Hypothesis Testing > KPSS Test

## What is the KPSS Test?

The Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test figures out if a time series is stationary around a mean or linear trend, or is non-stationary due to a unit root. A stationary time series is one where statistical properties — like the mean and variance — are constant over time.

- The null hypothesis for the test is that the data is stationary.
- The alternate hypothesis for the test is that the data is
*not*stationary.

## Overview of How The Test is Run

The KPSS test is based on linear regression. It breaks up a series into three parts: a deterministic trend (βt), a random walk (r_{t}), and a stationary error (ε_{t}), with the regression equation:

**x**

_{t}= r_{t}+ βt + ε_{1}.If the data is stationary, it will have a fixed element for an intercept or the series will be stationary around a fixed level (Wang, p.33). The test uses OLS find the equation, which differs slightly depending on whether you want to test for level stationarity or trend stationarity (Kocenda & Cerný). A simplified version, without the time trend component, is used to test level stationarity.

Data is normally log-transformed before running the KPSS test, to turn any exponential trends into linear ones.

## Running the KPSS Test

**Note**: At the time of writing, SPSS doesn’t have an option for this test.

**In R:** kpss.test(x, null = c(“Level”, “Trend”), lshort = TRUE)

Where:

- x is a numeric vector or univariate time series,
- null is either “Level” or “Trend” (you can specify just “L” or “T”).
- lshort indicates if the short version (TRUE) or long version (FALSE) should be used.

Full details can be found in the r documentation.

**In Stata:**

^kpss^ varname [^if^ exp] [^in^ range] [^,^ ^m^axlag^(^#^)^ ^notrend^ ]

Note: You must ^tsset^ your data before using ^kpss^; see help @[email protected] or the full Stata description for the command here.

## Interpreting the Results

The KPSS test authors derived **one-sided LM statistics** for the test. If the LM statistic is greater than the critical value (given in the table below for alpha levels of 10%, 5% and 1%), then the null hypothesis is rejected; the series is non-stationary.

You can also look at the p-value returned by the test and compare it to your chosen alpha level. For example, a p-value of 0.02 (2%) would cause the null hypothesis to be rejected at an alpha level of 0.05 (5%).

## Cautions

A major disadvantage for the KPSS test is that it has a high rate of Type I errors (it tends to reject the null hypothesis too often). If attempts are made to control these errors (by having larger p-values), then that negatively impacts the test’s power.

One way to deal with the potential for high Type I errors is to combine the KPSS with an ADF test. If the result from both tests suggests that the time series in stationary, then it probably is.

**References**:

Kocenda, E. & Cerný, A. (2017). Elements of Time Series Econometrics: An Applied Approach. Karolinum Press.

Kwiatowski et. al (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54 159-178. Retrieved November 22, 2106 from here.

Wang, W. (2006). Stochasticity, Nonlinearity and Forecasting of Streamflow Processes.

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