Feasible invertibility conditions and maximum likelihood estimation for observation-driven models. (English) Zbl 1473.62301

Summary: Invertibility conditions for observation-driven time series models often fail to be guaranteed in empirical applications. As a result, the asymptotic theory of maximum likelihood and quasi-maximum likelihood estimators may be compromised. We derive considerably weaker conditions that can be used in practice to ensure the consistency of the maximum likelihood estimator for a wide class of observation-driven time series models. Our consistency results hold for both correctly specified and misspecified models. We also obtain an asymptotic test and confidence bounds for the unfeasible “true” invertibility region of the parameter space. The practical relevance of the theory is highlighted in a set of empirical examples. For instance, we derive the consistency of the maximum likelihood estimator of the Beta-\(t\)-GARCH model under weaker conditions than those considered in previous literature.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators


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[1] Bec, F., Rahbek, A., and Shephard, N. (2008). The ACR model: a multivariate dynamic mixture autoregression., Oxford Bulletin of Economics and Statistics, 70, 583-618.
[2] Berkes, I., Horváth, L., and Kokoszka, P. (2003). GARCH processes: structure and estimation., Bernoulli, 9, 201-227. · Zbl 1064.62094
[3] Blasques, F., Gorgi, P., Koopman, S. J., and Wintenberger, O. (2015). A note on ‘Continuous invertibility and stable QML estimation of the EGARCH(1,1) model’., Tinbergen Institute Discussion Paper 15-131/III.
[4] Blasques, F., Koopman, S. J., and Lucas, A. (2014). Maximum likelihood estimation for correctly specified generalized autoregressive score models: feedback effects, contraction conditions and asymptotic properties., Tinbergen Institute Discussion Paper 14-074/III.
[5] Blasques, F., Koopman, S. J., and Lucas, A. (2014). Maximum likelihood estimation for generalized autoregressive score models., Tinbergen Institute Discussion Paper 14-029/III. · Zbl 1309.60034
[6] Blasques, F., Koopman, S. J., and Lucas, A. (2014). Optimal formulations for nonlinear autoregressive processes., Tinbergen Institute Discussion Paper 14-103/III. · Zbl 1309.60034
[7] Blasques, F., Koopman, S. J., and Lucas, A. (2014). Stationarity and ergodicity of univariate generalized autoregressive score processes., Electronic Journal of Statistics, 8, 1088-1112. · Zbl 1309.60034
[8] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity., Journal of Econometrics, 31, 307-327. · Zbl 0616.62119
[9] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions., SIAM Journal on Control and Optimization, 31, 942-959. · Zbl 0785.93040
[10] Cox, D. R. (1981). Statistical analysis of time series: some recent developments., Scandinavian Journal of Statistics, 8, 93-115. · Zbl 0468.62079
[11] Creal, D., Koopman, S. J., and Lucas, A. (2013). Generalized autoregressive score models with applications., Journal of Applied Econometrics, 28, 777-795.
[12] Davis, R. A., Dunsmuir, W. T. M., and Streett, S. B. (2003). Observation-driven models for Poisson counts., Biometrika, 90, 777-790. · Zbl 1436.62418
[13] Delle Monache, D. and Petrella, I. (2017). Adaptive models and heavy tails with an application to inflation forecasting., International Journal of Forecasting, 33, 482-501.
[14] Domowitz, I. and White, H. (1982). Misspecified models with dependent observations., Journal of Econometrics, 20, 35-58. · Zbl 0512.62114
[15] Douc, R., Fokianos, K. and Moulines, E. (2017). Asymptotic properties of quasi-maximum likelihood estimators in observation-driven time series models., Electronic Journal of Statistics, 11, 2707-2740. · Zbl 1366.62173
[16] Durrett, R. (1996)., Probability: theory and examples. Duxbury Press. · Zbl 1202.60002
[17] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation., Econometrica, 50, 987-1007. · Zbl 0491.62099
[18] Engle, R. F. (2002). Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models., Journal of Business & Economic Statistics, 20, 339-350.
[19] Engle, R. F. and Manganelli, S. (2004). CAViaR: conditional autoregressive value at risk by regression quantiles., Journal of Business & Economic Statistics, 22, 367-381.
[20] Engle, R. F. and Russell, J. R. (1998). Autoregressive conditional duration: a new model for irregularly spaced transaction data., Econometrica, 66, 1127-1162. · Zbl 1055.62571
[21] Fan, J., Qi, L. and Xiu, D. (2014). Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods., Journal of Business & Economic Statistics, 32, 178-191.
[22] Francq, C. and Zakoïan, J. M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes., Bernoulli, 10, 605-637. · Zbl 1067.62094
[23] Francq, C. and Zakoïan, J. M. (2006). Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process., Econometric Theory, 22, 815-834. · Zbl 1100.62083
[24] Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks., The Journal of Finance, 48, 1779-1801.
[25] Granger, C. and Andersen, A. (1978). On the invertibility of time series models., Stochastic Processes and their Applications, 8, 87-92. · Zbl 0387.62076
[26] Harvey, A. (2013)., Dynamic models for volatility and heavy tails. Cambridge University Press. · Zbl 1326.62001
[27] Harvey, A. and Luati, A. (2014). Filtering with heavy tails., Journal of the American Statistical Association, 109, 1112-1122. · Zbl 1368.62251
[28] Ito, R. (2016). Asymptotic theory for Beta-t-GARCH., Cambridge Working Papers in Economics 1607,
[29] Jensen, S. T. and Rahbek, A. (2004). Asymptotic inference for nonstationary GARCH., Econometric Theory, 20, 1203-1226. · Zbl 1069.62067
[30] Lee, S. and Hansen, B. (1994). Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator., Econometric Theory, 10, 29-52.
[31] Lumsdaine, R. L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in \(\textIGARCH(1,1)\) and covariance stationary \(\textGARCH(1,1)\) models., Econometrica, 64, 575-596. · Zbl 0844.62080
[32] Maasoumi, E. (1990). How to live with misspecification if you must., Journal of Econometrics, 44, 67-86.
[33] Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach., Econometrica, 59, 347-370. · Zbl 0722.62069
[34] Newey, W. and West, K. (1987). A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix., Econometrica, 55, 703-708. · Zbl 0658.62139
[35] Patton, A. J. (2006). Modelling asymmetric exchange rate dependence., International Economic Review, 47, 527-556.
[36] Pfanzagl, J. (1969). On the measurability and consistency of minimum contrast estimates., Metrika, 14, 249-272. · Zbl 0181.45501
[37] Potscher, B. M. and Prucha, I. R. (1997)., Dynamic nonlinear econometric models. Asymptotic theory. Springer-Verlag, Berlin. · Zbl 0923.62121
[38] Rao, R. R. (1962). Relations between weak and uniform convergence of measures with applications., The Annals of Mathematical Statistics, 33, 659-680. · Zbl 0117.28602
[39] Robinson, P. M. and Zaffaroni, P. (2006). Pseudo-maximum likelihood estimation of \(\textARCH(∞ )\) models., The Annals of Statistics, 34, 1049-1074. · Zbl 1113.62107
[40] Russell, J. R. (2001). Econometric modeling of multivariate irregularly-spaced high-frequency data., University of Chicago, Graduate School of Business.
[41] Sorokin, A. (2011). Non-invertibility in some heteroscedastic models., Arvix preprint 1104.3318.
[42] Straumann, D. (2005)., Estimation in conditionally heteroscedastic time series models. Springer-Verlag, Berlin. · Zbl 1086.62103
[43] Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach., The Annals of Statistics, 34, 2449-2495. · Zbl 1108.62094
[44] Wald, A. (1949). Note on the consistency of the maximum likelihood estimate., The Annals of Mathematical Statistics, 20, 595-601. · Zbl 0034.22902
[45] White, H. (1980). Using least squares to approximate unknown regression functions., International Economic Review, 21, 149-170. · Zbl 0444.62119
[46] White, H. (1982). Maximum likelihood estimation of misspecified models., Econometrica, 50, 1-25. · Zbl 0478.62088
[47] Wintenberger, O. (2013). Continuous invertibility and stable QML estimation of the EGARCH(1,1) model., Scandinavian Journal of Statistics, 40, 846-867. · Zbl 1283.62190
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