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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

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

$${y}_{t}=c+{\varphi}_{1}{y}_{t-1}+\dots +{\varphi}_{p}{y}_{t-p}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\dots +{\theta}_{q}{\epsilon}_{t-q},$$ | (6-10) |

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

$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (6-11) |

The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 6-10. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 6-10.

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

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

$${y}_{t}=\mu +\frac{\theta (L)}{\varphi (L)}{\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (6-12) |

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

The `Constant`

property of an `arima`

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

By Wold’s decomposition [2], Equation 6-12 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.

[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.

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

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