Bootstrap M Unit Root Tests

Abstract

In this article we propose wild bootstrap implementations of the local generalized least squares (GLS) de-trended M and ADF unit root tests of Stock (1999 Stock , J. H. ( 1999 ). A class of tests for integration and cointegration . In: Engle , R. F. , White , H. eds. Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger . Oxford : Oxford University Press , pp. 137 – 167 . [Google Scholar]), Ng and Perron (2001 Ng , S. , Perron , P. ( 2001 ). Lag length selection and the construction of unit root tests with good size and power . Econometrica 69 : 1519 – 1554 .[Crossref], [Web of Science ®] , [Google Scholar]), and Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]), respectively. The bootstrap statistics are shown to replicate the first-order asymptotic distributions of the original statistics, while numerical evidence suggests that the bootstrap tests perform well in small samples. A recolored version of our bootstrap is also proposed which can further improve upon the finite sample size properties of the procedure when the shocks are serially correlated, in particular ameliorating the significant under-size seen in the M tests against processes with autoregressive or moving average roots close to −1. The wild bootstrap is used because it has the desirable property of preserving in the resampled data the pattern of heteroskedasticity present in the original shocks, thereby allowing for cases where the series under test is driven by martingale difference innovations.

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