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ASYMPTOTIC THEORY FOR KERNEL ESTIMATORS UNDER MODERATE DEVIATIONS FROM A UNIT ROOT, WITH AN APPLICATION TO THE ASYMPTOTIC SIZE OF NONPARAMETRIC TESTS

Published online by Cambridge University Press:  21 October 2019

James A. Duffy
James A. Duffy*
Affiliation:
University of Oxford
*
Address correspondence to James A. Duffy, Corpus Christi College and Department of Economics, University of Oxford, Oxford, UK; e-mail: james.duffy@economics.ox.ac.uk.
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Abstract

We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered only nearly integrated and stationary autoregressive processes. These results have applications to nonparametric predictive regression models. In particular, we show that the null rejection probability of a nonparametric t test is controlled uniformly in the degree of persistence of the regressor. This provides a rigorous justification for the validity of the usual nonparametric inferential procedures, even in cases where regressors may be highly persistent.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 4 , August 2020 , pp. 559 - 582
Copyright
© Cambridge University Press 2019

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Footnotes

I thank V. Berenguer-Rico, S. Mavroeidis, B. Nielsen, and participants at seminars at Cambridge, Princeton, Vienna and UCL for comments on an earlier version of this article. The manuscript was prepared with LYX 2.3.1 and JabRef 3.8.2.

References

REFERENCES

Abadir, K.M., Distaso, W., Giraitis, L., & Koul, H.L. (2014) Asymptotic normality for weighted sums of linear processes. Econometric Theory 30, 252–284.CrossRefGoogle Scholar
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., Cheng, X., & Guggenberger, P. (2011) Generic Results for Establishing the Asymptotic Size of Confidence Sets and Tests. Cowles Foundation Discussion paper 1813, Yale University.CrossRefGoogle Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81, 27–60.CrossRefGoogle Scholar
Cavanagh, C.L., Elliott, G., & Stock, J.H. (1995) Inference in models with nearly integrated regressors. Econometric Theory 11, 1131–1147.CrossRefGoogle Scholar
Elliott, G., Müller, U.K., & Watson, M.W. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83, 771–811.CrossRefGoogle Scholar
Giraitis, L. & Phillips, P.C.B. (2006) Uniform limit theory for stationary autoregression. Journal of Time Series Analysis 27, 51–60.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Jansson, M. & Moreira, M.J. (2006) Optimal inference in regression models with nearly integrated regressors. Econometrica 74, 681–714.CrossRefGoogle Scholar
Jeganathan, P. (2004) Convergence of functionals of sums of r.v.s to local times of fractional stable motions. Annals of Probability 32, 1771–1795.CrossRefGoogle Scholar
Kasparis, I., Andreou, E., & Phillips, P.C.B. (2015) Nonparametric predictive regression. Journal of Econometrics 185, 468–494.CrossRefGoogle Scholar
Lehmann, E.L. & Romano, J.P. (2005) Testing Statistical Hypotheses, 3rd ed.Springer.Google Scholar
Magdalinos, T. & Phillips, P.C.B. (2009) Econometric Inference in the Vicinity of Unity. CoFie Working paper 7, Singapore.Google Scholar
Mikusheva, A. (2007) Uniform inference in autoregressive models. Econometrica 75, 1411–1452.CrossRefGoogle Scholar
Phillips, P.C.B. & Lee, J.H. (2013) Predictive regression under various degrees of persistence and robust long-horizon regression. Journal of Econometrics 177, 250–264.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Romano, J.P. (2004) On nonparametric testing, the uniform behaviour of the t-test, and related problems. Scandinavian Journal of Statistics 31, 567–584.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 nonlinear cointegrating regression. Econometric Theory 30, 509–535.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 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.CrossRefGoogle Scholar
Wu, W.B., Huang, Y., & Huang, Y. (2010) Kernel estimation for time series: An asymptotic theory. Stochastic Processes and their Applications 120, 2412–2431.CrossRefGoogle Scholar
Wu, W.B. & Mielniczuk, J. (2002) Kernel density estimation for linear processes. Annals of Statistics 30, 1441–1459.Google Scholar
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