Bootstrap Score Tests for Fractional Integration in Heteroskedastic ARFIMA Models, with an Application to Price Dynamics in Commodity Spot and Futures Markets

We propose bootstrap implementations of the asymptotic Wald, likelihood ratio and Lagrange multiplier tests for the order of integration of a fractionally integrated time series. Our main purpose in doing so is to develop tests which are robust to both conditional and unconditional heteroskedasticity of a quite general and unknown form in the shocks. We show that neither the asymptotic tests nor the analogues of these which obtain from using a standard i.i.d. bootstrap admit pivotal asymptotic null distributions in the presence of heteroskedasticity, but that the corresponding tests based on the wild bootstrap principle do. An heteroskedasticity-robust Wald test, based around a sandwich estimator of the variance, is also shown to deliver asymptotically pivotal inference under the null, and we show that it can be successfully bootstrapped using either i.i.d. resampling or the wild bootstrap. We quantify the dependence of the asymptotic size and local power of the asymptotic tests on the degree of heteroskedasticity present. An extensive Monte Carlo simulation study demonstrates that signi.cant improvements in .nite sample behaviour can be obtained by the bootstrap vis-à-vis the corresponding asymptotic tests in both heteroskedastic and homoskedastic environments. The results also suggest that a bootstrap algorithm based on model estimates obtained under the null hypothesis is preferable to one which uses unrestricted model estimates.