The moving average (MA) model captures serial autocorrelation in a time series *y _{t}* by expressing the conditional mean of

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) |

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

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

By Wold’s decomposition [2], an MA(*q*) process is always stationary because $$\theta (L)$$ 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 $$\theta (L)$$ 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.

[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.