Published
Additive Outliers in Open-Loop Threshold Autoregressive Models: A Simulation Study
Datos atípicos aditivos en modelos autorregresivos de umbrales: un estudio de simulación
DOI:
https://doi.org/10.15446/rce.v45n1.92965Keywords:
Nonlinear time series (en)Modelos open-loop TAR, Estimadores GM (es)
Downloads
The effect of additive outliers is studied on an adapted non-linearity test and a robust estimation method for autoregressive coefficients in open-loop TAR (threshold autoregressive) models. Through a Monte Carlo experiment, the power and size of the non-linearity test are studied. Regarding the estimation method, the bias and ratio of mean squared errors are compared between the robust estimator and least squares. Simulation exercises are carried out for different percentages of contamination and the proportion of observations on each model regime. Furthermore, the approximation of the univariate normal distribution to the empirical distribution of estimated coefficients is analyzed along with the coverage level of asymptotic confidence intervals for the parameters. Results show that the adapted non-linearity test does not have size distortions, and it has a superior power than its least-squares counterpart when additive outliers are present. On the other hand, the robust estimation method for the autoregressive coefficients has a better mean squared error than least-squares when this type of observations are present. Lastly, the use of the non-linearity test and the estimation method are illustrated through a real example.
Se investiga el efecto de observaciones atípicas aditivas en la adaptación de una prueba de no linealidad y un método de estimación robusto para los coeficientes autoregresivos en modelos open-loop TAR (threshold autoregres- sive). A través de un experimento Monte Carlo se estudia la potencia y el tamaño de la prueba de no linealidad. Respecto a la estimación, se compara el sesgo y la razón de error cuadrático medio entre el estimador robusto y el de mínimos cuadrados. Adicionalmente, se llevan a cabo ejercicios de simulación para diferentes porcentajes de contaminación, proporción de observaciones en cada régimen del modelo y se evalúa la aproximación de la distribución empírica de los coeficientes estimados por medio de la distribución normal univariada junto a los niveles de cobertura de los intervalos de confianza asintóticos para los parámetros. Los resultados indican que la prueba de no linealidad adaptada presenta una potencia superior a la basada en mínimos cuadrados y no presenta distorsiones en el tamaño bajo la presencia de datos atípicos aditivos. Por otro lado, el método de estimación robusto para los coeficientes autoregresivos supera al de mínimos cuadrados en términos de error cuadrático medio bajo la presencia de este tipo de observaciones. Finalmente, se ilustra a través de un ejemplo real el uso de la prueba de no linealidad y el método de estimación en la práctica.
References
Battaglia, F. & Orfei, L. (2005), ‘Outlier detection and estimation in nonlinear time series’, Journal of Time Series Analysis 26(1), 107–121. DOI: https://doi.org/10.1111/j.1467-9892.2005.00392.x
Chan, K. & Tong, H. (1990), ‘On likelihood ratio tests for threshold autoregression’, Journal of the Royal Statistical Society. Series B: Statistical Methodology 52(3), 469–476. DOI: https://doi.org/10.1111/j.2517-6161.1990.tb01800.x
Chan, W.-S. & Cheung, S.-H. (1994), ‘On robust estimation of threshold autoregressions’, Journal of Forecasting 13(1), 37–49. DOI: https://doi.org/10.1002/for.3980130106
Chan, W.-S. & Ng, M.-W. (2004), ‘Robustness of alternative non-linearity tests for setar models’, Journal of Forecasting 23(3), 215–231. DOI: https://doi.org/10.1002/for.915
Chen, C. & Liu, L.-M. (1993), ‘Joint estimation of model parameters and out- lier effects in time series’, Journal of the American Statistical Association 88(421), 284–297. DOI: https://doi.org/10.1080/01621459.1993.10594321
Denby, L. & Martin, R. D. (1979), ‘Robust estimation of the first-order autoregressive parameter’, Journal of the American Statistical Association 74(365), 140– 146. DOI: https://doi.org/10.1080/01621459.1979.10481630
Franses, P. H., Van Dijk, D. et al. (2000), Non-linear time series models in empirical finance, Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511754067
Giordani, P. (2006), ‘A cautionary note on outlier robust estimation of threshold models’, Journal of Forecasting 25(1), 37–47. DOI: https://doi.org/10.1002/for.972
Granger, C. & Teräsvirta, T. (1993), Modelling Non-Linear Economic Relationships, Oxford University Press.
Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. & Stahel, W. A. (1986), Robust statistics: the approach based on influence functions, Vol. 196, John Wiley & Sons.
Hansen, B. E. (2011), ‘Threshold autoregression in economics’, Statistics and its Interface 4(2), 123–127. DOI: https://doi.org/10.4310/SII.2011.v4.n2.a4
Hung, K. C., Cheung, S. H., Chan, W.-S. & Zhang, L.-X. (2009), ‘On a robust test for setar-type nonlinearity in time series analysis’, Journal of forecasting 28(5), 445–464. DOI: https://doi.org/10.1002/for.1122
Luukkonen, R., Saikkonen, P. & Teräsvirta, T. (1988), ‘Testing linearity against smooth transition autoregressive models’, Biometrika 75(3), 491–499. DOI: https://doi.org/10.1093/biomet/75.3.491
Maronna, R. A., Martin, D. R. & Yohai, V. J. (2006), Robust Statistics: Theory and Methods, John Wiley and Sons. DOI: https://doi.org/10.1002/0470010940
Mohammadi, H. & Jahan-Parvar, M. R. (2012), ‘Oil prices and exchange rates in oil-exporting countries: evidence from tar and m-tar models’, Journal of Economics and Finance 36(3), 766–779. DOI: https://doi.org/10.1007/s12197-010-9156-5
Ordoñez-Callamad, D. (2019), Desarrollo de la prueba de no linealidad y estimación de los coeficientes autoregresivos en modelos TAR bajo la presencia de datos atípicos aditivos, Master’s thesis, Universidad NAcional de Colombia,
Petruccelli, J. (1990), ‘A comparison of tests for setar-type non-linearity in time series’, Journal of Forecasting 9(1), 25–36. DOI: https://doi.org/10.1002/for.3980090104
Petruccelli, J. & Davies, N. (1986), ‘A portmanteau test for self-exciting threshold autoregressive-type nonlinearity in time series’, Biometrika 73(3), 687–694. DOI: https://doi.org/10.1093/biomet/73.3.687
Rousseeuw, P. J. (1984), ‘Least median of squares regression’, Journal of the American statistical association 79(388), 871–880. DOI: https://doi.org/10.1080/01621459.1984.10477105
Rousseeuw, P. J. & Van Zomeren, B. C. (1990), ‘Unmasking multivariate out- liers and leverage points’, Journal of the American Statistical association 85(411), 633–639. DOI: https://doi.org/10.1080/01621459.1990.10474920
Shapiro, S. S. & Wilk, M. B. (1965), ‘An analysis of variance test for normality (complete samples)’, Biometrika 52(3/4), 591–611. DOI: https://doi.org/10.1093/biomet/52.3-4.591
Tiao, G. C. & Tsay, R. S. (1994), ‘Some advances in non-linear and adaptive modelling in time-series’, Journal of Forecasting 13(2), 109–131. DOI: https://doi.org/10.1002/for.3980130206
Tong, H. (1978), ‘On a threshold model’, Pattern Recognition and Signal Processing . DOI: https://doi.org/10.1007/978-94-009-9941-1_24
Tong, H. (1990), Non-linear Time Series: A Dynamical System Approach, Dynam- ical System Approach, Clarendon Press.
Tsay, R. & Chen, R. (2019), Nonlinear Time Series Analysis, Wiley Interscience. Tsay, R. S. (1988), ‘Outliers, level shifts, and variance changes in time series’, Journal of forecasting 7(1), 1–20. DOI: https://doi.org/10.1002/for.3980070102
Tsay, R. S. (1989), ‘Testing and modeling threshold autoregressive processes’,Journal of the American statistical association 84(405), 231–240. DOI: https://doi.org/10.1080/01621459.1989.10478760
Vargas, H. (2011), Monetary policy and the exchange rate in Colombia, in B. for International Settlements, ed., ‘Capital flows, commodity price movements and foreign exchange intervention’, Vol. 57 of BIS Papers, Bank for International Settlements, pp. 129–153.
Zhang, L.-X., Chan, W.-S., Cheung, S.-H. & Hung, K.-C. (2009), ‘A note on the consistency of a robust estimator for threshold autoregressive processes’, Statistics & Probability Letters 79(6), 807–813. DOI: https://doi.org/10.1016/j.spl.2008.10.036
How to Cite
APA
ACM
ACS
ABNT
Chicago
Harvard
IEEE
MLA
Turabian
Vancouver
Download Citation
License
Copyright (c) 2022 Revista Colombiana de Estadística
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).