Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test

A Kwiatkowski-Phillips-Schmidt-Shin (KPSS) Test is a statistical test for the null hypothesis of stationarity against the alternative hypothesis of unit root.

• Context:
• It may be defined as:

1. [math]\displaystyle{ H_0 }[/math]: An observable time series is stationarity around a deterministic trend.
2. [math]\displaystyle{ H_a }[/math]: An unit root is present in the time series.

• See: Unit Root Test, Dickey–Fuller Test, Phillips–Perron Test, Breusch–Godfrey Test, Ljung–Box Test, Durbin–Watson Test, Augmented Dickey–Fuller Test, Time Series Analysis.

---

---

References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/KPSS_test Retrieved 2016-08-07
  • In econometrics, Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests are used for testing a null hypothesis that an observable time series is stationary around a deterministic trend (i.e. trend-stationary) against the alternative of a unit root.

Contrary to most unit root tests, the presence of a unit root is not the null hypothesis but the alternative. Additionally, in the KPSS test, the absence of a unit root is not a proof of stationarity but, by design, of trend-stationarity. This is an important distinction since it is possible for a time serie to be non-stationary, have no unit root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time serie will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).

Such models were proposed in 1982 by Alok Bhargava in his Ph.D. thesis where several John von Neumann- or Durbin–Watson-type finite sample tests for unit roots were developed (see Bhargava, 1986). Later, Denis Kwiatkowski, Peter C. B. Phillips, Peter Schmidt and Yongcheol Shin (1992) proposed a test of the null hypothesis that an observable series is trend stationary (stationary around a deterministic trend). The series is expressed as the sum of deterministic trend, random walk, and stationary error, and the test is the Lagrange multiplier test of the hypothesis that the random walk has zero variance. KPSS-type tests are intended to complement unit root tests, such as the Dickey–Fuller tests. By testing both the unit root hypothesis and the stationarity hypothesis, one can distinguish series that appear to be stationary, series that appear to have a unit root, and series for which the data (or the tests) are not sufficiently informative to be sure whether they are stationary or integrated.

2004

• (Hobijin et al., 2004) ⇒ Hobijn, B., Franses, P. H., & Ooms, M. (2004). Generalizations of the KPSS‐test for stationarity. Statistica Neerlandica, 58(4), 483-502. http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9574.2004.00272.x/full
  • Tests for the null hypothesis of stationarity have not yet become part of the standard tools of empirical time series analysts. In many cases, however, the hypothesis of stationarity is more likely than the more frequently used hypothesis of (autoregressive) unit root nonstationarity. If one only uses autoregressive unit root (Dickey-Fuller) type tests the hypothesis of stationarity is only chosen if one rejects the null hypothesis of a unit root. Most unit root tests have low power against stationary, and highly autoregressive alternatives. This standard approach, therefore, entails that stationarity is not often found.

An important argument against the use of tests for the null hypothesis of stationarity is the difficulty to control their size when the process is stationary, but highly autoregressive. Probably the best known test for stationarity in econometrics, the so-called KPSS test introduced by Kwiatkowski, Phillips, Schmidt and Shin (1992) is oversized in that case: it rejects the true hypothesis of stationarity too often, again leading to undue preference for the hypothesis if unit root nonstationarity.

1992

• (Kwiatkowski et al., 1992) ⇒ Kwiatkowski, D., Phillips, P. C., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?. Journal of econometrics, 54(1-3), 159-178. http://www.ccee.edu.uy/ensenian/catmetec/material/KPSS.pdf
  • We propose a test of the null hypothesis that an observable series is stationary around a deterministic trend. The series is expressed as the sum of deterministic trend, random walk, and stationary error, and the test is the LM test of the hypothesis that the random walk has zero variance. The asymptotic distribution of the statistic is derived under the null and under the alternative that the series is difference-stationary. Finite sample size and power are considered in a Monte Carlo experiment. The test is applied to the Nelson-Plosser data, and for many of these series the hypothesis of trend stationarity cannot be rejected.

---