Note: This page has been translated by MathWorks. Click here to see

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

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

Many observed time series exhibit serial autocorrelation; that is, linear
association between lagged observations. This suggests past observations might
predict current observations. The autoregressive (AR) process models the conditional
mean of *y _{t}* as a function of past
observations, $${y}_{t-1},{y}_{t-2},\dots ,{y}_{t-p}$$. An AR process that depends on

The form of the AR(*p*) model in Econometrics
Toolbox™ is

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

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

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

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

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

From this expression, you can see that

$${y}_{t}=\mu +{\varphi}^{-1}(L){\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (3) |

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

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

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.

Econometrics
Toolbox enforces stability of the AR polynomial. When you specify an AR model
using `arima`

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

imposes stationarity 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.