KPSS Test - rileywheadon/ffa-framework GitHub Wiki

The KPSS Test is used to identify if an autoregressive time series has a unit root.

• Null hypothesis: The time series has does not have a unit root.
• Alternative hypothesis: The time series has a unit root.

Precisely, the autoregressive time series shown below has unit root if $\sigma ^2 > 0$:

$$ \begin{align} y_{t} &= \mu_{t} + \alpha t + \epsilon_{t} \\ \mu_{t} &= \mu_{t-1} + v_{t} \\ \end{align} $$

Here's what each term in this formulation represents:

• $\mu_{t}$ is the drift, or the deviation of $y_{t}$ from $0$.
  • Under the null hypothesis, $\mu_{t}$ is constant (since $v_{t}$ is constant).
  • Under the alternative hypothesis, $\mu_t$ is a stochastic process with unit root.
• $\alpha t$ is a linear trend, which represents deterministic non-stationarity (i.e. climate change).
• $\epsilon_{t}$ is stationary noise, corresponding to reversible fluctuations in $y_{t}$.
  • In hydrology, $\epsilon_{t}$ represents fluctuations in streamflow due to random events (i.e. weather).
• $v_{t}$ is random walk innovation, or irreversible fluctuations in $\mu_{t}$.
  • In hydrology, $v_{t}$ could represent randomness in industrial activity causing climate change.

This test is implemented using R package aTSA with the following settings:

• lag.short = TRUE, since AMS data has minimal autocorrelation.
• We consider type3 results since we are assuming the presence of a trend.

For more information, see the documentation.

Warning: The implementation of the KPSS test in the aTSA package interpolates the p-value using a table from Hobjin et al. (2004). This table only contains significance thresholds for $0.01$, $0.05$ and $0.10$. Therefore, p-values below $0.01$ and above $0.10$ will be truncated.