Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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

Was this topic helpful?