Abstract
This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions on the design matrix and some moment conditions on the error distributions. It is shown that most of the error distributions encountered in practice satisfy these moment conditions. Some examples of the asymptotic covariance matrices are also given.
Similar content being viewed by others
References
Carroll, R. J. (1982). Adapting for heteroscedasticity in linear models, Ann. Statist., 10, 1224–1233.
Fahrmeir, L. and Kaufmann, H. (1985). Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models, Ann. Statist., 13, 342–368.
Fuller, W. A. and Rao, J. N. K. (1978). Estimation for a linear regression model with unknown diagonal covariance matrix, Ann. Statist., 6, 1149–1158.
Gourieroux, C. and Monfort, A. (1981). Asymptotic properties of the maximum likelihood estimator in dichotomous logit models, J. Econometrics, 17, 83–97.
Hartley, H. O., Rao, J. N. K. and Kiefer, G. (1969). Variance estimation with one unit per stratum, J. Amer. Statist. Assoc., 64, 841–851.
Horn, S. D., Horn, R. A. and Duncan, D. B. (1975). Estimating heteroscedastic variances in linear models, J. Amer. Statist. Assoc., 70, 380–385.
Rao, C. R. (1970). Estimation of heteroscedastic variances in linear models, J. Amer. Statist. Assoc., 65, 161–172.
Rao, J. N. K. (1973). On the estimation of heteroscedastic variances, Biometrics, 29, 11–24.
Shao, J. (1987). Estimating heteroscedastic variances in linear models I and II, Tech. Report #87–42, 43, Dept. of Statistics, Purdue University, Indiana.
Author information
Authors and Affiliations
About this article
Cite this article
Shao, J. Asymptotic distribution of the weighted least squares estimator. Ann Inst Stat Math 41, 365–382 (1989). https://doi.org/10.1007/BF00049402
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049402