Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-12T21:17:48.941Z Has data issue: false hasContentIssue false
  • English
  • Français

WEAK DEPENDENCE: MODELS AND APPLICATIONS TO ECONOMETRICS

Published online by Cambridge University Press:  01 December 2004

Patrick Ango Nze  and
Paul Doukhan
Patrick Ango Nze
Affiliation:
Université Lille 3
Paul Doukhan
Affiliation:
Université de Cergy Pontoise
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH(∞) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.

We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 6 , December 2004 , pp. 995 - 1045
Copyright
© 2004 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.)

References

REFERENCES

Alsmeyer, G. & C.D. Fuh (2001) Limit theorems for iterated random functions by regenerative methods. Stochastic Processes and Their Applications 96, 123–142.Google Scholar
Andrews, D.W.K. (1984) Non strong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.Google Scholar
Andrews, D.W.K. (1988) Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory 4, 458–467.Google Scholar
Andrews, D.W.K. (2002) Higher-order improvements of a computationally attractive k-step bootstrap for extremum estimators. Econometrica 70, 119–162.Google Scholar
Andrews, D.W.K. & D. Pollard (1994) An introduction to functional central limit theorems for dependent stochastic processes. International Statistical Review 62, 119–132.Google Scholar
Ango Nze, P. (1995) Critères d'ergodicité de modèles markoviens: Estimation non paramétrique sous des hypothèses de dépendance. Ph.D. Dissertation, Université Dauphine, Paris.
Ango Nze, P. (1998) Critères d'ergodicité géométrique ou arithmétique de modèles linéaires perturbés à représentation markovienne. Comptes Rendus de l'Académie des Sciences, Serie 1, Paris t. 326, 371–376.Google Scholar
Ango Nze, P., P. Bühlmann, & P. Doukhan (2002) Weak dependence beyond mixing and asymptotics for nonparametric regression. Annals of Statistics 30, 397–430.Google Scholar
Arcones, M.A. & B. Yu (1994) Central limit theorems for empirical and u-processes of stationary sequences. Journal of Theoretical Probability 7, 47–71.Google Scholar
Berbee, H.C.P. (1979) Random Walks with Stationary Increments and Renewal Theory. Mathematical centre tracts, Amsterdam.
Bickel, P. & P. Bühlmann (1999) A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5, 413–446.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Bradley, R.C. & W. Bryc (1985) Multilinear forms and measures of dependence between random variables. Journal of Multivariate Analysis 16, 335–367.Google Scholar
Breuer, P. & P. Major (1983) Central limit theorems for non-linear functionals of Gaussian random fields. Journal of Multivariate Analysis 13, 425–441.Google Scholar
Chambers, D. & D. Slud (1989) Central limit theorems for non-linear functionals of stationary Gaussian processes. Probability Theory and Related Fields 80, 323–346.Google Scholar
Chen, X. & Y. Fan (1999) Consistent hypotheses testing in semiparametric and nonparametric models for econometric time series. Journal of Econometrics 91, 373–401.Google Scholar
Cline, D. & H. Pu (1998) Verifying irreducibility and continuity of a nonlinear time series. Statistics and Probability Letters 40(2), 139–148.Google Scholar
Collomb, G. (1984) Propriétés de convergence presque complète du prédicteur à noyau. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 66, 441–460.Google Scholar
Coulon-Prieur, C. & P. Doukhan (2000) A triangular CLT for weakly dependent sequences. Statistics and Probability Letters 47, 61–68.Google Scholar
Csörgő, M. & J. Mielniczuk (1996) The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probability Theory and Related Fields 104, 15–25.Google Scholar
Davydov, Y. (1973) Mixing conditions for Markov chains. Theory of Probability and Its Applications 18, 313–328.Google Scholar
Dedecker, J. & E. Rio (2000) On the functional central limit theorem for stationary processes. Annales de l'institut Henri Poincaré, série B 36, 1–34.Google Scholar
Diaconis, P. & D. Freedman (1999) Iterated random functions. SIAM Review 41, 45–76Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer Verlag.
Doukhan, P. (2002) Models, inequalities and limit theorems for stationary sequences. In P. Doukhan, G. Oppenheim, & M. Taqqu (eds.), Long Range Dependence, Theory and Applications. Birkhäuser.
Doukhan, P., P. Massart, & E. Rio (1995) Invariance principles for absolutely regular empirical processes. Annales de l'institut Henri Poincaré, série B 31, 393–427.Google Scholar
Doukhan, P. & S. Louhichi (1999) A new weak dependence condition and application to moment inequalities. Stochastic Processes and Their Applications 84, 313–343.Google Scholar
Doukhan, P. & D. Surgailis (1998) Functional central limit theorem for the empirical process of short memory linear processes. Comptes Rendus de l'Académie des Sciences, Paris 326, 87–92.Google Scholar
Duflo, M. (1990) Méthodes récursives aléatoires. Masson. (English edition, Springer, 1996).
Esary, J., F. Proschan, & D. Walkup (1967) Association of random variables with applications. Annals of Mathematical Statistics 38, 1466–1476.Google Scholar
Fort, G. & E. Moulines (2002) Computable bounds for polynomial ergodicity. Stochastic Processes and Their Applications. Forthcoming.
Fortuin, C., P. Kastelyn, & J. Ginibre (1971) Correlation inequalities on some ordered sets. Communications in Mathematical Physics 22, 89–103.Google Scholar
Giraitis, L., P. Kokoszka, & R. Leipus (2000) Stationary ARCH models: Dependence structure and CLT. Econometric Theory 16, 3–22.Google Scholar
Giraitis, L. & D. Surgailis (1986) Multivariate Appell polynomials and the central limit theorem. In E. Eberlein & M. Taqqu (eds.), Dependence in Probability and Statistics, pp. 27–51. Birkhaüser.
Giraitis, L. & D. Surgailis (1994) A Central Limit Theorem for the Empirical Process of a Long Memory Linear Sequence. Preprint.
Granger, C.W.J. & A.P. Andersen (1978) An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht.
Hall, P. & C.C. Heyde (1980) Martingale Limit Theory and Its Applications. Academic Press.
Hall, P. & J. Horowitz (1996) Bootstrap critical values for tests based on generalized method of moment estimators. Econometrica 64, 891–916.Google Scholar
He, C. & T. Teräsvirta (1999) Properties of moments of a family of GARCH models. Journal of Econometrics 92, 173–192.Google Scholar
Ho, H.-C. & T. Hsing (1997) Limit theorems for functionals of moving averages. Annals of Probability 25, 1636–1669.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349–382.Google Scholar
Ibragimov, I.A. & Y.V. Linnik (1971) Independent and stationary sequences of random variables. Wolters-Noordhoff.
Ibragimov, R. & S. Sharakhmentov (2001) The exact constant in the Rosenthal inequality for random variables with mean zero. Theory of Probability and Its Applications 46, 127–132.Google Scholar
Jarner, S.F. & G.O. Roberts (2002) Polynomial convergence rates of Markov chains. Annals of Applied Probability 12, 224–247.Google Scholar
Jarner, S.F. & R.L. Tweedie (2001) Locally contacting iterated functions and stability of Markov chains. Journal of Applied Probability 38, 494–507.Google Scholar
Kallenberg, O. (1997) Foundations of Modern Probability. Springer-Verlag.
Kolmogorov, A.N. & Y.A. Rozanov (1960) On the strong mixing conditions for stationary Gaussian sequences. Theory of Probability and Its Applications 5, 204–207.Google Scholar
Künsch, H.R. (1989) The jacknife and bootstrap for general stationary observations. Annals of Statistics 17, 1217–1241.Google Scholar
Liebscher, E. (1996) Strong convergence of sums of α-mixing random variables with applications to density estimation. Stochastic Processes and Their Applications 65, 69–98.Google Scholar
Lin, Z.Y. & C.R. Lu (1996) Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Publishers.
Louhichi, S. (2000) Weak convergence for empirical processes of associated sequences. Annales de l'Institut Henri Poincaré, série B 36, 547–567.Google Scholar
McLeish, D.L. (1975) A maximal inequality and dependent strong laws. Annals of Probability 3, 829–839.Google Scholar
Meyn, S.P. & R.L. Tweedie (1993) Markov chains and stochastic stability. Communications in Control and Engineering.
Mills, T.C. (1999) The Econometric Modelling of Financial Time Series. Cambridge University Press.
Mokkadem, A. (1990) Propriétés de mélange des modèles autorégressifs polynômiaux. Annales de l'Institut Henri Poincaré, série B 26, 219–260.Google Scholar
Moricz, F.A., R.J. Serfling, & W.F. Stout (1982) Moment and probability bounds with quasi-superadditive structure for the maximum partial sum. Annals of Probability 10, 1032–1040.Google Scholar
Newman, C.M. (1984) Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y.L. Tong (ed.), Inequalities in Statistics and Probability, IMS Hayward, CA Lecture Notes-Monograph Series 5, pp. 127–140.
Newman, C.M. & A.L. Wright (1981) An invariance principle for certain dependent sequences. Annals of Probability 9, 671–675.Google Scholar
Ornstein, D.S. & G. Weiss (1990) Statistical properties of chaotic systems. Bulletin of the American Mathematical Society 24, 11–116.Google Scholar
Peligrad, M. (1987) The convergence of moments in ρ-mixing sequences of random variables. Proceedings of the American Mathematical Society 101, 142–148.Google Scholar
Peligrad, M. (1991) Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumptions. Stochastics and Stochastic Reports 40, 147–168.Google Scholar
Peligrad, M. (1996) On the asymptotic normality of weak dependent random variables. Journal of Theoretical Probabilities 9, 703–715.Google Scholar
de la Peña, V.H., R. Ibragimov, & S. Sharakhmentov (2003) On extremal distributions and sharp Lp-bounds for sums of multilinear forms. Annals of Probability 31, 630–675.Google Scholar
Petruccelli, J. & S.W. Woolford (1984) A threshold AR(1) model. Journal of Applied Probability 21, 270–286.Google Scholar
Pham, T.D. (1986) The mixing property of bilinear and generalized random coefficient autoregressive models. Stochastic Processes and Their Applications 23, 291–300.Google Scholar
Pham, T.D. & L.T. Tran (1985) Some mixing properties of time series models. Stochastic Processes and Their Applications 19, 297–303.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Pollard, D. (1981) Limit theorems for empirical processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 181–195.Google Scholar
Pötscher, B.M. & I.R. Prucha (1991a) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Reviews 10, 125–216.Google Scholar
Pötscher, B.M. & I.R. Prucha (1991b) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: Asymptotic normality. Econometric Reviews 10, 253–326.Google Scholar
Prakasa, R. (1983) Nonparametric Functional Estimation. Academic Press.
Rio, E. (1994) Inégalités de moments pour les suites stationnaires et fortement mélangeantes. Comptes Rendus de l'Académie des Sciences, Paris 318, 355–360.Google Scholar
Rio, E. (2000) Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications 31.Google Scholar
Rios, R. (1996) Utilisation des techniques non paramétriques et semi paramétriques en statistique des données dépendantes. Ph.D. Dissertation, Université Paris sud, Paris.
Robinson, P.M. (1983) Nonparametric estimators for time series. Journal of Time Series Analysis 4, 185–207.Google Scholar
Robinson, P.M. (1989) Hypothesis testing in semiparametric and nonparametric models for econometric time series. Review of Economic Studies 56, 511–534.Google Scholar
Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proceedings of National Academy of Sciences, U.S.A. 42, 43–47.Google Scholar
Rosenblatt, M. (1985) Stationary Sequences and Random Fields. Birkhaüser.
Rosenblatt, M. (1991) Stochastic curve estimation. NSF-CBMS Regional Conference Series in Probability and Statistics 3.
Shao, Q.M. (1988) Remark on the invariance principle for ρ-mixing sequences of random variables. Chinese Annals of Mathematics 9A, 409–412.Google Scholar
Shao, Q.M. (1995) Maximal inequalities for partial sums of ρ-mixing sequences. Annals of Probability 23, 948–965.Google Scholar
Shao, Q.M. (2000) A comparison theorem on moment inequalities between negatively associated and independent random variables. Journal of Theoretical Probability 13, 343–356.Google Scholar
Shao, Q.M. & H. Yu (1996) Weak convergence for weighted empirical processes of dependent sequences. Annals of Probability 24, 2052–2078.Google Scholar
Tjøstheim, D. (1986) Some doubly stochastic time series models. Journal of Time Series Analysis 7, 51–72.Google Scholar
Tjøstheim, D. (1990) Non-linear time series and Markov chains. Advances in Applied Probability 22, 587–611.Google Scholar
Tong, H. (1981) A note on a Markov bilinear stochastic process in discrete time. Journal of Time Series Analysis 2 (4), 279–284.Google Scholar
van der Vaart, A.W. & J.A. Wellner (1996) Weak convergence and empirical processes. Springer-Verlag.
Veretennikov, A.Y. (1999) On polynomial mixing and convergence rate for stochastic difference and differential equation. Theory of Probability and Its Applications 44, 361–374.Google Scholar
Withers, C.S. (1981) Central limit theorem for dependent variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 509–534.Google Scholar
Wolkonski, V.A. & Y.A. Rozanov (1959) Some limit theorems for random functions, Part I. Theory of Probability and Its Applications 4, 178–197.Google Scholar
Wolkonski, V.A. & Y.A. Rozanov (1961) Some limit theorems for random functions, Part II. Theory of Probability and Its Applications 6, 186–198.Google Scholar
Yoshihara, K. (1973) Billingsley's theorems on empirical processes of strong mixing sequences. Yokohama Mathematical Journal 23, 1–7.Google Scholar
Yoshihara, K. (1976) Limiting behaviour of U-statistics for stationary, absolutely regular processes. Zeitung Wahrscheinlichkeitstheorie und verwandte Gebiete 35, 237–252.Google Scholar
Yu, H. (1993) A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences. Probability Theory and Related Fields 95, 357–370.Google Scholar
Zhao, L. & Z. Fang (1985) Strong convergence of kernel estimates of nonparametric regression functions. Chinese Annals of Mathematics 6B, 147–155.Google Scholar