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On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders

Received: 26 April 2016     Accepted: 11 May 2016     Published: 25 May 2016
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Abstract

The most important assumptions about econometrics and time series data is stationarity, This study therefore suggests that, in trying to decide by classical methods whether economic data are stationary or not, it would be useful to perform tests of the null hypothesis of stationarity as well as tests of the null hypothesis of a unit root. The study compared power and type I error of Augmented Dickey-Fuller (ADF), Kwiatkowski, Phillips, Schmidt and Shin (KPSS) and Phillips and Perron (PP) to test the null hypothesis of stationarity against the alternative of a unit root at different order of autoregressive and moving average and various sample sizes. Simulation studies were conducted using R statistical package to investigate the performance of the tests of stationarity and unit root at sample size 20, 40, ..., 200 at first, second and third orders of autoregressive (AR), moving average (MA) and mixed autoregressive and moving average (ARMA) models. The relative performance of the tests was examined by their percentage of their powers and type I errors. The study concluded that PP is the best over all the conditions considered for the models, sample sizes and orders. However, in terms of type 1 error rate PP still is the best.

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 3)
DOI 10.11648/j.ajtas.20160503.20
Page(s) 146-153
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

ADF, KPSS, Stationarity, Simulation

References
[1] Park, J. Y. and B. Choi, (1988), A new approach to testing for a unit root, Working paper no. 88 23(Center for Analytical Economics, Cornell University, Ithaca, NY).
[2] Rudebusch, G. D., (1990), Trends and random walks in macroeconomic time series: A reexamination, Working paper no. 105 (Economic Activity Section, Board of Governors of the Federal Reserve System, Washington, DC).
[3] Yule, G. U. (1926). “Why do we sometimes get nonsense-correlations between time-series? — A study in sampling and the nature of time-series.” Journal of the Royal Statistical Society, 89, 1, 1–63.
[4] Tsay, R. S. (2008). Analysis of Financial Time Series, Second Edition, A John Wiley & Sons, Inc., Publication, pp 75-77.
[5] Tsay, R. S. (2010). Analysis of Financial Time Series; 3rd Edition, Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada
[6] Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis, Forecasting and Control, 2nd ed. New York: Prentice-Hall.
[7] Jonathan D. C. and Kung-Sik C. (2008). Time Series Analysis with Applications in R Second Edition, Springer Science+Business Media, LLC.
[8] Choi I (2006). \Nonstationary Panels. In TC Mills, K Patterson (eds.), Econometric Theory, volume 1 of Palgrave Handbook of Econometrics, chapter 13, pp. 511{539. Palgrave MacMillan, New York.
[9] Akeyede, I., Adeleke, B. L. and Yahya, W. B., (2015), Forecast strength of some linear and nonlinear time series data, American Journal of Mathematics and Statistics, 5(4): 37-41.
[10] Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press, Princeton, NJ.
[11] Dickey, D. A., Fuller, W. A (1981). Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root." Econometrica, 49(4), 1057-1072.
[12] Choi, I., (1990), Most of the US economic time series do not have unit roots: Nelson and Plosser’s results reconsidered, Discussion paper (Department of Economics, Ohio State University, Columbus, OH).
[13] Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., Shin, Y., (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159–178
[14] Davidson and Mackinnon (2004). Report that the Phillips Perron test performs worse in finite samples than the augmented Dickey–Fuller test, Journal of Economic Dynamics and Control 12, 297-332
[15] Phillips, P. C. B. and P. Perron. (1988), Testing for a unit root in time series regression, Biometrika 75, 335-346.
Cite This Article
  • APA Style

    Akeyede Imam, Danjuma Habiba, Bature Tajudeen Atanda. (2016). On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders. American Journal of Theoretical and Applied Statistics, 5(3), 146-153. https://doi.org/10.11648/j.ajtas.20160503.20

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    ACS Style

    Akeyede Imam; Danjuma Habiba; Bature Tajudeen Atanda. On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders. Am. J. Theor. Appl. Stat. 2016, 5(3), 146-153. doi: 10.11648/j.ajtas.20160503.20

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    AMA Style

    Akeyede Imam, Danjuma Habiba, Bature Tajudeen Atanda. On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders. Am J Theor Appl Stat. 2016;5(3):146-153. doi: 10.11648/j.ajtas.20160503.20

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  • @article{10.11648/j.ajtas.20160503.20,
      author = {Akeyede Imam and Danjuma Habiba and Bature Tajudeen Atanda},
      title = {On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {5},
      number = {3},
      pages = {146-153},
      doi = {10.11648/j.ajtas.20160503.20},
      url = {https://doi.org/10.11648/j.ajtas.20160503.20},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20160503.20},
      abstract = {The most important assumptions about econometrics and time series data is stationarity, This study therefore suggests that, in trying to decide by classical methods whether economic data are stationary or not, it would be useful to perform tests of the null hypothesis of stationarity as well as tests of the null hypothesis of a unit root. The study compared power and type I error of Augmented Dickey-Fuller (ADF), Kwiatkowski, Phillips, Schmidt and Shin (KPSS) and Phillips and Perron (PP) to test the null hypothesis of stationarity against the alternative of a unit root at different order of autoregressive and moving average and various sample sizes. Simulation studies were conducted using R statistical package to investigate the performance of the tests of stationarity and unit root at sample size 20, 40, ..., 200 at first, second and third orders of autoregressive (AR), moving average (MA) and mixed autoregressive and moving average (ARMA) models. The relative performance of the tests was examined by their percentage of their powers and type I errors. The study concluded that PP is the best over all the conditions considered for the models, sample sizes and orders. However, in terms of type 1 error rate PP still is the best.},
     year = {2016}
    }
    

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    DO  - 10.11648/j.ajtas.20160503.20
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
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    AB  - The most important assumptions about econometrics and time series data is stationarity, This study therefore suggests that, in trying to decide by classical methods whether economic data are stationary or not, it would be useful to perform tests of the null hypothesis of stationarity as well as tests of the null hypothesis of a unit root. The study compared power and type I error of Augmented Dickey-Fuller (ADF), Kwiatkowski, Phillips, Schmidt and Shin (KPSS) and Phillips and Perron (PP) to test the null hypothesis of stationarity against the alternative of a unit root at different order of autoregressive and moving average and various sample sizes. Simulation studies were conducted using R statistical package to investigate the performance of the tests of stationarity and unit root at sample size 20, 40, ..., 200 at first, second and third orders of autoregressive (AR), moving average (MA) and mixed autoregressive and moving average (ARMA) models. The relative performance of the tests was examined by their percentage of their powers and type I errors. The study concluded that PP is the best over all the conditions considered for the models, sample sizes and orders. However, in terms of type 1 error rate PP still is the best.
    VL  - 5
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Author Information
  • Department of Mathematics, Federal University, Lafia, Nigeria

  • Department of Statistics, Federal Polytechnic Bali, Taraba State, Nigeria

  • Department of Mathematics, Statistics Kwara State Polytechnic Ilorin, Kwara State, Nigeria

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