## Abstract

We analyse the properties of the conventional Gaussian-based co-integrating

rank tests of Johansen (1996) in the case where the vector of series under test

is driven by globally stationary, conditionally heteroskedastic (martingale differ-

ence) innovations. We first demonstrate that the limiting null distributions of the

rank statistics coincide with those derived by previous authors who assume either

i.i.d. or (strict and covariance) stationary martingale difference innovations. We

then propose wild bootstrap implementations of the co-integrating rank tests and

demonstrate that the associated bootstrap rank statistics replicate the first-order

asymptotic null distributions of the rank statistics. We show the same is also true

of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006).

The wild bootstrap, however, has the important property that, unlike the i.i.d.

bootstrap, it preserves in the re-sampled data the pattern of heteroskedasticity

present in the original shocks. Consistent with this, numerical evidence sug-

gests that, relative to tests based on the asymptotic critical values or the i.i.d.

bootstrap, the wild bootstrap rank tests perform very well in small samples un-

der a variety of conditionally heteroskedastic innovation processes. An empirical

application to the term structure of interest rates is given.

rank tests of Johansen (1996) in the case where the vector of series under test

is driven by globally stationary, conditionally heteroskedastic (martingale differ-

ence) innovations. We first demonstrate that the limiting null distributions of the

rank statistics coincide with those derived by previous authors who assume either

i.i.d. or (strict and covariance) stationary martingale difference innovations. We

then propose wild bootstrap implementations of the co-integrating rank tests and

demonstrate that the associated bootstrap rank statistics replicate the first-order

asymptotic null distributions of the rank statistics. We show the same is also true

of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006).

The wild bootstrap, however, has the important property that, unlike the i.i.d.

bootstrap, it preserves in the re-sampled data the pattern of heteroskedasticity

present in the original shocks. Consistent with this, numerical evidence sug-

gests that, relative to tests based on the asymptotic critical values or the i.i.d.

bootstrap, the wild bootstrap rank tests perform very well in small samples un-

der a variety of conditionally heteroskedastic innovation processes. An empirical

application to the term structure of interest rates is given.

Original language | English |
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Place of publication | Aarhus |

Publisher | Institut for Økonomi, Aarhus Universitet |

Number of pages | 46 |

Publication status | Published - 2009 |

## Keywords

- Co-integration, trace and maximum eigenvalue rank tests, conditional heteroskedasticity, i.i.d. bootstrap, wild bootstrap