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Finite Sample Lag Adjusted Critical Values and Probability Values for the Fourier Wavelet Unit Root Test

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Abstract

Inferences from tests for non-stationarity depend critically on whether and how breaks and/or non-linearities are specified. Recent work has shown that wavelet transformations that separate a variable’s high and low frequency components can enhance the performance of unit root and stationarity tests. This note provides response surface estimates of finite sample, lag-adjusted critical values and approximate probability values for an Augmented Dickey–Fuller type wavelet test that includes a Fourier term allowing for smooth breaks in the series. Applications highlight the practical benefits.

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Data availability

Matlab programs to replicate the analysis and calculate finite sample lag adjusted critical values and probability values are available as electronic supplementary material and also at http://web.business.queensu.ca/faculty/psephton under MatlabFiles (FWADF.zip) and on request to the author (Peter.Sephton@queensu.ca).

Notes

  1. An entire issue of Econometrics (2017) was devoted to unit roots and structural breaks—see Perron (2017).

  2. Aydin and Pata (2020) show the Haar wavelet transform led to the greatest test power and is employed here.

  3. See Eroglu and Soybilgen (2018) and Aydin and Pata (2020).

  4. For example, the FWADF constant and trend test adds two regressors to the testing equation, reducing the degrees of freedom, which in small samples, could affect the distribution of the test statistic. Cook (2001) showed that using lag-adjusted critical values can increase the power of unit root tests.

  5. Experimentation led to g set to 3 and h set to 2.5. Fractional powers of the lag adjusted regressors increased the explanatory power of the response surface regressions.

  6. The wavelet test decomposes the series into low and high frequency components, using only half of the data contained in the low frequency component to estimate the test statistic. The sample size associated with the length of the low frequency component was employed as the “driver” of the error variance.

  7. Variations in the goodness of fit across quantiles is not unusual: Kripfganz and Schneider (2020) report adjusted R-squared values as low as 0.77 for some response surfaces.

  8. See Kripfganz and Schneider (2020) and Sephton (2022) for specifics.

  9. Here the t distribution was employed to calculate the approximate probability values. Kripfganz and Schneider (2020) followed a similar approach.

  10. See Shahbaz and Sinha (2019), Erdogan and Solarin (2021), and Solarin et al. (2022)

  11. Tests that include only a constant and the Fourier term did not reject the null of a unit root for all series at five percent with both the AIC and BIC (with the exception Brazilian Naturals and the BIC). The coefficients on the trend terms were significantly different from zero at five percent.

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Acknowledgements

I thank Mucahit Aydin and Tolga Omay for helpful comments.

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  1. Smith School of Business, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Peter S. Sephton

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Sephton, P.S. Finite Sample Lag Adjusted Critical Values and Probability Values for the Fourier Wavelet Unit Root Test. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10458-4

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