## Abstract

Bootstrap likelihood ratio tests of cointegration rank are commonly used because they tend to have rejection probabilities that are closer to the nominal level than the rejection probabilities of asymptotic tests. The effect of bootstrapping the test on its power is largely unknown. We show that a new computationally inexpensive procedure can be applied to the estimation of the power function of the bootstrap test of cointegration rank. The bootstrap test is found to have a power function close to that of the level-adjusted asymptotic test. The bootstrap test therefore estimates the level-adjusted power of the asymptotic test highly accurately. The bootstrap test may have low power to reject the null hypothesis of cointegration rank zero, or underestimate the cointegration rank. An empirical application to Euribor interest rates is provided as an illustration of the findings.

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## Notes

In DGP 1 with cointegration rank \(r_{0}=1\) the condition 2 (ii) for testing \( r=0\) is that \({\varvec{\rho }} (\varvec{\Phi }^{(0)})<1\), where \( {\varvec{\Phi }}^{(0)}= \varvec{\Gamma }_{1}\). Since \(\rho (\varvec{\Phi }^{(0)})=\rho \ (\varvec{\Gamma }_{1})\), the condition is satisfied for \(\gamma =0.8\) and \(\delta =0\), \(0.1\), but not for \(\gamma =0.8\) and \(\delta =0.2\). In DGP 2 with cointegration rank \(r_{0}=2\) the condition 2 (ii) for testing \( r=0\) is that \(\rho (\varvec{\Phi }^{(0)})<1\) and the condition for testing \( r=1\) is that \(\rho (\varvec{\Phi }^{(0)})<1\) and \(\rho (\varvec{\Phi } ^{(1)})<1\). The condition \(\rho (\varvec{\Phi }^{(1)})<1\) is satisfied for \( \gamma =0.8\) and \(\delta =0,\,0.1\), but not for \(\gamma =0.8\) and \(\delta =0.2\).

The maximum number of replications is set to \(10\)m. Consequently, if the failure rate of the bootstrap algorithm exceeds \(99\,\%\) no results are obtained for the rejection probability of the test.

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## Acknowledgments

The authors want to thank two anonymous referees for their comments, and the Editor. Their comments have greatly improved the paper. The paper was presented in the 57th Session of the International Statistical Institute, Durban, 2009, and the 3rd International Conference on Computational and Financial Econometrics, 2009, Limassol. Niklas Ahlgren acknowledges financial support from Magnus Ehrnrooths Stiftelse.

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## Appendices

### Appendix 1: Computational details

To compute the rejection probability of a bootstrap likelihood ratio test of cointegration rank by (8) in model (10) with \(p=5,\,k=2\) and \(T=100\) when \(M=100000\) and \(B=1000\) requires about \(18\) h \(14\) min CPU time on the Core \(2\) Duo CPU, \(2.4\) GHz, \(2.00\) GB memory machine used to estimate the power functions in the paper. The total computational time for estimating a power function with \(100\) points would be about \(76\) days CPU time, making the computational time intractable. The computational time increases with the series length and the numbers of lags in the VAR model. Models with conditionally heteroskedastic errors require more computational time than models with normal errors.

The method in Davidson and MacKinnon (2006) reduces the computational burden by a factor of \(B/2\) LR statistics, which for \(B=1000\) is \(500\). The computational time to compute the rejection probability of a bootstrap likelihood ratio test of cointegration rank by (9) is about \(2\) min and \(21\) s CPU time. The estimation of a power function with \( 100\) points takes about \(3\) h \(55\) min CPU time.

### Appendix 2: Checking the approximation

In order to check the accuracy of the approximation in (9), we estimate the rejection probabilities of the bootstrap tests by (8) with \(M=100000\) and \(B=1000\) for some points on the power functions in Fig. 1. Table 9 compares the estimated rejection probabilities of the bootstrap tests with \(B=1000\) and \(B=1\). The differences in the rejection probabilities are smaller than \(0.02\,\%\) for size and smaller than \(0.5\,\%\) for power.

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Ahlgren, N., Antell, J. The power of bootstrap tests of cointegration rank.
*Comput Stat* **28**, 2719–2748 (2013). https://doi.org/10.1007/s00180-013-0425-6

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DOI: https://doi.org/10.1007/s00180-013-0425-6