# ARIMA and linear regression

I have a linear regression model that is used to forecast the 'afluent natural energy' (ANE) of some region.

The predictors for this model are:

• the previous month ANE (ANE0)
• the previous month rain volume (PREC0)
• the current month forecast for rain volume (PREC1)

We have 7 years of historical data for all of these variables, for each month. The current model just runs a OLS linear regression. I feel there's a lot of improvements to be done, but i'm not a time series specialist.

The first thing I notice is that the predictors are highly correlated (multicollinearity). I'm not certain of the impacts of multicollinearity on prediction confidence.

I decided to try a time series approach, so I ran a ACF and PACF on the historic data: The ACF shows a sine wave pattern, and the PACF has a spike at 1 and 2. So I tried both ARIMA (2, 0, 0) and ARIMA(2,0,1) to predict 20 periods ahead.

The ARIMA(2,0,1) shows good results, but I'm not certain as to how to compare it to the linear regression model.

What's the best way to test the performance of these model? I'm using R as analysis tool (together with the forecast package).

The best way to test out of sample prediction is to do a pseudo out of sample forecasting experiment. Use about 75 percent of the data to train the models, make the prediction, record the forecast error, update the information set, and repeat. At the end, you can use all of the forecast errors to get the mean-squared forecast error (MSFE). The model with the lowest MSFE is the best predictor. In general the out-of-sample (OOS) r-squared can help you compare a model to some benchmark.

$$\text{OOS } R^2 = 1 - \frac{MSFE_{model}}{MSFE_{benchmark}}$$

If the OOS r-squared is greater than zero, the model beats the benchmark. A typical choice for the benchmark is the historical mean. The higher the OOS r-squared, the better.

A couple of other thoughts: If you're not sure which model to choose, just average the forecasts from both models. Average forecasts usually perform better than individual forecasts. Also, since your time series looks like a sine wave you may want to check for seasonality and use seasonal time series models.

• The 'forecast' package has a function called 'accuracy' that does this kind of test automatically. I'll use 0.7 of the data to train the model, so i can get some indicators. I'll also look at ETS models. Thanks for you answer! – Fernando Jul 16 '12 at 1:19

If you have data beyond the range of the fit to test the forecasts then that would be good to compare the models. If you don't have that then fit criteria that don't require nested models and penalize for overparameterization would be best. I recommend using information criteria for that (versions of AIC or BIC). But if both models seem to have virtues you might consider a combined model such as a harmonic regression, which fits a sine wave plus autoregressive residuals.

• Thanks for your answer, i'll take a look at harmonic regression, so much stuff to learn! – Fernando Jul 16 '12 at 1:20