Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T23:57:27.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Least Absolute Deviation Estimation of a Shift

Published online by Cambridge University Press:  11 February 2009

Jushan Bai
Jushan Bai
Affiliation:
Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper develops the asymptotic theory for least absolute deviation estimation of a shift in linear regressions. Rates of convergence and asymptotic distributions for the estimated regression parameters and the estimated shift point are derived. The asymptotic theory is developed both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. Asymptotic distributions are also obtained for trending regressors and for dependent disturbances. The analysis is carried out in the framework of partial structural change, allowing some parameters not to be influenced by the shift. Efficiency relative to least-squares estimation is also discussed. Monte Carlo analysis is performed to assess how informative the asymptotic distributions are.

Type
Articles
Information
Econometric Theory , Volume 11 , Issue 3 , June 1995 , pp. 403 - 436
Copyright
Copyright © Cambridge University Press 1995

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.)

References

REFERENCES

Amemiya, T. (1982) Two stage least absolute deviations estimators. Econometrica 50, 689–711.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1992) Optimal Tests of Parameter Constancy. Manuscript, Cowles Foundation for Research in Economics, Yale University.Google Scholar
Babu, G.J. (1989) Strong representations for LAD estimators in linear models. Probability Theory and Related Fields 83, 547–558.CrossRefGoogle Scholar
Bai, J. (1993) LAD Estimation of a Shift. Manuscript, Department of Economics, Massachusetts Institute of Technology.Google Scholar
Bai, J. (1994a) Estimation of Structural Change Based on Wald Type Statistics. Working paper 94–6, Department of Economics, Massachusetts Institute of Technology.Google Scholar
Bai, J. (1994b) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15, 453–472.CrossRefGoogle Scholar
Banerjee, A., Lumsdaine, R.L., & Stock, J.H. (1992) Recursive and sequential tests of the unit root and trend break hypothesis, theory and international evidence. Journal of Business & Economic Statistics 10, 271–287.Google Scholar
Bassett, G. & Koenker, K. (1978) Asymptotic theory of least absolute error regression. Journal of the American Statistical Association 73, 618–622.CrossRefGoogle Scholar
Bhattacharya, P.K. (1987) Maximum likelihood estimation of a change–point in the distribution of independent random variables, general multiparameter case. Journal of Multivariate Analysis 23, 183–208.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.Google Scholar
Birnbaum, Z.W. & Marshall, A.W. (1961) Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Annals of Mathematical Statistics 32, 687–703.CrossRefGoogle Scholar
Bloomfield, P. & Steiger, W. (1983) Least Absolute Deviations, Theory, Applications, and Algorithms. Boston: Birkhauser.Google Scholar
Bradley, R.C. (1986) Basic properties of strong mixing conditions. In Eberlein, E. and Taqque, M.S. (eds.), Dependence in Probability and Statistics, pp. 165–192. Boston: Birkhauser.CrossRefGoogle Scholar
Broemeling, L.D. & Tsurumi, H. (1987) Econometrics and Structural Change. New York: Marcel Dekker.Google Scholar
Buchinsky, M. (1994) Changes in the U.S. wage structure 1963–1987, application of quantile regression. Econometrica 62, 405–458.CrossRefGoogle Scholar
Carlstein, E. (1988) Nonparametric change point estimation. Annals of Statistics 16, 188–197.CrossRefGoogle Scholar
Chamberlain, G. (1991) Quantile Regression, Censoring, and Structure of Wage. Discussion paper 1558, Harvard Institute of Economic Research.Google Scholar
Christiano, L.J. (1992) Searching for a break in GNP. Journal of Business & Economic Statistics 10, 237–250.Google Scholar
Chu, C.-S.J. & White, H. (1992) A direct test for changing trend. Journal of Business & Economic Statistics 10, 289–300.Google Scholar
Duembgen, L. (1991) The asymptotic behavior of some nonparametric change point estimators. Annals of Statistics 19, 1471–1495.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications. New York: Wiley.Google Scholar
Gastwirth, J.L. & Rubin, H. (1975) The behavior of robust estimators on dependent data. Annals of Statistics 3, 1070–1100.CrossRefGoogle Scholar
Gyorfi, L., Hardle, W., Sarda, P., & Vieu, P. (1990) Nonparametric Curve Estimation from Time Series. New York: Springer-Verlag.Google Scholar
Hackl, P. & Westlund, A.H. (1989) Statistical analysis of “structural change”, an annotated bibliography. In Kramer, W. (ed.), Econometrics of Structural Change, pp. 103–128. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
Hahn, J. (1992) Bootstrapping Quantile Regression Models. Manuscript, Department of Economics, Harvard University.Google Scholar
Hampel, F.R. (1974) The influence curve and its role in robust estimation. Journal of the American Statistical Association 62, 1179–1186.Google Scholar
Hansen, B.E. (1992) Tests for parameter instability in regressions with I(1) processes. Journal of Business & Economic Statistics 10, 321–335.Google Scholar
Hawkins, D.L. (1986) A simple least square method for estimating a change in mean. Communications in Statistics–Simulations 15, 655–679.Google Scholar
Hawkins, D.M. (1977) Testing a sequence of observations for a shift in location. Journal of the American Statistical Association 72, 180–186.Google Scholar
Hinkley, D. (1970) Inference about the change point in a sequence of random variables. Biometrika 57, 1–17.CrossRefGoogle Scholar
Honore, B.E. (1992) Trimmed LAD and least squares estimation of truncated and censored regression models with fixed effects. Econometrica 60, 533–565.CrossRefGoogle Scholar
Hsu, D.A. (1982a) A Bayesian robust detection of shift in the risk structure of stock market returns. Journal of the American Statistical Association 77, 29–39.CrossRefGoogle Scholar
Hsu, D.A. (1982b) Robust inferences for structural shift in regression models. Journal of Econometrics 19, 89–107.CrossRefGoogle Scholar
Huber, P.J. (1981) Robust Statistics. New York: Wiley.Google Scholar
Ibragimov, I.A. & Linnik, Y.A. (1971) Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.Google Scholar
James, B., James, K., & Siegmund, D. (1987) Tests for a change point. Biometrika 74, 71–83.Google Scholar
Kim, J. & Pollard, D. (1990) Cube root asymptotics. Annals of Statistics 18, 191–219.CrossRefGoogle Scholar
Kim, H.J. & Siegmund, D. (1989) The likelihood ratio test for a change point in simple linear regression. Biometrika 76, 409–423.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. & Bassett, G. (1982) Robust tests for heteroskedasticity based on regression quantiles. Econometrica 56, 43–61.Google Scholar
Kramer, W., Ploberger, W., & Alt, R. (1988) Testing for structural changes in dynamic models. Econometrica 56, 1355–1370.CrossRefGoogle Scholar
Krishnaiah, P.R. & Miao, B.Q. (1988) Review about estimation of change points. In Krishnaiah, P.R. & Rao, C.R. (eds.), Handbook of Statistics, vol. 7, pp. 375–402. New York: Elsevier.Google Scholar
Mokkadem, A. (1988) Mixing properties of ARMA processes. Stochastic Processes and Their Applications 29, 309–315.CrossRefGoogle Scholar
Newey, W.K. & Powell, J.L. (1990) Efficient estimation of linear and type I censored regression models under conditional quantile restrictions. Econometric Theory 6, 295–317.Google Scholar
Newey, W.K. & West, K. (1987) A simple, positive definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708.Google Scholar
Perron, P. (1993) Trend, Unit Root and Structural Change in Macroeconomic Time Series. Manuscript, C.R.E.D., University of Montreal.Google Scholar
Perron, P. & Vogelsang, T. (1992) Nonstationarity and level shifts with an application to purchasing power parity. Journal of Business & Economic Statistics 10, 321–335.Google Scholar
Pham, T.D. & Tran, L.T. (1985) Some mixing properties of time series models. Stochastic Processes and Their Applications 19, 297–303.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) A shortcut to LAD estimator asymptotics. Econometric Theory 7, 450–463.CrossRefGoogle Scholar
Picard, D. (1985) Testing and estimating change-points in time series. Advances in Applied Probability 17, 841–867.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
Pollard, D. (1990) Empirical Processes: Theory and Applications, vol. 2. CBMS Conference Series in Probability and Statistics. Hayward, CA: Institute of Mathematical Statistics.CrossRefGoogle Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.CrossRefGoogle Scholar
Powell, J.L. (1984) Least absolute deviations estimation for the censored regression model. Journal of Econometrics 25, 303–325.Google Scholar
Quandt, R.E. (1958) The estimation of parameters of a linear regression system obeying two separate regimes. Journal of the American Statistical Association 53, 873–880.CrossRefGoogle Scholar
Rosenblatt, M. (1956) A central limit theory and a strong mixing condition. Proceedings of the National Academy of Sciences USA 42, 43–47.CrossRefGoogle Scholar
Sen, A. & Srivastava, M.S. (1975a) On tests for detecting change in mean. Annals of Statistics 3, 96–103.Google Scholar
Sen, A. & Srivastava, M.S. (1975b) Some one-sided tests for change in level. Technometrics 17, 61–64.CrossRefGoogle Scholar
Serfling, R.J. (1970) Moment inequalities for the maximum cumulative sum. Annals of Mathematical Statistics 41, 1227–1234.Google Scholar
Weiss, A. (1991) Estimating nonlinear dynamic models using least absolute error estimation. Econometric Theory 7, 46–68.Google Scholar
Withers, C.S. (1981) Conditions for linear processes to be strong–mixing. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57, 477–480.CrossRefGoogle Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems for dependent and heterogeneous processes. Econometric Theory 4, 210–230.CrossRefGoogle Scholar
Worsley, K.J. (1979) On the likelihood ratio test for a shift in locations of normal populations. Journal of the American Statistical Association 72, 180–186.Google Scholar
Worsley, K.J. (1986) Confidence regions and tests for a change–point in a sequence of exponential family random variables. Biometrika 73, 91–104.CrossRefGoogle Scholar
Yao, Yi-Ching (1987) Approximating the distribution of the ML estimate of the change–point in a sequence in independent r.v.'s. Annals of Statistics 15, 1321–1328.Google Scholar
Zacks, S. (1983) Survey of classical and Bayesian approaches to the change-point problem, fixed sample and sequential procedures of testing and estimation. In Rivzi, M.H., Rustagi, J.S., & Siegmund, D. (eds.), Recent Advances in Statistics, pp. 245–269. New York: Academic Press.CrossRefGoogle Scholar
Zivot, E. & Phillips, P.C.B. (1994) A Bayesian analysis of trend determination in economics time series. Econometric Reviews 13, 291–336.Google Scholar