This article considers the modelling of short-term interest rates with the ARFIMA model in six European countries based on daily data in the 1990s using the Modified Profile Likelihood estimation method. This allows one to study the different convergence processes that have been followed in each case. Empirical evidence shows that, even with this estimation method, the standard AIC tends to select models that in some cases are in accordance with traditional inference but in other cases may not be so. Analysing these results, the series for Switzerland appears to be an I(1) series, which conflicts with the findings in previous literature.