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WEAK CONVERGENCE TO STOCHASTIC INTEGRALS FOR ECONOMETRIC APPLICATIONS

Published online by Cambridge University Press:  24 July 2015

Hanying Liang ,
Peter C.B. Phillips ,
Hanchao Wang  and
Qiying Wang
Hanying Liang
Affiliation:
Tongji University
Peter C.B. Phillips
Affiliation:
Yale University
Hanchao Wang
Affiliation:
Zhejiang University
Qiying Wang*
Affiliation:
The University of Sydney
*
*Address correspondence to Qiying Wang, School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia; e-mail: qiying@maths.usyd.edu.au.
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Abstract

Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on functional weak convergence. In establishing such convergence, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications involve a cointegration framework where endogeneity and nonlinearity play major roles and complicate the limit theory. This paper explores weak convergence limit theory to stochastic integral functionals in such settings. We use a novel decomposition of sample covariances of functions of I (1) and I (0) time series that simplifies the asymptotics and our limit results for such covariances hold for linear process, long memory, and mixing variates in the innovations. These results extend earlier findings in the literature, are relevant in many applications, and involve simple conditions that facilitate practical implementation. A nonlinear extension of FM regression is used to illustrate practical application of the methods.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 6 , December 2016 , pp. 1349 - 1375
Copyright
Copyright © Cambridge University Press 2015 

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References

REFERENCES

Aldous, D.J. & Eagleson, G.K. (1978) On mixing and stability of limit theorems. Annals of Probability 2, 325–331.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Chan, N. & Wang, Q. (2015) Nonlinear regression with nonstationary time series. Journal of Econometrics 185, 182–195.Google Scholar
Chan, N.H. & Wei, C.Z. (1988) Limit distributions of least-squares estimates of unstable auto-regressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2011) Endogeneity in nonlinear regressions with integrated time series. Econometric Reviews 30, 51–87.Google Scholar
Chang, Y., Park, J.Y., & Phillips, P.C.B. (2001) Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometrics Journal 4, 1–36.Google Scholar
Cheng, T.-L. & Chow, Y.-S. (2002) On stable convergence in the central limit theorem. Statistics and Probability Letters 57, 307–313.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory: An Introduction for Econometricians. Oxford University Press.Google Scholar
de Jong, R. & Davidson, J. (2000a) The functional central limit theorem and weak convergence to stochastic integral I: Weak dependent processes. Econometric Theory 16, 621–642.Google Scholar
de Jong, R. & Davidson, J. (2000b) The functional central limit theorem and weak convergence to stochastic integral II: Fractionally integrated processes. Econometric Theory 16, 643–666.Google Scholar
de Jong, R. (2002) Nonlinear Regression with Integrated Regressors but Without Exogeneity. Mimeograph, Department of Economics, Michigan State University.Google Scholar
Duffie, D. & Protter, P. (1992) From discrete to continuous time finance: weak convergence of the financial gain process. Mathematical finance 2, 1–15.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and its Application. Academic Press.Google Scholar
Hansen, B.E. (1992). Covergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.Google Scholar
Ibragimov, R. & Phillips, P.C.B. (2008) Regression asymptotics using martingale convergence methods. Econometric Theory 24, 888–947.Google Scholar
Jacod, J. & Shiryaev, A.N. (1987/2003) Limit Theorems for Stochastic Processes. Springer–Verlag.CrossRefGoogle Scholar
Jakubowski, A. (1996) Convergence in various topologies for stochastic integrals driven by semimartingales. Annals of Probability 24, 2141–2153.Google Scholar
Jeganathan, P. (1995) Some aspects of asymptotic theory with applications to time series models. Econometric Theory 11, 818–867.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.Google Scholar
Lin, Z. & Wang, H. (2010) On convergence to stochastic integrals. arXiv:1006.4693 [math.PR].Google Scholar
McLeish, D.L. (1975) A maximal inequality and dependent strong laws. Annals of Probability 3, 829–839.Google Scholar
Mykland, P.A. & Zhang, L. (2006) ANOVA for diffusions and Ito processes. Annals of Statistics 34, 1931–1963.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468–497.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5, 95–131.Google 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. (2000) Nonstationary binary choice. Econometrica 68, 1249–1280.Google Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–302.Google Scholar
Phillips, P.C.B. (1988a) Multiple regression with integrated processes. In Prabhu, N.U. (ed.), Statistical Inference from Stochastic Processes, Contemporary Mathematics, vol. 80. American Mathematical Society, pp. 79–106.Google Scholar
Phillips, P.C.B. (1988b) Weak convergence to sample covariance matrices to stochastic integrals via martingale approximation. Econometric Theory 4, 528–533.Google Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.Google Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283–306.Google Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.Google Scholar
Phillips, P.C.B & Outliaris, S. (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.Google Scholar
Phillips, P.C.B & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Rényi, A. (1963) On stable sequences of events. Sankhya Series A 25, 293–302.Google Scholar
Strasser, H. (1986) Martingale difference arrays and stochastic integrals. Probabity Theory and Related Fields 72, 83–89.Google Scholar
Wang, Q. (2014) Martingale limit theorem revisited and nonlinear cointegrating regression. Econometric Theory 30, 509–535.CrossRefGoogle Scholar
Wang, Q., Lin, Y.X., & Gulati, C.M. (2003) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19, 143–164.Google Scholar
Wang, Q. & Phillips, P.C.B. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.Google Scholar
Wang, Q. & Phillips, P.C.B. (2009b) Structural nonparametric cointegrating regression. Econometrica 77, 1901–1948.Google Scholar
Wang, Q. & Phillips, P.C.B. (2011) Asymptotic theory for zero energy functionals with nonparametric regression applications. Econometric Theory 27, 235–259.Google Scholar
Wang, Q. & Phillips, P.C.B. (2012) A specification test for nonlinear nonstationary models. Annals of Statistics 40, 727–758.Google Scholar