Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-10T15:32:08.226Z Has data issue: false hasContentIssue false
  • English
  • Français

COUNT AND DURATION TIME SERIES WITH EQUAL CONDITIONAL STOCHASTIC AND MEAN ORDERS

Published online by Cambridge University Press:  17 March 2020

Abdelhakim Aknouche  and
Christian Francq
Abdelhakim Aknouche*
Affiliation:
USTHB and Qassim University
Christian Francq
Affiliation:
CREST and University of Lille
*
Address correspondence to Abdelhakim Aknouche, Faculty of Mathematics, University of Science and Technology Houari Boumediene, Bab Ezzouar, Algeria; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

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 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 this paper.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 2 , April 2021 , pp. 248 - 280
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are deeply grateful to the Editor-in-Chief Peter Phillips, the Co-Editor Dennis Kristensen, and two anonymous referees for their careful reading, very stimulating comments, and helpful suggestions that have led to significant improvements of this manuscript. C.F. is grateful to the Agence Nationale de la Recherche (ANR), which supported this work via the Project MultiRisk (ANR-16-CE26-0015-02). He also thanks the labex ECODEC.

References

REFERENCES

Agosto, A., Cavaliere, G., Kristensen, D. & Rahbek, A. (2016) Modeling corporate defaults: Poisson autoregressions with exogenous covariates (PARX). Journal of Empirical Finance 38, 640–663.CrossRefGoogle Scholar
Ahmad, A. & Francq, C. (2016) Poisson QMLE of count time series models. Journal of Time Series Analysis 37, 291–314.CrossRefGoogle Scholar
Aknouche, A., Bendjeddou, S. & Touche, N. (2018) Negative binomial quasi-likelihood inference for general integer-valued time series models. Journal of Time Series Analysis 39, 192–211.CrossRefGoogle Scholar
Al-Osh, M.A. & Alzaid, A.A. (1987) First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8, 261–275.CrossRefGoogle Scholar
Billingsley, P. (2008) Probability and Measure, 3rd ed. Wiley.Google Scholar
Bougerol, P. (1993) Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31, 942–959.CrossRefGoogle Scholar
Cameron, A.C. & Trivedi, P.K. (2001) Essentials of count data regression. In Baltagi, B.H. (ed.), A Companion to Theoretical Econometrics, pp. 331–348. Blackwell.Google Scholar
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.CrossRefGoogle Scholar
Christou, V. & Fokianos, K. (2014) Quasi-likelihood inference for negative binomial time series models. Journal of Time Series Analysis 35, 55–78.CrossRefGoogle Scholar
Davis, R.A., Holan, S.H., Lund, R. & Ravishanker, N. (2016) Handbook of Discrete-Valued Time Series. Chapman and Hall.Google Scholar
Davis, R.A. & Liu, H. (2016) Theory and inference for a class of nonlinear models with application to time series of counts. Statistica Sinica 26, 1673–1707.Google Scholar
Davis, R.A., Matsui, M., Mikosch, T. & Wan, P. (2018) Applications of distance correlation to time series. Bernoulli 24, 3087–3116.CrossRefGoogle Scholar
Douc, R., Doukhan, P. & 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.CrossRefGoogle Scholar
Douc, R., Roueff, F. & Sim, T. (2015) Handy sufficient conditions for the convergence of the maximum likelihood estimator in observation-driven models. Lithuanian Mathematical Journal 55, 367–392.CrossRefGoogle Scholar
Douc, R., Roueff, F. & 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.CrossRefGoogle Scholar
Doukhan, P., Fokianos, K. & Tjøstheim, D. (2012) On weak dependence conditions for Poisson autoregressions. Statistics and Probability Letters 82, 942–948.Google Scholar
Doukhan, P., Fokianos, K. & 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.CrossRefGoogle Scholar
Doukhan, P. & Neumann, M.H. (2019) Absolute regularity of semi-contractive GARCH-type processes. Journal of Applied Probability 56, 91–115.Google Scholar
Engle, R. (2002) New frontiers for ARCH models. Journal of Applied Econometrics 17, 425–446.CrossRefGoogle Scholar
Engle, R. & Russell, J. (1998) Autoregressive conditional duration: A new model for irregular spaced transaction data. Econometrica 66, 1127–1162.CrossRefGoogle Scholar
Ferland, R., Latour, A. & Oraichi, D. (2006) Integer-valued GARCH process. Journal of Time Series Analysis 27, 923–942.CrossRefGoogle Scholar
Fokianos, K., Rahbek, A. & Tjøstheim, D. (2009) Poisson autoregression. Journal of the American Statistical Association 140, 1430–1439.CrossRefGoogle Scholar
Francq, C., Jiménez-Gamero, M.D. & Meintanis, S.G. (2017) Tests for conditional ellipticity in multivariate GARCH models. Journal of Econometrics 196, 305–319.CrossRefGoogle Scholar
Francq, C. & Thieu, L. (2019) QML inference for volatility models with covariates. Econometric Theory 35, 37–72.CrossRefGoogle Scholar
Francq, C. & Zakoian, J.-M. (2019) GARCH Models: Structure, Statistical Inference and Financial Applications, 2nd ed. Wiley.CrossRefGoogle Scholar
Franke, J. (2010) Weak dependence of functional INGARCH processes. Technical report, University of Kaiserslautern.Google Scholar
Gonçalves, E., Mendes-Lopes, N. & Silva, F. (2015) Infinitely divisible distributions in integer-valued GARCH models. Journal of Time Series Analysis 36, 503–527.CrossRefGoogle Scholar
Gouriéroux, C., Monfort, A. & Trognon, A. (1984) Pseudo maximum likelihood methods: Theory. Econometrica 52, 681–700.CrossRefGoogle Scholar
Gurmu, S. & Trivedi, P.K. (1996) Excess zeros in count models for recreational trips. Journal of Business & Economic Statistics 14, 469–477.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications.Academic Press.Google Scholar
Jain, G.C. & Consul, P.C. (1971) A generalized negative binomial distribution. SIAM Journal on Applied Mathematics 21, 501–513.CrossRefGoogle Scholar
Lehmann, E.L. (1955). Ordered families of distributions. Annals of Mathematical Statistics 26, 399–419.CrossRefGoogle Scholar
Liboschik, T., Fokianos, K. & Fried, R. (2017) tscount: An R package for analysis of count time series following generalized linear models. Journal of Statistical Software 82, 1–51.CrossRefGoogle Scholar
Lucas, D.D., Kwok, C.Yver, Cameron-Smith, P., Graven, H., Bergmann, D., Guilderson, T.P., Weiss, R. & Keeling, R. (2015) Designing optimal greenhouse gas observing networks that consider performance and cost. Geoscientific Instrumentation Methods and Data Systems 4, 12–137.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (2009) Markov Chains and Stochastic Stability, 2nd ed. Springer Science & Business Media.CrossRefGoogle Scholar
Neumann, M.H. (2011) Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268–1284.CrossRefGoogle Scholar
Ridout, M., Demétrio, C.G. & Hinde, J. (1998) Models for count data with many zeros. Proceedings of the XIXth International Biometric Conference 19, 179–192.Google Scholar
Rizzo, M.L. & Székely, G.J. (2016) Energy distance. Wiley Interdisciplinary Reviews: Computational Statistics 8, 27–38.CrossRefGoogle Scholar
Siakoulis, V. (2015) ACP: Autoregressive Conditional Poisson. R package version 2.1.Google Scholar
Sim, T., Douc, R. & Roueff, F. (2016) General-order observation-driven models. Hal preprint, Nb hal-01383554.Google Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Székely, G.J., Rizzo, M.L. & Bakirov, N.K. (2007) Measuring and testing dependence by correlation of distances. Annals of Statistics 35, 2769–2794.CrossRefGoogle Scholar
Tjøstheim, D. (2012) Some recent theory for autoregressive count time series. Test 21, 413–438.CrossRefGoogle Scholar
Wedderburn, R.W. (1974) Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61, 439–447.Google Scholar
Wooldridge, J.M. (1999) Quasi-likelihood methods for count data. In Pesaran, M.H. & Schmidt, P. (eds.), Handbook of Applied Econometrics, Volume 2: Microeconomics, pp. 35–406. Blackwell.Google Scholar
Yu, Y. (2009) Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability 46, 244–254.CrossRefGoogle Scholar
Zhu, F. (2011) A negative binomial integer-valued GARCH model. Journal of Time Series Analysis 32, 54–67.Google Scholar
Zhu, F. (2012) Zero-inflated Poisson and negative binomial integer-valued GARCH models. Journal of Statistical Planning and Inference 142, 826–839.CrossRefGoogle Scholar
Supplementary material: PDF

Aknouche and Francq Supplementary Materials

Aknouche and Francq Supplementary Materials 1

Download Aknouche and Francq Supplementary Materials(PDF)
PDF 70.8 KB
Supplementary material: File

Aknouche and Francq Supplementary Materials

Aknouche and Francq Supplementary Materials 2

Download Aknouche and Francq Supplementary Materials(File)
File 14.3 KB