Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T05:47:21.921Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A FAMILY OF CONTRASTS FOR PARAMETRIC INFERENCE IN DEGENERATE ARCH MODELS

Published online by Cambridge University Press:  16 April 2014

Lionel Truquet
Lionel Truquet*
Affiliation:
University of Rennes 1 and CREST-ENSAI
*
*Address correspondence to Lionel Truquet, Campus de Ker-Lann, rue Blaise Pascal, BP 37203, 35172 Bruz cedex, France; e-mail: lionel.truquet@ensai.fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we introduce a family of contrasts for parametric inference in ARCH models the volatility of which exhibits some degeneracy. We focus specifically on ARCH processes with a linear volatility (called LARCH processes), for which the Gaussian quasi-likelihood estimator may be inconsistent. Our approach generalizes that of Beran and Schützner (2009) and gives an interesting alternative to the WLSE used by Francq and Zakoïan (2010) for an autoregressive process with LARCH errors. The family of contrasts is indexed by a single parameter that controls the smoothness of an approximated quasi-likelihood function. Under mild conditions, the resulting estimators are shown to be strongly consistent and asymptotically normal. The optimal asymptotic variance is also given. For LARCH processes, an atypical result is obtained: under assumptions, we show that the limiting distribution of the estimators can be arbitrarily close to a Gaussian distribution supported on a line. Extensions to multivariate processes are also discussed.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 6 , December 2014 , pp. 1165 - 1206
Copyright
Copyright © Cambridge University Press 2014 

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

Bardet, J.-M. & Wintenberger, O. (2009) Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes. Annals of Statistics 37(5B), 2730–2759.CrossRefGoogle Scholar
Beran, J. & Schützner, M. (2009) On approximate pseudo-maximum likelihood estimation for LARCH-processes. Bernoulli 15(4), 1057–1081.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P.S. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. John Wiley.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Doukhan, P., Teyssière, G., & Winant, P. (2006) Vector valued ARCH infinity processes. In Bertail, P., Doukhan, P., & Soulier, P. (eds.), Dependence in Probability and Statistics. Springer.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.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., Makarova, S., Zakoïan, J.-M. (2008) A class of stochastic unit-root bilinear processes. Mixing properties and unit-root test. Journal of Econometrics 142(1), 312–326.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2010) Inconsistency of the QMLE and inference based on weighted LS for LARCH models. Journal of Econometrics 159(1), 151–165.CrossRefGoogle Scholar
Genon-Catalot, V. & Jacod, J. (1993) On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Annales de l’institut Henri Poincar (B) Probabilits et Statistiques 29, 119–151.Google Scholar
Giraitis, L., Robinson, P., & Surgailis, D. (2000) A model for long memory conditional heteroscedasticity. Annals of Applied Probability 10(3), 1002–1024.CrossRefGoogle Scholar
Giraitis, L., Leipus, R., Robinson, P., & Surgailis, D. (2004) LARCH, leverage, and long memory. Journal of Financial Econometrics 2(2), 177–210.CrossRefGoogle Scholar
Horowitz, J.L. (1992) A smoothed maximum score estimator for the binary response model. Econometrica 60, 505–531.CrossRefGoogle Scholar
Ionides, E.L. (2005) Maximum smoothed likelihood estimation. Statistica Sinica 15, 1003–1014.Google Scholar
Jeantheau, T. (1998) Strong Consistency of Estimators for Multivariate ARCH Models. Econometric Theory 14(1), 70–86.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
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
Mikosch, T. & Straumann, D. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34(5), 2449–2495.Google Scholar
Pfanzagl, J. (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–272.CrossRefGoogle Scholar
Sentana, E. (1995) Quadratic ARCH models. Review of Economic Studies 62, 639–661.CrossRefGoogle Scholar
Seo, M. & Linton, O. (2007) A smoothed least squares estimator for threshold regression models. Journal of Econometrics 141, 704–735.CrossRefGoogle Scholar
Sorensen, H. (2004) Parametric inference for diffusion processes observed at discrete points in time: A survey. International Statistical Review 72(3), 337–354.CrossRefGoogle Scholar
Straumann, D. (2004) Estimation in Conditionally Heteroscedastic Time Series Models. Lecture Notes in Statistics. Springer Verlag.Google Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar