Least absolute deviations estimation for the censored regression model. (English) Zbl 0571.62100

Powell, James L.

J. Econom. 25, 303-325 (1984).

This paper deals with the model \(Y_ t=\max (0\), \(x'_ t\beta_ 0+u_ t)\), \(t=1,...,T\). \(u_ t\) are unobservable disturbance terms with median zero and continuous distribution, mutually independent. Then \(\max(0,x'_ t\beta_ 0)\) is the median of \(y_ t\), given the exogenous variables \(x_ t\). \(\beta_ 0\) is estimated by minimizing \(\sum^{T}_{t=1}| y_ t-\max (0,x'_ t\beta)|\) over a compact set B.
The author formulates conditions on \(\{x_ t\}\) and \(\{u_ t\}\) in order that the estimators \({\hat \beta}_ T\) of \(\beta\) converge to \(\beta_ 0\) almost surely as \(T\to \infty\). Also conditions are formulated that \(\sqrt{T}({\hat\beta}_ T-\beta_ 0)\) is asymptotically normally distributed. The asymptotic covariance matrix is determined and estimated consistently.

Reviewer: H.Drygas

MSC:

62P20	Applications of statistics to economics
62F12	Asymptotic properties of parametric estimators
62J99	Linear inference, regression

Keywords:

least absolute deviations estimation; censored regression model; restricted regressors; asymptotic normality; Tobit model; robust to heteroscedasticity; consistent estimators; median; asymptotic covariance matrix

References:

[1]	Amemiya, T., Regression analysis when the dependent variable is truncated normal, Econometrica, 41, 997-1016 (1973) · Zbl 0282.62061
[2]	Amemiya, T., Two stage least absolute deviations estimators, Econometrica, 50, 689-711 (1982) · Zbl 0493.62098
[3]	Arabmazar, A.; Schmidt, P., Further evidence on the robustness of the Tobit estimator to heteroskedasticity, Journal of Econometrics, 17, 253-258 (1981)
[4]	Arabmazar, A.; Schmidt, P., An investigation of the robustness of the Tobit estimator to non-normality, Econometrica, 50, 1055-1063 (1982) · Zbl 0497.62036
[5]	Bassett, G.; Koenker, R., Asymptotic theory of least absolute error regression, Journal of the American Statistical Association, 73, 667-677 (1978)
[6]	Bickel, P. J., One-step Huber estimation in the linear model, Journal of the American Statistical Association, 70, 428-433 (1975) · Zbl 0322.62038
[7]	Buckley, J.; James, I., Linear regression with censored data, Biometrika, 66, 429-436 (1979) · Zbl 0425.62051
[8]	Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society B, 39, 1-22 (1977) · Zbl 0364.62022
[9]	Goldberger, A. S., Abnormal selection bias, (Workshop paper no. 8006 (1980), Social Systems Research Institute, University of Wisconsin: Social Systems Research Institute, University of Wisconsin Madison, WI) · Zbl 0568.62099
[10]	Hausman, J. A., Specification tests in econometrics, Econometrica, 46, 1251-1271 (1978) · Zbl 0397.62043
[11]	Heckman, J., The common structure of statistical models of truncation, sample selection, and limited dependent variables and a simple estimator for such models, Annals of Economic and Social Measurement, 5, 475-492 (1976)
[12]	Heckman, J., Sample bias as a specification error, Econometrica, 47, 153-162 (1979) · Zbl 0392.62093
[13]	Himmelblau, D. M., (Applied nonlinear programming (1972), McGraw-Hill: McGraw-Hill New York)
[14]	Huber, P. J., Robust estimation of a location parameter, Annals of Mathematical Statistics, 35, 73-101 (1964) · Zbl 0136.39805
[15]	Huber, P. J., The behavior of maximum likelihood estimates under nonstandard conditions, Proceedings of the Fifth Berkeley Symposium, 1, 221-233 (1965)
[16]	Jurečková, J., Asymptotic relations of \(M\)-estimates and \(R\)-estimates in the linear model, Annals of Statistics, 5, 464-472 (1977) · Zbl 0365.62034
[17]	Kalbfleisch, J. O.; Prentice, R. L., (The statistical analysis of failure time data (1980), Wiley: Wiley New York) · Zbl 0504.62096
[18]	Kaplan, E. L.; Meier, P., Nonparametric estimation from incomplete observations, Journal of the American Statistical Association, 53, 457-481 (1958) · Zbl 0089.14801
[19]	Koenker, R.; Bassett, G., Robust tests for heteroscedasticity based on regression quantiles, Econometrica, 50, 43-61 (1982) · Zbl 0482.62023
[20]	Koenker, R.; Bassett, G., Tests of linear hypotheses and \(L_1\) estimation, Econometrica, 50, 1577-1583 (1982) · Zbl 0497.62057
[21]	Maddala, G. S.; Nelson, F. D., Specification errors in limited dependent variable models, National Bureau for Economic Research working paper no. 96 (1975)
[22]	Nelder, J. A.; Mead, R., A simplex method for function minimization, The Computer Journal, 7, 308-313 (1965) · Zbl 0229.65053
[23]	Obenhofer, W., The consistency of nonlinear regression minimizing the \(L_1\) norm, Annals of Statistics, 10, 316-319 (1982)
[24]	Parzen, E., On estimation of a probability density function and its mode, Annals of Mathematical Statistics, 33, 1065-1076 (1962) · Zbl 0116.11302
[25]	Powell, J. L., The asymptotic normality of two-stage least absolute deviations estimators, Econometrica, 51, 1569-1575 (1983) · Zbl 0537.62095
[26]	Powell, M. J.D, An efficient method for finding the minimum of a function of several variables without calculating derivatives, The Computer Journal, 7, 155-162 (1964) · Zbl 0132.11702
[27]	Robinson, P. M., On the asymptotic properties of estimators of models containing limited dependent variables, Econometrica, 50, 27-41 (1982) · Zbl 0524.62117
[28]	Ruppert, D.; Carroll, R. J., Trimmed least squares estimation in the linear model, Journal of the American Statistical Association, 75, 828-838 (1980) · Zbl 0459.62055
[29]	Taylor, L. D., Estimation by minimizing the sum of absolute errors, (Zarembka, P., Frontiers in econometrics (1974), Academic Press: Academic Press New York)
[30]	Tobin, J., Estimation of relationships for limited dependent variables, Econometrica, 26, 24-36 (1958) · Zbl 0088.36607
[31]	White, H., Nonlinear regression on cross-section data, Econometrica, 48, 721-746 (1980) · Zbl 0442.62050
[32]	White, H., A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48, 817-838 (1980) · Zbl 0459.62051