Efficient Wald Tests for Fractional Unit Roots

Abstract

In this article we introduce efficient Wald tests for testing the null hypothesis of unit root against the alternative of fractional unit root. In a local alternative framework, the proposed tests are locally asymptotically equivalent to the optimal Robinson (1991Robinson ( , 1994a) Lagrange Multiplier tests. Our results contrast with the tests for fractional unit roots introduced by Dolado, Gonzalo and Mayoral (2002) which are inefficient. In the presence of short range serial correlation, we propose a simple and efficient two-step test that avoids the estimation of a nonlinear regression model. In addition, the first order asymptotic properties of the proposed tests are not affected by the pre-estimation of short or long memory parameters * Acknowledgements: We thank the co-editor and two referees for very useful comments, J. AbstractIn this article we introduce e¢ cient Wald tests for testing the null hypothesis of unit root against the alternative of fractional unit root. In a local alternative framework, the proposed tests are locally asymptotically equivalent to the optimal Robinson (1991Robinson ( , 1994a Lagrange Multiplier tests. Our results contrast with the tests for fractional unit roots introduced by Dolado, Gonzalo and Mayoral (2002) which are ine¢ cient. In the presence of short range serial correlation, we propose a simple and e¢ cient two-step test that avoids the estimation of a nonlinear regression model. In addition, the …rst order asymptotic properties of the proposed tests are not a¤ected by the pre-estimation of short or long memory parameters.We thank the co-editor and two referees for very useful comments,

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“…Table 3 7 As in other standard large-sample tests, Wald and LR test statistics against fractional alternatives have the same null and limit theory as the LM test of Robinson (1994). Lobato and Velasco (2007) essentially employed such a Wald testing procedure, and, although this and other recent methods such as the one developed by Demetrescu et al (2008) have even been shown to be robust with respect to unconditional heteroscedasticity (Kew and Harris, 2009) [Insert Table 3 about here]…”

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“…Other parametric methods for estimating d based on the frequency domain were proposed, among others, by Fox and Taqqu (1986) and Dahlhaus (1989) (see also Robinson, 1994 andLobato andVelasco, 2008 …”

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“…Speci…cally, we focus on a modi…cation of the Fractional Dickey-Fuller (FDF) test by Dolado, Gonzalo and Mayoral (2002;DGM hereafter) recently proposed by Lobato and Velasco (2007;LV hereafter) which achieves a slight improvement in e¢ ciency over the former. This test, henceforth denoted as the EFDF (e¢ cient FDF) test, generalizes the traditional DF test of I(1) against I(0) processes without deterministic components to the broader framework of testing I(1) against I(d) with 0 d < 1.…”