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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 yt 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

 ${y}_{t}=c+{\varphi }_{1}{y}_{t-1}+\dots +{\varphi }_{p}{y}_{t-p}+{\epsilon }_{t},$ (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 $\varphi \left(L\right)=\left(1-{\varphi }_{1}L-\dots -{\varphi }_{p}{L}^{p}\right)$ . You can write the AR(p) model as

 $\varphi \left(L\right){y}_{t}=c+{\epsilon }_{t}.$ (6-7)
The signs of the coefficients in the AR lag operator polynomial, $\varphi \left(L\right)$, 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.

Stationarity of the AR Model

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

`$\varphi \left(L\right){y}_{t}=c+{\epsilon }_{t}.$`
From this expression, you can see that
 ${y}_{t}=\mu +{\varphi }^{-1}\left(L\right){\epsilon }_{t}=\mu +\psi \left(L\right){\epsilon }_{t},$ (6-8)
where
`$\mu =\frac{c}{\left(1-{\varphi }_{1}-\dots -{\varphi }_{p}\right)}$`
is the unconditional mean of the process, and $\psi \left(L\right)$ is an infinite-degree lag operator polynomial, $\left(1+{\psi }_{1}L+{\psi }_{2}{L}^{2}+\dots \right)$.

Note

The `Constant` property of an `arima` model object corresponds to c, and not the unconditional mean 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, $\varphi \left(L\right)$, 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.