Electronic Journal of Statistics

Stationarity and ergodicity of univariate generalized autoregressive score processes

Francisco Blasques, Siem Jan Koopman, and André Lucas

Full-text: Open access

Abstract

We characterize the dynamic properties of generalized autoregressive score models by identifying the regions of the parameter space that imply stationarity and ergodicity of the corresponding nonlinear time series process. We show how these regions are affected by the choice of parameterization and scaling, which are key features for the class of generalized autoregressive score models compared to other observation driven models. All results are illustrated for the case of time-varying means, variances, or higher-order moments.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 1088-1112.

Dates
First available in Project Euclid: 5 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1407243244

Digital Object Identifier
doi:10.1214/14-EJS924

Mathematical Reviews number (MathSciNet)
MR3263114

Zentralblatt MATH identifier
1309.60034

Subjects
Primary: 60G10: Stationary processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 91B84: Economic time series analysis [See also 62M10]

Keywords
Nonlinear dynamics observation driven time-varying parameter models stochastic recurrence equations contracting properties

Citation

Blasques, Francisco; Koopman, Siem Jan; Lucas, André. Stationarity and ergodicity of univariate generalized autoregressive score processes. Electron. J. Statist. 8 (2014), no. 1, 1088--1112. doi:10.1214/14-EJS924. https://projecteuclid.org/euclid.ejs/1407243244


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References

  • [1] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31(3), 307–327.
  • [2] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31(4), 942–959.
  • [3] Cox, D. R. (1981). Statistical analysis of time series: some recent developments. Scandinavian Journal of Statistics 8, 93–115.
  • [4] Creal, D., Koopman, S. J., and Lucas, A. (2011). A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations. Journal of Business and Economic Statistics 29(4), 552–563.
  • [5] Creal, D., Koopman, S. J., and Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28(5), 777–795.
  • [6] Creal, D., Schwaab, B., Koopman, S. J., and Lucas, A. (2013). Observation driven mixed-measurement dynamic factor models. Review of Economics and Statistics, forthcoming.
  • [7] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM review, 45–76.
  • [8] Durbin, J. and Koopman, S. J. (2012). Time Series Analysis by State Space Methods (2nd ed.). Oxford University Press.
  • [9] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflations. Econometrica 50, 987–1008.
  • [10] Engle, R. F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics 17(5), 425–446.
  • [11] Engle, R. F. and Russell, J. R. (1998). Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66, 1127–1162.
  • [12] Francq, C. and Zakoian, J.-M. (2013). Inference in non stationary asymmetric GARCH models. Annals of Statistics 41(4), 1970–1998.
  • [13] Granger, C. W. J. and Terasvirta, T. (1993). Modelling Non-Linear Economic Relationships. Oxford University Press.
  • [14] Hamilton, J. D. (1986). State-space models. In R. F. Engle and D. McFadden (Eds.), Handbook of Econometrics, Volume 4 of Handbook of Econometrics, Chapter 50, pp. 3039–3080. Elsevier.
  • [15] Harvey, A. C. (2013). Dynamic Models for Volatility and Heavy Tails. Cambridge University Press.
  • [16] Harvey, A. C. and Chakravarty, T. (2008). Beta-t-(E)GARCH. University of Cambridge, Faculty of Economics, Working paper CWPE 08340.
  • [17] Harvey, A. C. and Luati, A. (2014). Filtering with heavy tails. Journal of the American Statistical Association, forthcoming.
  • [18] Janus, P., Koopman, S. J., and Lucas, A. (2011). Long memory dynamics for multivariate dependence under heavy tails. Tinbergen Institute Discussion Papers 11-175/DSF28.
  • [19] Jensen, S. T. and Rahbek, A. (2004). Asymptotic inference for nonstationary garch. Econometric Theory 20(6), 1203–1226.
  • [20] Koopman, S. J., Lucas, A., and Scharth, M. (2012). Predicting time-varying parameters with parameter-driven and observation-driven models. Tinbergen Institute Discussion Papers 12-020/4.
  • [21] Krengel, U. (1985). Ergodic Theorems. De Gruyter studies in Mathematics, Berlin.
  • [22] Lucas, A., Schwaab, B., and Zhang, X. (2014). Conditional euro area sovereign default risk. Journal of Business and Economic Statistics 32(2), 271–284.
  • [23] Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.
  • [24] Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370.
  • [25] Rydberg, T. H. and Shephard, N. (2003). Dynamics of trade-by-trade price movements: decomposition and models. Journal of Financial Econometrics 1(1), 2.
  • [26] Stock, J. H. and Watson, M. W. (2006). Why has u.s. inflation become harder to forecast? NBER Working Papers 12324, National Bureau of Economic Research, Inc.
  • [27] Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroeskedastic time series: A stochastic recurrence equations approach. The Annals of Statistics 34(5), 2449–2495.
  • [28] Wintenberger, O. (2013). Continuous invertibility and stable QML estimation of the EGARCH(1, 1) model. Scandinavian Journal of Statistics 40(4), 846–867.
  • [29] Wu, W. and Shao, X. (2004). Limit theorems for iterated random functions. Journal of Applied Probability 41(2), 425–436.
  • [30] Zhang, X., Creal, D., Koopman, S. J., and Lucas, A. (2011). Modeling dynamic volatilities and correlations under skewness and fat tails. Tinbergen Institute Discussion Papers 11-078/DSF22.