I know that a random walk is an AR(1) with a unit root, but there are also higher order autoregressive processes with unit roots. Does the unit root in such a higher order autoregressive process also imply unpredictability (or at least that the best forecast is just the current state as for a random walk) of the time series? Are forecasts of processes with unit roots possible if it is a higher order AR-process?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
Presence of a unit root in a higher-order autoregressive process does not imply unpredictability as in the case of a random walk. Here is a counterexample. If the first-differenced process is AR(1) $$ \Delta x_{t}=\varphi\Delta x_{t-1}+\varepsilon_t, $$ then the original process is AR(2) with a unit root $$ x_{t}=(1+\varphi)x_{t-1}-\varphi x_{t-2}+\varepsilon_t. $$ The optimal 1-step-ahead point forecast under square loss is \begin{aligned} \mathbb{E}(x_t|I_{t-1}) &= (1+\varphi)x_{t-1}-\varphi x_{t-2} \\ &\not\equiv x_{t-1}. \end{aligned} This is an example of a higher-order AR process with a unit root for which the best prediction is not the last observed value (as would be the case for random walk).
answered Jun 8, 2021 at 15:12
-
$\begingroup$ In context of an augmented dickey fuller test, if the lag amount that exhibits the lowest information criteria is >=2 this means that the process is not necessarily unpredictable, is that correct? $\endgroup$ Commented Jun 8, 2021 at 15:24
-
$\begingroup$ @J3lackkyy, yes, I think so. $\endgroup$ Commented Jun 8, 2021 at 15:32