Q:

Forecasting adjustment factor in the gas consumption formula supplied by industry

I am trying to model gas consumption in France. The industry publishes a formula to use for this. A simplified version looks like this consumption = K * f(x), where K is an adjustment factor, where they try to correct themselves for the error on the model f(x). These K values are published every day.
Using this formula it is simply a matter of plugging the data into f(x). However, I dont know what the K values will be. I have the data for the published K values for the past 9 years, and they look like this:

Hence I would like to model the K values, at least for the short term though long term would be useful too.
What is the best way of tackling this problem? 
I am thinking ARIMA though I think it may be correlated with weather too.
Useful info:

The data is auto-correlated
You can see seasonality in the plot 
The data is stationary
Suspect correlation with weather
I have data for a 14 day weather forecast

A:

I would form a daily model between consumption (the k values ) and weather . Arima models that include predictor variables both user-specified and identified from the data can be very useful ... they are sometimes referred to ARMAX models or Transfer Function Models or Dynamic Regression models. Data analysis might suggest monthly effects , holiday effects , day-of-the-week effects , day-of-the-month effects , week=of-the-month effects , long-weekend effects et al. and perhaps some memory effects.
I would search for "daily data" in SE for example Simple method of forecasting number of guests given current and historical data to get a flavor of this kind of approach

