Share / Export Citation / Email / Print / Text size:

Statistics in Transition New Series

Polish Statistical Association

Central Statistical Office of Poland

Subject: Economics, Statistics & Probability


ISSN: 1234-7655
eISSN: 2450-0291





Volume / Issue / page

Related articles

VOLUME 19 , ISSUE 3 (September 2018) > List of articles


Katarzyna Jabłońska

Keywords : heteroskedasticity, linear regression, HC-estimators, quality of life

Citation Information : Statistics in Transition New Series. Volume 19, Issue 3, Pages 423-452, DOI:

License : (BY-NC-ND 4.0)

Published Online: 13-December-2018



Using the estimation method of ordinary least squares leads to unreliable results in the case of heteroskedastic linear regression model. Other estimation methods are described, including weighted least squares, division of the sample and heteroskedasticity-consistent covariance matrix estimators, all of which can give estimators with better properties than ordinary least squares. The methods are presented giving the example of modelling quality of life of older people, based on a data set from the first wave of the COURAGE – Poland study. The comparison of estimators and their practical application may teach how to choose methodologically the most appropriate estimation tool after detection of heteroscedasticity. 

Content not available PDF Share



BEAL, C., (2006). Loneliness in older women: a review of the literature. Issues in Mental Health Nursing, 27 (7), pp. 795–813. 
BOX, G.E.P., COX, D. R., (1964). An Analysis of Transformations. Journal of the Royal Statistical Society, Series B (26), pp. 211–252. 
BREUSCH, T. S., PAGAN, A. R., (1979). A Simple Test for Heteroscedasticity and Random Coefficient Variation, Econometrica, 47 (5), pp. 1287–1294. 
CABALLERO, F. F., MIRET, M., POWER, M., CHATTERJI, S., TOBIASZADAMCZYK, B., KOSKINEN, S., AYUSO-MATEOS, J. L., (2013). Validation of an instrument to evaluate quality of life in the aging population: WHOQOLAGE. Health Qual Life Outcomes, 11 (177). 
CARROLL, R. J., RUPPERT, D., (1984). Power Transformations When Fitting Theoretical Models to Data. Journal of the American Statistical Association, 79 (386). 
CHESHER, A., JEWITT, I., (1987). The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica, 55 (5), pp. 1217–1222. 
CRAGG, J. G., (1983). More efficient estimation in the presence of heteroscedasticity of unknown form. Econometrica, 51 (3), pp. 751–763. 
CRIBARI-NETO, F., (2004). Asymptotic inference under heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 45, pp. 215–233. 
CRIBARI-NETO, F., DA SILVA, W. B., (2011). A new heteroskedasticityconsistent covariance matrix estimator for the linear regression model. Advances in Statistical Analysis, 95, pp. 129–146. 
CRIBARI-NETO, F., FERRARI, S. L. P., CORDEIRO, H. M., (2000). Improved heteroscedasticity-consistent covariance matrix estimators. Biometrika, 87 (4), pp. 907–918. 
CRIBARI–NETO, F., SOUZA, T. C., VASCONCELLOS, K. L. P., (2007). Inference Under Heteroskedasticity and Leveraged Data. Communications in Statistics—Theory and Methods, 36, pp. 1877–1888. 
CRIBARI-NETO, F., ZARKOS, S. P., (2001). Heteroskedasticity-consistent covariance matrix estimation: White’s estimator and the bootstrap. Journal of Statistical Computation and Simulation, 68, pp. 391–411. 
DALGARD, O. S., (1996). Community health profile: a tool for psychiatric prevention, in Promotion of Mental Health. D. R. Trent, C.A. Reed (eds). Aldershot: Avebury Press, pp. 681–695. 
DAVIDIAN, M., CARROLL, R. J., (1987). Variance Function Estimation. Journal of the American Statistical Association. 82 (400), pp. 1079-1091. 

DODGE, Y., (2008). Gauss-Markov Theorem, in The Concise Encyclopedia of Statistics. Springer New York, pp. 217–218. 
EFRON, B., (1982). The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia. 
GOLDFELD, S. M., QUANDT, R. E., (1965). Some Tests for Homoscedasticity. Journal of the American Statistical Association, 6 (310), pp. 539–547. 
GREENE, W. H., (2012). Econometric Analysis (7th Edition). Pearson Education Limited. 
HAYES, A. F. CAI, L., (2007). Using heteroskedasticity-consistent standard error estimators in OLS regression: An introduction and software implementation. Behavior Research Methods, 39 (4), pp. 709–722. 
HAYES, A. F., CAI, L., (2008). A New Test of Linear Hypotheses in OLS Regression Under Heteroscedasticity of Unknown Form. Journal of Educational and Behavioral Statistics, 33 (1), pp. 21–40. 
HINKLEY, D. V., (1977). Jackknifing In Unbalanced Situations. Technometrics, 19 (3). 
HUGHES, M. E., WAITE, L. J., HAWKLEY, L. C., CACIOPPO, J. T., (2004). A Short Scale for Measuring Loneliness in Large Surveys: Results From Two Population-Based Studies. Research on aging, 26 (6), pp. 655–672. 
KAUERMAN, G., CARROLL, R. J., (2001). A note on the efficiency of sandwich covariance matrix estimation. Journal of the American Statistical Association, 96, pp. 1387–1396. 
KOJIMA, M., KOJIMA, T., ISHIGURO, N., OGUCHI, T., OBA, M., TSUCHIYA, H., TOKUDOME, S., (2009). Psychosocial factors, disease status, and quality of life in patients with rheumatoid arthritis. Journal of Psychosomatic Research, 67, pp. 425–431. 
KIVIET, J.F., FENG, Q., (2015). Efficiency Gains by Modifying GMM Estimation in Linear Models under Heteroskedasticity. Econometrics Working Papers, 14 (06). 
LI, S., ZHANG, N., ZHANG, X., WANG, G., (2017). A new heteroskedasticityconsistent covariance matrix estimator and inference under heteroscedasticity. Journal Of Statistical Computation And Simulation, 87 (1) pp. 198–210. 
LONG, J. S., ERVIN, L.H., (2000). Using Heteroscedasticity Consistent Standard Errors in the Linear Regression Model. The American Statistician, 54, pp. 217–224. 
MACKINNON, J. G., WHITE, H., (1985). Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. Queen’s Economics Department Working Paper, 537. 

MILLER, S., STARTZ, R., (2017). Feasible Generalized Least Squares Using Machine Learning. Social Science Research Network [online] (6th February 2018). Available at: [Accessed on 27 March 2018]. 
MUSICH, S., WANG, S. S., HAWKINS, K., YEH, C. S., (2015). The Impact of Loneliness on Quality of Life and Patient Satisfaction Among Older. Sicker Adults. Gerontology & Geriatric Medicine, 1. 
SAKIA, R. M., (1992). The Box-Cox transformation technique: a review. The Statistician, 41, pp. 169–178. 
SMYTH, G. K., (2002). An Efficient Algorithm for REML in Heteroscedastic Regression. Journal of Graphical and Computational Statistics, 11, pp. 836– 847. 
STARTZ, R., (2017). Bayesian Heteroskedasticity-Robust Regression, 2015. The Economics Department of UCSB, [online] (Revised February 2015). Available  
 at: [Accessed on 27 March 2018]. 
THOMOPOULOU, I., THOMOPOULOU, D., KOUTSOUKI, D., (2010). The differences at quality of life and loneliness between elderly people. Biology of Exercise, 6 (2), pp. 13–28. 
TOBIASZ-ADAMCZYK, B., GALAS, A., ZAWISZA, K., CHATTERJI, S., HARO, J. M., AYUSO-MATEOS, J. L., LEONARDI, M., (2017). Gender-related differences in the multi-pathway effect of social determinants on quality of life in older age—the COURAGE in Europe project. Quality of Life Research, 26 (7), pp. 1865–1878. 
VERBEEK, M., (2004). A Guide to Modern Econometrics (2nd edition), John Wiley & Sons Ltd. 
VILHENA, E., PAIS-RIBEIRO, J., SILVA, I., PEDRO, L., MENESES, R., CARDOSO, H., MENDONÇA, D., (2014). Psychosocial factors as predictors of quality of life in chronic Portuguese patients. Health and Quality of Life Outcomes, 12 (3). 
WAGENER, J., DETTE, H., (2012). Bridge Estimators and the Adaptive Lasso under Heteroscedasticity. Mathematical Methods of Statistics, 21 (2),  pp. 109–126. 
WHITE, H., (1980). A heteroskedasticity consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica, 48 (4), pp. 817–838. 
WIECZOROWSKA-TOBIS, K., TALARSKA, D., (2010). Geriatria i pielęgniarstwo geriatryczne [Geriatrics and geriatric nursing]. Wydawnictwo Lekarskie PZWL, Warszawa. 

ZAWISZA, K., GAŁAŚ, A., TOBIASZ-ADAMCZYK, B., (2014). Polska wersja Courage Social Network Index — skali do oceny poziomu sieci społecznych [Polish version of The Courage Social Network Index — the scale for the assessment of social networks]. Gerontologia Polska, 22 (1), pp. 31–41. 
ZAWISZA, K., GAŁAŚ, A., TOBIASZ-ADAMCZYK, B., (2016). Walidacja polskiej wersji skali oceny jakości życia WHOQOL-AGE w populacji osób starszych [Validation of the Polish version of the WHOQOL-AGE scale in older population]. Gerontologia Polska, 1, pp. 7–16.