Unit Root - Real Interest Rate I want to find out whether the real interest rate of different countries are non-stationary. The real interest rate is defined as the difference between the nominal interest rate and the inflation rate. 
My first approach was to calculate the real interest rate and use it directly for in an augmented Dickey-Fuller-Test. Based on the results I can not reject the null hypothesis of non-stationarity for all countries. Therefore my first instinct was to assume that the real interest rate is non-stationary. 
Another approach would be to test the nominal interest rate and the inflation rate separately (this approach is the one mostly used in the literature). If only one of them contains a unit root the real interest rate is also non-stationary. But this is not the case for me. Looking at the nominal rate and the inflation rate  I can never reject the null hypothesis using an augmented Dickey-Fuller-Test for all countries, i. e. nominal interest rate and inflation are non-stationary. This could mean that if the nominal interest rate and the inflation rate are conintegrated the real interest rate would be stationary. Therefore I use the Johansen approach to test for cointegration. The test always indicates no cointegration which means that the real interest rate is non-stationary which confirms my previous results.
And here is my question: Why not use the real interest rate directly and which approach is "better"?
 A: Finding cointegration between (I'm assuming everything in natural logs) $r =$ (nominal interest rate) and $\pi = $ (inflation rate) -- make sure you know whether you want ex ante or ex post real rates -- wouldn't imply stationarity of $r-\pi$, merely that there's some $\alpha,\beta$ such that $\alpha r + \beta \pi$ is stationary; $\alpha = 1, \beta = -1$ being only one such linear combination. 
Otherwise, your approach with the ADF test appears to be in line with papers in the literature such as

Rose, A. K. (1988). Is the real interest rate stable?. The Journal of Finance, 43(5), 1095-1112.

and 

Lai, Kon S. "On structural shifts and stationarity of the ex ante real interest rate." International Review of Economics & Finance 13.2 (2004): 217-228.

Most of the literature however is concerned with detecting or filtering the effect of regime shifts (such as those seen in Latin America) rather than studying very stable countries like the US/UK/EU. So maybe the first thing is to examine whether there are strong exogenous shocks in policy that would either shake interest rates violently or provoke unexpected inflation (remember again the difference between ex ante and ex post rates).
