Testing for codependence of cointegrated variables

Abstract

We analyse nonstationary time series that do not only trend together in the long run, but restore the equilibrium immediately in the period following a deviation. While this represents a common serial correlation feature, the framework is extended to codependence, allowing for delayed adjustment. We show which restrictions are implied for the Moving Average (MA) and Vector Error Correction Model (VECM) representations and put forward a Generalized Method of Moments (GMM) test. In addition, for cases where the constraints can be uniquely imposed on a VECM a likelihood ratio test is proposed. We apply the concept to US and European interest rate data, examining the capability of the Federal Reserve Bank (Fed) and European Central Bank (ECB) to control overnight money market rates.

Keywords:

• serial correlation common features
• codependence
• cointegration
• overnight interest rates
• central banks

JEL Classification::

• C32
• E52

Acknowledgements

We are grateful to Kyusang Yu and participants of seminars at the Central Bank Norway, the Humboldt University Berlin, the Leibniz University Hannover, and the University of Mannheim for very helpful comments. Of course, all remaining errors are our own. The research was supported by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 884 ‘Political Economy of Reforms’.

Notes

¹The full row rank assumptions on and γ′ ensure that the polynomials , , are of full-column rank in the sense of Franchi and Paruolo (2011).

²Note that δ′x _t can be given a univariate MA() representation with according to Lütkepohl (2005, Proposition 11.1). However, the error term in this representation is not equal to δ′ϵ _t .

³Hence, δ′x _t does not need to be I(1) if δ is a codependence vector with respect to Δx _t in contrast to the statement in Schleicher (2007, Appendix A).

⁴For joint imposition of several codependence vectors on a VECM see the Section ‘Test for minimal codependence orders’.

⁵As an alternative to the GMM test, one may use a test based on canonical correlations that was suggested by Tiao and Tsay (1989). However, this procedure does not allow to impose the specific composition of codependence orders tested under the null hypothesis. Therefore, we do not consider the procedure of Tiao and Tsay (1989).

⁶On 15 October 2008, the ECB switched to fixed rate tenders. So, the last few, but indecisive, MBR observations in our sample effectively equal the fixed rate. The end point is chosen as in the US case for reasons of comparability.

⁷The GMM test cannot be adjusted to the restricted VAR model.

⁸Note that the upper bound for the codependence order in a VECM is given by q _max = 2 in the current setup. Therefore, we did not apply the GMM test to orders beyond two, see Section ‘VAR representation and level codependence’.