SciELO - Scientific Electronic Library Online

 
vol.34 issue1Socioeconomic Determinants of Infant Mortality in Colombia{,} 1993Bayesian Analysis for the Generalized Lognormal Distribution Applied to Failure Time Analysis author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.34 no.1 Bogotá Jan./June 2011

 

Testing Linearity against a Univariate TAR Specification in Time Series with Missing Data

Sobre una prueba de linealidad en presencia de datos faltantes contra la alternativa de no linealidad especificada por un modelo TAR

FABIO H. NIETO1, MILENA HOYOS2

1Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. Profesor titular. Email: fhnietos@unal.edu.co
2Universidad Nacional de Colombia, Facultad de Economía, Bogotá, Colombia. Profesora auxiliar. Email: nmhoyosg@unal.edu.co


Abstract

Nowadays, procedures for testing the null hypothesis of linearity of a (univariate or multivariate) stochastic process are well known, almost all of them based on the assumption that their paths (i.e. observed time series) are complete. This paper describes an approach for testing this null hypothesis in the presence of missing data, using an extension of one of the test statistics used in the literature. The alternative hypothesis is that the univariate stochastic process of interest follows a threshold autoregressive (TAR) model. It is found that if the missing-data percentage is low, the null distribution of the proposed test statistic is maintained; while if it is high, it is not. A threshold value for the missing-data percentage is detected, which can be utilized in practice.

Key words: Linearity test, Missing data, Nonlinear time series, Threshold autoregressive model.


Resumen

Las pruebas estadísticas que se conocen actualmente para examinar la hipótesis nula de linealidad de un proceso estocástico (univariado o multivariado) están basadas, casi todas, en el supuesto de que las series temporales observadas son completas. En este trabajo, se presenta un nuevo procedimiento para examinar esta hipótesis nula, en presencia de datos faltantes, el cual es una extensión de un método muy citado en la literatura. La hipótesis alternativa especifica que el proceso estocástico de interés obedece a un modelo autoregresivo de umbrales (TAR). Se encuentra que si el porcentaje de observaciones faltantes es bajo, la distribución nula de la estadística de prueba se mantiene; en otro caso no. El estudio arroja un valor umbral para este porcentaje, el cual puede ser usado en la práctica.

Palabras clave: datos faltantes, modelos autoregresivos de umbrales, prueba de linealidad, series de tiempo no linales.


Texto completo disponible en PDF


References

1. Brockwell, P. J. (1994), `On continuous-time threshold ARMA processes´, Journal of Statistical Planning and Inference 39, 291-303.         [ Links ]

2. Brockwell, P. J. & Davis, R. A. (1991), Time Series: Theory and Methods, Springer-Verlag, New York.         [ Links ]

3. Caporello, G. & Maravall, A. (2003), Software TSW, Banco de Espa\~na, Madrid.         [ Links ]

4. Carter, C. K. & Kohn, R. (1994), `On Gibbs sampling for state space models´, Biometrika 81, 541-553.         [ Links ]

5. Carter, C. K. & Kohn, R. (1996), `Markov chain Monte Carlo in conditionally gaussian state space models´, Biometrika 83, 589-601.         [ Links ]

6. Catlin, D. (1989), Estimation, Control, and the Discrete Kalman Filter, Springer-Verlag, New York.         [ Links ]

7. G\'omez, V. & Maravall, A. (1994), `Estimation, prediction, and interpolation for nonstationary series with the Kalman filter´, Journal of the American Statistical Association 89, 611-624.         [ Links ]

8. Hansen, B. E. (1996), `Inference when a nuisance parameter is not identified under the null hypothesis´, Econometrica 64, 413-460.         [ Links ]

9. Harvey, A. C. (1989), Forecasting, Structural Time Series, and the Kalman filter, Cambridge University Press, Cambridge.         [ Links ]

10. Kim, C. & Nelson, C. R. (1999), State Space Models with Regime Switching, The MIT Press, Cambridge.         [ Links ]

11. Nieto, F. H. (2005), `Modeling bivariate threshold autoregressive processes in the presence of missing data´, Communications in Statistics - Theory and Methods 34(4), 905-930.         [ Links ]

12. Shumway, R. H. & Stoffer, D. S. (1991), `Dynamic linear models with switching´, Journal of the American Statistical Association 86, 411-430.         [ Links ]

13. Tong, H. (1978), On a threshold model in pattern recognition and signal processing, `Pattern recognition and signal processing´, Sijhoff & Noordhoff, Amsterdam.         [ Links ]

14. Tong, H. & Lim, K. S. (1980), `Threshold autoregression, limit cycles, and cyclical data´, Journal of the Royal Statistical Society, Series B 42, 245-292.         [ Links ]

15. Tong, H. & Yeung, I. (1991a), `On tests for self-exciting threshold autoregressive non-linearity in partially observed time series´, Applied Statistics 40, 43-62.         [ Links ]

16. Tong, H. & Yeung, I. (1991b), `Threshold autoregressive modeling in continuous time´, Statistica Sinica 1, 411-430.         [ Links ]

17. Tsai, H. & Chan, K. S. (2000), `Testing for nonlinearity with partially observed time series´, Biometrika 87, 805-821.         [ Links ]

18. Tsay, R. S. (1998), `Testing and modeling multivariate threshold models´, Journal of the American Statistical Association 93, 1188-1202.         [ Links ]

[Recibido en enero de 2008. Aceptado en febrero de 2011]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv34n1a04,
    AUTHOR  = {Nieto, Fabio H. and Hoyos, Milena},
    TITLE   = {{Testing Linearity against a Univariate TAR Specification in Time Series with Missing Data}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2011},
    volume  = {34},
    number  = {1},
    pages   = {73-94}
}

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License