¹Universidad Nacional de Colombia, Facultad de Ciencias, Medellín, Colombia. Universidad de Antioquia, Facultad de Ciencias Económicas, Medellín, Colombia. Profesor asociado, profesor titular. Email: elkincv@gmail.com
²Universidad Nacional de Colombia, Facultad de Ciencias Humanas y Económicas, Departamento de Economía, Medellín, Colombia. Profesor auxiliar. Email: kgomezp@unal.edu.co
³Universidad Nacional de Colombia, Facultad de Ciencias Humanas y Económicas, Departamento de Economía, Medellín, Colombia. Profesor auxiliar. Email: sgallong@unal.edu.co ]]>

Este documento presenta una nueva prueba para el parámetro de diferenciación fraccional de un modelo ARFIMA, basada en una aproximación autorregresiva de su componente a corto plazo. El comportamiento de la prueba se estudia por medio de experimentos Monte Carlo en una distribución normal, y se compara con el comportamiento de algunas de las pruebas más utilizadas. Para los casos estudiados, se concluye que la nueva prueba tiene generalmente potencias superiores, conservando un tamaño adecuado. A partir de la estimación del parámetro de diferenciación fraccional usando el modelo aproximado, es posible identificar el modelo correcto para la componente a corto plazo, lo cual permite mejorar la inferencia sobre dicho parámetro. Una ventaja adicional del procedimiento propuesto es que permite probar la existencia de larga memoria en presencia de errores dependientes, como en el caso de modelos de volatilidad de la familia ARCH. Se ilustra su aplicación en un procedimiento de identificación y estimación de un modelo ARFIMA--ARCH usando datos simulados.

Palabras clave: memoria larga, modelo ARFIMA, aproximación autorregresiva, identificación, prueba de hipótesis, diferencia fraccional.

This paper presents a new test for the fractional differencing parameter of an ARFIMA model, based on an autoregressive approximation of its short-range component. The tests behavior is studied using Monte Carlo simulations under a normal distribution and is compared to results found for others well--known long memory tests. In general, the results show that the new test has a superior power while maintaining an adequate size of the test. From the estimation of the fractional differencing parameter using the approximate model, it is possible to identify the correct model for the short--term component, which allows improving the inference on the above mentioned parameter. An additional advantage of the proposed procedure is the possibility of testing long memory in the presence of dependent errors such as in the volatility models of ARCH family. The identification and estimation procedure is applied to simulated data from an ARFIMA--ARCH model

Key words: Long memory, Arfima model, Autoregressive process, Identification, Testing hypothesis, Fractional differencing.

Texto completo disponible en PDF

1. Baillie, R. (1996), `Long Memory Processes and Fractional Integration in Econometrics´, Journal of Econometrics 73, 5-59.         [ Links ]

2. Beran, J. (1994), Statistics for Long-Memory Processes, Chapman & Hall/CRC, New York, United States.         [ Links ]

3. Bhardwaj, G. & Swanson, N. R. (2004), `An Empirical Investigation of the Usefulness of ARFIMA Models for Predicting Macroeconomic and Financial Time Series´, Journal of Econometrics 131, 539-578.         [ Links ]

4. Bos, C. S., P. H. Franses, & Ooms, M. (2002), `Inflation, Forecast Intervals and Long Memory Regression Models´, International Journal of Forecasting 110, 167-185.         [ Links ]

5. Brockwell, P. J. & Davies, R. (2006), Time Series: Theory and Methods, Second edn, Springer-Verlag, New York, United States.         [ Links ]

6. Cheung, Y. W. (1993), `Long Memory in Foreign-Exchanges Rates´, Journal of Business and Economic Statistics 11, 93-101.         [ Links ]

7. Cheung, Y. W. & Lai, K. (1995), `A Search of Long Memory in International Stock Market Returns´, Journal of International Money and Finance 14, 597-615.         [ Links ]

8. Chio, K. & Zivot, E. (2007), `Long Memory and Structural Changes in the Forward Discount: An Empirical Investigation´, Journal of International Money and Finance 26, 342-363.         [ Links ]

9. Chow, K. V., Denning, K. C., Ferris, S. & Noronha, G. (1995), `Long-Term and Short-Term Price Memory in the Stock Market´, Economics Letters 49, 287-293.         [ Links ]

10. Davidson, J. (2007), `Time Series Modelling Version 4.24´. Tomado en diciembre de 2007 de la página web. *         [ Links ]http://www.timeseriesmodelling.com

11. Diebold, F. & Rudebush, G. (1989), `Long Memory and Persistence in Aggregate Output´, Journal of Monetary Economics 24, 189-209.         [ Links ]

12. Geweke, J. & Porter-Hudak, S. (1983), `The Estimation and Application of Long-Memory Time Series Models´, Journal of Time Series Analysis 4, 221-238.         [ Links ]

13. Granger, C. W. J. (1980), `Long Memory Relationships and the Aggregation of Dynamic Models´, Journal of Econometrics 14, 227-238.         [ Links ]

14. Granger, C. W. J. & Joyeux, R. (1980), `An Introduction to Long-Memory Time Series Models and Fractional Differencing´, Journal of Time Series Analysis 1, 15-39.         [ Links ]

15. Harris, D., McCabe, B. & Leybourne, S. (2008), `Testing for Long Memory´, Forthcoming in Econometric Theory 24, 143-175.         [ Links ]

16. Hassler, U. & Wolters, J. (1995), `Long Memory in Inflation Rates: International Evidence´, Journal of Business and Economic Statistics 13, 37-45.         [ Links ]

17. Hauser, M. (1997), `Semiparametric and Nonparametric Testing for Long Memory: A Monte Carlo Study´, Empirical Economics 22, 247-271.         [ Links ]

18. Hosking, J. R. M. (1981), `Fractional Differencing´, Biometrika 68, 165-176.         [ Links ]

19. Hurst, H. E. (1951), `Long-Term Storage Capacity of Reservoirs´, Transactions of the American Society of Civil Engineers 116, 770-799.         [ Links ]

20. Hyung, N. & Franses, P. H. (2001), Structural Breaks and Long Memory in US Inflation Rates: Do They Matter for Forecasting?, Econometric Institute Reports, 13 , Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam, Netherlands.         [ Links ]

21. 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.         [ Links ]

22. Lee, D. & Schmidt, P. (1996), `On the Power of the KPSS Test of Stationarity against Fractionally Integrated Alternatives´, Journal of Econometrics 73(1), 285-302.         [ Links ]

23. Lobato, I. N. & Robinson, P. M. (1998), `A Nonparametric Test for I(0)´, Review of Economic Studies 65, 475-495.         [ Links ]

24. Lobato, I. & Robinson, P. M. (1996), `Averaged Periodogram Estimation of Long Memory´, Journal of Econometrics 73, 303-324.         [ Links ]

25. Mandelbrot, B. (1962), `Sur certains prix spéculatifs: faits empiriques et modèle basé sur les processus stables additifs de Paul Lévy´, Comptes Rendus 254, 3968-3970.         [ Links ]

26. Newey, W. & West, K. (1987), `A Simple, Positive Semi-Definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix´, Econometrica 55, 703-708.         [ Links ]

27. Robinson, P. M. (1994), `Semiparametric Analysis of Long-Memory Time Series´, Annals of Statistics 22, 515-539.         [ Links ]

28. Robinson, P. (2003), Time Series with Long Memory, Oxford University Press, London, United Kingdom.         [ Links ]

29. Said, S. & Dickey, D. (1984), `Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order´, Biometrika 71, 599-607.         [ Links ]

30. Schwert, G. W. (1989), `Tests for Unit Roots: A Monte Carlo Investigation´, Journal of Business and Economics Statistics 7, 147-159.         [ Links ]

31. Soofi, A., Wang, S. & Zhang, Y. (2006), `Testing for Long Memory in the Asian Foreign Exchange Rates´, Journal of Systems Science and Complexity 19, 182-190.         [ Links ]

32. Sowell, F. (1992), `Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models´, Journal of Econometrics 53, 165-188.         [ Links ]

33. Stock, J. & Watson, M. (2002), `Macroeconomic Forecasting Using Diffusion Indexes´, Journal of Business and Economic Statistics 20, 147-162.         [ Links ]

34. Tanaka, K. (1999), `The Non-stationary Fractional Unit Root´, Econometric Theory 15, 549-582.         [ Links ]

35. Tschernig, R. (1994), `Long Memory in Foreign Exchange Rates Revisited´, Institute of Statistics and Econometrics. Humboldt University of Berlin.         [ Links ]

36. Velasco, C. (1999), `Gaussian Semiparametric Estimation of Non-stationary Time Series´, Journal of Time Series Analysis 20, 87-127.         [ Links ]

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