Does the ARFIMA really shift?

Short memory models contaminated by level shifts have long-memory features similar to those associated to processes generated under fractional integration. In this paper, we propose a robust testing procedure, based on an encompassing parametric specification, that allows to disentangle the level shift term from the ARFIMA component. The estimation is carried out via a state-space methodology and it leads to a robust estimate of the fractional integration parameter also in presence of level shifts.The Monte Carlo simulations show that this approach produces unbiased estimates of the fractional integration parameter when shifts in the mean, or in other slowly varying trends, are present in the data. Once the fractional integration parameter is estimated, the KPSS test statistic is adopted to assess if the level shift component is statistically significant. The test has correct size and generally the highest power compared to other existing tests for spurious long-memory. Finally, we illustrate the usefulness of the proposed approach on the daily series of bipower variation and share turnover and on the monthly inflation series of G7 countries.