Abstract
We propose a double frequency fourier Dickey–Fuller (DF) unit root test. The asymptotic theory of the newly proposed test is first presented in this study. We conduct a series of simulations which suggest the proposed test statistic has correct size performance and gains more power when breaks are located at the beginning and end of the sample and in smooth type. In empirical analysis, we utilize the new test to examine the unit root hypothesis of relative commodity prices measured by Harvey et al. (Rev Econ Stat 92(2):367–377, 2010). The empirical results show that more relative commodity prices are stationary around a deterministic trend generated from double frequency Fourier function.
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Notes
Another method to approximate smooth breaks can be seen by using exponential functions. Omay and Emirmahmutoğlu (2017) and Omay et al. (2018) utilize such method in panel unit root test by solving cross-sectional dependence in the mean time. In this paper, our proposed method of using double frequency Fourier function can be extended to panel unit root tests as well.
Although Kellard and Wohar (2006), Ghoshray (2011, 2018) and Winkelried (2016) all test for the same dataset proposed by Grilli and Yang (1988), the conclusions are mixed across different periods and econometric models. The aims of their studies are to examine the Prebisch and Singer hypothesis (Prebisch 1962; Singer 1975). The common procedures are to test for the unit root in relative commodity price at first, and then measure the prevalence of trend. Therefore, a powerful unit root test is crucial to get correct inference.
In Online Appendix, we also provide empirical results by using the popular Grilli and Yang (1988) dataset.
Enders and Lee (2012a) suggest that only small frequencies can be used to approximate structural breaks. Therefore, we set the maximum frequency to be 2. The method about how to select the optimal double frequency will be introduced in the following sections.
In this study, we derive the asymptotic theory when the specification including both intercept and trend. However, the case when including only intercept can be derived as the same method.
Becker et al. (2006) propose that a large frequency cannot be used to approximate the breaks. They further show that larger frequencies always associate to stochastic parameter variability (not structural breaks).
The main text only presents critical values by using integer frequencies. The results obtained from fractional values are available in Online Appendix.
We find a power deterioration after incorporating time trend into the model, which is also presented in Enders and Lee (2012a).
See Harvey et al. (2010) for more details about the nominal commodity price.
See Harvey et al. (2010) for more details about the historical manufacturing value-added price index.
In this study, we set the maximum frequency \(k_{max}=5\) with the searching precision \(\Delta k=0.1\) when using fractional frequency.
Since the critical values are dependent upon frequency, we do not show all critical values for saving space. The results are upon request.
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Acknowledgement
We thank the editor, three anonymous referees, and participants of a Work-in-Progress seminar at UWA Business School for valuable comments and suggestions. Data used in this study were kindly provided by Neil Kellard. Work on this paper is supported by Australian Government International Research Training Program Scholarship and University Postgraduate Award.
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Cai, Y., Omay, T. Using Double Frequency in Fourier Dickey–Fuller Unit Root Test. Comput Econ 59, 445–470 (2022). https://doi.org/10.1007/s10614-020-10075-5
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DOI: https://doi.org/10.1007/s10614-020-10075-5