Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T11:41:40.536Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDEDNESS OF M-ESTIMATORS FOR LINEAR REGRESSION IN TIME SERIES

Published online by Cambridge University Press:  04 September 2018

Søren Johansen  and
Bent Nielsen
Søren Johansen
Affiliation:
University of Copenhagen, and CREATES Aarhus University
Bent Nielsen*
Affiliation:
University of Oxford
*
*Address correspondence to Bent Nielsen, Nuffield College & Department of Economics, University of Oxford & Programme for Economic Modelling Nuffield College, Oxford, OX1 1NF, UK; e-mail: bent.nielsen@nuffield.ox.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

We show boundedness in probability uniformly in sample size of a general M-estimator for multiple linear regression in time series. The positive criterion function for the M-estimator is assumed lower semicontinuous and sufficiently large for large argument. Particular cases are the Huber-skip and quantile regression. Boundedness requires an assumption on the frequency of small regressors. We show that this is satisfied for a variety of deterministic and stochastic regressors, including stationary and random walks regressors. The results are obtained using a detailed analysis of the condition on the regressors combined with some recent martingale results.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 3 , June 2019 , pp. 653 - 683
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author is grateful to CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation, and to Steffen Lauritzen for useful comments.

References

REFERENCES

Abadir, K.M. & Lucas, A. (2000) Quantiles for t-statistics based on M-estimators and unit roots. Economic Letters 67, 131–137.CrossRefGoogle Scholar
Bercu, B. & Touati, A. (2008) Exponential inequalities for self-normalized martingales with applications. Annals of Applied Probability 18, 1848–1869.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Chen, X.R. & Wu, Y.H. (1988) Strong consistency of M-estimates in linear models. Journal of Multivariate Analysis 27, 116–130.CrossRefGoogle Scholar
Chow, Y.S. (1965) Local convergence of martingales and the law of large numbers. Annals of Mathematical Statistics 36, 552–558.CrossRefGoogle Scholar
Čižek, P. (2005) Least trimmed squares in nonlinear regression under dependence. Journal of Statistical Planning and Inference 136, 3967–3988.CrossRefGoogle Scholar
Davies, L. (1990) The asymptotics of S-estimators in the linear regression model. Annals of Statistics 18, 1651–1675.CrossRefGoogle Scholar
Fasano, M.V., Maronna, R.A., Sued, M., & Yohai, V.J. (2012) Continuity and differentiability of regression M functionals. Bernoulli 18, 1284–1309.CrossRefGoogle Scholar
Huber, P.J. (1964) Robust estimation of a location parameter. Annals of Mathematical Statistics 35, 73–101.CrossRefGoogle Scholar
Huber, P.J. & Ronchetti, E.M. (2009) Robust Statistics. Wiley.CrossRefGoogle Scholar
Jennrich, R.I. (1969) Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40, 633–643.CrossRefGoogle Scholar
Johansen, S. & Nielsen, B. (2016) Analysis of the forward search using some new results for martingales and empirical processes. Bernoulli 22, 1131–1183.CrossRefGoogle Scholar
Jurečková, J., Sen, P.K., & Picek, J. (2012) Methodological Tools in Robust and Nonparametric Statistics. Chapman & Hall/CRC Press.CrossRefGoogle Scholar
Knight, K. (1989) Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261–278.CrossRefGoogle Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Koenker, R. & Xiao, Z. (2004) Unit root quantile autoregression inference. Journal of the American Statistical Association 99, 775–787.CrossRefGoogle Scholar
Lai, T.L. & Wei, C.Z. (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Annals of Statistics 10, 154–166.CrossRefGoogle Scholar
Liese, F. & Vajda, I. (1994) Consistency of M-estimates in general regression models. Journal of Multivariate Analysis 50, 93–114.CrossRefGoogle Scholar
Lucas, A. (1995) Unit root tests based on M estimators. Econometric Theory 11, 331–346.CrossRefGoogle Scholar
Maronna, R.A., Martin, D.R., & Yohai, V.J. (2006) Robust Statistics: Theory and Methods. Wiley.CrossRefGoogle Scholar
Víšek, J.Á. (2006) The least trimmed squares. Part I: Consistency. Kybernetika 42, 1–36.Google Scholar