Skip to main content


  • Søren Johansen (a1) and Bent Nielsen (a2)

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.

Corresponding author
*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:
Hide All

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.

Hide All
Abadir, K.M. & Lucas, A. (2000) Quantiles for t-statistics based on M-estimators and unit roots. Economic Letters 67, 131137.
Bercu, B. & Touati, A. (2008) Exponential inequalities for self-normalized martingales with applications. Annals of Applied Probability 18, 18481869.
Billingsley, P. (1968) Convergence of Probability Measures . Wiley.
Chen, X.R. & Wu, Y.H. (1988) Strong consistency of M-estimates in linear models. Journal of Multivariate Analysis 27, 116130.
Chow, Y.S. (1965) Local convergence of martingales and the law of large numbers. Annals of Mathematical Statistics 36, 552558.
Čižek, P. (2005) Least trimmed squares in nonlinear regression under dependence. Journal of Statistical Planning and Inference 136, 39673988.
Davies, L. (1990) The asymptotics of S-estimators in the linear regression model. Annals of Statistics 18, 16511675.
Fasano, M.V., Maronna, R.A., Sued, M., & Yohai, V.J. (2012) Continuity and differentiability of regression M functionals. Bernoulli 18, 12841309.
Huber, P.J. (1964) Robust estimation of a location parameter. Annals of Mathematical Statistics 35, 73101.
Huber, P.J. & Ronchetti, E.M. (2009) Robust Statistics. Wiley.
Jennrich, R.I. (1969) Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40, 633643.
Johansen, S. & Nielsen, B. (2016) Analysis of the forward search using some new results for martingales and empirical processes. Bernoulli 22, 11311183.
Jurečková, J., Sen, P.K., & Picek, J. (2012) Methodological Tools in Robust and Nonparametric Statistics . Chapman & Hall/CRC Press.
Knight, K. (1989) Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261278.
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7, 200212.
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 3350.
Koenker, R. & Xiao, Z. (2004) Unit root quantile autoregression inference. Journal of the American Statistical Association 99, 775787.
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, 154166.
Liese, F. & Vajda, I. (1994) Consistency of M-estimates in general regression models. Journal of Multivariate Analysis 50, 93114.
Lucas, A. (1995) Unit root tests based on M estimators. Econometric Theory 11, 331346.
Maronna, R.A., Martin, D.R., & Yohai, V.J. (2006) Robust Statistics: Theory and Methods. Wiley.
Víšek, J.Á. (2006) The least trimmed squares. Part I: Consistency. Kybernetika 42, 136.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed