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In this paper, we show that the central limit theorem (CLT) satisfied by the data-driven Multidimensional Increment Ratio (MIR) estimator of the memory parameter  d  established in Bardet and Dola (2012. Adaptive Estimator of the Memory Parameter and Goodness-of-Fit Test Using a Multidimensional Increment Ratio Statistic.”  Journal of Multivariate Analysis  105:222–40) for  dϵ (–0.5, 0.5) can be extended to a semiparametric class of Gaussian fractionally integrated processes with memory parameter  d ϵ(–0.5, 1.25). Since the asymptotic variance of this CLT can be estimated, by data-driven MIR tests for the two cases of stationarity and non-stationarity, so two tests are constructed distinguishing the hypothesis  d  &lt; 0.5 and  d ≥0.5, as well as a fractional unit roots test distinguishing the case  d =1 from the case  d  &lt; 1. Simulations done on numerous kinds of short-memory, long-memory and non-stationary processes, show both the high accuracy and robustness of this MIR estimator compared to those of usual semiparametric estimators. They also attest of the reasonable efficiency of MIR tests compared to other usual stationarity tests or fractional unit roots tests.

Semiparametric Stationarity and Fractional Unit Roots Tests Based on Data-Driven Multidimensional Increment Ratio Statistics