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 dif-
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+ Address correspondence to Peter C+B+ Phillips, Department of
Economics, Yale University, P+O+ Box 208268, New Haven, CT 06520-8268, USA; e-mail: peter+phillips@yale+edu+
Econometric Theory, 24, 2008, 888–947+ Printed in the United States of America+
doi:10+10170S0266466608080365
888 © 2008 Cambridge University Press 0266-4666008 $15+00
ferent 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+
