# Seasonal data deemed stationary by ADF and KPSS tests

I have got two time series and I want to evaluate a VAR model. For this, it is necessary that both time series are stationary.

Using R, I have found periodicity with the function spectrum in the lag 16 and 98 in both time series and lots of others in the goal-function y-Data you see in the second picture.

Obviously, both time series are seasonal. In my opinion, the consequence of this is, that the time series both are nonstationary, because the expected value of the time series depends on time.

Now I check stationarity with the ADF and the KPSS tests, and both seem to suggest stationarity.

adf.test(Data)

Augmented Dicke y-Fuller Test

data:  Data
Dickey-Fuller = -3.4722, Lag order = 7, p-value = 0.04498
alternative hypothesis: stationary


KPSS:

kpss.test(Data, null="L", lshort="F")

KPSS Test for Level Stationarity

data:  Data
KPSS Level = 0.03706, Truncation lag parameter = 15, p-value = 0.1


Question: Why do they indicate stationarity?

Both the augmented Dickey-Fuller (ADF) test and the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test are tailored for detecting nonstationarity in the form of a unit root in the process. (The test equations explicitly allow for a unit root; see the refence below.) However, they are not tailored for detecting other forms of nonstationarity. Therefore, it is not surprising that they do not detect nonstationarity of the seasonal kind.

• The result of the ADF test ($p$-value below 0.05) suggests that the null hypothesis of presence of a unit root can be rejected at 95% confidence level.
• The result of the KPSS test ($p$-value above 0.05) suggests that the null hypothesis of absence of a unit root presence of unit root cannot be rejected at 95% confidence level.

(The bullet points are there just to confirm what you implied.)

For an accessible and intuitive yet technically precise treatment of the ADF and the KPSS tests I suggest Eric Zivot's "Modelling Financial Time Series with S-PLUS" (2nd ed., 2006) Chapter 4 "Unit Root Tests" (especially sections 4.3 and 4.4).

• Thanks for your answer, especially the link. However, there remains one question. Is there a test which tests on seasonality? My problem is, a VAR-model expects two or more stationary time series, but my time series show multiple seasonality - so differencing doesn't work. I've tried to difference the time series in the main lags and estimates a VAR-model thereafter (although there are still some cyclical effects). I hope that both time series are "stationary enough" to interpret the results and therefor I need a test, which tests for seasonality. Do you know a test that could help? Commented Jul 25, 2016 at 7:01
• See this (recent) thread which asks the same question. In practice you could either (1) seasonally adjust the data before using the VAR model or (2) include some seasonal terms such as dummy variables or Fourier terms as exogenous regressors in the VAR model (see, for example, functions seasonaldummy and fourier in "forecast" package in R). If you neglect seasonality, you may find spurious relationships purely due to coinciding seasonal patterns. Commented Jul 25, 2016 at 7:11

You can use HEGY test for seasonality or CH test for seasonality to check for seasonal unit roots. Better to use HEGY test. ADF and KPSS test for non seasonal unit roots. Since your seasonality is strong, take the seasonal difference and proceed to test for non seasonal unit roots. Probably seasonal differencing will remove all forms of non-stationarity and give you a stationary series. This results in a seasonally differenced series.