Solved – P-value of Augmented Dickey-Fuller test and KPSS test

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I would like to test if the time series of the US 3-month treasury bills (monthly data from 1934 to 2015) is stationary. I'm using the ADF test in R (from the package tseries), but I get contradictory results: if I change the alternative hypothesis the p-value is always high.

Augmented Dickey-Fuller Test

data: x Dickey-Fuller = -2.0698, Lag order = 9, p-value = 0.5488
alternative hypothesis: stationary

Augmented Dickey-Fuller Test

data: x Dickey-Fuller = -2.0698, Lag order = 9, p-value = 0.4512
alternative hypothesis: explosive

The same thing happen if I use the KPSS test (p-value is always low):

KPSS Test for Level Stationarity

data: x KPSS Level = 3.3275, Truncation lag parameter = 7, p-value =
0.01

KPSS Test for Trend Stationarity

data: x KPSS Trend = 2.1869, Truncation lag parameter = 7, p-value =
0.01

What am I doing wrong?

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Best Answer

Two things:

With ADF, what you do is to test both the null of a unit root against a stationary process as well as against an explosive process, i.e., in a model like $y_t=rho y_{t-1}+epsilon_t$, that $rho=1$ against $|rho|<1$ or against $rho>1$.

There is no reason whatsoever that inability to reject a null against an alternative in one direction should automatically imply that we will be able to reject in the opposite direction. This is not specific to unit root tests at all: it is perfectly possible that the data is not sufficiently informative to reject the null that a regression coefficient is zero against a positive or against a negative coefficient.

With KPSS you are not looking at the same types of alternatives. Instead, you are using two different specifications for the deterministic trend part of the process, level and trend. You first test the null that the process is stationary around some constant mean, and in the second case, that the process is stationary around some time trend.

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