Computational Statistics

, Volume 32, Issue 4, pp 1533–1568 | Cite as

Numerical distribution functions for seasonal unit root tests with OLS and GLS detrending

  • Tomás del Barrio Castro
  • Andrii Bodnar
  • Andreu Sansó
Original Paper


This paper implements the approach introduced by MacKinnon (J Bus Econ Stat 12:167–176, 1994, J Appl Econom 11:601–618, 1996) to estimate the response surface of the test statistics of seasonal unit root tests with OLS and GLS detrending for quarterly and monthly time series. The Gauss code that is available in the supplementary material of the paper produces p values for five test statistics depending on the sample size, deterministic terms and frequency of the data. A comparison with previous studies is undertaken, and an empirical example using airport passenger arrivals to a tourist destination is carried out. Quantile function coefficients are reported for simple computation of critical values for tests at 1, 5 and 10 % significance levels.


HEGY test GLS detrending Response surfaces p values 



We would like to thank an anonymous referee for his/her helpful comments. Tomas del Barrio Castro and Andreu Sansó knowledge financial support from Spanish Ministerio de Educación, Cultura y Deporte under Grant ECO2014-58991-C3-3-R.


  1. Beaulieu JJ, Miron JA (1993) Seasonal unit roots in aggregate US data. J Appl Econom 55:305–328MATHMathSciNetGoogle Scholar
  2. Burridge P, Taylor AMR (2001) On the properties of regression-based tests for seasonal unit roots in the presence of higher-order serial correlation. J Bus Econ Stat 19:374–379CrossRefMathSciNetGoogle Scholar
  3. Cragg JG (1983) More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica 58:751–763CrossRefMATHMathSciNetGoogle Scholar
  4. del Barrio Castro T, Osborn DR (2011) HEGY tests in the presence of moving averages. Oxf Bull Econ Stat 73:691–704CrossRefGoogle Scholar
  5. del Barrio Castro T, Osborn DR, Taylor AMR (2012) On augmented HEGY tests for seasonal unit roots. Econom Theory 28:1121–1143CrossRefMATHMathSciNetGoogle Scholar
  6. del Barrio Castro T, Osborn DR, Taylor AMR (2016) The performance of lag selection and detrending methods for HEGY seasonal unit root tests. Econom Rev 35:122–168CrossRefMathSciNetGoogle Scholar
  7. Diaz-Emparanza I (2014) Numerical distribution functions for seasonal unit root tests. Comput Stat Data Anal 76:237–247CrossRefMATHMathSciNetGoogle Scholar
  8. Elliott G, Rothenberg TJ, Stock JH (1996) Efficient tests for an autoregressive unit root. Econometrica 64:813–836CrossRefMATHMathSciNetGoogle Scholar
  9. Franses PH, Hobijn B (1997) Critical values for unit root tests in seasonal time series. J Appl Stat 24:25–48CrossRefMathSciNetGoogle Scholar
  10. Ghysels E, Lee HS, Noh J (1994) Testing for unit roots in seasonal time series. J Econom 62:414–442CrossRefGoogle Scholar
  11. Gregoir S (2006) Efficient tests for the presence of a pair of complex conjugate unit roots in real time series. J Econom 130:45–100Google Scholar
  12. Harvey DI, van Dijk D (2006) Sample size, lag order and critical values of seasonal unit root tests. Comput Stat Data Anal 50:2734–2751CrossRefMATHMathSciNetGoogle Scholar
  13. Hylleberg S, Engle RF, Granger CWJ, Yoo BS (1990) Seasonal integration and cointegration. J Econom 44:215–238CrossRefMATHMathSciNetGoogle Scholar
  14. MacKinnon JG (1994) Approximate asymptotic distribution functions for unit-root and cointegration tests. J Bus Econ Stat 12:167–176MathSciNetGoogle Scholar
  15. MacKinnon JG (1996) Numerical distribution functions for unit root and cointegration tests. J Appl Econom 11:601–618CrossRefGoogle Scholar
  16. Rodrigues PMM, Taylor AMR (2007) Efficient tests of the seasonal unit root hypothesis. J Econom 141:548–573CrossRefMATHMathSciNetGoogle Scholar
  17. Smith RJ, Taylor AMR (1998) Additional critical values and asymptotic representations for seasonal unit root tests. J Econom 85:269–288CrossRefMATHMathSciNetGoogle Scholar
  18. Smith RJ, Taylor AMR, del Barrio Castro T (2009) Regression-based seasonal unit root tests. Econom Theory 25:527–560CrossRefMATHMathSciNetGoogle Scholar
  19. Smith RJ, Taylor AMR (1999) Likelihood ratio tests for seasonal unit roots. J Time Ser Anal 20:453–476CrossRefMATHMathSciNetGoogle Scholar
  20. Taylor AMR (1998) Testing for unit roots in monthly time series. J Time Ser Anal 19(3):349–368CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Tomás del Barrio Castro
    • 1
  • Andrii Bodnar
    • 1
  • Andreu Sansó
    • 1
  1. 1.Department of Applied EconomicsUniversity of the Balearic IslandsPalma de MallorcaSpain

Personalised recommendations