# Concluding Remarks

Recent sampling theory research on heteroskedastic models seems to be concentrated on methods for estimation and hypothesis testing that do not require

specification of a particular parametric form of heteroskedasticity. They are motivated by our inability to be certain about the most appropriate variance specification. However, methodology suggested along these lines is generally asymptotic and may not perform well in finite samples. What is likely to be important, and what seems to have been neglected, is whether the types of inferences we make in practice are very sensitive to the assumed form of hetero- skedasticity. If they are not, then efforts to develop alternative methods, that do not require an explicit variance function, may be misplaced.

Bayesian estimation has several advantages. Results are presented in terms of intuitively meaningful posterior pdfs. Marginal posterior pdfs reflect all the parameter uncertainty in a model and do not condition on point estimates of nuisance parameters. Predictive pdfs for future values can also be constructed without conditioning on point estimates (Boscardin and Gelman, 1996). The advent of MCMC techniques means that many more practical applications of Bayesian inference to heteroskedastic models are now possible.

Note

* The author acknowledges valuable comments on an earlier version by three anonymous reviewers.

References

Amemiya, T. (1973). Regression analysis when the variance of the dependent variable is proportional to the square of its expectation. Journal of the American Statistical Association 68, 928-34.

Amemiya, T. (1977). A note on a heteroscedastic model. Journal of Econometrics 6, 365-70;

and Corrigenda. Journal of Econometrics 8, 275.

Amemiya, T. (1983). Partially generalized least squares and two-stage least squares estimators. Journal of Econometrics 23, 275-83.

Baltagi, B. H. (1998). Econometrics. New York: Springer-Verlag.

Boscardin, W. J., and A. Gelman (1996). Bayesian computation for parametric models of heteroscedasticity in the linear model. In R. C. Hill (ed.) Advances in Econometrics Volume 11A: Bayesian Computational Methods and Applications. Greenwich: JAI Press.

Breusch, T. S., and A. R. Pagan (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica 47, 1287-94.

Brown, B. W., and M. B. Walker (1989). The random utility hypothesis and inference in demand systems. Econometrica 57, 815-29.

Brown, B. W., and M. B. Walker (1995). Stochastic specification in random production models of cost minimizing firms. Journal of Econometrics 66, 175-205.

Carroll, R. J. (1982). Adapting for heteroscedasticity in linear models. Annals of Statistics 10, 1224-33.

Carroll, R. J., and D. Ruppert (1988). Transformation and Weighting in Regression. New York: Chapman and Hall.

Cragg, J. G. (1992). Quasi-Aitken estimation for heteroskedasticity of unknown form. Journal of Econometrics 54, 179-202.

Davidson, R., and J. G. MacKinnon (1993). Estimation and Inference in Econometrics. New York: Oxford University Press.

Delgado, M. A. (1992). Semiparametric generalized least squares in the multivariate nonlinear regression model. Econometric Theory 8, 203-22.

Donald, S. G. (1995). Two-step estimation of heteroskedastic sample selection models. Journal of Econometrics 65, 347-80.

Evans, M. A., and M. L. King (1988). A further class of tests for heteroscedasticity. Journal of Econometrics 37, 265-76.

Farebrother, R. W. (1987). The statistical foundations of a class of parametric tests for heteroskedasticity. Journal of Econometrics 36, 359-68.

Geweke, J. (1999). Using simulation methods for Bayesian econometric models: inference, development and communication. Econometric Reviews 18, 1-74.

Godfrey, L. G. (1978). Testing for multiplicative heteroskedasticity. Journal of Econometrics 8, 227-36.

Godfrey, L. G. (1996). Some results on the Glejser and Koenker tests for heteroskedasticity. Journal of Econometrics 72, 275-99.

Godfrey, L. G., and C. D. Orme (1999). The robustness, reliability and power of heteroskedasticity tests. Econometric Reviews 18, 169-94.

Goldfeld, S. M., and R. E. Quandt (1965). Some tests for homoscedasticity. Journal of the American Statistical Association 60, 539-47.

Goldfeld, S. M., and R. E. Quandt (1972). Nonlinear Methods in Econometrics. Amsterdam: North-Holland.

Greene, W. (1997). Econometric Analysis, 3rd edn. Upper Saddle River: Prentice Hall.

Griffiths, W. E. (1972). Estimation of actual response coefficients in the Hildreth – Houck random coefficient model. Journal of the American Statistical Association 67, 633-5.

Griffiths, W. E., and J. R. Anderson (1982). Using time-series and cross-section data to estimate a production function with positive and negative marginal risks. Journal of the American Statistical Association 77, 529-36.

Griffiths, W. E., and G. G. Judge (1992). Testing and estimating location vectors when the error covariance matrix is unknown. Journal of Econometrics 54, 121-38.

Griffiths, W. E., and K. Surekha (1986). A Monte Carlo evaluation of the power of some tests for heteroscedasticity. Journal of Econometrics 31, 219-31.

Griffiths, W. E., R. G. Drynan, and S. Prakash (1979). Bayesian estimation of a random coefficient model. Journal of Econometrics 10, 201-20.

Harvey, A. C. (1976). Estimating regression models with multiplicative heteroscedasticity. Econometrica 44, 461-5.

Hidalgo, J. (1992). Adaptive estimation in time series regression models with hetero – skedasticity of unknown form. Econometric Theory 8, 161-87.

Hildreth, C., and J. P. Houck (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association 63, 584-95.

Hill, R. C., J. R. Knight, and C. F. Sirmans (1997). Estimating capital asset price indexes. Review of Economics and Statistics 80, 226-33.

Hooper, P. M. (1993). Iterative weighted least squares estimations in heteroscedastic linear models. Journal of the American Statistical Association 88, 179-84.

Jobson, J. D., and W. A. Fuller (1980). Least squares estimation when the covariance matrix and parameter vector are functionally related. Journal of the American Statistical Association 75, 176-81.

Judge, G. G., W. E. Griffiths, R. C. Hill, and T.-C. Lee (1985). The Theory and Practice of Econometrics. New York: John Wiley and Sons.

Judge, G. G., R. C. Hill, W. E. Griffiths, H. Lutkepohl, and T.-C. Lee (1988). An Introduction to the Theory and Practice of Econometrics. New York: John Wiley and Sons.

Keener, R. W., J. Kmenta, and N. C. Weber (1991). Estimation of the covariance matrix of the least-squares regression coefficients when the disturbance covariance matrix is of unknown form. Econometric Theory 7, 22-43.

Koenker, R., and G. Bassett, Jr. (1982). Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50, 43-61.

Lee, B.-J. (1992). A heteroskedasticity test robust to conditional mean specification. Econometrica 60, 159-72.

Li, Q., and T. Stengos (1994). Adaptive estimation in the panel data error model with heteroskedasticity of unknown form. International Economic Review 35, 981-1000.

Linton, O. B. (1996). Second order approximation in a linear regression with the heteroskedasticity of unknown form. Econometric Reviews 15, 1-32.

Mandy, D. M., and C. Martins-Filho (1993). Seemingly unrelated regressions under additive heteroscedasticity: theory and share equation applications. Journal of Econometrics 58, 315-46.

Orme, C. (1992). Efficient score tests for heteroskedasticity in microeconometrics. Econometric Reviews 11, 235-52.

Pagan, A., and Y. Pak (1993). Testing for heteroskedasticity. In G. S. Maddala, C. R. Rao, and H. D. Vinod (eds.) Handbook of Statistics II: Econometrics. Amsterdam: North-Holland, 489-518.

Rilestone, P. (1991). Some Monte Carlo evidence on the relative efficiency of parametric and semiparameteric EGLS estimators. Journal of Business and Economic Statistics 9, 179-87.

Robinson, P. M. (1987). Asymptotically efficient estimation in the presence of hetero – scedasticity of unknown form. Econometrica 55, 875-91.

Rutemiller, H. C., and D. A. Bowers (1968). Estimation in a heteroscedastic regression model. Journal of the American Statistical Association 63, 552-7.

Surekha, K., and W. E. Griffiths (1984). A Monte Carlo comparison of some Bayesian and sampling theory estimators in two heteroscedastic error models. Communications in Statistics B 13, 85-105.

Szroeter, J. (1978). A class of parametric tests for heteroskedasticity in linear econometric models. Econometrica 46, 1311-28.

Szroeter, J. (1994). Exact finite-sample relative efficiency of sub-optimality weighted least squares estimators in models with ordered heteroscedasticity. Journal of Econometrics 64, 29-44.

Welsh, A. H., R. J. Carroll, and D. Ruppert (1994). Fitting heteroscedastic regression models. Journal of the American Statistical Association 89, 100-16.

White, H. (1980). A heteroscedasticity-consistent covariance matrix estimators and a direct test for heteroscedasticity. Econometrica 48, 817-38.

Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. New York: John Wiley and Sons.

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