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AR and MA together

1. AR and MA together

You've gotten to know the AR model and the MA model separately;

2. AR and MA Together: ARMA

now it's time to put them together to make one AWESOME model. Displayed is an ARMA(1,1) model. Think of it as

3. AR and MA Together: ARMA

auto-regression with correlated errors. This is natural for time series because time series data are typically correlated.

4. AR and MA Together: ARMA

The graphic shows a plot of data generated from an ARMA(1,1) model. Notice that this particular series looks like some of the actual time series we have already seen, for example, the detrended price of chicken series.

5. ACF and PACF of ARMA Models

Recall the table of ACF and PACF for the various models. You have seen the

6. ACF and PACF of ARMA Models

pure AR and pure MA. For a mixed ARMA model, both the ACF and PACF tail off. (The PACF here is not cutting off, it is just so small you can't see it after lag 5). Note that you can't tell the order of the model, just that it is ARMA. The best thing to do is to start small; that is, try an ARMA(1,1) model, and then add parameters as needed. You will see how to do this as we move along.

7. Estimation

Estimation for ARMA is basically no different than for pure models. I generated some data from this ARMA(1,1) model and then estimated the parameters using sarima. You see that the estimates are close to the actual values.

8. Let's practice!