# 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?

• 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.
• 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. – Richard Hardy Jul 25 '16 at 7:11