This paper proposes goodness-of-fit tests for the class of covariance stationary FARIMA processes. They are based on functionals of weighted empirical processes, say Sn C.), where the weights are the relative error between the periodogram and the fitted spectral density function under the null specification of the data. Two examples of such functionals are the Tp - Barlett and the Cramer-Von Mises standardized ro - statistics. We show that the tests are able to detect contiguous alternatives converging to the null at the rate n-JI2 • However, because the cumbersome covariance structure of the limiting process of Sn C.), tests based on its asymptotic distribution are difficult to implement in practice_ To circumvent this problem, we propose a bootstrap test, showing its consistency, and studying its small sample performance by a Monte Carlo experiment. _________________________________________________
