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Jul 3, 2014 at 6:49 history edited javlacalle CC BY-SA 3.0
I have simplied the answer (only one regression as example) since more relevant information has been given in comments.
Jul 2, 2014 at 20:58 comment added javlacalle @whuber Very interesting. Knowing the context of the data is always helpful to propose a good model.
Jul 2, 2014 at 20:17 comment added whuber If you contemplate how heating actually works, you would modify your model and find it can be a much better fit. There will be a temperature threshold above which no heating takes place: that needs to be explicitly in the model (which makes its fitting a nonlinear regression exercise). For temperatures near that threshold, some heating will occur due to daily variations in temperature: this causes the apparent quadratic effect, but it's localized near the threshold temperature. When you account for these in your model, I expect you will find little or no need to introduce seasonal dummies.
Jul 2, 2014 at 20:06 history answered javlacalle CC BY-SA 3.0