# Correctness of regression with ARIMA errors model and coefficient interpretation issues

I am trying to forecast electricity consumption in GWh for 2 years ahead (from June 2013 ahead), using R (the forecast package). For that purpose, I tried regression with ARIMA errors. I fitted the model using the auto.arima function, and I used the following variables in the xreg argument in the forecast.Arima function:

- Heating and Cooling Degree Days,
- Dummies for all 12 months and
- Moving holidays dummies (Easter and Ramadan)

I have several questions regarding the model:

1) Is it correct to use all 12 dummies for monthly seasonality, since when I tried to include 11, the function returned error. The Auto.arima function returned the model ARIMA(0,1,2)

2)The model returned the following coefficients (I won't specify all of them as there are too many coefficients):

ma1      ma2     HDD     CDD   January  February  March     April
-0.52 -0.16      0.27    0.12  525.84   475.13    472.57    399.01


I am trying to determine the influence of the temperature component over electricity load. In percentages, (interpreting the coefficients just as with the usual regression) the temperature components (HDD+CDD) account for 11,3% of the electricity consumption. Isn't this too little, considering the fact that the electricity consumption is mostly influenced by the weather component? On the other hand, taking look at the dummies' coefficients, it turns out that the seasonality accounts for the greater part of the load. Why is this? Is the model completely incorrect?

I tried linear regression, and the temperature component accounts for 20%, but it is still a low percentage. Why is this?

3) I am obviously making some mistakes in the use of forecast.Arima or the plot function parameters since when I plot the forecasts, I get a picture of the original time series which is continued (merged) with the forecasts for the whole time series period (from 2004 until 2015). I don't know how to explain this better, I tried to paste the picture, but it seems I cannot paste pictures here.

• If you upload your plot image(s) somewhere (I suggest imgur.com, since that's what is used here), and give the link (url) in your post, someone will put it in your post for you. – Glen_b Jun 2 '13 at 11:10
• if you consider the seasonal cycle (one dummy variable for each month) you'll "mask" the influence of temperature, because it is already 'contained' into the dummy variables. I suggest you to use only one dummy variable for August, to model the industrial closure. That's a common approach, in order to underline the weather influence on electricity demand. – Matteo De Felice Jun 2 '13 at 14:47
• @MatteoDeFelice i tried with excluding the monthly dummy variables, and include only the moving holiday dummy variables, but then the overall forecasted annual load shape is distorted and does not even approximately resemble the annual load shape usually observed during the for the previous 6 years. With such model definition, auto.arima returns ARIMA (0,1,1)(0,0,2), and the weather components contributes with 18%, which is still low percentage. – Mari Jun 2 '13 at 15:20

1. In any regression model, including a regression with ARMA errors, you must specify one less dummy variable than the number of categories. Intuitively, this is because if you know the value of 11 monthly dummy variables, then you know the value of the 12th. So it provides no new information.

2. There are two problems here. First, seasonality is confounded with the weather, so you cannot separate out their effects. Second, it is not possible to allocate a percentage contribution from each predictor unless the predictors are all orthogonal.

3. The plotting method for forecast objects shows the historical data and the forecasts along with prediction intervals. Look at the help file to see how to modify the plot to your own purposes.

• Thank you very much Mr. Hyndman!...I am total beginner in the field of forecasting and i started studying forecasting recently, and i am currently trying to understand how regression with arima errors works. Talking about the interpretation of the weather component regression coefficients of this model, what would be the appropriate and correct approach to emphasize the weather impact over electricity load? I searched literature on such models, but i haven't found one that is clearly interpreting the coefficients related to such issue, except some general reference to their values – Mari Jun 3 '13 at 18:43
• Try reading otexts.com/fpp/9/1 – Rob Hyndman Jun 3 '13 at 23:34
• Thank you very much Professor Hyndman! I have already read that in details, and it had helped me a lot, especially as a beginner...in fact I wouldn't have started studying forecasting if i didn't found your online book, as well as the one named little book of r for time series analysis. Anyway, I just asked for more detailed reasoning, if possible, regarding the weather component since i am not able to emphasize the effect of the weather component. – Mari Jun 4 '13 at 12:13