Local asymptotic powers of nonparametric and semiparametric tests for fractional integration

The paper concerns testing long memory for fractionally integrated nonlinear processes. We show that the exact local asymptotic power is of order O [(log n)^{- 1}] for four popular nonparametric tests and is O (m^{- 1 / 2}), where m is the bandwidth which is allowed to grow as fast as n^κ, κ ∈ (0, 2 / 3), for the semiparametric Lagrange multiplier (LM) test proposed by Lobato and Robinson [I. Lobato, P.M. Robinson, A nonparametric test for I (0), Rev. Econom. Stud. 68 (1998) 475-495]. Our theory provides a theoretical justification for the empirical findings in finite sample simulations by Lobato and Robinson [I. Lobato, P.M. Robinson, A nonparametric test for I (0), Rev. Econom. Stud. 68 (1998) 475-495] and Giraitis et al. [L. Giraitis, P. Kokoszka, R. Leipus, G. Teyssiére, Rescaled variance and related tests for long memory in volatility and levels, J. Econometrics 112 (2003) 265-294] that nonparametric tests have lower power than LM tests in detecting long memory.