## Moving Average Model

### MA(q) Model

The moving average (MA) model captures serial autocorrelation in a time series yt by expressing the conditional mean of yt as a function of past innovations, ${\epsilon }_{t-1},{\epsilon }_{t-2},\dots ,{\epsilon }_{t-q}$. An MA model that depends on q past innovations is called an MA model of degree q, denoted by MA(q).

The form of the MA(q) model in Econometrics Toolbox™ is

 ${y}_{t}=c+{\epsilon }_{t}+{\theta }_{1}{\epsilon }_{t-1}+\dots +{\theta }_{q}{\epsilon }_{t-q},$ (1)
where ${\epsilon }_{t}$ is an uncorrelated innovation process with mean zero. For an MA process, the unconditional mean of yt is μ = c.

In lag operator polynomial notation, ${L}^{i}{y}_{t}={y}_{t-i}$. Define the degree q MA lag operator polynomial $\theta \left(L\right)=\left(1+{\theta }_{1}L+\dots +{\theta }_{q}{L}^{q}\right).$ You can write the MA(q) model as

`${y}_{t}=\mu +\theta \left(L\right){\epsilon }_{t}.$`

### Invertibility of the MA Model

By Wold’s decomposition [2], an MA(q) process is always stationary because $\theta \left(L\right)$ is a finite-degree polynomial.

For a given process, however, there is no unique MA polynomial—there is always a noninvertible and invertible solution [1]. For uniqueness, it is conventional to impose invertibility constraints on the MA polynomial. Practically speaking, choosing the invertible solution implies the process is causal. An invertible MA process can be expressed as an infinite-degree AR process, meaning only past events (not future events) predict current events. The MA operator polynomial $\theta \left(L\right)$ is invertible if all its roots lie outside the unit circle.

Econometrics Toolbox enforces invertibility of the MA polynomial. When you specify an MA model using `arima`, you get an error if you enter coefficients that do not correspond to an invertible polynomial. Similarly, `estimate` imposes invertibility constraints during estimation.

## References

[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[2] Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.