Efficient inference in multivariate fractionally integrated time series models

Summary A computationally simple maximum likelihood procedure for multivariate fractionally integrated time series models is introduced. This allows, e.g., efficient estimation of the memory parameters of fractional models or efficient testing of the hypothesis that two or more series are integrated of the same possibly fractional order. In particular, we show the existence of a local time domain maximum likelihood estimator and its asymptotic normality, and under Gaussianity asymptotic efficiency. The likelihood‐based test statistics (Wald, likelihood ratio and Lagrange multiplier) are derived and shown to be asymptotically equivalent and chi‐squared distributed under local alternatives, and under Gaussianity locally most powerful. The finite sample properties of the likelihood ratio test are evaluated by Monte Carlo experiments, which show that rejection frequencies are very close to the asymptotic local power for samples as small as n= 100 .     

Residual-based block bootstrap unit root testing in the presence of trend breaks

Summary In this paper, we build on the ‘residual-based block bootstrap unit root testing’ (RBB) method, proposed by Paparoditis and Politis (2003) . We develop an extension of this method to allow for bootstrap unit root testing in a model defined by an augmented Dickey–Fuller (ADF) equation that contains linear combinations of arbitrary dummies as its deterministic part. The main application of such an extension is that it allows for unit root testing in the presence of arbitrary multiple trend breaks, such as jumps or changes of the slope of a linear trend. The model framework used here is an extension of Perron's ‘innovational outlier model’, and allows as well for gradual transitions of the expectation of the series when such breaks or outliers occur; this assumption is particularly appealing in the context of analysing economic time series. Our extension of the bootstrap method involves specifying a drift term and adjustments for the expectation of the residuals and the pseudo-differences, which all appropriately take into account the dependency structure. We prove asymptotic validity of the proposed modified bootstrap procedure in the case of a single break in slope. The small sample behaviour of the proposed methodology is studied in a simulation experiment.