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

$${y}_{t}=c+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\dots +{\theta}_{q}{\epsilon}_{t-q},$$ | (5-9) |

where $${\epsilon}_{t}$$ is an uncorrelated
innovation process with mean zero. For an MA process, the unconditional
mean of *y _{t}* is

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 [1], 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 [2]. 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] Wold, H. *A Study in the Analysis
of Stationary Time Series*. Uppsala, Sweden: Almqvist &
Wiksell, 1938.

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

- Specify Conditional Mean Models Using arima
- MA Model Specifications
- Plot the Impulse Response Function

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