Offline and online weighted least squares estimation of nonstationary power ARCH processes. (English) Zbl 1219.62131

Summary: This paper proposes two estimation methods based on a weighted least squares criterion for non-(strictly) stationary power ARCH models. The weights are the squared volatilities evaluated at a known value in the parameter space. The first method is adapted for fixed sample size data while the second one allows for online data available in real time. It will be shown that these methods provide consistent and asymptotically Gaussian estimates having asymptotic variance equal to that of the quasi-maximum likelihood estimate (QMLE) regardless of the value of the weighting parameter. Finite-sample performances of the proposed WLS estimates are shown via a simulation study for various sub-classes of power ARCH models.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
65C60 Computational problems in statistics (MSC2010)
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[1] Aknouche, A., Al-Eid, E., 2011. Asymptotic inference of unstable periodic \(A R C H\) processes, Statistical Inference for Stochastic Processes (in press). · Zbl 1236.62010
[2] Aue, A., Horváth, L., 2011. Quasi-likelihood estimation in stationary and nonstationary autoregressive models with random coefficients, Statistica Sinica (in press).
[3] Benveniste, A.; Métivier, M.; Priouret, P., Adaptive algorithms and stochastic approximation, (1990), Springer-Verlag Berlin
[4] Berkes, I.; Horváth, L.; Ling, S., Estimation in nonstationary random coefficient autoregressive models, Journal of time series analysis, 30, 395-416, (2009) · Zbl 1224.62046
[5] Brown, B.M., Martingale central limit theorems, Annals of mathematical statistics, 42, 59-66, (1971) · Zbl 0218.60048
[6] Chan, N.H.; Wei, C.Z., Limiting distributions of least-squares estimates of unstable autoregressive processes, Annals of statistics, 16, 367-401, (1988) · Zbl 0666.62019
[7] Dickey, D.A.; Fuller, W.A., Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057-1071, (1981) · Zbl 0471.62090
[8] Ding, Z.; Granger, C.W.J.; Engle, R.F., A long memory property of stock market returns and a new model, Journal of empirical finance, 1, 83-106, (1993)
[9] Engle, R.F., Autoregressive conditional heteroskedasticity with estimates of variance of UK inflation, Econometrica, 50, 987-1008, (1982)
[10] Francq, C., Zakoïan, J.M., 2010, Strict stationarity testing and estimation of explosive ARCH models, Preprint MPRA, 22414.
[11] Higgins, M.L.; Bera, A.K., A class of nonlinear \(A R C H\) models, International economic review, 33, 137-158, (1992) · Zbl 0744.62152
[12] Hwang, S.Y.; Basawa, I.V., Stationarity and moment structure for box-Cox transformed threshold \(G A R C H(1, 1)\) processes, Statistics & probability letters, 68, 209-220, (2004) · Zbl 1075.62080
[13] Hwang, S.Y.; Basawa, I.V., Explosive random-coefficient \(A R(1)\) processes and related asymptotics for least squares estimation, Journal of time series analysis, 26, 807-824, (2005) · Zbl 1097.62081
[14] Jensen, S.T.; Rahbek, A., Asymptotic normality of the \(Q M L\) estimator of \(A R C H\) in the nonstationary case, Econometrica, 72, 641-646, (2004) · Zbl 1091.62074
[15] Jensen, S.T.; Rahbek, A., Asymptotic inference for nonstationary \(G A R C H\), Econometric theory, 20, 1203-1226, (2004) · Zbl 1069.62067
[16] Kushner, H.J.; Yin, G.G., Stochastic approximation algorithms and applications, (1997), Springer New York · Zbl 0914.60006
[17] Ling, S.; Li, D., Asymptotic inference for a nonstationary double \(A R(1)\) model, Biometrika, 95, 257-263, (2008) · Zbl 1437.62533
[18] Linton, O.; Pan, J.; Wang, H., Estimation for a non-stationary semi-strong \(G A R C H(1, 1)\) model with heavy-tailed errors, Econometric theory, 26, 1-28, (2010)
[19] Ljung, L.; Söderström, T., Theory and practice of recursive identification, (1983), MIT Press Cambridge, Massachusetts · Zbl 0548.93075
[20] Lumsdaine, R.L., Consistency and asymptotic normality of the quasi-maximum likelihood estimator in \(I G A R C H(1, 1)\) and covariance stationary \(G A R C H(1, 1)\) models, Econometrica, 64, 575-596, (1996) · Zbl 0844.62080
[21] Nelson, D.B., Stationarity and persistence in the \(G A R C H(1, 1)\) model, Econometric theory, 6, 318-334, (1990)
[22] Pan, J.; Wang, H.; Tong, H., Estimation and tests for power transformed and threshold GARCH models, Journal of econometrics, 142, 352-378, (2008) · Zbl 1418.62345
[23] Plackett, R.L., Some theorems in least squares, Biometrika, 32, 149, (1950) · Zbl 0041.46803
[24] White, J.S., The limiting distribution of the serial correlation coefficient in explosive case, Annals of mathematical statistics, 29, 1188-1197, (1958) · Zbl 0099.13004
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