  This work considers the problem of modified portmanteau tests for testing the
adequacy of FARIMA models under the assumption that the errors are uncorrelated
but not necessarily independent (i.e. weak FARIMA). We first study the joint
distribution of the least squares estimator and the noise empirical
autocovariances. We then derive the asymp-totic distribution of residual
empirical autocovariances and autocorrelations. We deduce the asymptotic
distribution of the Ljung-Box (or Box-Pierce) modified portmanteau statistics
for weak FARIMA models. We also propose another method based on a
self-normalization approach to test the adequacy of FARIMA models. Finally some
simulation studies are presented to corroborate our theoretical work. An
application to the Standard \& Poor's 500 and Nikkei returns also illustrate
the practical relevance of our theoretical results. AMS 2000 subject
classifications: Primary 62M10, 62F03, 62F05; secondary 91B84, 62P05.
