A Fractional Dickey–Fuller Test for Unit Roots

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Instrumental variable and variable addition based inference in predictive regressions (2015, methodology): “and note that Lemma 2 (17) in (Dolado et al., 2002) implies that 1 T ∑( ∆−d ∗ + vt−1 )2 p → σ(2) v Γ (1− 2d∗) Γ2 (1− d∗) .”
NUMERICAL DISTRIBUTION FUNCTIONS OF FRACTIONAL UNIT ROOT AND COINTEGRATION TESTS (2010, background): “Dolado et al. (2002) also concludes that a fractional unit root is present with very similar estimated values of d.”
THE ROLE OF INITIAL VALUES IN CONDITIONAL SUM-OF-SQUARES ESTIMATION OF NONSTATIONARY FRACTIONAL TIME SERIES MODELS (2015, methodology): “That is, (10) is an approximation to the conditional likelihood (9), where (10) has the advantage that it can be calculated based on available data from t = 1 to T0 = N + T .”
Multivariate Lagrange Multiplier Tests for Fractional Integration (2005, unknown): “Instead, (5) is a regression variant along the lines of the Dickey-Fuller test and the fractional Dickey-Fuller test, see Dolado et al. (2002).”
Modelling structural breaks, long memory and stock market volatility: an overview (2005, methodology): “Finally, a new test for fractionally integrated processes is proposed by Dolado et al. (2002), where the testing procedure is in the time domain and extends the well-known Dickey–Fuller approach.”