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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, $$(1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots )$$.

**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.

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
- AR Model Specifications
- Plot the Impulse Response Function of Conditional Mean Model