Abstract
This paper studies asymptotic properties of the quasi maximum likelihood and weighted least squares estimates (QMLE and WLSE) of the conditional variance slope parameters of a strictly unstable ARCH model with periodically time varying coefficients (PARCH in short). The model is strictly unstable in the sense that its parameters lie outside the strict periodic stationarity domain and its boundary. Obtained from the regression form of the PARCH, the WLSE is a variant of the least squares method weighted by the square of the conditional variance evaluated at any fixed value in the parameter space. In calculating the QMLE and WLSE, the conditional variance intercepts are set to any arbitrary values not necessarily the true ones. The theoretical finding is that the QMLE and WLSE are consistent and asymptotically Gaussian with the same asymptotic variance irrespective of the fixed conditional variance intercepts and the weighting parameters. So because of its numerical complexity, the QMLE may be dropped in favor of the WLSE which enjoys closed form.
Similar content being viewed by others
References
Aknouche A (2011) Multi-stage weighted least squares estimation of ARCH processes in the stable and unstable cases. Statistical inference for stochastic processes (forthcoming)
Aknouche A, Bentarzi M (2008) On the existence of higher-order moments of periodic GARCH models. Stat Probab Lett 78: 3262–3268
Aknouche A, Bibi A (2009) Quasi-maximum likelihood estimation of periodic GARCH and periodic ARMA–GARCH processes. J Time ser Anal 30: 19–46
Aue A, Horváth L (2011) Quasi-likelihood estimation in stationary and nonstationary autoregressive models with random coefficients. Stat Sin (in press)
Berkes I, Horváth L, Ling S (2009) Estimation in nonstationary random coefficient autoregressive models. J Time Ser Anal 30: 395–416
Bollerslev T, Ghysels E (1996) Periodic autoregressive conditional heteroskedasticity. J Bus Econ Stat 14: 139–152
Bose A, Mukherjee K (2003) Estimating the ARCH parameters by solving linear equations. J Time Ser Anal 24: 127–136
Boyles RA, Gardner WA (1983) Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE Trans Inform Theory 29: 105–114
Breiman L (1968) Probability. Addison Wesley, Menlo Park
Brown BM (1971) Martingale central limit theorems. Ann Math Stat 42: 59–66
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50: 987–1008
Francq C, Zakoïan JM (2004) Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10: 605–637
Francq C, Zakoïan JM (2010a) Inconsistency of the MLE and inference based on weighted LS for LARCH models. J Econom 159: 151–165
Francq C, Zakoïan JM (2010b) Strict stationarity testing and estimation of explosive ARCH models. Preprint MPRA, 22414
Franses P, Paap R (2000) Modeling day-of-the-week seasonality in the S&P 500 index. Appl Finan Econ 10: 483–488
Franses P, Paap R (2004) Periodic time series models. Oxford University Press, Oxford
Ghysels E, Osborn D (2001) The econometric analysis of seasonal time series. Cambridge University Press, Cambridge
Goldie C, Maller R (2000) Stability of perpetuities. Ann Probab 28: 1195–1218
Hwang SY, Basawa IV (2005) Explosive random-coefficient AR(1) processes and related asymptotics for least squares estimation. J Time Ser Anal 26: 807–824
Jensen ST, Rahbek A (2004a) Asymptotic normality of the QML estimator of ARCH in the nonstationary case. Econometrica 72: 641–646
Jensen ST, Rahbek A (2004b) Asymptotic inference for nonstationary GARCH. Econom Theory 20: 1203–1226
Klüppelberg C, Lindner A, Maller R (2004) A continuous time GARCH Process driven by a Lévy process: stationarity and second order behavior. J Appl Probab 41: 601–622
Ling S (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA–GARCH/IGARCH models. J Econom 140: 849–873
Ling S, Li D (2008) Asymptotic inference for a nonstationary double AR(1) model. Biometrika 95: 257–263
Linton O, Pan J, Wang H (2010) Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econom Theory 26: 1–28
Martens M, Chang Y, Taylor SJ (2002) A comparison of seasonal adjustment methods when forecasting intraday volatility. J Finan Res 25: 283–299
Nelson DB (1990) Stationarity and persistence in the GARCH(1,1) model. Econom Theory 6: 318–334
Pantula SG (1988) Estimation of autoregressive models with ARCH errors. Sankhya 50: 119–138
Taylor JW (2006) Density forecasting for the efficient balancing of the generation and consumption of electricity. Int J Forecast 22: 707–724
Taylor N (2004) Modeling discontinuous periodic conditional volatility: evidence from the commodity futures market. J Futur Mark 24: 805–834
Weiss AA (1986) Asymptotic theory for ARCH models: estimation and testing. Econom Theory 2: 107–131
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aknouche, A., Al-Eid, E. Asymptotic inference of unstable periodic ARCH processes. Stat Inference Stoch Process 15, 61–79 (2012). https://doi.org/10.1007/s11203-011-9063-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11203-011-9063-1
Keywords
- Periodic ARCH process
- Strict periodic Stationarity
- Quasi maximum likelihood estimate
- Weighted least squares estimate
- Asymptotic normality