# How do you test persistence in an AR(p) regression?

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.

• Look into unit roots? Nov 18, 2020 at 7:11

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).