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LEAST SQUARES ESTIMATION FOR NONLINEAR REGRESSION MODELS WITH HETEROSCEDASTICITY

Published online by Cambridge University Press:  11 January 2021

Qiying Wang
Qiying Wang*
Affiliation:
The University of Sydney
*
Address correspondence to Qiying Wang, School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia; e-mail: qiying.wang@sydney.edu.au
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Abstract

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This paper develops an asymptotic theory of nonlinear least squares estimation by establishing a new framework that can be easily applied to various nonlinear regression models with heteroscedasticity. As an illustration, we explore an application of the framework to nonlinear regression models with nonstationarity and heteroscedasticity. In addition to these main results, this paper provides a maximum inequality for a class of martingales, which is of interest in its own right.

Information

Type
MISCELLANEA
Information
Econometric Theory , Volume 37 , Issue 6 , December 2021 , pp. 1267 - 1289
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

The author thanks Professor Peter Phillips (Editor), Professor Giuseppe Cavaliere (Co-Editor) and four referees for their very helpful comments on the original version. The author also acknowledges the research support given by the Australian Research Council.

References

REFERENCES

Andrews, D.W.K. (1994) Empirical process methods in econometrics. Chapter 37. In Handbook of Econometrics, vol. 5, pp. 2248–2294. Elsevier Amsterdam.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, 767–793.CrossRefGoogle Scholar
Bae, Y., Kakkar, V., & Ogaki, M. (2004) Money Demand in Japan and the Liquidity Trap. Working Paper #04–06. Ohio State University Department of Economics.Google Scholar
Bae, Y., Kakkar, V., & Ogaki, M. (2006) Money demand in Japan and nonlinear cointegration. Journal of Money, Credit, and Banking 38, 1659–1667.CrossRefGoogle Scholar
Bercu, B. & Touati, A. (2008) Exponential inequalities for self-normalized martingales with applications. Annals of Applied Probability 18, 1848–1869.CrossRefGoogle Scholar
Boswijk, H.P., Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2016) Inference on co-integration parameters in heteroskedastic vector autoregressions. Journal of Econometrics 192, 64–85.CrossRefGoogle Scholar
Buchmann, B. & Chan, N.H. (2007) Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence. Annals of Statistics 35, 2001–2017.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140, 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) Heteroskedastic time series with a unit root. Econometric Theory 25, 1228–1276.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2010) Testing for co-integration in vector autoregressions with non-stationary volatility. Journal of Econometrics 158, 7–24.CrossRefGoogle 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., & Phillips, P.C.B. (2001) Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometric Journal 4, 1–36.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) Index models with integrated time series. Journal of Econometrics 114, 73–106.CrossRefGoogle Scholar
De Jong, R. & Hu, L. (2011) A note on nonlinear models with integrated regressors and convergence order results. Econometric Letters 111, 23–25.CrossRefGoogle Scholar
Dong, C. & Linton, O. (2018) Additive nonparametric models with time variables and both stationary and nonstationary regressors. Journal of Econometrics 207, 212–236.CrossRefGoogle Scholar
Dong, C., Gao, J., & Tjötheim, D. (2016) Estimation for single-index and partially linear single-index integrated model. Annals of Statistics 44, 425–453.CrossRefGoogle Scholar
Duffy, J.A. (2016) A uniform law for convergence to the local times of linear fractional stable motions. The Annals of Applied Probability 25, 45–72.Google Scholar
Duffy, J.A. (2017). Uniform convergence rates over maximal domains in structural nonparametric cointegrating regression. Econometric Theory 33, 1387–1417.CrossRefGoogle Scholar
Gao, J. & Phillips, P.C.B. (2013) Semiparametric estimation in triangular system equations with nonstatioanrity. Journal of Econometrics 176, 59–79.CrossRefGoogle Scholar
Gao, J., Kanaya, S., Li, D., & Tjöstheim, D. (2015) Uniform consistency for nonparametric estimators in null recurrent time series. Econometric Theory 31, 911–952.CrossRefGoogle Scholar
Goncalves, S. & Kilian, L. (2004) Bootstrapping autoregressions with conditional heteroscedasticity of unknown form. Journal of Econometrics 123, 89–120.CrossRefGoogle Scholar
Hansen, B.E. (1995) Regression with nonstationary volatility. Econometrica 63, 1113–1132.CrossRefGoogle Scholar
Jacob, C. (2010) Conditional least squares estimation in nonstationary nonlinear stochastic regression models. Annals of Statistics 38, 566–597.CrossRefGoogle Scholar
Jennrich, R.I. (1969) Asymptotic properties of nonlinear least squares estimation. Annals of Mathematical Statistics 40, 633–643.CrossRefGoogle Scholar
Jones, D.A. (1976) Non-linear autoregressive processes. PhD dissertation, University of London.Google Scholar
Kim, S.K. & Kim, I.M. (2012) Partial parametric estimation for nonstationary nonlinear regressions. Journal of Econometrics 167, 448–457.CrossRefGoogle Scholar
Lai, T.L. (1994) Asymptotic theory of nonlinear least square estimations. The Annals of Statistics 22, 1917–1930.CrossRefGoogle Scholar
Lanne, M. & Saikkonen, P. (2005) Nonlinear GARCH models for highly persistency volatility. Econometric Theory 8, 251–276.Google Scholar
Nishiyama, Y. (2000a) Weak convergence of some classes of martingales with jumps. Annals of Probability 28, 685–712.CrossRefGoogle Scholar
Nishiyama, Y. (2000b) Entropy Methods for Martingale. CWI Amsterdam.Google Scholar
Nishiyama, Y. (2007) On the paper “Weak convergence of some classes of martingales with jumps”. Annals of Probability 35, 1194–1200.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
Pollard, D. & Radchenko, P. (2006) Nonlinear least squares estimation. Journal of Multi-Criteria Decision Analysis 97, 548–562.CrossRefGoogle Scholar
Shao, X. & Wu, W.-B. (2007) Asymptotic spectral theory for nonlinear time series. Annals of Statistics 35, 1773-1801.CrossRefGoogle Scholar
Skouras, K. (2000) Strong consistency in nonlinear stochastic regression models. The Annals of Statistics 28, 871–879.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. & 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
Wang, Q., Phillips, P.C.B., & Kasparis, I. (2020) Latent variable nonparametric cointegrating regression. Econometric Theory 1–31. DOI: 10.1017/S02664666200001221.CrossRefGoogle Scholar
Wooldridge, J. (1994) Estimation and inference for dependent processes. In Engle, R.F., McFadden, D.L. (eds.), Handbook of Econometrics, vol. IV, pp. 2639–2738. Elsevier Amsterdam.Google Scholar
Wu, C.F. (1981) Asymptotic theory of nonlinear least square estimations. Annals of Statistics 9, 501–513.CrossRefGoogle Scholar
Wu, W.B. & Min, W. (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar