This paper proposes bootstrap assisted specification tests for the autoregressive frac-tionally integrated moving average model based on the Bartlett Tp-process with estimated parameters whose limiting distribution under the null depends on the esti-mated model and the estimation method employed. The computation of the asymp-totic critical values is not easy if at all possible under these circumstances. To circumvent this problem Delgado, Hidalgo, and Velasco (2005, Annals of Statistics 33, 2568–2609) proposed an asymptotically pivotal transformation of the Tp-process with estimated parameters. The aim of this paper is twofold. First, to examine alter-native methods based on bootstrap algorithms for estimating the distribution of the test under the null, showing its validity. And second, to study the finite-sample per-formance of the different alternative procedures via Monte Carlo simulation. 1
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