Weak Convergence to Stochastic Integrals for Econometric Applications

30 Pages Posted: 3 Dec 2014

See all articles by Hanying Liang

Hanying Liang

Tongji University - School of Economics and Management

Peter C. B. Phillips

University of Auckland Business School; Yale University - Cowles Foundation; Singapore Management University - School of Economics

Hanchao Wang

Zhejiang University - College of Economics; Shandong University - Zhongtai Securities Institute for Financial Studies

Qiying Wang

University of Sydney

Date Written: December 2, 2014

Abstract

Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on function space weak convergence. In establishing weak convergence of sample covariances to stochastic integrals, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications in econometrics involve a cointegration framework where endogeneity and nonlinearity play a major role and lead to complications in 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 asymptotic development and we provide limit results for such covariances when linear process, long memory, and mixing variates are involved in the innovations. The limit results extend earlier findings in the literature, are relevant in many econometric applications, and involve simple conditions that facilitate implementation in practice. A nonlinear extension of FM regression is used to illustrate practical application of the methods.

Keywords: Decomposition, FM regression, Linear process, Long memory, Stochastic integral, Semimartingale, α-mixing

JEL Classification: C22, C65

Suggested Citation

Liang, Hanying and Phillips, Peter C. B. and Wang, Hanchao and Wang, Qiying, Weak Convergence to Stochastic Integrals for Econometric Applications (December 2, 2014). Cowles Foundation Discussion Paper No. 1971, Available at SSRN: https://ssrn.com/abstract=2533003 or http://dx.doi.org/10.2139/ssrn.2533003

Hanying Liang

Tongji University - School of Economics and Management ( email )

1239 Si Ping Lu
Shanghai 200092
China

Peter C. B. Phillips (Contact Author)

University of Auckland Business School ( email )

12 Grafton Rd
Private Bag 92019
Auckland, 1010
New Zealand
+64 9 373 7599 x7596 (Phone)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3695 (Phone)
203-432-5429 (Fax)

Singapore Management University - School of Economics

90 Stamford Road
178903
Singapore

Hanchao Wang

Zhejiang University - College of Economics ( email )

Yuquan Campus 38 Zheda Road
Hangzhou, Zhejiang 310027
China

Shandong University - Zhongtai Securities Institute for Financial Studies ( email )

27 Shanda South Road
Jinan City, Shandong 250100
China

Qiying Wang

University of Sydney ( email )

University of Sydney
Sydney, NSW 2006
Australia

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