Francq, Christian and Zakoian, JeanMichel (2009): Inconsistency of the QMLE and asymptotic normality of the weighted LSE for a class of conditionally heteroscedastic models.

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Abstract
This paper considers a class of finiteorder autoregressive linear ARCH models. The model captures the leverage effect, allows the volatility to be zero and to reach its minimum for nonzero innovations, and is appropriate for longmemory modeling when infinite orders are allowed. It is shown that the quasimaximum likelihood estimator is, in general, inconsistent. To solve this problem, we propose a selfweighted leastsquares estimator and show that this estimator is asymptotically normal. Furthermore, a score test for conditional homoscedasticity and diagnostic portmanteau tests are developed. The latter have an asymptotic distribution which is far from the standard chisquare. Simulation experiments are carried out to assess the performance of the proposed estimator.
Item Type:  MPRA Paper 

Original Title:  Inconsistency of the QMLE and asymptotic normality of the weighted LSE for a class of conditionally heteroscedastic models. 
Language:  English 
Keywords:  Conditional homoscedasticity testing; Inconsistent estimator; Leverage effect; Linear ARCH; Quasimaximum likelihood; Weighted leastsquares 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 
Item ID:  15147 
Depositing User:  Christian Francq 
Date Deposited:  11. May 2009 01:45 
Last Modified:  11. Feb 2013 19:50 
References:  Beran, J. (2006) On location estimation for LARCH processes. Journal of Multivariate Analysis 97, 17661782. Beran, J. and M. Schützner (2008) Estimation of dependence parameters for linear ARCH processes. ???. Berkes, I. and L. Horv\'ath (2003) Asymptotic results for long memory LARCH sequences. Annals of Applied Probability 13, 641668. Berkes, I., Horv\'ath, L. and P. S. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201227. Billingsley, P.(1961) The LindebergLevy theorem for martingales. Proceedings of the American Mathematical Society 12, 788792. Billingsley, P.(1995) Probability and Measure. John Wiley, New York. Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327. Bougerol, P. and N. Picard (1992a) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 17141729. Bougerol, P. and N. Picard (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115127. Boyles, R.A., Marshall, A.W., and F. Proschan (1985) Inconsistency of the maximum likelihood estimator of a distribution having increasing failure rate average. The Annals of Statistics 13, 413417. Doukhan, P., Teyssière, G. and P. Winant. (2006) Vector valued ARCH infinity processes, in Dependence in Probability and Statistics. P. Bertail, P. Doukhan and P. Soulier Editors, Springer, New York. Engle, R. F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 9871007. Ferguson, T.S. (1982) An inconsistent maximum likelihood estimate. Journal of the American Statistical Association, 77, 380. Francq, C., Makarova, S. and JM. Zakoïan (2007) A class of stochastic unitroot bilinear processes. Mixing properties and unitroot test. Journal of Econometrics 142, 312326. Francq, C., Roy, R. and JM. Zakoïan (2005) Diagnostic checking in ARMA Models with Uncorrelated Errors, Journal of the American Statistical Association 100, 532544. Francq, C., and JM. Zakoïan (2004) Maximum Likelihood Estimation of Pure GARCH and ARMAGARCH Processes. Bernoulli 10, 605637. Francq, C., and JM. Zakoïan (2008) Testing the nullity of GARCH coefficients : correction of the standard tests and relative efficiency comparisons. Unpublished document. Ghosh, M. (1985) Inconsistent maximum likelihood estimators for the Rasch model. Statistic and Probability Letters 23, 165170. Giraitis, L., Leipus, R., Robinson, P.M. and D. Surgailis (2004) LARCH, Leverage, and Long Memory. Journal of Financial Econometrics 2, 177210. Giraitis, L., Robinson, P.M. and D. Surgailis (2000) A model for long memory conditional heteroscedasticity. The Annals of Applied Probability 10, 10021024. Giraitis, L. and D. Surgailis (2002) ARCHtype bilinear models with double long memory. Stochastic Processes and their Applications 100, 275300. Godfrey, L.G. (1988) Misspecification tests in econometrics: the Lagrange multiplier principle and other approaches. Cambridge University Press, Cambridge. Hall, P. and Q. Yao (2003) Inference in ARCH and GARCH models with heavytailed errors. Econometrica 71, 285317. Higgins, M.L. and A.K. Bera (1992) A class of nonlinear ARCH models. International Economic Review 33, 137158. Horv\'ath, L. and F. Liese (2004) $L_p$estimators in ARCHmodels. Journal of Statistical Planning and Inference 119, 277309. Kiefer, J. and J. Wolfowitz (1956) Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters. The Annals of Mathematical Statistics 27, 887906. Li, W.K. and T.K. Mak (1994) On the squared residual autocorrelations in nonlinear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627636. Ling, S. (2005) Selfweighted LSE and MLE for ARMAGARCH models. Unpublished paper, HongKong University of Science and Technology. Ling, S. (2007) Selfweighted and local quasimaximum likelihood estimators for ARMAGARCH/IGARCH models. Journal of Econometrics 140, 849873. McLeod, A.I. and W.K. Li (1983) Diagnostic Checking ARMA Time Series Models Using SquaredResidual Autocorrelations. Journal of Time Series Analysis 4, 269273. Neyman, J. and E.L. Scott (1948) Consistent estimates based on partially consistent observations. Econometrica 16, 132. Redner, R. and H. Walker (1984) Mixture densities, maximum likelihood and the EM algorithm. SIAM revue 26, 195239. Robinson, P.M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 6784. Sentana, E. (1995) Quadratic ARCH models. Review of Economic Studies 62, 639661. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/15147 