Confidence Region for long memory based on Inverting Bootstrap Tests: an application to Stock Market Indices

In the context of long memory, the finite-sample distorsion of statistic distributions is so large, that bootstrap confidence intervals (percentile and percentile-t) for the long memory parameter do not perform better than the corresponding asymptotic confidence interval. In this paper, we propose confidence intervals based on inverting bootstrap tests proposed by Davidson and MacKinnon [2001] and adapted to the long memory parameter in the ARFIMA model. We show that classical confidence intervals have very poor performances, even the percentile-t interval, whereas confidence intervals based on inverting bootstrap tests have quite satisfactory performance. For this purpose, we use techniques for measuring effectiveness of confidence regions and for the graphical display of simulation evidence concerning the coverage and effectiveness of confidence regions in finite sample. These graphical techniques are inspired by the graphical techniques of Davidson and MacKinnon [1998] in the case of tests. Monte Carlo results on the confidence intervals for various situations are presented. These intervals are then applied on stock market indices.