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ROBUST ESTIMATION AND INFERENCE FOR THRESHOLD MODELS WITH INTEGRATED REGRESSORS

Published online by Cambridge University Press:  27 October 2014

Haiqiang Chen
Haiqiang Chen*
Affiliation:
Xiamen University
*
*Address correspondence to Haiqiang Chen, Xiamen University, Xiamen, Fujian 361005, China; e-mail: [email protected].
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Abstract

This paper studies the robust estimation and inference of threshold models with integrated regressors. We derive the asymptotic distribution of the profiled least squares (LS) estimator under the diminishing threshold effect assumption that the size of the threshold effect converges to zero. Depending on how rapidly this sequence converges, the model may be identified or only weakly identified and asymptotic theorems are developed for both cases. As the convergence rate is unknown in practice, a model-selection procedure is applied to determine the model identification strength and to construct robust confidence intervals, which have the correct asymptotic size irrespective of the magnitude of the threshold effect. The model is then generalized to incorporate endogeneity and serial correlation in error terms, under which, we design a Cochrane–Orcutt feasible generalized least squares (FGLS) estimator which enjoys efficiency gains and robustness against different error specifications, including both I(0) and I(1) errors. Based on this FGLS estimator, we further develop a sup-Wald statistic to test for the existence of the threshold effect. Monte Carlo simulations show that our estimators and test statistics perform well.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 778 - 810
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

I would like to thank Professor Peter C.B. Phillips and the referees for helpful comments. Special thanks also go to Zongwu Cai, Jiti Gao, Wolfgang Härdle, Yongmiao Hong, and seminar participants at Cornell University and the 3rd XMU-Humboldt Workshop. The research was supported in part by the National Nature Science Foundation of China grants #11201390, #71201137, #71131008, and #71271179 as well as Deutsche Forschungsgemeinschaft (DFG) through the SFB649 “Economic Risk”.

References

REFERENCES

Andrews, D.W.K. & McDermott, C.J. (1995) Nonlinear econometric models with deterministically trending variables. Review of Economic Studies 62(3), 343–360.CrossRefGoogle Scholar
Balke, N. & Fomby, T. (1997) Threshold cointegration. International Economics Review 8, 627–645.Google Scholar
Bierens, H. & Martins, L. (2010) Time-varying cointegration. Econometric Theory 26, 1453–1490.Google Scholar
Cai, Z., Li, Q., & Park, J.Y. (2009) Functional-coefficient models for nonstationary time series data. Journal of Econometrics 148, 101–113.Google Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81, 27–60.CrossRefGoogle Scholar
Caner, M. & Hansen, B.E. (2001) Threshold autoregression with a unit root. Econometrica 69, 1555–1596.Google Scholar
Chan, K.S. (1993) Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Annals of Statistics 21, 520–533.CrossRefGoogle Scholar
Chen, H. (2013) Robust Estimation and Inference for Threshold Models with Integrated Regressors. Working papers, Xiamen University. Available at SSRN: http://ssrn.com/abstract=2287442.CrossRefGoogle Scholar
Cheng, X. (2008) Robust Confidence Intervals in Nonlinear Regression under Weak Identification. Manuscript, Department of Economics, Yale University.Google Scholar
Choi, C.Y., Hu, L., & Ogaki, M. (2008) Robust estimation for structural spurious regressions and a Hausman-type cointegration test. Journal of Econometrics 142, 327–351.CrossRefGoogle Scholar
Choi, I. & Saikkonen, P. (2010) Test of nonlinear cointegration. Econometric Theory 26, 682–709.CrossRefGoogle Scholar
Durlauf, S.N. & Johnson, P.A. (1995) Multiple regimes and cross-country growth behavior. Journal of Applied Econometrics 10, 365–384.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2007) Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141(2), 1196–1218.CrossRefGoogle Scholar
Gao, J., King, M.L., Lu, Z., & Tjøstheim, D. (2009a) Specification testing in nonstationary time series autoregression. Annals of Statistics 37, 3893–3928.CrossRefGoogle Scholar
Gao, J., King, M.L., Lu, Z., & Tjøstheim, D. (2009b) Nonparametric specification testing for nonlinear time series with nonstationarity. Econometric Theory 25, 1869–1892.Google Scholar
Gonzalo, J. & Pitarakis, J. (2002) Estimation and model selection based inference in single and multiple threshold models. Journal of Econometrics 110, 319–352.Google Scholar
Gonzalo, J. & Pitarakis, J.Y. (2006) Threshold effects in cointegrating regressions. Oxford Bulletin of Economics and Statistics 68, 813–833.CrossRefGoogle Scholar
Gonzalo, J. & Pitarakis, J.Y. (2012) Regime specific predictability in predictive regressions. Journal of Business and Economic Statistics 30(2), 229–241.Google Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (1997) Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics 2, 1–14.Google Scholar
Hansen, B.E. (2000) Sample splitting and threshold estimation. Econometrica 68, 575–603.Google Scholar
Li, D. & Ling, S.Q. (2012) On the least squares estimation of multiple-regime threshold autoregressive models. Journal of Econometrics 167, 240–253.CrossRefGoogle Scholar
Marmer, V. (2008) Nonlinearity, nonstationarity, and spurious forecasts. Journal of Econometrics 142(1), 1–27.Google Scholar
Park, J.Y. (1992) Canonical cointegrating regressions. Econometrica 60, 119–143.CrossRefGoogle Scholar
Park, J.Y. & Hahn, S. (1999) Cointegration regression with time varying coefficients. Econometric Theory 15, 664–703.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69(1), 117–161.CrossRefGoogle Scholar
Perron, P. & Yabu, T. (2009) Testing for shifts in trend with an integrated or stationary noise component. Journal of Business and Economics Statistics 27(3), 369–396.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 474–95.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.CrossRefGoogle Scholar
Phillips, P.C.B. & Hodgson, D.J. (1994) Spurious regression and generalized least squares. Econometric Theory 10, 957–958.CrossRefGoogle Scholar
Phillips, P.C.B. & Lee, J.H. (2013) Predictive regression under varying degrees of persistence and robust long-horizon regression. Journal of Econometrics 177(2), 250–264.CrossRefGoogle Scholar
Phillips, P.C.B. & Park, J.Y. (1988) Asymptotic equivalence of ordinary least squares and generalized least squares in regressions with integrated regressors. Journal of the American Statistical Association 83, 111–115.CrossRefGoogle Scholar
Potter, S.M. (1995) A nonlinear approach to US GNP. Journal of Applied Econometrics 10, 109–125.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–20.Google Scholar
Seo, M. & Linton, O. (2007) A smoothed least squares estimator for threshold regression models. Journal of Econometrics 141, 704–735.CrossRefGoogle Scholar
Shao, Q. & Lu, C. (1987) Strong approximation for partial sums of weakly dependent random variables. Scientia Sinica 15, 576–587.Google Scholar
Shi, X. & Phillips, P.C.B. (2012) Nonlinear cointegration regression under weak identification. Econometric Theory 28(1), 1–39.CrossRefGoogle Scholar
Wang, Q. & Phillips, P.C.B. (2009) Structural nonparametric cointegrating regression. Econometrica 77(6), 1901–1948.Google Scholar
Yu, P. (2012) Likelihood estimation and inference in threshold regression. Journal of Econometrics 167, 274–294.Google Scholar