How can this time series be stationary?

Question

I have the following irregularly spaced time series.

The related autocorrelogram is:

and I run the following tests:

> adf.test(x)

Augmented Dickey-Fuller Test

data:  x
Dickey-Fuller = -70, Lag order = 70, p-value = 0.01
alternative hypothesis: stationary

Warning message:
In adf.test(x) : p-value smaller than printed p-value


> kpss.test(x)

KPSS Test for Level Stationarity

data:  x
KPSS Level = 30, Truncation lag parameter = 100, p-value = 0.01

Warning message:
In kpss.test(x) : p-value smaller than printed p-value


> Box.test(x, type = "Ljung-Box")

Box-Ljung test

data:  x
X-squared = 20000, df = 1, p-value <2e-16

How can this time series be stationary? I suppose problems arise because my time series is irregularly spaced. In this case, there exist methods to assess stationarity with irregularly spaced time series?

Answer 1

While the data does not appear to have a unit root (which is also confirmed by the ADF test results), KPSS test rejects stationarity. Recall that the null hypothesis of KPSS is stationarity (in contrary to the ADF test). That is quite intuitive when you look at the graph.

Edit (to respond to edited post)
Of course, irregular spacing will be a problem. My answer only addresses the apparently incorrect interpretation of the KPSS test results.