1. Compare AR and MA models
2. MA and AR processes
The MA and AR models are similar in many ways. Both have mean zero white noise terms epsilon with a variance parameter sigma-squared. Both include a mean parameter mu. In the MA model, Today's observation Y_t is regressed on Yesterday's noise epsilon_t minus 1. In the AR model, the Today's observation Y_t is regressed on Yesterday's observation Y_t minus 1.
However, the MA model only has autocorrelation at one lag, but the AR model has autocorrelation at many lags.
3. MA and AR processes: autocorrelations
Consider an MA model with theta equal to 0-point-75. It's lag 1 autocorrelation is 0-point-48, with zero autocorrelation at other lags, as shown top left. An AR model with phi equal to 0-point-48 also has lag 1 autocorrelation 0-point-48. However, its autocorrelation decreases to zero at higher lags, as shown top right.
An MA model with theta equal to -0-point-75 has lag 1 autocorrelation -0-point-48, and zero autocorrelation otherwise, as shown bottom left. An AR model with phi equal to -0-point-48 also has lag 1 autocorrelation -0-point-48. However, its autocorrelation is positive at lag 2, negative at lag three, and overall decreasing to zero in magnitude at higher lags, as shown bottom right.
4. MA and AR processes: simulations
Simulating 100 observations from each of these four models using the same white noise epsilon provides another comparison.
The top figures both have lag 1 autocorrelation of 0-point-48. The local variation is slightly larger for the AR model on the right, but the two series appear quite similar. The bottom figures both have lag 1 autocorrelation of -0-point-48. The oscillating pattern is a little more pronounced for the AR model on the right, but again the two series appear quite similar.
5. MA and AR processes: fitted values
You can fit both an MA and AR model to the changes in one-month US inflation dataset. 24 observations from the dataset are shown by the dotted black line.
The solid red line is from an MA fit and the solid blue line is from an AR fit. The fitted values are quite similar. Similar fits are expected when the estimated lag 1 autocorrelation is below 0-point-5 in magnitude and other estimated autocorrelations are small or zero. Here the lag one estimate was -0-point-38, and all others were below 0-point-12 in magnitude.
6. MA and AR processes: forecasts
With similar fitted values, the 1-step through 6-step forecasts should also be similar. The last 12 observations from the inflation changes series is shown in black. Forecasts from the MA model are in red and the AR model are in blue. The biggest difference is at 1-step ahead, they are nearly identical after that.
7. Forcasting
Backtesting or forecast updating are more advanced concepts that can be used to determine which of the two models produces better forecasts for a given dataset.
Another approach to selecting a model is using a goodness of fit measure such as an information criterion. Specifically, the AIC and BIC measures are commonly used for time series models.
Information criterion is a more advanced concept, but for either measure, a lower value indicates a relatively better fitting model. For each model the AIC and BIC functions were applied. AIC was about 2467 for the MA model, which was much smaller than the AIC of 2542 for the AR model.
Similarly, BIC was about 2480 and 2555 for the MA and AR models, respectively, and again the MA model had the smaller value. Based on both the AIC and BIC measures the MA model provides a better fit to the inflation changes series.
8. Let's practice!