# Augmented Dickey Fuller and Nonlinear Time Series

Since the Augmented Dickey Fuller test assumes the time series to be an autoregressive-moving average model, I was wondering if it would be possible to explicitly construct a nonlinear time series that is stationary but that does not reject the null hypothesis of the ADF test.

More precisely, is it possible to define a function $$f:R^d \to R$$ and a time series $$(x_t)$$ given by $$x_{t+1} = f(x_{t-d+1},...,x_{t-d}) + \epsilon_t$$ where $$\epsilon_t$$ is white noise such that $$(x_t)$$ is stationary and the ADF test does not reject $$H_0$$?

Alternatively, is it possible to define a function $$f$$ and a time series $$(x_t)$$ as above, such that the time series is mean reverting and the ADF test does not reject $$H_0$$?

• Welcome to CV, Juju. Can you say a little more about "such that $(x_t)$ is stationary and the ADF test does not reject $H_0$?" You want the process to (i) be stationary, but (2) fail to test as stationary over $\text{H}_{0}\text{: }x_{t}\text{ has unit root}$? – Alexis Aug 5 '20 at 20:10
• Hi Alexis, thanks for the quick reply! Yes, I was wondering if this would be possible given that $(x_t)$ would not be an autoregressive-moving average time series as is assumed in the original paper (academic.oup.com/biomet/article-abstract/71/3/599/…). I understand that in practice the ADF is usually applied to the time series data without worrying about this assumption, but was wondering if a particular nonlinear time series could be constructed such that the above scenario is obtained. – Juju Aug 5 '20 at 21:27