econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...

, Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests are used for testing a

null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

that an observable

time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

is stationary around a deterministic trend (i.e. trend-stationary) against the alternative of a

unit root In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is ...

. Contrary to most

unit root test In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either Stationary process, s ...

s, 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 series to be non-stationary, have no

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 series 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). Later,

Denis Kwiatkowski Denis may refer to: People * Saint Denis of Paris, 3rd-century Christian martyr and first bishop of Paris * Denis the Areopagite, Biblical figure * Denis, son of Ampud (died 1236), baron in the Kingdom of Hungary * Denis the Carthusian (1402–1 ...

Peter C. B. Phillips Peter Charles Bonest Phillips (born 23 March 1948) is an econometrician. Since 1979 he has been Professor of Economics and Statistics at Yale University. He also holds positions at the University of Auckland, Singapore Management University and t ...

, Peter Schmidt and

Yongcheol Shin Yongcheol Shin (born 24 December 1960) is a South Korean-born British economist at the University of York. He has previously held positions at leading academic institutions such as University of Cambridge, University of Edinburgh and Universit ...

(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 In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

, and stationary error, and the test is the

Lagrange multiplier test In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the ''score''—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the ...

of the hypothesis that the random walk has zero variance. KPSS-type tests are intended to complement

s, such as the

Dickey–Fuller test In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationa ...

s. 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.

References

* {{Cite journal , doi = 10.1016/0304-4076(92)90104-Y, title = Testing the null hypothesis of stationarity against the alternative of a unit root, journal = Journal of Econometrics, volume = 54 , issue=1–3, pages = 159–178, year = 1992, last1 = Kwiatkowski , first1 = D., last2 = Phillips , first2 = P. C. B., last3 = Schmidt , first3 = P., last4 = Shin , first4 = Y. (
free pdf of the 1992 paper is hosted here
on deu.edu.tr) Time series statistical tests