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STATISTICAL INFERENCE FOR MEASUREMENT EQUATION SELECTION IN THE LOG-REALGARCH MODEL

  • Yu-Ning Li (a1), Yi Zhang (a1) and Caiya Zhang (a2)
Abstract

This article investigates the statistical inference problem of whether a measurement equation is self-consistent in the logarithmic realized GARCH model (log-RealGARCH). First, we provide the sufficient and necessary conditions for the strict stationarity of both the log-RealGARCH model and the log-GARCH-X model. Under these conditions, strong consistency and asymptotic normality of the quasi-maximum likelihood estimators of these two models are obtained. Then, based on the asymptotic results, we propose a Hausman-type self-consistency test for diagnosing the suitability of the measurement equation in the log-RealGARCH model. Finally, the results of simulations and an empirical study are found to accord with the theoretical results.

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Corresponding author
*Address correspondence to Yi Zhang, School of Mathematical Sciences, Zhejiang University, Hangzhou, China; e-mail: zhangyi63@zju.edu.cn
Caiya Zhang, Department of Statistics, Zhejiang University City College, Hangzhou, China; e-mail: zhangcy@zucc.edu.cn.
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We especially thank two anonymous referees and Pentti Saikkonen, the co-editor, for helpful suggestions that greatly improved the article. We are also grateful for the comments from seminar and conference participants at the 2nd International Symposium on Interval Data Modelling: Theory and Applications (SIDM2016) and Contributed Sessions in Statistics (CSS09) of 2017 IMS-China International Conference on Statistics and Probability. This research is partly supported by the Zhejiang Provincial Natural Science Foundation (No. LY18A010005), the Research Project of Humanities and Social Science of Ministry of Education of China (No. 17YJA910003), the Fundamental Research Funds for the Central Universities and Major Project of the National Social Science Foundation of China (No.13&ZD163).

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References
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Andreou, E. & Werker, B.J.M. (2015) Residual-based rank specification tests for AR-GARCH type models. Journal of Econometrics 185(2), 305331.
Barndorff-Nielsen, O.E. & Shephard, N. (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(2), 253280.
Billingsley, P. (1995) Probability and Measure. Wiley.
Bloomfield, P. & Watson, G.S. (1975) The inefficiency of least squares. Biometrika 62(1), 121128.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31(3), 307327.
Bougerol, P. & Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52(1–2), 115127.
Brockwell, P.J. & Davis, R.A. (2009) Time Series: Theory and Methods. Springer.
Corsi, F. (2009) A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7(2), 174196.
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4), 9871007.
Engle, R.F. & Ng, V.K. (1993) Measuring and testing the impact of news on volatility. The Journal of Finance 48(5), 17491778.
Francq, C. & Sucarrat, G. (2017) An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns. Journal of Multivariate Analysis 153, 1632.
Francq, C., Wintenberger, O., & Zakoïan, J.-M. (2013) GARCH models without positivity constraints: Exponential or log GARCH? Journal of Econometrics 177(1), 3446.
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH. Bernoulli 10(4), 605637.
Geweke, J. (1986) Modeling the persistence of conditional variances: A comment. Econometric Reviews 5(1), 5761.
Han, H. (2015) Asymptotic properties of GARCH-X processes. Journal of Financial Econometrics 13(1), 188221.
Han, H. & Kristensen, D. (2014) Asymptotic theory for the QMLE in GARCH-X models with stationary and nonstationary covariates. Journal of Business and Economic Statistics 32(3), 416429.
Hansen, P.R. & Huang, Z. (2016) Exponential GARCH modeling with realized measures of volatility. Journal of Business and Economic Statistics 34(2), 269287.
Hansen, P.R., Huang, Z., & Shek, H.H. (2012) Realized GARCH: A joint model for returns and realized measures of volatility. Journal of Applied Econometrics 27(6), 877906.
Hausman, J.A. (1978) Specification tests in econometrics. Econometrica 46(6), 12511271.
Halunga, A.G. & Orme, C.D. (2009) First-order asymptotic theory for parametric misspecification tests of GARCH models. Econometric Theory 25(2), 364410.
Huang, Z., Liu, H., & Wang, T. (2016) Modeling long memory volatility using realized measures of volatility: A realized HAR GARCH model. Economic Modelling 52, 812821.
Lee, S.W. & Hansen, B.E. (1994) Asymptotic theory for the GARCH (1, 1) quasi-maximum likelihood estimator. Econometric Theory 10(1), 2952.
Lundbergh, S. & Teräsvirta, T. (2002) Evaluating GARCH models. Journal of Econometrics 110(2), 417435.
Mcleod, A.I. & Li, W.K. (1983) Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis 4(4), 269273.
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59(2), 347370.
Pantula, S.G. (1986) Modeling the persistence of conditional variances: A comment. Econometric Reviews 5(1), 7174.
Straumann, D. (2005) Estimation in Conditionally Heteroscedastic Time Series Model. Lecture Notes in Statistics, vol. 181. Springer.
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34(5), 24492495.
Sucarrat, G., Grønneberg, S., & Escribano, A. (2016) Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown. Computational Statistics and Data Analysis 100, 582594.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
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