainties are easily formed by using monte-carlo techniques. Daily models enable rapid updating of weekly forecasts. Additionally the 53 week leap years distorts weekly effects. Avoid weekly when you can !