Re-examining the Fisher Effect: An Application of Small Sample Distributions of the Covariate Unit Root Test

Abstract

This article employs the covariate unit root test proposed by Elliott and Jansson to investigate the stationarity properties of real interest rates. Instead of blindly trusting the asymptotic distribution of the test, we extend Rudebusch's method to estimate its finite sample distributions under the null and alternative hypotheses. With these distributions, we can obtain the probabilities that the test statistic comes from the null and alternative hypotheses, and quantify the asymptotic size as well as the test power for each specific series. Our simulation experiments show that first, due to the higher power raised by the inclusion of covariates, the test can overwhelmingly reject the unit root null for the 16 industrialized countries; secondly, the Ng and Perron tests deliver lower powers in most countries, and thus lead to the false conclusion of non-stationary real interest rates. Finally, allowing for multiple endogenous breaks in the real interest rates provides only stationary evidence in half of the 16 countries.

Keywords:

• Real interest rate
• fisher effect
• covariate unit root test
• small sample distribution
• stationarity

Keywords:

• C32
• E44
• G12

Acknowledgements

The authors are deeply grateful to the editor and the anonymous referee for their insightful comments and suggestions on an earlier version of this article. All errors are ours. This research is financially supported by the National Science Council of Taiwan (NSC 99-2410-H-260-019).

Notes

1. We also select the lag with BIC and obtain similar results.

2. For more information about the EGLS, please refer to Lütkepohl (1991), 5.2.1 and 5.2.2.

3. See Lütkepohl (1991), 4.4.21 and 4.4.23 for details.

4. We thank the referee for this helpful comment.

5. The case that both pv–H ₀ and pv–H ₁ are not greater than 0.05 never happens in our empirical study and is omitted for brevity.

6. The moving block bootstrap (MBB), resampling the time series from the data-set, can also be used to obtain the finite sample distribution of the CPT test under the null hypothesis. However, the performance of the MBB is crucially dependent on the block length, and there is no effective method to choose it (e.g. Hall and Jing, 1996). Hence, we do not pursue this method and leave this issue for future research.

7. All the results are obtained by using a common desktop with Pentium 4 3.4 GHz CPU and 1 GB RAM.

8. The exceptions to the 1957:1 start date are Austria (1970:1) and Japan (1966:4). The exception to the 2008:4 end date is Ireland (1998:4).

9. Lai (2008) also obtained similar weak evidence for the existence of the Fisher hypothesis using the univariate unit root tests with asymptotic values.

10. The insights conveyed through Figure 1 are similar in other cases. Therefore, figures for other countries are omitted for brevity.

11. Strictly speaking, these powers can not be directly compared across tests, since they depend on their own estimated models under the alternative hypothesis and are not the same.

12. Note that the data-set and the econometric method we used are different from those in Lai (2008).

13. The results can be obtained from the authors on request.