REGRESSION ASYMPTOTICS USING MARTINGALE CONVERGENCE METHODS

By Rustam Ibragimov, Peter C. B. Phillips. Econometric Theory, 2008

81 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/d04697d6211c9a5c5baec8d8912682c919573e4b

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Citations in the corpus

Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series (2006, methodology): “The authors are grateful to Peter Phillips, two anonymous referees, the editor, and the participants at the Prospectus Workshop at the Department of Economics, Yale University, in 2002-2003 for helpful comments and suggestions.”
The Link between Economic Complexity and Carbon Emissions in the European Union Countries: A Model Based on the Environmental Kuznets Curve (EKC) Approach (2019, unknown): “The martingale theory framework of Ibragimov and Phillips (2008) [73] was used by Wagner and Hong (2016) [71] in order to test the errors’ dynamics, namely if they behaviour as martingale (i.e., a sequence of random variables for which at a given time, the conditional expectation of the next value…”
Testing Linearity in Cointegrating Relations With an Application to Purchasing Power Parity (2010, unknown): “…examined nonlinear sample covariance asymptotics under conditions that are less strict on the innovation processes, but more restrictive in terms of functional forms; and a general semimartingale approach to establishing limit results of this type was developed in Ibragimov and Phillips (2004).”
COINTEGRATING POLYNOMIAL REGRESSIONS: FULLY MODIFIED OLS ESTIMATION AND INFERENCE (2011, methodology): “In terms of asymptotic theory our work relies heavily upon important contributions by Chang et al. (2001), Park and Phillips (1999, 2001), and Ibragimov and Phillips (2008), with a main difference being that the regressors are allowed to be endogenous rather than only predetermined.”
Rank − 1 / 2: A Simple Way to Improve the OLS Estimation of Tail Exponents (2007, unknown): “Following the pioneering work by Mandelbrot (1963) (see also the papers in Mandelbrot, 1997; Fama, 1965), numerous studies have documented that time series encountered in many fields in economics and finance are typically thick-tailed and can be well approximated using distributions with tails exhib”