Autoregressive Model

AR(p) Model

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 p past observations is called an AR model of degree p, denoted by AR(p).

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)

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(p) model as

$$\varphi (L)y_t=c+{\epsilon }_t.$$

(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 of the AR Model

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)

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, $$(1+{\psi }_{1}L+{\psi }_2{L}^2+\dots )$$.

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.

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.