Count and duration time series with equal conditional stochastic and mean orders

Aknouche, Abdelhakim and Francq, Christian (2018): Count and duration time series with equal conditional stochastic and mean orders.

Abstract

We consider a positive-valued time series whose conditional distribution has a time-varying mean, which may depend on exogenous variables. The main applications concern count or duration data. Under a contraction condition on the mean function, it is shown that stationarity and ergodicity hold when the mean and stochastic orders of the conditional distribution are the same. The latter condition holds for the exponential family parametrized by the mean, but also for many other distributions. We also provide conditions for the existence of marginal moments and for the geometric decay of the beta-mixing coefficients. We give conditions for consistency and asymptotic normality of the Exponential Quasi-Maximum Likelihood Estimator (QMLE) of the conditional mean parameters. Simulation experiments and illustrations on series of stock market volumes and of greenhouse gas concentrations show that the multiplicative-error form of usual duration models deserves to be relaxed, as allowed in the present paper.

• Item Type: MPRA Paper
• Language: English
• Keywords: Absolute regularity, Autoregressive Conditional Duration, Count time series models, Distance correlation, Ergodicity, Exponential QMLE, Integer-valued GARCH, Mixing.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Agosto, A., Cavaliere, G., Kristensen, D. and Rahbek, A. (2016) Modeling corporate defaults: Poisson autoregressions with exogenous covariates (PARX). Journal of Empirical Finance 38, 640--663.

Ahmad, A. and Francq, C. (2016) Poisson qmle of count time series models. Journal of Time Series analysis 37, 291--314.

Aknouche, A., Bendjeddou, S. and Touche, N. (2018) Negative Binomial Quasi-Likelihood Inference for General Integer-Valued Time Series Models. Journal of Time Series Analysis 39, 192--211.

Al-Osh, M.A. and Alzaid, A.A. (1987) First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8, 261--275.

Billingsley P. (2008) Probability and measure. John Wiley & Sons, Third Edition.

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

Cameron, A.C. and Trivedi, P.K. (2001) Essentials of count data regression, in H. Baltagi, B. Hani, A companion to theoretical econometrics, 331-348, Blackwell.

Chou, R.Y. (2005) Forecasting financial volatilities with extreme values: The conditional autoregressive range (CARR) Model. Journal of Money, Credit, and Banking 37, 561--582.

Christou, V. and Fokianos, K. (2014) Quasi-likelihood inference for negative binomial time series models. Journal of Time Series Analysis 35, 55--78.

Davis, R.A., Holan, S.H., Lund, R. and Ravishanker, N. (2016) Handbook of discrete-valued time series. Chapman and Hall.

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.

Davis, R.A., Matsui, M., Mikosch, T. and Wan, P. (2018) Applications of distance correlation to time series. Bernoulli 24, 3087--3116.

Douc, R., Doukhan, P. and Moulines, E. (2013) Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator. Stochastic Processes and their Applications 123, 2620--2647.

Douc, R., Roueff, F. and Sim, T. (2015) Handy sufficient conditions for the convergence of the maximum likelihood estimator in observation-driven models. Lithuanian Mathematical Journal 55, 367--392.

Douc, R., Roueff, F. and Sim, T. (2016) The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains. The Annals of Applied Probability 26, 2357--2383.

Doukhan, P. and Neumann, M.H. (2019) Absolute regularity of semi-contractive GARCH-type processes. Journal of Applied Probability 56, 91-115.

Doukhan, P., Fokianos, K. and Tjøstheim, D. (2012) On weak dependence conditions for Poisson autoregressions. Statistics and Probability Letters 82, 942--948.

Doukhan, P., Fokianos, K. and Tjøstheim, D. (2013) Correction to "On weak dependence conditions for Poisson autoregressions" [Statist. Probab. Lett. 82 (2012) 942--948]. Statistics and Probability Letters 83, 1926--1927.

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.

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

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

Francq, C., Jiménez-Gamero, M.D., and Meintanis, S.G. (2017) Tests for conditional ellipticity in multivariate GARCH models. Journal of Econometrics 196, 305--319.

Francq, C. and Zakoian, J.-M. (2019) GARCH models: structure, statistical inference and financial applications. John Wiley & Sons, Second Edition.

Franke J. (2010) Weak dependence of functional INGARCH processes. Technical report, University of Kaiserslautern.

Gouriéroux, C., Monfort, A. and Trognon, A. (1984) Pseudo maximum likelihood methods: Theory. Econometrica 52, 681--700.

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.

Gurmu, S. and Trivedi, P.K. (1996) Excess Zeros in Count Models for Recreational Trips. Journal of Business & Economic Statistics 14, 469--477.

Hall P, Heyde CC. (1980) Martingale Limit Theory and its Applications. Academic Press, New York.

Jain, G.C. and Consul, P.C. (1971) A generalized negative binomial distribution. SIAM Journal on Applied Mathematics 21, 501--513.

Lehmann, E. L. (1955). Ordered Families of Distributions. Ann. Math. Statist. 26, 399-419.

Liboschik T., Fokianos K. and Fried, R. (2017) tscount: An R Package for Analysis of Count Time Series Following Generalized Linear Models. Journal of Statistical Software 82, 1--51.

Lucas, D.D., Yver Kwok, C., Cameron-Smith, P., Graven, H., Bergmann, D., Guilderson, T.P, Weiss, R. and Keeling, R. (2015) Designing optimal greenhouse gas observing networks that consider performance and cost. Geoscientific Instrumentation Methods and Data Systems 4, 121-137.

Meyn, S.P. and Tweedie, R.L. (2009) Markov chains and stochastic stability. Springer Science & Business Media, Second Edition.

Neumann, M.H. (2011) Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268--1284.

Ridout, M., Demétrio, C.G. and Hinde, J. (1998) Models for count data with many zeros. In Proceedings of the XIXth international biometric conference 19, 179-192.

Rizzo, M.L. and Székely, G.J. (2016) Energy distance. Wiley Interdisciplinary Reviews: Computational Statistics 8, 27--38.

Siakoulis, V. (2015) acp: Autoregressive Conditional Poisson. R package version 2.1.

Sim, T., Douc, R. and Roueff, F. (2016) General-order observation-driven models. Hal preprint, Nb hal-01383554.

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.

Székely, G.J., Rizzo, M.L. and Bakirov, N.K. (2007) Measuring and testing dependence by correlation of distances. Annals of Statististics 35, 2769--2794.

Tjøstheim D. (2012) Some recent theory for autoregressive count time series. Test 21, 413--438.

Yu, Y. (2009) Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability 46, 244--254.

Wedderburn, R.W. (1974) Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61, 439--447.

Wooldridge, J.M. (1999) Quasi-Likelihood Methods for Count Data. In M.H. Pesaran and P. Schmidt (ed.), Handbook of Applied Econometrics, Volume 2: Microeconomics, (pp. 352-406). Oxford: Blackwell.

Zhu, F. (2011) A negative binomial integer-valued GARCH model. Journal of Time Series Analysis 32, 54--67.

Zhu, F. (2012) Zero-inflated Poisson and negative binomial integer-valued GARCH models. Journal of Statistical Planning and Inference 142, 826--839.