Econometric Theory

Research Article


Rustam Ibragimova1 and Peter C.B. Phillipsa2 c1

a1 Yale University

a2 Cowles Foundation for Research in Economics, Yale University, University of Auckland and University of York


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.


c1 Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268, USA; e-mail:


An extended version of this work is available as Ibragimov and Phillips (2004). We are particularly grateful to three anonymous referees and Pentti Saikkonen for many helpful comments and suggestions on earlier versions. We also thank Victor Chernozhukov, Jin-Chuan Duan, Xavier Gabaix, Patrik Guggenberger, Alex Maynard, Roger Moon, Marcelo Moreira, and seminar participants at Yale University, the University of California at Los Angeles, the University of Southern California, the University of Toronto, and the 9th World Congress of the Econometric Society (London, 2005) for commenting on this work. The paper has also benefited from discussions with colleagues at the Departments of Economics at the University of British Columbia, the University of California at San Diego, Harvard University, the London School of Economics and Political Science, Massachusetts Institute of Technology, the Université de Montréal, McGill University and New York University, the Division of the Humanities and Social Sciences at California Institute of Technology, and Nuffield College, University of Oxford. Rustam Ibragimov gratefully acknowledges financial support from a Yale University Dissertation Fellowship and a Cowles Foundation Prize. Peter C.B. Phillips acknowledges partial research support from a Kelly Fellowship and the NSF under grant SES 04-142254.