Advertisement

# Heteroscedasticity and serial correlation

• G. Barrie Wetherill
• P. Duncombe
• M. Kenward
• J. Köllerström
• S. R. Paul
• B. J. Vowden
Chapter
Part of the Monographs on Statistics and Applied Probability book series (MSAP)

## Abstract

There are many situations occurring in practice when the simple structure of random variation assumed in (1.1) does not hold; examples will be given below. Suppose that instead of (1.1) we have
$$\left. {{}_{V\left( Y \right)\, = \,V{\sigma ^2},}^{E\left( Y \right)\, = \,a\theta ,}} \right\}$$
(9.1)
where V is a known n × n positive definite matrix, then the appropriate method of estimation is generalized least squares (GLS), leading to
$$\hat \theta g = {\left( {a'{V^{ - 1}}a} \right)^{ - 1}}a'{V^{ - 1}}Y.$$
(9.2)
If V = 1, then this reduces to the ordinary least squares (OLS) estimator
$$\hat \theta = a{\left( {a'a} \right)^{ - 1}}a'Y.$$
(9.3)
If (9.3) is used when (9.2) is appropriate, then (9.3) is still unbiased, but it is not efficient, and the OLS estimator of σ 2 will in general be biased. All tests will therefore be invalid, Clearly, further research needs to be carried out on how serious the effects are of using OLS when (9.2) ought to be used, but this argument indicates the desirability of testing for departures from the assumption of V = I.

## Preview

Unable to display preview. Download preview PDF.

## References

1. Anscombe, J. F. (1961) Examination of residuals. Proceedings of the Fourth Berkely Symposium on Mathematical Statistics and Probability, Vol. 1. University of California Press, Berkeley, CA, pp. 1–36.Google Scholar
2. Bickel, P. J. (1978) Using residuals robustly. I: Tests for heteroscedasticity, nonlinearity. Ann. Statist., 6, 266–291.
3. Bradley, E. L. (1973) The equivalence of maximum likelihood and weighted least squares estimates in the exponential family. J. Amer. Statist. Assoc., 68, 199–200.Google Scholar
4. Carroll, R. J. and Ruppert, D. (1981) On robust test for heteroscedasticity. Ann. Statist., 9, 205–209.
5. Chambers, J. M. (1977) Computational Methods for Data Analysis. Wiley, New York.Google Scholar
6. Charnes, A., Frome, E. L. and Yu, P. L. (1976) The equivalence of generalized least squares and maximum likelihood estimates in the exponential family. J. Amer. Statist. Assoc., 71, 169–171.
7. Cook, R. D. and Weisberg, S. (1983) Diagnostics for heteroscedasticity in regression. Biometrika, 70, 1–10.
8. Durbin, J. and Watson, G. S. (1950) Testing for serial correlation in least squares regression, I. Biometrika, 37, 409–428.Google Scholar
9. Durbin, J. and Watson, G. S. (1971) Testing for serial correlation in least squares regression, III. Biometrika, 58, 1–42.Google Scholar
10. Farebrother, R. W. (1980) AS 153 Pan’s procedure for the tail probabilities of the Durbin-Watson statistic. Appl. Statist., 29, 224–227.
11. Glejser, H. (1969) A new test for heteroscedasticity. J. Amer. Statist. Assoc., 64, 316–323.
12. Godfrey, L. G. (1978) Testing for multiplicative heteroscedasticity. J. Econ., 8, 227–236.
13. Goldfeld, S. M. and Quandt, R. E. (1965) Some tests for homoscedasticity. J. Amer. Statist. Assoc., 60, 539–547.
14. Harrison, M. J. and McCabe, B. P. M. (1979) A test for heteroscedasticity based on ordinary least squares residuals. J. Amer. Statist. Assoc., 74, 494–499.Google Scholar
15. Harvey, A. C. (1976) Estimating regression models with multiplicative heteroscedasticity. Econometrica, 44, 461–465.
16. Harvey, A. C. and Phillips, G. D. A. (1974) A comparison of the power of some tests for heteroscedasticity in the general linear model. J. Econ. 2, 307–316.
17. Hedayat, A. and Robson, D. S. (1970) Independent stepwise residuals for testing homoscedasticity. J. Amer. Statist. Assoc., 65, 1573–1581.
18. Holt, D., Smith, T. M. F. and Winter, P. D. (1980) Regression analysis of data from complex surveys. J. Roy. Statist. Soc. A, 143, 474–487.Google Scholar
19. Horn, P. (1981) Heteroscedasticity: a non-parametric alterative to the Goldfeld-Quandt peak test. Commun. Statist. A, 10, 795–808.Google Scholar
20. Judge, G. G., Griffiths, W. E., Hill, R. C. and Lee, Tsoung-Chao (1980) The Theory and Practice of Econometrics. Wiley, New York.Google Scholar
21. Koenker, R. (1981) A note on studentizing a test for heteroscedasticity. J. Econ., 17, 107–112.
22. L’Esperance, W. L. Chall, D. and Taylor, D. (1976) An algorithm for determining the distribution function of the Durbin-Watson statistic. Econometrika, 44, 1325–1346.Google Scholar
23. Nelder, J. A. and Wedderburn, R. W. M. (1972) Generalized linear models. J. Roy. Statist. Soc. A, 135, 370–384.Google Scholar
24. von Neumann, J. (1941) Distribution of the ratio of the mean square successive difference to the variance. Ann. Math. Statist., 12, 367–395.
25. Park, R. E. (1966) Estimation with heteroscedastic error terms. Econometrica, 34, 888.
26. Ramsey, J. B. (1969) Tests for specification errors in classical linear least squares regression analysis. J. Roy. Statist. Soc. B, 31, 350–371.Google Scholar
27. Rutemiller, H. C. and Bowers, D. A. (1968) Estimation in a heteroscedastic regression model. J. Amer. Statist. Assoc., 63, 552–557.
28. Szroeter, J. (1978) A class of parametric tests for heteroscedasticity in linear econometric models. Econometrica, 46, 1311–1327.
29. Theil, H. (1971) Principles of Econometrics. Wiley, New York.Google Scholar
30. Williams, E. J. (1959) Regression Analysis. Wiley, New York.Google Scholar

## Further reading

1. Abdullah, M. (1979) Tests for heteroscedasticity in regression: a review. M.Sc. Dissertation, University of Kent.Google Scholar
2. Draper, N. R. and Smith, H. (1981) Applied Regression Analysis. Wiley, New York.Google Scholar
3. Nelder, J. A. (1968) Regression, model building and invariance. J. Roy. Statist. Soc. A, 131, 303–315.Google Scholar

## Copyright information

© G. Barrie Wetherill 1986

## Authors and Affiliations

• G. Barrie Wetherill
• 1
• P. Duncombe
• 2
• M. Kenward
• 3
• J. Köllerström
• 3
• S. R. Paul
• 4
• B. J. Vowden
• 3
1. 1.Department of StatisticsThe University of Newcastle upon TyneUK
2. 2.Applied Statistics Research UnitUniversity of Kent at CanterburyUK
3. 3.Mathematical InstituteUniversity of Kent at CanterburyUK
4. 4.Department of Mathematics and StatisticsUniversity of WindsorCanada