# 8.4 Moving average models

Rather than use past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model.

$$y_{t} = c + e_t + \theta_{1}e_{t-1} + \theta_{2}e_{t-2} + \dots + \theta_{q}e_{t-q},$$

where $e_t$ is white noise. We refer to this as an **MA($q$) model**. Of course,
we do not *observe* the values of $e_t$, so it is not really
regression in the usual sense.

Notice that each value of $y_t$ can be
thought of as a weighted moving average of the past few forecast errors.
However, moving average *models* should not be confused with moving
average *smoothing* we discussed in Chapter 6.
A moving average model is used for forecasting future values while moving average smoothing is used for
estimating the trend-cycle of past values.

Figure 8.6 shows some data from an MA(1) model and an MA(2) model. Changing the parameters $\theta_1,\dots,\theta_q$ results in different time series patterns. As with autoregressive models, the variance of the error term $e_t$ will only change the scale of the series, not the patterns.

It is possible to write any stationary AR($p$) model as an MA($\infty$) model. For example, using repeated substitution, we can demonstrate this for an AR(1) model :

Provided $-1 < \phi_1 < 1$, the value of $\phi_1^k$ will get smaller as $k$ gets larger. So eventually we obtain

$$y_t = e_t + \phi_1 e_{t-1} + \phi_1^2 e_{t-2} + \phi_1^3 e_{t-3} + \cdots, $$

an MA($\infty$) process.

The reverse result holds if we impose some constraints on the MA parameters. Then the MA model is called “invertible”. That is, that we can write any invertible MA($q$) process as an AR($\infty$) process.

Invertible models are not simply to enable us to convert from MA models to AR models. They also have some mathematical properties that make them easier to use in practice.

The invertibility constraints are similar to the stationarity constraints.

- For an MA(1) model: $-1<\theta_1<1$.
- For an MA(2) model: $-1<\theta_2<1$, $\theta_2+\theta_1 >-1$, $\theta_1 -\theta_2 < 1$.

More complicated conditions hold for $q\ge3$. Again, R will take care of these constraints when estimating the models.