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NONLINEAR COINTEGRATING POWER FUNCTION REGRESSION WITH ENDOGENEITY

Published online by Cambridge University Press:  26 January 2021

Zhishui Hu ,
Peter C.B. Phillips  and
Qiying Wang
Zhishui Hu
Affiliation:
University of Science and Technology of China
Peter C.B. Phillips
Affiliation:
Yale University The University of Auckland University of Southampton Singapore Management 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.wang@sydney.edu.au.
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Abstract

This paper develops an asymptotic theory for nonlinear cointegrating power function regression. The framework extends earlier work on the deterministic trend case and allows for both endogeneity and heteroskedasticity, which makes the models and inferential methods relevant to many empirical economic and financial applications, including predictive regression. A new test for linear cointegration against nonlinear departures is developed based on a simple linearized pseudo-model that is very convenient for practical implementation and has standard normal limit theory in the strictly exogenous regressor case. Accompanying the asymptotic theory of nonlinear regression, the paper establishes some new results on weak convergence to stochastic integrals that go beyond the usual semimartingale structure and considerably extend existing limit theory, complementing other recent findings on stochastic integral asymptotics. The paper also provides a general framework for extremum estimation limit theory that encompasses stochastically nonstationary time series and should be of wide applicability.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 6 , December 2021 , pp. 1173 - 1213
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

The authors thank the Co-Editor, Pentti Saikkonen, and two referees for helpful comments on the original manuscript, which have led to many improvements. Hu acknowledges research support from NSFC (No. 11671373); Wang acknowledges research support from the Australian Research Council; Phillips acknowledges research support from the NSF under Grant No. 18-50860 and the Kelly Fund at The University of Auckland.

References

REFERENCES

Andrews, D.W.K. & Cheng, X. (2012) Estimation and inference with weak, semi-strong, and strong identification. Econometrica 80, 2153–2211.Google Scholar
Andrews, D.W.K. & Sun, Y. (2004) Adaptive local polynomial Whittle estimation of long-range dependence. Econometrica 72, 569–614.CrossRefGoogle Scholar
Bae, Y. & de Jong, R. (2007) Money demand function estimation by nonlinear cointegration. Journal of Applied Econometrics 22(4), 767–793.CrossRefGoogle Scholar
Baek, Y.I., Cho, H.S., & Phillips, P.C.B. (2015) Testing linearity using power transforms of regressors. Journal of Econometrics 187, 376–384.CrossRefGoogle Scholar
Berkes, I. & Horváth, L. (2006) Convergence of integral functionals of stochastic processes. Econometric Theory 22, 304–322.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Chan, N. & Wang, Q. (2014) Uniform convergence for nonparametric estimators with nonstationary data. Econometric Theory 30, 1110–1133.CrossRefGoogle Scholar
Chan, N. & Wang, Q. (2015) Nonlinear regression with nonstationary time series. Journal of Econometrics 185, 182–195.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) Index models with integrated time series. Journal of Econometrics 114, 73–106.CrossRefGoogle Scholar
Chang, Y., Park, J.Y., & Phillips, P.C.B. (2001) Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometric Journal 4, 1–36.CrossRefGoogle Scholar
Cho, J.S. & Phillips, P.C.B. (2018) Sequentially testing polynomial model hypotheses using power transforms of regressors. Journal of Applied Econometrics 33, 141–159.CrossRefGoogle Scholar
de Jong, R. (2002) Nonlinear Regression with Integrated Regressors but Without Exogeneity. Mimeograph, Department of Economics, Michigan State University.Google Scholar
Ethier, S.N. & Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. Wiley.CrossRefGoogle Scholar
Gao, J. & Phillips, P.C.B. (2013) Semiparametric estimation in triangular system equations with nonstationarity. Journal of Econometrics 176, 59–79.CrossRefGoogle Scholar
Hansen, B. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Kim, S.K. & Kim, I.M. (2012) Partial parametric estimation for nonstationary nonlinear regressions. Journal of Econometrics 167, 448–457.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Liang, H., Phillips, P.C.B., Wang, H., & Wang, Q. (2016) Weak convergence to stochastic integrals for econometric applications. Econometric Theory 32, 1349–1375.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2000) Nonstationary binary choice. Econometrica 68, 1249–1280.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001). Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Peng, J. & Wang, Q. (2018) Weak convergence to stochastic integrals under primitive conditions in nonlinear econometric models. Econometric Theory 34, 1132–1157.CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557–614.CrossRefGoogle Scholar
Phillips, P.C.B. (2015) Pitfalls and possibilities in predictive regression. Journal of Financial Econometrics 13, 521–555.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 1–22.CrossRefGoogle Scholar
Shi, X. & Phillips, P.C.B. (2012) Nonlinear cointegrating regression under weak identification. Eco- nometric Theory 28, 509–547.CrossRefGoogle Scholar
Stein, E.M. & Shakarchi, R. (2005) Real Analysis, Measure Theory, Integration and Hilbert Spaces. Princeton University Press.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics. Springer.CrossRefGoogle Scholar
Wang, Q. (2014) Martingale limit theorems revisited and non-linear cointegrating regression. Econometric Theory 30, 509–535.CrossRefGoogle Scholar
Wang, Q. (2015) Limit Theorems for Nonlinear Cointegrating Regression. World Scientific.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.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25, 710–738.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009b) Structural nonparametric cointegrating regression. Econometrica 25, 1901–1948.Google Scholar
Wang, Q. & Phillips, P.C.B. (2016) Nonparametric cointegrating regression with endogeneity and long memory. Econometric Theory 32, 359–401.CrossRefGoogle Scholar
Wooldridge, J.M. (1994) Estimation and inference for dependent processes. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. IV. North Holland.Google Scholar