## Autoregressive Moving Average Model

### ARMA(*p*,*q*) Model

For some observed time series, a very high-order AR or MA model is needed to model the underlying process well. In this case, a combined autoregressive moving average (ARMA) model can sometimes be a more parsimonious choice.

An ARMA model expresses the conditional mean of *y _{t}* as a function of both past observations, $${y}_{t-1},\dots ,{y}_{t-p}$$, and past innovations, $${\epsilon}_{t-1},\dots ,{\epsilon}_{t-q}.$$The number of past observations that

*y*depends on,

_{t}*p*, is the AR degree. The number of past innovations that

*y*depends on,

_{t}*q*, is the MA degree. In general, these models are denoted by ARMA(

*p*,

*q*).

The form of the ARMA(*p*,*q*) model in Econometrics Toolbox™ is

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

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

$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (2) |

The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 1. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 1.

### Stationarity and Invertibility of the ARMA Model

Consider the ARMA(*p*,*q*) model in lag operator notation,

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

From this expression, you can see that

$${y}_{t}=\mu +\frac{\theta (L)}{\varphi (L)}{\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (3) |

$$\mu =\frac{c}{\left(1-{\varphi}_{1}-\dots -{\varphi}_{p}\right)}$$

is the unconditional mean of the process, and $$\psi (L)$$ is a rational, infinite-degree lag operator polynomial, $$(1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots )$$.

**Note**

The `Constant`

property of an `arima`

model object corresponds to *c*, and not the unconditional mean *μ*.

By Wold’s decomposition [2], Equation 3 corresponds to a stationary stochastic process provided the coefficients $${\psi}_{i}$$ are absolutely summable. This is the case when the AR polynomial, $$\varphi (L)$$, is *stable*, meaning all its roots lie outside the unit circle. Additionally, the process is *causal* provided the MA polynomial is *invertible*, meaning all its roots lie outside the unit circle.

Econometrics Toolbox enforces stability and invertibility of ARMA processes. When you specify an ARMA model using `arima`

, you get an error if you enter coefficients that do not correspond to a stable AR polynomial or invertible MA polynomial. Similarly, `estimate`

imposes stationarity and invertibility constraints during estimation.

## References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. *Time Series Analysis: Forecasting and Control*. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

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

## See Also

## Related Examples

- Specify Conditional Mean Models
- ARMA Model Specifications
- Plot the Impulse Response Function of Conditional Mean Model