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"?