Published

2015-01-01

TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution

Modelamiento TAR con datos faltantes cuando el proceso del ruido blanco tiene una distribución t de Student

DOI:

https://doi.org/10.15446/rce.v38n1.48813

Keywords:

Bayesian Statistics, Gibbs Sampler, Missing Data, Forecasting, Time Series, Threshold Autoregressive Model (en)
Datos faltantes, Estadística Bayesiana, Modelo autoregresivo de umbrales, Muestreador de Gibbs, Pronóstico, Series de tiempo (es)

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Authors

  • Hanwen Zhang Universidad Santo Tomás
  • Fabio H. Nieto Universidad Nacional de Colombia - Sede Bogotá - Facultad de Ciencias - Departamento de Estadística

This paper considers the modeling of the threshold autoregressive (TAR) process, which is driven by a noise process that follows a Student’s t-distribution. The analysis is done in the presence of missing data in both the threshold process {Zt} and the interest process {Xt}. We develop a three-stage procedure based on the Gibbs sampler in order to identify and estimate the model. Additionally, the estimation of the missing data and the forecasting procedure are provided. The proposed methodology is illustrated with simulated and real-life data.

En este trabajo consideramos el modelamiento de los modelos autoregresivos de umbrales (TAR) con datos faltantes tanto en la serie de umbrales como la serie de interés cuando el proceso del ruido blanco sigue una distribución t de student. Desarrollamos un procedimiento de tres etapas basado en el muestreador de Gibbs para identificar y estimar el modelo, además de la estimación de los datos faltantes y el procedimiento para el pronóstico. La metodología propuesta fue aplicada a datos simulados y datos reales.

https://doi.org/10.15446/rce.v38n1.48813

TAR Modeling with Missing Data when the White Noise Process Follows a Student's t-Distribution

Modelamiento TAR con datos faltantes cuando el proceso del ruido blanco tiene una distribución t de Student

HANWEN ZHANG1, FABIO H. NIETO2

1Universidad Santo Tomás, Facultad de Estadística, Bogotá, Colombia. Professor. Email: hanwenzhang@usantotomas.edu.co
2Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. Professor. Email: fhnietos@unal.edu.co

Abstract

This paper considers the modeling of the threshold autoregressive (TAR) process, which is driven by a noise process that follows a Students t-distribution. The analysis is done in the presence of missing data in both the threshold process {Zt} and the interest process {Xt}. We develop a three-stage procedure based on the Gibbs sampler in order to identify and estimate the model. Additionally, the estimation of the missing data and the forecasting procedure are provided. The proposed methodology is illustrated with simulated and real-life data.

Key words: Bayesian Statistics, Gibbs Sampler, Missing Data, Forecasting, Time Series, Threshold Autoregressive Model.

Resumen

En este trabajo consideramos el modelamiento de los modelos autoregresivos de umbrales (TAR) con datos faltantes tanto en la serie de umbrales como la serie de interés cuando el proceso del ruido blanco sigue una distribución t de student. Desarrollamos un procedimiento de tres etapas basado en el muestreador de Gibbs para identificar y estimar el modelo, además de la estimación de los datos faltantes y el procedimiento para el pronóstico. La metodología propuesta fue aplicada a datos simulados y datos reales.

Palabras clave: datos faltantes, estadística Bayesiana, modelo autoregresivo de umbrales, muestreador de Gibbs, pronóstico, series de tiempo.

Texto completo disponible en PDF

References

1. Briñez, A. & Nieto, F. (2005), 'Fitting a nonlinear model to the precipitation variable in a Colombian Hydrological/Meteorological station', Revista Colombiana de Estadística 28, 113-124.

2. Carlin, B. P. & Chib, S. (1995), 'Bayesian model choice via Markov Chain Monte Carlo Methods', Journal of the Royal Statistical Society. Serie B 37(3), 473-484.

3. Chen, H., Chong, T. T. & Bai, J. (2012), 'Theory and applications of TAR model with two threshold variables', Econometric Reviews 31, 142-170.

4. Congdon, P. (2001), Bayesian Statistical Modeling, John Wiley & Sons, New York.

5. Geweke, J. (1992), Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, 'Bayesian Statistics', University Press, p. 169-193.

6. Moreno, E. (2010), Modelos TAR en series de tiempo financieras, Master's thesis, Universidad Nacional de Colombia.

7. Nieto, F. H. (2005), 'Modeling bivariate threshold autoregressive processes in the presence of missing data', Communications in Statistics, Theory and Methods. 34, 905-930.

8. Nieto, F. H. (2008), 'Forecasting with univariate TAR models', Statistical Methodology. 5, 263-276.

9. Nieto, F. H. & Moreno, E. (2013), A note on the specification of conditional heteroscedasticity using a TAR model, RI21, Universidad Nacional de Colombia.

10. Nieto, F. H., Zhang, H. & Li, W. (2013), 'Using the Reversible Jump MCMC Procedure for Identifying and Estimating Univariate TAR Models', Communications In Statistics. Simulation And Computation 42(4), 814-840.

11. Nieto, F. & Hoyos, M. (2011), 'Testing linearity against a univariate TAR specification in time series with missing data', Revista Colombiana de Estadística 34, 73-94.

12. Plummer, M., Best, N., Cowles, K. & Vines, K. (2006), 'Coda: convergence diagnosis and output analysis for mcmc', R News 6(1), 7-11. *http://CRAN.R-project.org/doc/Rnews/

13. Sáfadi, T. & Morettin, P. (2000), 'Bayesian analysis of thresholds autoregressive moving average models', The Indian Journal of Statistics 62, 353-371.

14. Tong, H. (1978), On a Threshold Model, 'Pattern Recognition and Signal Processing', Sijthoff & Noordhoff, Netherlands, p. 575-586.

15. Tsay, R. S. (1989), 'Testing and modeling threshold autoregressive processes', Journal of American Statistical Association 84, 231-240.

16. Tsay, R. S. (1998), 'Testing and modeling multivariate threshold models', Journal of American Statistical Association 93, 1188-1202.

17. Vargas, L. (2012), Cálculo de la distribución predictiva en un modelo TAR, Master's thesis, Universidad Nacional de Colombia.

18. Watanabe, T. (2001), 'On sampling the degree-of-freedom of Student's-t disturbances', Statistics & Probability Letters 52, 177-181.

19. Xia, Q., Liu, L., Pan, J. & Liang, R. (2012), 'Bayesian analysis of two-regime threshold autoregressive moving average model with exogenous inputs', Communications in Statistics - Theory and Methods 41, 1089-1104.

20. Zhang, H. (2012), 'Estimación de los modelos TAR cuando el proceso del ruido sigue una distribución t', Comunicaciones en Estadística 4(2), 109-119.

[Recibido en octubre de 2013. Aceptado en noviembre de 2014]

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

@ARTICLE{RCEv38n1a13,
    AUTHOR  = {Zhang, Hanwen and Nieto, Fabio H.},
    TITLE   = {{TAR Modeling with Missing Data when the White Noise Process Follows a Student's t-Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {1},
    pages   = {239-266}
}

References

Briñez, A. & Nieto, F. (2005), ‘Fitting a nonlinear model to the precipitation variable in a Colombian Hydrological/Meteorological station’, Revista Colombiana de Estadística 28, 113–124.

Carlin, B. P. & Chib, S. (1995), ‘Bayesian model choice via Markov Chain Monte Carlo Methods’, Journal of the Royal Statistical Society. Serie B 37(3), 473– 484.

Chen, H., Chong, T. T. & Bai, J. (2012), ‘Theory and applications of TAR model with two threshold variables’, Econometric Reviews 31, 142–170.

Congdon, P. (2001), Bayesian Statistical Modeling, John Wiley & Sons, New York.

Geweke, J. (1992), Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in ‘Bayesian Statistics’, University Press, pp. 169–193.

Moreno, E. (2010), Modelos TAR en series de tiempo financieras, Master’s thesis, Universidad Nacional de Colombia.

Nieto, F. H. (2005), ‘Modeling bivariate threshold autoregressive processes in the presence of missing data’, Communications in Statistics, Theory and Methods. 34, 905–930.

Nieto, F. H. (2008), ‘Forecasting with univariate TAR models’, Statistical Methodology. 5, 263–276.

Nieto, F. H. & Moreno, E. (2013), A note on the specification of conditional heteroscedasticity using a TAR model, Technical Report RI21, Universidad Nacional de Colombia.

Nieto, F. H., Zhang, H. & Li, W. (2013), ‘Using the Reversible Jump MCMC Procedure for Identifying and Estimating Univariate TAR Models’, Communications In Statistics. Simulation And Computation 42(4), 814–840.

Nieto, F. & Hoyos, M. (2011), ‘Testing linearity against a univariate TAR specification in time series with missing data’, Revista Colombiana de Estadística 34, 73–94.

Plummer, M., Best, N., Cowles, K. & Vines, K. (2006), ‘Coda: Convergence diagnosis and output analysis for mcmc’, R News 6(1), 7–11.

*http://CRAN.R-project.org/doc/Rnews/

Sáfadi, T. & Morettin, P. (2000), ‘Bayesian analysis of thresholds autoregressive moving average models’, The Indian Journal of Statistics 62, 353–371.

Tong, H. (1978), On a Threshold Model, in C. H. Chen, ed., ‘Pattern Recognition and Signal Processing’, Sijthoff & Noordhoff, Netherlands, pp. 575–586.

Tsay, R. S. (1989), ‘Testing and modeling threshold autoregressive processes’, Journal of American Statistical Association 84, 231–240.

Tsay, R. S. (1998), ‘Testing and modeling multivariate threshold models’, Journal of American Statistical Association 93, 1188–1202.

Vargas, L. (2012), Cálculo de la distribución predictiva en un modelo TAR, Master’s thesis, Universidad Nacional de Colombia.

Watanabe, T. (2001), ‘On sampling the degree-of-freedom of Student’s-t disturbances’, Statistics & Probability Letters 52, 177–181.

Xia, Q., Liu, L., Pan, J. & Liang, R. (2012), ‘Bayesian analysis of two-regime threshold autoregressive moving average model with exogenous inputs’, Communications in Statistics - Theory and Methods 41, 1089–1104.

Zhang, H. (2012), ‘Estimación de los modelos TAR cuando el proceso del ruido sigue una distribución t’, Comunicaciones en Estadística 4(2), 109–119.

How to Cite

APA

Zhang, H. and Nieto, F. H. (2015). TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution. Revista Colombiana de Estadística, 38(1), 239–265. https://doi.org/10.15446/rce.v38n1.48813

ACM

[1]
Zhang, H. and Nieto, F.H. 2015. TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution. Revista Colombiana de Estadística. 38, 1 (Jan. 2015), 239–265. DOI:https://doi.org/10.15446/rce.v38n1.48813.

ACS

(1)
Zhang, H.; Nieto, F. H. TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution. Rev. colomb. estad. 2015, 38, 239-265.

ABNT

ZHANG, H.; NIETO, F. H. TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution. Revista Colombiana de Estadística, [S. l.], v. 38, n. 1, p. 239–265, 2015. DOI: 10.15446/rce.v38n1.48813. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/48813. Acesso em: 29 jan. 2025.

Chicago

Zhang, Hanwen, and Fabio H. Nieto. 2015. “TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution”. Revista Colombiana De Estadística 38 (1):239-65. https://doi.org/10.15446/rce.v38n1.48813.

Harvard

Zhang, H. and Nieto, F. H. (2015) “TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution”, Revista Colombiana de Estadística, 38(1), pp. 239–265. doi: 10.15446/rce.v38n1.48813.

IEEE

[1]
H. Zhang and F. H. Nieto, “TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution”, Rev. colomb. estad., vol. 38, no. 1, pp. 239–265, Jan. 2015.

MLA

Zhang, H., and F. H. Nieto. “TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution”. Revista Colombiana de Estadística, vol. 38, no. 1, Jan. 2015, pp. 239-65, doi:10.15446/rce.v38n1.48813.

Turabian

Zhang, Hanwen, and Fabio H. Nieto. “TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution”. Revista Colombiana de Estadística 38, no. 1 (January 1, 2015): 239–265. Accessed January 29, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/48813.

Vancouver

1.
Zhang H, Nieto FH. TAR Modeling with Missing Data when the White Noise Process Follows a Student’s t-Distribution. Rev. colomb. estad. [Internet]. 2015 Jan. 1 [cited 2025 Jan. 29];38(1):239-65. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/48813

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CrossRef citations2

1. Joaquín González Borja, Fabio Humberto Nieto Sánchez. (2020). Bayesian Analysis of Multiplicative Seasonal Threshold Autoregressive Processes. Revista Colombiana de Estadística, 43(2), p.251. https://doi.org/10.15446/rce.v43n2.81261.

2. Lizet Viviana Romero Orjuela, Sergio Alejandro Calderón Villanueva. (2021). Bayesian estimation of a multivariate TAR model when the noise process follows aStudent-tdistribution. Communications in Statistics - Theory and Methods, 50(11), p.2508. https://doi.org/10.1080/03610926.2019.1669807.

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