Abstract
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.
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