Stationarity and ergodicity of Markov switching positive conditional mean models

Aknouche, Abdelhakim and Francq, Christian (2020): Stationarity and ergodicity of Markov switching positive conditional mean models.

Abstract

A general Markov-Switching autoregressive conditional mean model, valued in the set of nonnegative numbers, is considered. The conditional distribution of this model is a finite mixture of nonnegative distributions whose conditional mean follows a GARCH-like dynamics with parameters depending on the state of a Markov chain. Three different variants of the model are examined depending on how the lagged-values of the mixing variable are integrated into the conditional mean equation. The model includes, in particular, Markov mixture versions of various well-known nonnegative time series models such as the autoregressive conditional duration (ACD) model, the integer-valued GARCH (INGARCH) model, and the Beta observation driven model. Under contraction in mean conditions, it is shown that the three variants of the model are stationary and ergodic when the stochastic order and the mean order of the mixing distributions are equal. The proposed conditions match those already known for Markov-switching GARCH models. We also give conditions for finite marginal moments. Applications to various mixture and Markov mixture count, duration and proportion models are provided.

• Item Type: MPRA Paper
• Language: English
• Keywords: Autoregressive Conditional Duration, Count time series models, finite mixture models, Ergodicity, Integer-valued GARCH, Markov mixture models.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C25 - Discrete Regression and Qualitative Choice Models ; Discrete Regressors ; Proportions ; Probabilities

References

Abramson, A. and Cohen, I. (2007). On the stationarity of Markov-switching GARCH processes. Econometric Theory 23, 485--500.

Aknouche, A. and Demmouche, N. (2019). Ergodicity conditions for a double mixed Poisson autoregression. Statistics and Probability Letters, 147, 6--11.

Aknouche, A. and Francq, C. (2020). Count and duration time series with equal conditional stochastic and mean orders. Econometric Theory, DOI: https://doi.org/10.1017/S0266466620000134.

Bauwens, L., Preminger, A., and Rombouts, J.V.K. (2010). Theory and inference for a Markov switching GARCH model. Econometrics Journal 13, 218--244.

Benjamin, M.A., Rigby, R.A. and Stasinopoulos, D.M. (2003). Generalized autoregressive moving average models. Journal of the American Statistical Association 98, 214--223.

Berentsen, G.D., Bulla, J., Maruotti, A. and Støve, B. (2018). Modelling corporate defaults: A Markov-switching Poisson log-linear autoregressive model. ArXiv, https://arxiv.org/abs/1804.09252.

Bhogal, S.K. and Variyam, R.T. (2019). Conditional duration models for high-frequency data: A review on recent developments. Journal of Economic Surveys 33, 252--273.

Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31, 942--959.

Chen, F., Diebold, F.X. and Schorfheide, F. (2013). A Markov-switching multifractal inter-trade duration model, with application to US equities. Journal of Econometrics 177, 320--342.

Creal, D., Koopman, S.J. and Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28, 777--795.

Davis, R.A. and Liu, H. (2016). Theory and inference for a class of nonlinear models with application to time series of counts. Statistica Sinica 26, 1673--1707.

De Luca, G. and Zuccolotto, P. (2006). Regime-switching Pareto distributions for ACD models. Computational Statistics & Data Analysis 51, 2179--2191.

Diop, M.L., Diop, A. and Diongue, A.K. (2016). A mixture integer-valued GARCH model. Revstat - Statistical Journal 14, 245--271.

Diop, M.L., Diop, A. and Diongue, A.K. (2018). A negative binomial mixture integer-valued GARCH model. Statistika Afrika 13, 1645--1666.

Doukhan, P., Fokianos, K. and Rynkiewicz, J. (2018). Mixtures of Nonlinear Poisson Autoregressions. Working paper, ResearchGate 326478827.

Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory. Stochastic Processes and their Applications 118, 1997--2013.

Engle, R. (2002). New frontiers for Arch models. Journal of Applied Econometrics 17, 425--446.

Engle, R. and Russell, J. (1998). Autoregressive conditional duration: A new model for irregular spaced transaction data. Econometrica 66, 1127--1162.

Ferland, R., Latour, A., and Oraichi, D. (2006). Integer-valued GARCH process. Journal of Time Series Analysis 27, 923--942.

Fisher, R.A. (1953). Dispersion on a Sphere. Proceedings of the Royal Society of London A 217, 295--305.

Fong, W.M. and See, K.H. (2001). Modelling the conditional volatility of commodity index futures as a regime switching process. Journal of Applied Econometrics 16, 133--163.

Francq, C. and Roussignol, M. (1998). Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum-likelihood estimator, Statistics 32, 151--173.

Francq, C., and Thieu, L. (2019). Qml inference for volatility models with covariates. Econometric Theory 35, 37--72.

Francq, C. and Zakoian, J.-M. (2001). Stationarity of multivariate Markov-switching ARMA models. Journal of Econometrics 102, 339--364.

Francq, C. and Zakoian, J.-M. (2005). The L2-structures of standard and switching-regime GARCH models. Stochastic Processes and their Applications 115, 1557--1582.

Francq, C. and Zakoian, J.-M. (2008). Deriving the autocovariances of powers of Markov-switching GARCH models, with applications to statistical inference. Computational Statistics and Data Analysis 52, 3027--3046.

Francq, C. and Zakoian, J.-M. (2019). GARCH Models: Structure, Statistical Inference and Financial Applications, 2nd edition, John Wiley & Sons.

Fokianos, K., Rahbek, A., and Tjøstheim D. (2009). Poisson autoregression. Journal of the American Statistical Association 140, 1430--1439.

Gonçalves, E., Mendes-Lopes, N. and Silva, F. (2015). Infinitely divisible distributions in integer-valued GARCH models. Journal of Time Series Analysis 36, 503--527.

Gorgi, P. and Koopman, S.J. (2020). Beta observation-driven models with exogenous regressors: A joint analysis of realized correlation and leverage effects. Tinbergen Institute Discussion Paper 20-004/III.

Gray, S. (1996). Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 27--62.

Haas, M., Mittnik, S. and Paolella, M. (2004). A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, 493--530.

Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357--384.

Hamilton, J.D. (1994). Time Series Analysis. Princeton, New Jersey: Princeton University Press.

Hamilton, J.D. and Susmel, R. (1994). Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307--333.

Harvey, A.C. (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press.

Hautsch, N. (2012). Econometrics of financial high-frequency data. Springer-Verlag.

Heinen, A. (2003). Modelling time series count data: an autoregressive conditional Poisson model. Available at SSRN 1117187.

Horn, R. and Johnson, C.R. (2013). Matrix analysis, 2nd edition, Cambridge University Press.

Hujer, R. and Vuletic, S. (2007). Econometric analysis of financial trade processes by discrete mixture duration models. Journal of Economic Dynamics and Control 31, 635--667.

Hujer, R., Vuletic, S. and Kokot, S. (2002). The Markov Switching ACD Model. Finance and Accounting Working Paper 90, Johann Wofgang Goethe-University, Frankfurt.

Hujer, R., Kokot, S. and Vuletic, S. (2003). Comparison of MSACD models. Johann Wolfgang Goethe University, Frankfurt/Main.

Jorgensen, B. (1997). The theory of dispersion models. Chapman and Hall, London.

Klaassen, F. (2002). Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27, 363--94.

Lee, J. and Hwang, E. (2018). A generalized regime-switching integer-valued GARCH(1, 1) model and its volatility forecasting. Communications for Statistical Applications and Methods 25, 29--42.

Liu, J.-C. (2006). Stationarity of a Markov-switching GARCH model. Journal of Financial Econometrics 4, 573--593.

Mao, H., Zhu, F. and Cui, Y. (2019). Mixtures of Nonlinear Poisson Autoregressions. Statistical Methods & Applications, forthcoming. https://doi.org/10.1007/s10260-019-00498-2.

McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models (2nd edn). London: Chapman and Hall.

Nelder, J.A. and Wedderburn, R.W. (1972). Generalized Linear Models. Journal of the Royal Statistical Society, Series A 135, 370--384.

Rydberg, T.H. and Shephard, N. (2000). BIN models for trade-by-trade data. Modelling the number of trades in a fixed interval of time. In World Conference Econometric Society, 2000, Seattle. Contributed Paper 0740.

Zheng, T., Xiao, H. and Chen, R. (2015). Generalized ARMA with martingale difference errors. Journal of Econometrics 189, 492--506.

Zhu, F., Li, Q. and Wang, D. (2010). A mixture integer-valued ARCH model. Journal of Statistical Planning and Inference 140, 2025--2036.