1Universidad de Antioquia, Facultad de Ciencias Económicas, Departamento de Estadística y Matemáticas - Departamento de Economía, Medellín, Colombia. Universidad de Antioquia, Facultad de Ciencias Económicas, Grupo de Econometría Aplicada, Medellín, Colombia. Profesor asistente. Email: santiagog@udea.edu.co
2Universidad Nacional de Colombia, Facultad de Ciencias Humanas y Económicas, Departamento de Economía, Medellín, Colombia. Universidad de Antioquia, Facultad de Ciencias Económicas, Grupo de Econometría Aplicada, Medellín, Colombia. Profesor auxiliar. Email: kgomezp@unal.edu.co
The modeling and estimation of the conditional volatility associated with a stochastic process usually have been based on parametric ARCH-type and stochastic volatility models. These time series models are very powerful in representing the dynamic stochastic properties of the data generating process only if the parametric functions are correctly specified. The nonparametric approach acquires importance as a complementary and flexible method to explore these properties without imposing particular functional forms on the conditional moments of process. This paper presents an application of nonparametric time series methods to estimate the conditional volatility function of the COP/USD exchange rate returns. Additionally, we estimate the conditional mean function under this approach.
Key words: Nonparametric regression, Local polynomial regression, Nonlinear time series, Variance function estimation, Autoregressive conditional heteroscedasticity, Time series analysis.
La modelación y estimación de la volatilidad condicional asociada a un proceso estocástico ha estado basada en los modelos paramétricos tipo ARCH y de volatilidad estocástica. Estos modelos son muy poderosos para representar las propiedades dinámicas estocásticas del proceso generador de datos solo si las funciones paramétricas están correctamente especificadas. En este sentido, el enfoque no paramétrico adquiere importancia como un método complementario y flexible para explorar dichas propiedades al no imponer formas funcionales particulares en los momentos condicionales del proceso. Este documento presenta una aplicación de los métodos no paramétricos de series de tiempo para estimar la función de volatilidad condicional de los retornos de la tasa de cambio COP/USD. Además, se estima la función de media condicional bajo este enfoque.
Palabras clave: regresión no paramétrica, regresión polinomial local, series de tiempo no lineales, estimación de la función de varianza, heterocedasticidad condicional autorregresiva, análisis de series de tiempo.
Texto completo disponible en PDF
References
]]>1. Ait-Sahalia, Y. (1996a), 'Nonparametric Pricing of Interest Rate Derivative Securities', Econometrica 64, 527-560. [ Links ]
2. Ait-Sahalia, Y. (1996b), 'Testing Continuous-Time Models of the Spot Interest Rate', The Review of Financial Studies 9, 385-426. [ Links ]
3. Ait-Sahalia, Y. & Lo, A. (1998), 'Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices', Journal of Finance 53, 499-548. [ Links ]
4. Ait-Sahalia, Y. & Lo, A. (2000), 'Nonparametric Risk Management and Implied Risk Aversion', Journal of Econometrics 94, 9-51. [ Links ]
5. Andersen, T., Bollerslev, T. & Diebold, F. (2009), Parametric and Nonparametric Volatility Measurement, 'Handbook of Financial Econometrics', Vol. 1, North Holland, Amsterdam, chapter 2. [ Links ]
6. Andersen, T., Davis, R., Krei\ss, J. & Mikosch, T. (2009), Handbook of Financial Time Series, Springer, New York. [ Links ]
7. Baillie, R., Bollerslev, T. & Mikkelsen, H. (1996), 'Fractionally Integrated Generalized Autoregressive Conditional Heteroscedasticity', Journal of Econometrics 74, 3-30. [ Links ]
8. Bollerslev, T. (1986), 'Generalized Autoregressive Conditional Heteroskedasticity', Journal of Econometrics 31, 307-327. [ Links ]
9. Bollerslev, T., Chou, R. & Kroner, K. (1992), 'ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence', Journal of Econometrics 52, 5-59. [ Links ]
10. Bollerslev, T., Engle, R. & Nelson, D. (1994), ARCH Models, 'Handbook of Econometrics', Vol. 4, North-Holland: Amsterdam. [ Links ]
11. Bossaerts, P., Hardle, W. & Hafner, C. (1995), 'A New Method for Volatility Estimation with Applications in Foreign Exchange Rate Series', Proceedings of the 5th. Karlsruher Okonometrie-Workshop. Universit at Karlsruhe. [ Links ]
12. Bossaerts, P., Hardle, W. & Hafner, C. (1996), Foreign Exchange-Rates Have Surprising Volatility, 'Athens Conference on Applied Probability and Time Series Analysis', Vol. 2, Springer, p. 55-72. [ Links ]
13. Brock, W., Dechert, W., Scheinkman, J. & LeBaron, B. (1996), 'A Test for Independence Based on the Correlation Dimension', Econometric Reviews 15, 197-235. [ Links ]
14. Cabrera, J. (2008), The locpol Package: Kernel Local Polynomial Regression, R Foundation for Statistical Computing. *http://www.R-project.org [ Links ]
15. Castaño, E., Gómez, K. & Gallón, S. (2008), 'Pronóstico y estructuras de volatilidad multiperiodo de la tasa de cambio del peso colombiano', Cuadernos de Economía 48, 241-266. [ Links ]
16. Chen, R. & Tsay, R. (1993), 'Functional-Coefficient Autoregressive Models', Journal of the American Statistical Association 88, 298-308. [ Links ]
17. Cleveland, W. (1979), 'Robust Locally Weighted Regression and Smoothing Scatterplots', Journal of the American Statistical Association 74, 829-836. [ Links ]
18. Davidson, J. (2004), 'Moment and Memory Properties of Linear Conditional Heteroscedasticity Models, and a New Model', Journal of Business and Economics Statistics 22, 16-29. [ Links ]
19. Diabold, F. & Nason, J. (1990), 'Nonparametric Exchange Rate Prediction?', Journal of International Economics 28, 315-332. [ Links ]
20. Ding, Z., Granger, C. & Engle, R. (1993), 'A Long Memory Property of Stock Market Returns and A New Model', Journal of Empirical Finance 1, 83-106. [ Links ]
21. Engle, R. (1982), 'Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation', Econometrica 50, 987-1008. [ Links ]
22. Engle, R. & Bollerslev, T. (1986), 'Modelling the Persistence of Conditional Variances', Econometric Reviews 5, 1-50. [ Links ]
23. Engle, R. & Gonzalez-Rivera, G. (1991), 'Semi-Parametric ARCH Models', Journal of Business and Economic Statistics 9, 345-359. [ Links ]
24. Fan, J. (1993), 'Local Linear Regression Smoothers and their Minimax Efficiency', Annals of Statistics 21, 196-216. [ Links ]
25. Fan, J. & Gijbels, I. (1995), 'Data-Driven Bandwidth Selection in Local Polynomial Fitting: Variable Bandwidth and Spatial Adaptation', Journal of the Royal Statistical Society B57, 371-394. [ Links ]
26. Fan, J. & Gijbels, I. (1996), Local Polynomial Modelling and Its Applications, Chapman and Hall/CRC, London. [ Links ]
27. Fan, J. & Yao, Q. (1998), 'Efficient Estimation of Conditional Variance Functions in Stochastic Regression', Biometrika 85, 645-660. [ Links ]
28. Fan, J. & Yao, Q. (2005), Nonlinear Time Series, Nonparametric and Parametric Methods, Springer, New York. [ Links ]
29. Gao, J. (2007), Nonlinear Time Series: Semiparametric and Nonparametric Methods, Chapman and Hall/CRC, London. [ Links ]
30. Ghysels, E., Harvey, A. & Renault, E. (1996), Stochastic Volatility, 'Handbook of Statistics', Vol. 14, Butterworth-Heinemann, Amsterdam. [ Links ]
31. Hardle, W. (1990), Applied Nonparametric Regression, Cambridge University Press, Cambridge. [ Links ]
32. Hardle, W. & Linton, O. (1994), Applied Nonparametric Methods, 'Handbook of Econometrics', Vol. 4, Elsevier Science & Technology Books, Amsterdam. [ Links ]
33. Hardle, W., Lutkepohl, H. & Chen, R. (1997), 'A Review of Nonparametric Time Series Analysis', International Statistical Review 65, 49-72. [ Links ]
34. Hardle, W., Muller, M., Sperlich, S. & Werwatz, A. (2004), Nonparametric and Semiparametric Models, Springer, New York. [ Links ]
35. Hardle, W. & Tsybakov, A. (1997), 'Local Polynomial Estimators of the Volatility Function in Nonparametric Autoregression', Journal of Econometrics 81, 223-242. [ Links ]
36. Harvey, A., Ruiz, E. & Shephard, N. (1994), 'Multivariate Stochastic Variance Models', The Review of Economic Studies 61, 47-264. [ Links ]
37. Jones, D. A. (1978), 'Non-linear Autoregressive Processes', Journal of the Royal Statistical Society A360, 71-95. [ Links ]
38. Julio, J., Rodríguez, N. & Zárate, H. (2005), 'Estimating the COP Exchange Rate Volatility Smile and the Market Effect of Central Bank Interventions: A CHARN Approach', Borradores de Economía(347). Banco de la República. [ Links ]
39. LeBaron, B. (1990), Forecast Improvements using a Volatility Index, Working Paper , University of Wisconsin-Maddison. [ Links ]
40. Masry, E. & Tjostheim, D. (1995), 'Nonparametric Estimation and Identification of Nonlinear ARCH Time Series: Strong Convergence and Asymptotic Normality', Econometric Theory 11, 258-289. [ Links ]
41. Maya, C. & Gómez, K. (2008), 'What Exactly is Bad News in Foreign Exchange Markets? Evidence from Latin American Markets', Cuadernos de Economía 45, 161-183. [ Links ]
42. Meese, R. A. & Rose, A. K. (1990), 'Nonlinear, Nonparametric, Nonessential Exchange Rate Estimation', American Economic Review Paper and Proceedings 80, 192-196. [ Links ]
43. Mizrach, B. (1992), 'Multivariate Nearest-neighbour Forecasts of EMS Exchange Rates', Journal of Applied Econometrics 7, S151-63. [ Links ]
44. Nelson, D. (1991), 'Conditional Heteroskedasticity in Asset Returns: A New Approach', Econometrica 59, 347-370. [ Links ]
45. Pagan, A. & Ullah, A. (1988), 'The Econometric Analysis of Models with Risk Terms', Journal of Time Series Analysis 3, 87-105. [ Links ]
46. Robinson, P. M. (1983), 'Nonparametric Estimators for Time Series', Journal of Time Series Analysis 4, 185-207. [ Links ]
47. Shephard, N. (2005), Stochastic Volatility: Selected Readings, Oxford University Press, New York. [ Links ]
48. Straumann, D. (2005), Estimation in Conditionally Heteroscedastic Time Series Models, Vol. 181 of Lecture Notes in Statistics, Springer, Berlin. [ Links ]
49. Tjostheim, D. (1999) Nonparametric Specification Procedures for Time Series 'Asymptotics, Nonparametrics, and Time Series' Vol. 158, Dekker, Marcel Inc. New York p. 149-199. [ Links ]
50. Ziegelmann, F. (2002), 'Nonparametric Estimation of Volatility Functions: The Local Exponential Estimator', Econometric Theory 18, 985-991. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv33n1a03,
AUTHOR = {Gallón, Santiago and Gómez, Karoll},
TITLE = {{Nonparametric Time Series Analysis of the Conditional Mean and Volatility Functions for the COP/USD Exchange Rate Returns}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2010},
volume = {33},
number = {1},
pages = {25-41}
}