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An AR(p) process is defined as the regression of a variable against its p lags-

$Y_t=c+\sum_{i=1}^p\phi_iY_{t-i}+\epsilon_t$.

Persistence in an AR process can be defined as a measure of how much the old shocks matter for the current variable value. In AR(1), it can be tested by checking how close the $\phi$ is to 1.

$Y_t=\mu+\phi Y_{t-1}+\epsilon_t=\mu[\frac{1-\phi^t}{1-\phi}]+\phi^t Y_0+\sum_{i=1}^t\phi^{t-i}\epsilon_i\approx\frac{\mu}{1-\phi}+\sum_{i=1}^t\phi^{t-i}\epsilon_i$

Clearly, the closer $\phi$ is to 1, the stronger the dependence on the old shocks. For $\phi\geq1$, the mean would no longer be stationary and the AR(1) is not the tool to analyse such series.

What's the analogous condition for the higher order AR regressions? Intuitively it seems to be testing how close $\sum_{i=1}^p\phi_i$ is to 1. Is this correct, and is there a mathematical way to show this? I tried to expand the AR(p) expression in terms of its lags, but got a lot of interaction terms which don't lend themselves very well to the simple sum argument.

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  • $\begingroup$ Look into unit roots? $\endgroup$ Nov 18, 2020 at 7:11

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The persistence is equal to the largest eigenvalue of the companion matrix (in absolute value).

For example, an AR(3) process can be written as $$\left[ \begin{array}{c} Y_{t} \\ Y_{t-1} \\ Y_{t-2}% \end{array}% \right] =\left[ \begin{array}{c} c \\ 0 \\ 0% \end{array}% \right] +\left[ \begin{array}{ccc} \phi _{1} & \phi _{2} & \phi _{3} \\ 1 & 0 & 0 \\ 0 & 1 & 0% \end{array}% \right] \left[ \begin{array}{c} Y_{t-1} \\ Y_{t-2} \\ Y_{t-3}% \end{array}% \right] +\left[ \begin{array}{c} \varepsilon _{t} \\ 0 \\ 0% \end{array}% \right] $$

The $3 \times 3$ matrix in this expression is called companion matrix. If it has eigenvalues larger than 1 in absolute value (outside the unit circle), then the process is explosive / divergent (it goes to plus or minus infinity even if the variance of the error term is zero).

When people speak about the persistence of an AR(p) process, they usually refer to the largest eigenvalue (in modulus) of the companion matrix. It is the eigenvalue that determines whether the process is stationary or not.

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In a simple AR(p) process, the so-called cumulative impulse response is given by 1/(1-p), where p is the sum of the autoregressive coefficients. Therefore, your intuition can be justified. See: Andrews Donald W. K., Chen, Hong-Yuan, 1994, “Approximately median-unbiased estimation of autoregressive models”, Journal of Business and Economic Statistics, Vol. 12, No. 2, 187- 204

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