Fractional Gaussian noise (fGn) is characterized by the mean ($\mu$), the standard deviation ($\sigma$), and the Hurst index ($H$). It's my understanding that it is stationary, for the simple reason that the mean, variance, and correlation structure are constant over time.

I also understand that the KPSS test (Kwiatkowski et al., 1992) can be used to check if a time series is stationary. However, the null hypothesis (which, if accepted, corresponds to a decision of 'stationary') is that $\sigma^2 = 0$, versus the alternative that $\sigma^2 > 0$.

In my view, this leads to a contradiction. On one hand, the fGn time series is considered stationary because it has a constant variance, etc. On the other hand, the KPSS test considers a time series to be stationary only if no significant variance is present. Are two different meanings of stationarity involved here? Or have I misunderstood something else?


1 Answer 1


Looking at the Kwiatkowski paper, they assume the model $$ y_t = r_t + \varepsilon_t$$ (forgetting about the trend, i.e. setting $\xi = 0$ in their model). Here $(r_t)$ is a random walk $$ r_t = r_{t-1} + u_t$$ with $u_t$ i.i.d. $(0,\sigma_u^2)$ and $(\varepsilon_t)$ is a stationary time series. The assumption that $\sigma_u^2 = 0$ corresponds to $r_t = constant$ in which case $y_t = constant + \varepsilon_t$ is indeed a stationary time series.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.