Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T18:02:02.908Z Has data issue: false hasContentIssue false
  • English
  • Français

PARAMETER ESTIMATION IN NONLINEAR AR–GARCH MODELS

Published online by Cambridge University Press:  31 May 2011

Mika Meitz  and
Pentti Saikkonen
Mika Meitz*
Affiliation:
Koç University
Pentti Saikkonen*
Affiliation:
University of Helsinki
*
*Address correspondence to: Mika Meitz, Department of Economics, Koç University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey; e-mail: [email protected]; or to: Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FIN–00014 University of Helsinki, Finland; e-mail: [email protected].
*Address correspondence to: Mika Meitz, Department of Economics, Koç University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey; e-mail: [email protected]; or to: Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FIN–00014 University of Helsinki, Finland; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 6 , December 2011 , pp. 1236 - 1278
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Billingsley, P. (1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Billingsley, P. (1995) Probability and Measure, 3rd ed.Wiley.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T., Engle, R.F., & Nelson, D.B. (1994) ARCH models. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. IV, chap. 49. Elsevier.Google Scholar
Bougerol, P. (1993) Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31, 942–959.CrossRefGoogle Scholar
Chan, K.S. (1993) Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Annals of Statistics 21, 520–533.CrossRefGoogle Scholar
Chan, K.S. & Tong, H. (1986) On estimating thresholds in autoregressive models. Journal of Time Series Analysis 7, 179–190.CrossRefGoogle Scholar
Chen, R. & Tsay, R.S. (1993) Functional-coefficient autoregressive models. Journal of the American Statistical Association 88, 298–308.Google Scholar
Cline, D.B.H. (2007) Stability of nonlinear stochastic recursions with application to nonlinear AR–GARCH models. Advances in Applied Probability 39, 462–491.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Ding, Z., Granger, C.W.J., & Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Escanciano, J.C. (2009) Quasi-maximum likelihood estimation of semi-strong GARCH models. Econometric Theory 25, 561–570.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2007) Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and Their Applications 117, 1265–1284.CrossRefGoogle Scholar
Glosten, L.W., Jaganathan, R., & Runkle, D.E. (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
González-Rivera, G. (1998) Smooth-transition GARCH models. Studies in Nonlinear Dynamics and Econometrics 3, 61–78.Google Scholar
Haggan, V. & Ozaki, T. (1981) Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika 68, 189–196.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2005) Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2009) Asymptotics of the QMLE for non-linear ARCH models. Journal of Time Series Econometrics 1(1), article 2.CrossRefGoogle Scholar
Lai, T.L. (1994) Asymptotic properties of nonlinear least squares estimates in stochastic regression models. Annals of Statistics 22, 1917–1930.CrossRefGoogle Scholar
Lange, T., Rahbek, A., & Jensen, S.T. (2006) Estimation and Asymptotic Inference in the First Order AR–ARCH Model. Preprint No. 4, University of Copenhagen.Google Scholar
Lanne, M. & Saikkonen, P. (2005) Nonlinear GARCH models for highly persistent volatility. Econometrics Journal 8, 251–276.CrossRefGoogle Scholar
Lee, S.-W. & Hansen, B.E. (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Liebscher, E. (2005) Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes. Journal of Time Series Analysis 26, 669–689.CrossRefGoogle Scholar
Ling, S. (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA–GARCH/ IGARCH models. Journal of Econometrics 140, 849–873.CrossRefGoogle Scholar
Ling, S. & Li, W.K. (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184–1194.CrossRefGoogle Scholar
Ling, S. & Li, W.K. (1998) Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors. Annals of Statistics 26, 84–125.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Linton, O., Pan, J., & Wang, H. (2010) Estimation for a nonstationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 26, 1–28.CrossRefGoogle Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575–596.CrossRefGoogle Scholar
Lundbergh, S. & Teräsvirta, T. (2002) Evaluating GARCH models. Journal of Econometrics 110, 417–435.CrossRefGoogle Scholar
Meitz, M. & Saikkonen, P. (2008a) Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory 24, 1291–1320.CrossRefGoogle Scholar
Meitz, M. & Saikkonen, P. (2008b) Parameter Estimation in Nonlinear AR–GARCH Models. Department of Economics Discussion Paper Series no. 396, University of Oxford.CrossRefGoogle Scholar
Meitz, M. & Saikkonen, P. (2008c) Stability of nonlinear AR–GARCH models. Journal of Time Series Analysis 29, 453–475.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.CrossRefGoogle Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Pan, J., Wang, H., & Tong, H. (2008) Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics 142, 352–378.CrossRefGoogle ScholarPubMed
Pantula, S.G. (1988) Estimation of autoregressive models with ARCH errors. Sankhya B 50, 119–138.Google Scholar
Pfanzagl, J. (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–272.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1991a) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Reviews 10, 125–216.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1991b) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: Asymptotic normality. Econometric Reviews 10, 253–325.CrossRefGoogle Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Teräsvirta, T. (1994) Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association 89, 208–218.Google Scholar
Tjøstheim, D. (1986) Estimation in nonlinear time series models. Stochastic Processes and Their Applications 21, 251–273.CrossRefGoogle Scholar
van Dijk, D., Teräsvirta, T., & Franses, P.H. (2002) Smooth transition autoregressive models — A survey of recent developments. Econometric Reviews 21, 1–47.CrossRefGoogle Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar
Zakoïan, J.-M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar