Vector error correction model and trending series

Question

I am struggling with VECM pretests.

1. I have conducted correlogram and it shows that all of my variables are nonstationary, except the log of adjusted GDP.

2. So I decide to double check and conduct ADF test. It shows that all variables are nonstationary when I conduct the ADF test with a constant. But when I add time trend to the regression it shows that null hypothesis of unit root is rejected for log of adjusted GDP. It means that instead of differencing I should eliminate the trend, right? My question is whether my VECM will take care of it and I can proceed further to JJ procedure and further estimation. PP tests show the same results regarding the log of adjusted GDP.

3. When i take the first difference of the variables and conduct the ADF including the trend and intercept I again reject the null hypothesis of stationarity which again makes me eliminate the trend, right? Which I guess is good. But isn't that weird that the level of log of adjusted GDP and the differences of logs of other variables (all of my variables are in log form) show the same results, when I include trend in the ADF?

4. Will my VECM be estimated properly if my log of adjusted GDP is nonstationary when tested with ADF, but stationary when trend included? Can I at all use VECM in that case?

Answer 1

First, what is adjusted GDP? Does that mean the seasonally-adjusted GDP? Or the deviations of GDP from its potential level (e.g. deviations of the actual GDP from a Hodrick-Prescott-smoothed-version of GDP)? Or something else?

I will assume it is seasonally adjusted GDP, and I will skip the word "adjusted" going forward.

1. log(GDP) is normally found or assumed to be an integrated process. You may just follow that tradition.

2. Nominal log(GDP) may be a random walk with a drift, the drift being the result of inflation, productivity growth, population growth etc. Real log(GDP) would be the same aside from inflation. However, it is less natural to expect log(GDP) to be a random walk with a drift and a trend, which would imply quadratic growth in GDP (level, not log). Therefore, including a trend in the ADF test need not be a good idea. Meanwhile, I understand that log(GDP) is found to be an integrated process if trend is not included in the ADF specification; this result does make sense.

3. Including a trend when applying the ADF test on $\Delta$log(GDP) makes even less sense than including a trend when testing log(GDP). A trend in the ADF test of $\Delta$log(GDP) would imply a cubic trend in GDP. That does not sound sensible. Normally, if you conduct the ADF test on growth rates such as $\Delta$log(GDP), you would exclude both the drift and the trend. Typically, the data would reject the null hypothesis that $\Delta$log(GDP) has a unit root.

4. You can do the cointegration analysis using the Johansen procedure (which is the most common) on variables that are all integrated of the same order. In economics, it is most often order 1 (with rare exceptions where order 2 could be relevant). If any of the variables has a time trend, the test specification can be altered (in R package "urca" you would use function ca.jo and its argument ecdet). You can use a VECM if you find that the variables are cointegrated. If they are not cointegrated, difference them and use a VAR model instead.