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.

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

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

where $${\epsilon}_{t}$$ is an uncorrelated innovation process with mean zero.

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(

$$\varphi (L){y}_{t}=c+{\epsilon}_{t}.$$ | (6-7) |

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

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

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

From this expression, you can see that

$${y}_{t}=\mu +{\varphi}^{-1}(L){\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (6-8) |

where

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

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

.μ |

By Wold's decomposition [2], Equation 6-8 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.

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

Was this topic helpful?