After reading about AR(p) processes I have one question regarding the characteristic polynomial of AR(p) processes and its roots.

Let's say that you want to determine whether the time series

$$y_t = \frac{1}{5} y_{t-1} + 3\pi + \epsilon_t$$

is stationary. Then you would set up the characteristic polynomial of this AR(p) process and consider the roots. Formally, I know how it works with introducing some lag operator $L$.

My question is about the importance of the constants. Does $3\pi$ matter or could I just write $3\pi + 2017$ and the characteristic polynomial would still be the same? I thought it would be okay since you can define

$$\tilde{\epsilon}_t = 3\pi + \epsilon_t$$ but I am not sure this is alright.


Also, there is a similiar question with regard to the time series

$$y_t = \frac{1}{5} y_{t-1} + 3\pi + 2\epsilon_t$$

Does the doubled white noise have any influence with regard to stationarity?


1 Answer 1


$y_t = \frac{1}{5} y_{t-1} + 3\pi + \epsilon_t$ can be written as $$ \left(1 - \frac{1}{5}L\right)\left(y_t - \frac{15}{4}\pi\right) = \epsilon_t. $$ So your characteristic polynomial is $(1 - z/5)$. The single root to this is $5$, which is outside of the unit circle. So your model is causal and in particular stationary.

The constant $3\pi$ does not matter in the sense that it does not affect the characteristic polynomial. Substitute in another value, and you will see the latter remains the same. It will only change the mean or intercept of your process.

Scaling the error terms does not affect the characteristic polynomial, either. It will just affect the variance of the process. Moreover, adding scaled lagged error terms will affect the MA polynomial, and not the AR polynomial. This will not affect stationarity either. Only invertibility.

  • $\begingroup$ Could you elabote a bit more about how exactly scaling the error term affects the variance? $\endgroup$
    – Taufi
    Jul 29, 2017 at 7:23
  • 1
    $\begingroup$ @Taufi we've written down the conditional distribution. Remember that the stationary/marginal distribution is $x_t \sim \text{Normal}(15\pi/4, \sigma^2/(1- \phi^2))$, where $\sigma^2 = \text{Var}(\epsilon_t)$. If you use different error terms, you get different variances. But the characteristic polynomial doesn't change. $\endgroup$
    – Taylor
    Jul 29, 2017 at 16:10

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