Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-23T08:28:09.009Z Has data issue: false hasContentIssue false
  • English
  • Français

ITERATIONS OF DEPENDENT RANDOM MAPS AND EXOGENEITY IN NONLINEAR DYNAMICS

Published online by Cambridge University Press:  04 February 2021

Zinsou Max Debaly  and
Lionel Truquet
Zinsou Max Debaly
Affiliation:
Center for Research in Economics and Statistics
Lionel Truquet*
Affiliation:
Center for Research in Economics and Statistics
*
Address correspondence to Lionel Truquet, Center for Research in Economics and Statistics, National School of Statistics and Information Analysis, UMR CNRS 9194, Campus de Ker-Lann, rue Blaise Pascal, BP37203, 35172 Bruz Cedex, France; e-mail: Lionel.TRUQUET@ensai.Fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated into the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also discuss the dependence properties of these processes using the functional dependence measure of Wu (2005, Proceedings of the National Academy of Sciences 102, 14150–14154) that delivers a central limit theorem giving a wide range of applications. Our results are illustrated with conditional heteroscedastic autoregressive nonlinear models, Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) processes, count time series, binary choice models, and categorical time series for which we provide many extensions of existing results.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 6 , December 2021 , pp. 1135 - 1172
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

This work was funded by CY Initiative of Excellence (grant “Investissements d’Avenir” ANR-16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013.

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
Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21(4), 930–934.CrossRefGoogle Scholar
Borovkov, A.A. (1998) Ergodicity and Stability of Stochastic Processes. Wiley.Google Scholar
Bougerol, P. & Picard, N. (1992) Strict stationarity of generalized autoregressive processes. The Annals of Probability 20(4), 1714–1730.CrossRefGoogle Scholar
Brandt, A. (1986) The stochastic equation ${Y}_{n+1}={A}_n{Y}_n+{B}_n$ with stationary coefficients. Advances in Applied Probability 18(1), 211–220.Google Scholar
Chamberlain, G. (1982) The general equivalence of Granger and Sims causality. Econometrica: Journal of the Econometric Society 50(3), 569–581.CrossRefGoogle 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(4), 1673–1707.Google Scholar
de Jong, R.M. & Woutersen, T. (2011) Dynamic time series binary choice. Econometric Theory 27(4), 673–702.CrossRefGoogle Scholar
Diaconis, P. & Freedman, D. (1999) Iterated random functions. SIAM Review 41(1), 45–76.CrossRefGoogle Scholar
Doob, J.L. (2012) Measure Theory, vol. 143. Springer Science & Business Media.Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Lecture Notes in Statistics, vol. 85. Springer-Verlag.CrossRefGoogle Scholar
Elton, J.H. (1990) A multiplicative ergodic theorem for Lipschitz maps. Stochastic Processes and Their Applications 34(1), 39–47.CrossRefGoogle Scholar
Engle, R.F., Hendry, D.F., & Richard, J.-F. (1983) Exogeneity. Econometrica: Journal of the Econometric Society 51(2), 277–304.CrossRefGoogle Scholar
Fokianos, K. & Kedem, B. (2003) Regression theory for categorical time series. Statistical Science 18(3), 357–376.CrossRefGoogle Scholar
Fokianos, K. & Truquet, L. (2019) On categorical time series models with covariates. Stochastic Processes and Their Applications 129(9), 3446–3462.CrossRefGoogle Scholar
Francq, C. & Thieu, L.Q. (2019) QML inference for volatility models with covariates. Econometric Theory 35(1), 37–72.CrossRefGoogle Scholar
Härdle, W. & Tsybakov, A. (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81(1), 223–242.CrossRefGoogle Scholar
Härdle, W., Tsybakov, A., & Yang, L. (1998) Nonparametric vector autoregression. Journal of Statistical Planning and Inference 68(2), 221–245.CrossRefGoogle Scholar
Iosifescu, M. (2009) Iterated Function Systems: A Critical Survey. Mathematical Reports (Bucureşti). 3.Google Scholar
Kallenberg, O. (2006) Foundations of Modern Probability. Springer Science & Business Media.Google Scholar
Kauppi, H. & Saikkonen, P. (2008) Predicting US recessions with dynamic binary response models. The Review of Economics and Statistics 90(4), 777–791.CrossRefGoogle Scholar
Letac, G. (1986) A contraction principle for certain Markov chains and its applications. Contemporary Mathematics 50, 263–273.CrossRefGoogle Scholar
Liu, W. & Wu, W.B. (2010a) Asymptotics of spectral density estimates. Econometric Theory 26(4), 1218–1245.CrossRefGoogle Scholar
Liu, W. & Wu, W.B. (2010b) Simultaneous nonparametric inference of time series. Annals of Statistics 38(4), 2388–2421.CrossRefGoogle Scholar
Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis. Springer Science & Business Media.CrossRefGoogle Scholar
Pedersen, R.S. & Rahbek, A. (2018) Testing GARCH-X type models. Econometric Theory 35(5), 1012–1047.CrossRefGoogle Scholar
Propp, J.G. & Wilson, D.B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures & Algorithms 9(1–2), 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Russell, J.R. & Engle, R.F. (2005) A discrete-state continuous-time model of financial transactions prices and times: The autoregressive conditional multinomial–autoregressive conditional duration model. Journal of Business & Economic Statistics 23(2), 166–180.CrossRefGoogle Scholar
Sims, C.A. (1972) Money, income, and causality. The American Economic Review 62(4), 540–552.Google Scholar
Stenflo, Ö. (2001) Markov chains in random environments and random iterated function systems. Transactions of the American Mathematical Society 353(9), 3547–3562.CrossRefGoogle Scholar
Stenflo, Ö. (2012) A survey of average contractive iterated function systems. Journal of Difference Equations and Applications 18(8), 1355–1380.CrossRefGoogle Scholar
Wu, W.B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences 102(40), 14150–14154.CrossRefGoogle Scholar
Wu, W.B. (2007) Strong invariance principles for dependent random variables. The Annals of Probability 35(6), 2294–2320.CrossRefGoogle Scholar
Wu, W.B., Huang, Y., & Huang, Y. (2010) Kernel estimation for time series: An asymptotic theory. Stochastic Processes and Their Applications 120(12), 2412–2431.CrossRefGoogle Scholar
Wu, W.B. & Shao, X. (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41(2), 425–436.CrossRefGoogle Scholar
Wu, W.B. & Wu, Y.N. (2016) Performance bounds for parameter estimates of high-dimensional linear models with correlated errors. Electronic Journal of Statistics 10(1), 352–379.CrossRefGoogle Scholar
Xiao, H. & Wu, W.B. (2012) Covariance matrix estimation for stationary time series. The Annals of Statistics 40(1), 466–493.CrossRefGoogle Scholar