[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/a/wly/japmet/v21y2006i3p307-336.html
   My bibliography  Save this article

Normal mixture GARCH(1,1): applications to exchange rate modelling

Author

Listed:
  • Carol Alexander
  • Emese Lazar
Abstract
Some recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time variation in both conditional skewness and kurtosis. A main focus of the paper is to provide evidence that, for modelling exchange rates, generalized two‐component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student's t‐GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, euro and Japanese yen), we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates. Copyright © 2006 John Wiley & Sons, Ltd.

Suggested Citation

  • Carol Alexander & Emese Lazar, 2006. "Normal mixture GARCH(1,1): applications to exchange rate modelling," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(3), pages 307-336, April.
    • Handle: RePEc:wly:japmet:v:21:y:2006:i:3:p:307-336
    DOI: 10.1002/jae.849
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/jae.849
    Download Restriction: no
    File URL: https://libkey.io/10.1002/jae.849?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Engle, Robert F & Ito, Takatoshi & Lin, Wen-Ling, 1990. "Meteor Showers or Heat Waves? Heteroskedastic Intra-daily Volatility in the Foreign Exchange Market," Econometrica, Econometric Society, vol. 58(3), pages 525-542, May.
    3. Bollerslev, Tim & Ghysels, Eric, 1996. "Periodic Autoregressive Conditional Heteroscedasticity," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 139-151, April.
    4. Carol Alexandra & Emese Lazar, 2004. "The Equity Index Skew, Market Crashes and Asymmetric Normal Mixture GARCH," ICMA Centre Discussion Papers in Finance icma-dp2004-13, Henley Business School, University of Reading.
    5. Bauwens, L. & Bos, C.S. & van Dijk, H.K., 1999. "Adaptive Polar Sampling with an Application to a Bayes Measure of Value-at-Risk," Econometric Institute Research Papers TI 99-082/4, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    6. V. L. Martin & G. M. Martin & G. C. Lim, 2005. "Parametric pricing of higher order moments in S&P500 options," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(3), pages 377-404.
    7. Muller, Ulrich A. & Dacorogna, Michel M. & Dave, Rakhal D. & Olsen, Richard B. & Pictet, Olivier V. & von Weizsacker, Jacob E., 1997. "Volatilities of different time resolutions -- Analyzing the dynamics of market components," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 213-239, June.
    8. Huisman, Ronald, et al, 2001. "Tail-Index Estimates in Small Samples," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 208-216, April.
    9. Harvey, Campbell R. & Siddique, Akhtar, 1999. "Autoregressive Conditional Skewness," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(4), pages 465-487, December.
    10. Baillie, Richard T & Bollerslev, Tim, 2002. "The Message in Daily Exchange Rates: A Conditional-Variance Tale," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 60-68, January.
    11. Klaassen, F.J.G.M., 1998. "Improving Garch Volatility Forecasts," Other publications TiSEM f5bcd096-7744-4137-aabc-6, Tilburg University, School of Economics and Management.
    12. Markus Haas, 2004. "A New Approach to Markov-Switching GARCH Models," Journal of Financial Econometrics, Oxford University Press, vol. 2(4), pages 493-530.
    13. Hsieh, David A, 1989. "Testing for Nonlinear Dependence in Daily Foreign Exchange Rates," The Journal of Business, University of Chicago Press, vol. 62(3), pages 339-368, July.
    14. repec:bla:jecsur:v:16:y:2002:i:3:p:447-85 is not listed on IDEAS
    15. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    16. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    17. Nelson, Daniel B., 1996. "Asymptotic filtering theory for multivariate ARCH models," Journal of Econometrics, Elsevier, vol. 71(1-2), pages 1-47.
    18. repec:bla:jfinan:v:59:y:2004:i:2:p:755-793 is not listed on IDEAS
    19. Engle, Robert F. & White (the late), Halbert (ed.), 1999. "Cointegration, Causality, and Forecasting: Festschrift in Honour of Clive W. J. Granger," OUP Catalogue, Oxford University Press, number 9780198296836.
    20. Ding, Zhuanxin & Granger, Clive W. J., 1996. "Modeling volatility persistence of speculative returns: A new approach," Journal of Econometrics, Elsevier, vol. 73(1), pages 185-215, July.
    21. Kai-Li Wang & Christopher Fawson & Christopher B. Barrett & James B. McDonald, 2001. "A flexible parametric GARCH model with an application to exchange rates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(4), pages 521-536.
    22. Gurdip Bakshi & Nikunj Kapadia & Dilip Madan, 2003. "Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options," The Review of Financial Studies, Society for Financial Studies, vol. 16(1), pages 101-143.
    23. Melick, William R. & Thomas, Charles P., 1997. "Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(1), pages 91-115, March.
    24. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    25. Engle, Robert F & Gonzalez-Rivera, Gloria, 1991. "Semiparametric ARCH Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(4), pages 345-359, October.
    26. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2002. "Parametric and Nonparametric Volatility Measurement," NBER Technical Working Papers 0279, National Bureau of Economic Research, Inc.
    27. Boothe, Paul & Glassman, Debra, 1987. "The statistical distribution of exchange rates: Empirical evidence and economic implications," Journal of International Economics, Elsevier, vol. 22(3-4), pages 297-319, May.
    28. Alexander, Carol, 2004. "Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects," Journal of Banking & Finance, Elsevier, vol. 28(12), pages 2957-2980, December.
    29. S»bastien Laurent and Jean-Philippe Peters, 2001. "G@RCH 2.0: An Ox Package for Estimating and Forecasting Various ARCH Models," Computing in Economics and Finance 2001 123, Society for Computational Economics.
    30. Fernández, C. & Steel, M.F.J., 1996. "On Bayesian Modelling of Fat Tails and Skewness," Other publications TiSEM 0991c197-c9e8-4904-8119-3, Tilburg University, School of Economics and Management.
    31. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    32. Vlaar, Peter J G & Palm, Franz C, 1993. "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(3), pages 351-360, July.
    33. Bates, David S, 1991. "The Crash of '87: Was It Expected? The Evidence from Options Markets," Journal of Finance, American Finance Association, vol. 46(3), pages 1009-1044, July.
    34. Epps, Thomas W & Epps, Mary Lee, 1976. "The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis," Econometrica, Econometric Society, vol. 44(2), pages 305-321, March.
    35. Gray, Stephen F., 1996. "Modeling the conditional distribution of interest rates as a regime-switching process," Journal of Financial Economics, Elsevier, vol. 42(1), pages 27-62, September.
    36. Chris Brooks, 2005. "Autoregressive Conditional Kurtosis," Journal of Financial Econometrics, Oxford University Press, vol. 3(3), pages 399-421.
    37. K. Newey, Whitney, 1985. "Generalized method of moments specification testing," Journal of Econometrics, Elsevier, vol. 29(3), pages 229-256, September.
    38. Ball, Clifford A. & Torous, Walter N., 1983. "A Simplified Jump Process for Common Stock Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 18(1), pages 53-65, March.
    39. Bai, Xuezheng & Russell, Jeffrey R. & Tiao, George C., 2003. "Kurtosis of GARCH and stochastic volatility models with non-normal innovations," Journal of Econometrics, Elsevier, vol. 114(2), pages 349-360, June.
    40. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-730, August.
    41. Markus Haas, 2004. "Mixed Normal Conditional Heteroskedasticity," Journal of Financial Econometrics, Oxford University Press, vol. 2(2), pages 211-250.
    42. Richard T. Baillie & Tim Bollerslev, 1991. "Intra-Day and Inter-Market Volatility in Foreign Exchange Rates," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 58(3), pages 565-585.
    43. Hsieh, David A, 1989. "Modeling Heteroscedasticity in Daily Foreign-Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(3), pages 307-317, July.
    44. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
    45. Hsieh, David A., 1988. "The statistical properties of daily foreign exchange rates: 1974-1983," Journal of International Economics, Elsevier, vol. 24(1-2), pages 129-145, February.
    46. Cai, Jun, 1994. "A Markov Model of Switching-Regime ARCH," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(3), pages 309-316, July.
    47. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-547, August.
    48. Hamilton, James D. & Susmel, Raul, 1994. "Autoregressive conditional heteroskedasticity and changes in regime," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 307-333.
    49. De Vries, C.G. & Leuven, K.U., 1994. "Stylized Facts of Nominal Exchange Rate Returns," Papers 94-002, Purdue University, Krannert School of Management - Center for International Business Education and Research (CIBER).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Carol Alexander & Emese Lazar, 2009. "Modelling Regime‐Specific Stock Price Volatility," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 71(6), pages 761-797, December.
    2. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    3. Degiannakis, Stavros & Xekalaki, Evdokia, 2004. "Autoregressive Conditional Heteroskedasticity (ARCH) Models: A Review," MPRA Paper 80487, University Library of Munich, Germany.
    4. BAUWENS, Luc & HAFNER, Christian & LAURENT, Sébastien, 2011. "Volatility models," LIDAM Discussion Papers CORE 2011058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
      • Bauwens, L. & Hafner, C. & Laurent, S., 2012. "Volatility Models," LIDAM Reprints ISBA 2012028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
      • Bauwens, L. & Hafner C. & Laurent, S., 2011. "Volatility Models," LIDAM Discussion Papers ISBA 2011044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Franses,Philip Hans & Dijk,Dick van, 2000. "Non-Linear Time Series Models in Empirical Finance," Cambridge Books, Cambridge University Press, number 9780521779654.
    6. Font, Begoña, 1998. "Modelización de series temporales financieras. Una recopilación," DES - Documentos de Trabajo. Estadística y Econometría. DS 3664, Universidad Carlos III de Madrid. Departamento de Estadística.
    7. Cheung, Yin-Wong & Chung, Sang-Kuck, 2009. "A Long Memory Model with Mixed Normal GARCH for US Inflation Data," Santa Cruz Department of Economics, Working Paper Series qt2202s99q, Department of Economics, UC Santa Cruz.
    8. Yin-Wong Cheung & Sang-Kuck Chung, 2011. "A Long Memory Model with Normal Mixture GARCH," Computational Economics, Springer;Society for Computational Economics, vol. 38(4), pages 517-539, November.
    9. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2005. "Volatility forecasting," CFS Working Paper Series 2005/08, Center for Financial Studies (CFS).
    10. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    11. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    12. Chihwa Kao, 2001. "Geography, Industrial Organization, and Agglomeration Heteroskedasticity Models with Estimates of the Variances of Foreign Exchange Rates," Center for Policy Research Working Papers 34, Center for Policy Research, Maxwell School, Syracuse University.
    13. Alexander, Carol, 2004. "Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects," Journal of Banking & Finance, Elsevier, vol. 28(12), pages 2957-2980, December.
    14. Yang (Greg) Hou & Mark Holmes, 2020. "Do higher order moments of return distribution provide better decisions in minimum-variance hedging? Evidence from US stock index futures," Australian Journal of Management, Australian School of Business, vol. 45(2), pages 240-265, May.
    15. Luc, BAUWENS & Arie, PREMINGER & Jeroen, ROMBOUTS, 2006. "Regime switching GARCH models," Discussion Papers (ECON - Département des Sciences Economiques) 2006006, Université catholique de Louvain, Département des Sciences Economiques.
    16. LeBaron, Blake, 2003. "Non-Linear Time Series Models in Empirical Finance,: Philip Hans Franses and Dick van Dijk, Cambridge University Press, Cambridge, 2000, 296 pp., Paperback, ISBN 0-521-77965-0, $33, [UK pound]22.95, [," International Journal of Forecasting, Elsevier, vol. 19(4), pages 751-752.
    17. Mohamed Saidane & Christian Lavergne, 2009. "Optimal Prediction with Conditionally Heteroskedastic Factor Analysed Hidden Markov Models," Computational Economics, Springer;Society for Computational Economics, vol. 34(4), pages 323-364, November.
    18. Yongmiao Hong & Haitao Li & Feng Zhao, 2013. "Can the Random Walk Model be Beaten in Out-of-Sample Density Forecasts? Evidence from Intraday Forei," Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    19. Dinghai Xu & Tony S. Wirjanto, 2008. "An Empirical Characteristic Function Approach to VaR under a Mixture of Normal Distribution with Time-Varying Volatility," Working Papers 08008, University of Waterloo, Department of Economics.
    20. Eduardo Rossi, 2010. "Univariate GARCH models: a survey (in Russian)," Quantile, Quantile, issue 8, pages 1-67, July.

    More about this item

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:japmet:v:21:y:2006:i:3:p:307-336. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: http://www.interscience.wiley.com/jpages/0883-7252/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.