# What is the correct unit root/stationarity test for this variable? Why do different tests provide different conclusions?

This is something of a follow up question to a previous question I had here: Can over differencing cause a singular matrix in a VAR model?

A brief recap of what I am trying to accomplish: I want to forecast employment for a city using a VARS model. The major variable being forecast is total non-farm employment along with various employment shares (such as percent of nonfarm employment tied to software). Many of these variables are inherently non-stationary, so I first differenced the variables.

However, when testing the stationarity of nonfarm employment, I found some interesting results. Using the Augmented Dickey Fuller test on the log of nonfarm employment resulted in the test saying the data was stationary.

Augmented Dickey-Fuller Test

data:  ln_nonfarm
Dickey-Fuller = -4.1498, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary


I then did an ADF test on the first difference of nonfarm employment, which gave a result of the data being non-stationary.

Augmented Dickey-Fuller Test

data:  ln_nonfarm_d
Dickey-Fuller = -2.6703, Lag order = 4, p-value = 0.2991
alternative hypothesis: stationary


Puzzled by this, I then checked out the Phillips-Perron test on the log of nonfarm employment, which said the data was non-stationary (which I expected).

pp.test(ln_nonfarm)

Phillips-Perron Unit Root Test

data:  ln_nonfarm
Dickey-Fuller Z(alpha) = -5.4773, Truncation lag parameter = 3, p-value = 0.7989
alternative hypothesis: stationary


I then performed the Phillips-Perron test on the first difference of log of nonfarm employment, which provided a result that the data was now stationary.

pp.test(ln_nonfarm_d)

Phillips-Perron Unit Root Test

data:  ln_nonfarm_d
Dickey-Fuller Z(alpha) = -125.3459, Truncation lag parameter = 3, p-value =    0.01
alternative hypothesis: stationary


I then checked out the Kwiatkowski-Phillips-Schmidt-Shin test of the on the log of nonfarm employment and the first difference of nonfarm employment. The results of the two tests suggests that both the log and the first difference of nonfarm employment are not stationary.

Here is the KPSS test on log of nonfarm employment.

kpss.test(ln_nonfarm)

KPSS Test for Level Stationarity

data:  ln_nonfarm
KPSS Level = 2.9993, Truncation lag parameter = 2, p-value = 0.01


And here is the KPSS test on the first difference of logged nonfarm employment.

kpss.test(ln_nonfarm_d)

KPSS Test for Level Stationarity

data:  ln_nonfarm_d
KPSS Level = 0.1527, Truncation lag parameter = 2, p-value = 0.1


Of these tests, the only ones whose results make sense to me are the Phillips-Perron tests. I am curious as to why the other unit tests give contradictory results. What is the correct test and how should I transform the data so I can create an accurate forecast model? I have attached graphs of Logged Nonfarm Employment and First Differenced Logged Nonfarm Employment.