Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T09:51:13.680Z Has data issue: false hasContentIssue false
  • English
  • Français

REGRESSION ASYMPTOTICS USING MARTINGALE CONVERGENCE METHODS

Published online by Cambridge University Press:  04 April 2008

Rustam Ibragimov  and
Peter C.B. Phillips
Rustam Ibragimov
Affiliation:
Yale University
Peter C.B. Phillips*
Affiliation:
Cowles Foundation for Research in Economics, Yale University, University of Auckland and University of York
*
Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Weak convergence of partial sums and multilinear forms in independent random variables and linear processes and their nonlinear analogues to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and conceptually simple method for obtaining such forms of convergence. The method relies on the fact that the econometric quantities of interest involve discrete time martingales or semimartingales and shows how in the limit these quantities become continuous martingales and semimartingales. The limit theory itself uses very general convergence results for semimartingales that were obtained in the work of Jacod and Shiryaev (2003, Limit Theorems for Stochastic Processes). The theory that is developed here is applicable in a wide range of econometric models, and many examples are given. %One notable outcome of the new approach is that it provides a unified treatment of the asymptotics for stationary, explosive, unit root, and local to unity autoregression, and also some general nonlinear time series regressions. All of these cases are subsumed within the martingale convergence approach, and different rates of convergence are accommodated in a natural way. Moreover, the results on multivariate extensions developed in the paper deliver a unification of the asymptotics for, among many others, models with cointegration and also for regressions with regressors that are nonlinear transforms of integrated time series driven by shocks correlated with the equation errors. Because this is the first time the methods have been used in econometrics, the exposition is presented in some detail with illustrations of new derivations of some well-known existing results, in addition to the provision of new results and the unification of the limit theory for autoregression.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 4 , August 2008 , pp. 888 - 947
Copyright
Copyright © Cambridge University Press 2008

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

Akonom, J. ’ Gouriéroux, C. (1987) A functional limit theorem for fractional processes. Technical report 8801, CEPREMAP.Google Scholar
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676–687.CrossRefGoogle Scholar
Avram, F. (1988) Weak convergence of the variations, iterated integrals and Doléans-Dade exponentials of sequences of semimartingales. Annals of Probability 16, 246–250.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Borodin, A.N. ’ Ibragimov, I.A. (1995) Limit theorems for functionals of random walks. Proceedings of the Steklov Institute of Mathematics 195, no. 2.Google Scholar
Burkholder, D.L. (1973) Distribution function inequalities for martingales. Annals of Probability 1, 19–42.CrossRefGoogle Scholar
Chan, N.H. ’ Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Chan, N.H. ’ Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Coffman, E.G., Puhalskii, A.A., ’ Reiman, M.I. (1998) Polling systems in heavy traffic: A Bessel process limit. Mathematics of Operations Research 23, 257–304.CrossRefGoogle Scholar
Coquet, F. ’ Mémin, J. (1994) Vitesse de convergence en loi pour des solutions d'équations différentielles stochastiques vers une diffusion. (Rate of convergence in law of the solutions of stochastic differential equations to a diffusion.) In Azema, J., Meyer, P.-A., ’ Yor, M. (eds.), Séminaire de Probabilités, XXVIII, Lecture Notes in Mathematics 1583, pp. 279–292. Springer.CrossRefGoogle Scholar
Csörgő, M. ’ Horvàth, L. (1993) Weighted Approximations in Probability and Statistics. Wiley.Google Scholar
Dedecker, J. ’ Merlevède, F. (2002) Necessary and sufficient conditions for the conditional central limit theorem. Annals of Probability 30, 1044–1081.CrossRefGoogle Scholar
Dedecker, J. ’ Rio, E. (2000) On the functional central limit theorem for stationary processes. Annales de l'Institut H. Poincaré. Probabilités et Statistiques 36, 1–34.CrossRefGoogle Scholar
de Jong, R.M. (2002) Nonlinear Estimators with Integrated Regressors but without Exogeneity. Working paper, Michigan State University.Google Scholar
de la Peña, V.H., Ibragimov, R., ’ Sharakhmetov, S. (2003) On extremal distributions and sharp L p-bounds for sums of multilinear forms. Annals of Probability 31, 630–675.Google Scholar
Dharmadhikari, S.W., Fabian, V., ’ Jogdeo, K. (1968) Bounds on the moments of martingales. Annals of Mathematical Statistics 39, 1719–1723.CrossRefGoogle Scholar
Doukhan, P. (2003) Models, inequalities, and limit theorems for stationary sequences. In Doukhan, P., Oppenheim, G., ’ Taqqu, M.S. (eds.), Theory and Applications of Long-Range Dependence, pp. 43–100. Birkhäuser.Google Scholar
Doukhan, P., Oppenheim, G., ’ Taqqu, M.S., eds. (2003) Theory and Applications of Long-Range Dependence. Birkhäuser.Google Scholar
Durrett, R. (1996) Stochastic Calculus. A Practical Introduction. CRC Press.Google Scholar
Dynkin, E.B. ’ Mandelbaum, A. (1983) Symmetric statistics, Poisson point processes, and multiple Wiener integrals. Annals of Statistics 11, 739–745.CrossRefGoogle Scholar
Giraitis, L. ’ Phillips, P.C.B. (2006) Uniform limit theory for stationary autoregression. Journal of Time Series Analysis 27, 51–60.CrossRefGoogle Scholar
Hall, P. ’ Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
He, S.-W., Wang, J.-G., ’ Yan, J.-A. (1992) Semimartingale Theory and Stochastic Calculus. CRC Press.Google Scholar
Hu, L. ’ Phillips, P.C.B. (2004) Dynamics of the federal funds target rate: A nonstationary discrete choice approach. Journal of Applied Econometrics 19, 851–867.CrossRefGoogle Scholar
Ibragimov, R. (1997) Estimates for moments of symmetric statistics. Ph.D. (Kandidat Nauk) dissertation, Institute of Mathematics of Uzbek Academy of Sciences, Tashkent; http://www.economics.harvard.edu/faculty/ibragimov/files/IbragimovPh.D.Dissertation.pdf.Google Scholar
Ibragimov, R. ’ Phillips, P.C.B. (2004) Regression Asymptotics Using Martingale Convergence Methods. Cowles Foundation Discussion paper 1473; http://cowles.econ.yale.edu/P/cd/d14b/d1473.pdf.Google Scholar
Ibragimov, R. ’ Sharakhmetov, S. (2002) Bounds on moments of symmetric statistics. Studia Scientiarum Mathematicarum Hungarica 39, 251–275.CrossRefGoogle Scholar
Ikeda, N. ’ Watanabe, S. (1989) Stochastic Differential Equations and Diffusion Processes. North-Holland.Google Scholar
Jacod, J. ’ Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2nd ed., Springer-Verlag.CrossRefGoogle Scholar
Jakubowski, A., Mémin, J., ’ Pagès, G. (1989) Convergence en loi des suites d'intégrales stochastiques sur l'espace D 1 de Skorokhod. (Convergence in law of sequences of stochastic integrals on the Skorokhod space D 1.) Probability Theory and Related Fields 81, 111–137.CrossRefGoogle Scholar
Jeganathan, P. (2003) Second Order Limits of Functionals of Sums of Linear Processes That Converge to Fractional Stable Motions. Working paper, Indian Statistical Institute.CrossRefGoogle Scholar
Jeganathan, P. (2004) Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Annals of Probability 32, 1771–1795.CrossRefGoogle Scholar
Kurtz, T.G. ’ Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Liptser, R.Sh. ’ Shiryaev, A.N. (1989) Theory of Martingales. Kluwer.CrossRefGoogle Scholar
Mandelbaum, A. ’ Taqqu, M.S. (1984) Invariance principle for symmetric statistics. Annals of Statistics 12, 483–496.CrossRefGoogle Scholar
Mémin, J. (2003) Stability of Doob-Meyer decomposition under extended convergence. Acta Mathematicae Applicatae Sinica. >English Series 19, 177–190.Google Scholar
Mémin, J. ’ Słomiński, L. (1991) Condition UT et stabilité en loi des solutions d'équations différentielles stochastiques. Séminaire de Probabilités, XXV, Lecture Notes in Mathematics 1485, pp. 162–177. Springer.CrossRefGoogle Scholar
Nze, P.A. ’ Doukhan, P. (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20, 995–1045.CrossRefGoogle Scholar
Park, J.Y. ’ Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Park, J.Y. ’ Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a). Time-series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b). Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1999) Discrete Fourier Transforms of Fractional Processes. Cowles Foundation Discussion paper 1243, Yale University; http://cowles.econ.yale.edu/P/cd/d12a/d1243.pdf.Google Scholar
Phillips, P.C.B. (2007) Unit root log periodogram regression. Journal of Econometrics 138, 104–124.CrossRefGoogle Scholar
Phillips, P.C.B. ’ Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Phillips, P.C.B. ’ Perron, P. (1988) Testing for a unit root in time series regression Biometrika 75, 335–346.CrossRefGoogle Scholar
Phillips, P.C.B. ’ Ploberger, W. (1996) An asymptotic theory of Bayesian inference for time series. Econometrica 64, 381–413.CrossRefGoogle Scholar
Phillips, P.C.B. ’ Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Prigent, J.-L. (2003) Weak Convergence of Financial Markets, Springer Finance. Springer-Verlag.CrossRefGoogle Scholar
Saikkonen, P. ’ Choi, I. (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 301–340.CrossRefGoogle Scholar
Shorack, G.R. ’ Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley.Google Scholar
Stroock, D.W. ’ Varadhan, S.R.S. (1979) Multidimensional Diffusion Processes. Springer-Verlag.Google Scholar
White, J.S. (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29, 1188–1197.CrossRefGoogle Scholar