A bootstrap approximation for the distribution of the Local Whittle estimator

The asymptotic properties of the Local Whittle estimator of the memory parameter d have been widely analysed and its consistency and asymptotic distribution have been obtained for values of d∈(−1/2,1] in a wide range of situations. However, the asymptotic distribution may be a poor approximation of the exact one in several cases, e.g. with small sample sizes or even with larger samples when d>0.75. In other situations the asymptotic distribution is unknown, as for example in a noninvertible context or in some nonlinear transformations of long memory processes, where only consistency is obtained. For all these cases a bootstrap strategy based on resampling a (perhaps locally) standardised periodogram is proposed. A Monte Carlo analysis shows that this strategy leads to a good approximation of the exact distribution of the Local Whittle estimator in those situations where the asymptotic distribution is not reliable.