1. ARIMA - integrated ARMA
In this chapter, you will learn how to fit ARIMA (or integrated ARMA) models to non-stationary time series. You will also learn how to use the models for forecasting.
2. Identifying ARIMA
A time series exhibits ARIMA behavior if the differenced data has ARMA behavior.
Here is a simulation of an ARIMA(p equals 1,d equals 1,q equals 0). The d equals 1 means that if we difference once, we will get stationarity. So you see the ARIMA process on the left, and on the right is a plot of the differenced data, which looks stationary.
3. ACF and PCF of an Integrated ARMA
Although it's not necessary to the analysis, it is helpful to see what the ACF and PACF of integrated data look like. Notice the ACF decays in a linear fashion, whereas the PACF is almost 1 at lag 1.
4. ACF and PCF of a Differenced ARIMA
Now notice that, after differencing, the ACF and PACF look like what you saw for ARMA models in the previous chapter. In this case, the ACF and PACF
5. ACF and PCF of a Differenced ARIMA
indicate an AR(1) for the differenced data. This, of course, agrees with how the data were generated.
6. Weekly Oil Prices
Now let's look at some real data. Here are weekly oil prices in the US between
7. Weekly Oil Prices
2000 and 2006. The data are in the astsa package.
The raw data indicate a random walk, so differencing is appropriate.
The differenced data appear to be stationary.
The ACF and PACF of the differenced data both appear to be tailing off, so you would try an ARMA(1,1) on the differenced data, or an ARIMA(1,1,1) on the data itself.
8. Let's practice!