KPSS Tests With A Stationarity Null

Why Test Stationarity Directly?

Unit-root tests such as the augmented Dickey–Fuller and Phillips–Perron tests use a unit root as their null hypothesis. The KPSS test reverses the testing perspective: its null is that the series is stationary around specified deterministic terms. This distinction matters because a failure to reject a unit-root null and a rejection of a stationarity null provide different evidence than either result alone.

KPSS is not a test of whether the sample looks smooth or whether its estimated trend is statistically significant. It asks whether deviations from a chosen deterministic component are consistent with a stationary process, allowing for short-run serial dependence. The choice between removing only a level and removing a level plus a linear trend is part of the test specification.

Model, Null, and Statistic

A standard representation for the KPSS null and alternative is

Here \(d_t\) is the deterministic component: \(d_t=\mu\) for the level version, or \(d_t=\mu+\beta t\) for the trend version. The component \(u_t\) is a zero-mean, weakly stationary disturbance with a finite, positive long-run variance. The component \(r_t\) is a random walk, with innovations \(\eta_t\) of mean zero and variance \(\sigma_\eta^2\), independent of \(u_t\) under the standard setup. The null is \(H_0:\sigma_\eta^2=0\), so there is no stochastic trend; the alternative is \(H_a:\sigma_\eta^2>0\).

For the chosen deterministic specification, estimate \(d_t\) by ordinary least squares and let \(\hat{u}_t\) be the residual for observation \(t\). Define the partial sum \(S_t=\sum_{j=1}^{t}\hat{u}_j\). The KPSS statistic is

where \(\hat{\omega}^{\,2}\) estimates the long-run variance of the stationary disturbance. Under the null, the statistic has a nonstandard limiting distribution, and its critical values differ between the level and trend versions. It is not compared with the usual \(t\), \(F\), or chi-square critical values.

Estimating the Long-Run Variance

The long-run variance is the sum of the disturbance’s autocovariances across all lags. It accounts for serial dependence that would be missed by using only the variance at lag zero. A practical estimate truncates and weights the sample autocovariances. One common choice is the Bartlett, or Newey–West, estimate:

The nonnegative integer \(\ell\) is the bandwidth, or maximum included lag. The Bartlett weights decline to zero at the truncation point. Setting \(\ell=0\) gives \(\hat{\omega}^{\,2}=\hat{\gamma}_0\), which is useful for hand calculations but may be inadequate when the residuals are serially correlated. In asymptotic use, bandwidth rules are chosen so that the bandwidth grows with the sample while remaining small relative to it; in practice, software defaults and finite-sample choices can affect the result.

Worked Calculations

Why the Partial Sums Matter

The KPSS construction accumulates residuals over time. Under the stationarity null, residual partial sums fluctuate, but their scale is governed by the long-run variance. The factor \(n^2\) in the denominator and the estimate \(\hat{\omega}^{\,2}\) put the sum of squared partial sums on a scale with a nonstandard limiting distribution. If the series instead contains a stochastic trend, detrending does not generally remove its persistent movements: cumulative residuals tend to become large relative to the stationary-noise benchmark, pushing the statistic upward.

This explains both the direction of the test and its sensitivity to specification. A suitable deterministic component must be chosen before forming residuals. Omitting a genuine linear trend can make trend-stationary data look nonstationary; adding a trend when the intended null is level stationarity changes the test and its critical values. Serial dependence also matters: the lag-zero variance estimate alone can misrepresent the scale of cumulative fluctuations.

Using KPSS Alongside Unit-Root-Null Tests

KPSS is most informative when read alongside tests such as ADF or Phillips–Perron, which use a unit-root null. The tests have different null hypotheses, so their results are complementary rather than interchangeable. A common summary is:

This comparison does not create a mechanical decision rule. The tests may use different deterministic terms, lag or bandwidth choices, and finite-sample approximations. Results should be interpreted in light of those choices and the economic context. In particular, neither a KPSS rejection nor a unit-root-test nonrejection alone establishes the data-generating process.

Common Mistakes and Practical Checks

• Reversing the null. KPSS has a stationarity null, not a unit-root null. A large statistic is evidence against stationarity.
• Using the wrong deterministic specification. The level and trend versions use different residuals and different critical values. Do not compare a trend-version statistic with level-version critical values.
• Ignoring serial correlation. The long-run variance estimate is not automatically just the residual variance at lag zero. State the kernel and bandwidth or lag choice used.
• Calling nonrejection proof. Not rejecting stationarity means the test did not find sufficient evidence against it at the selected level; it does not prove the null true.
• Combining tests as if they had one null. ADF or Phillips–Perron nonrejection and KPSS rejection are statements about distinct null hypotheses. Report each result separately before drawing a qualified overall conclusion.