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

Akeyede Imam; Danjuma Habiba; Bature Tajudeen Atanda

doi:doi:10.11648/j.ajtas.20160503.20

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

Akeyede Imam, Danjuma Habiba, Bature Tajudeen Atanda

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 3)

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), 2024. Published by Science Publishing Group

Keywords

ADF, KPSS, Stationarity, Simulation

References

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[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; 3^rd 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.

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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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TY  - JOUR
T1  - On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders
AU  - Akeyede Imam
AU  - Danjuma Habiba
AU  - Bature Tajudeen Atanda
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PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20160503.20
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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EP  - 153
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20160503.20
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
IS  - 3
ER  -

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Author Information

Akeyede Imam

Department of Mathematics, Federal University, Lafia, Nigeria
Danjuma Habiba

Department of Statistics, Federal Polytechnic Bali, Taraba State, Nigeria
Bature Tajudeen Atanda

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - On Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders
AU  - Akeyede Imam
AU  - Danjuma Habiba
AU  - Bature Tajudeen Atanda
Y1  - 2016/05/25
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.20160503.20
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
SP  - 146
EP  - 153
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20160503.20
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
IS  - 3
ER  -

Copy | Download