Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T17:52:58.238Z Has data issue: false hasContentIssue false
  • English
  • Français

On Modeling Heteroskedasticity: The Student's t and Elliptical Linear Regression Models

Published online by Cambridge University Press:  11 February 2009

Aris Spanos
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes a new approach to modeling heteroskedastidty which enables the modeler to utilize information conveyed by data plots in making informed decisions on the form and structure of heteroskedasticity. It extends the well-known normal/linear/homoskedastic models to a family of non-normal/linear/heteroskedastic models. The non-normality is kept within the bounds of the elliptically symmetric family of multivariate distributions (and in particular the Student's t distribution) that lead to several forms of heteroskedasticity, including quadratic and exponential functions of the conditioning variables. The choice of the latter family is motivated by the fact that it enables us to model some of the main sources of heteroskedasticity: “thicktails,” individual heterogeneity, and nonlinear dependence. A common feature of the proposed class of regression models is that the weak exogeneity assumption is inappropriate. The estimation of these models, without the weak exogeneity assumption, is discussed, and the results are illustrated by using cross-section data on charitable contributions.

Type
Research Article
Information
Econometric Theory , Volume 10 , Issue 2 , June 1994 , pp. 286 - 315
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Amemiya, T. Regression analysis when the variance of the dependent variable is proportional to the square of its expectation. Journal of the American Statistical Association 68 (1973): 928–934.CrossRefGoogle Scholar
2. Barrett, K. Charitable giving and federal income tax policy. Ph.D. dissertation, Virginia Polytechnic Institute and State University, 1992.Google Scholar
3. Blattberg, R.C. & Gonedes, N.J.. A comparison of the stable and student distributions as statistical models for stock prices. Journal of Business 47 (1974): 244–280.10.1086/295634CrossRefGoogle Scholar
4. Bollerslev, T. A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69 (1987): 542–547.10.2307/1925546CrossRefGoogle Scholar
5. Breiman, L., Meisel, W. & Purcell, E.. Variable kernel estimates of multivariate densities. Techometrics 19 (1977): 135–144.10.1080/00401706.1977.10489521CrossRefGoogle Scholar
6. Chib, S., Osiewalski, J. & Steel, M.F.J.. Posterior inference on the degrees of freedom parameter in multivariate-r regression models. Economic Letters 37 (1991): 391–397.10.1016/0165-1765(91)90076-WCrossRefGoogle Scholar
7. Chu, K.-C. (1993). Estimation and decision for linear systems with elliptical random processes. IEEE Transactions on Automatic Control 18 (1973): 499–505.Google Scholar
8. Dagpunar, J. Principles of Random Number Generation. Oxford: Oxford University Press, 1988.Google Scholar
9. D'Agostino, R.B. & Stephens, M.A.. Goodness of Fit Techniques. New York: Marcel Dekker, 1986.Google Scholar
10. Engle, R.F., Hendry, D.F. & Richard, J.F.. Exogeneity. Econometrica 51 (1983): 277–304.10.2307/1911990CrossRefGoogle Scholar
11. Fang, K.-T., Kotz, S. & Ng, K-W.. Symmetric Multivariate and Related Distributions. London: Chapman and Hall, 1990.10.1007/978-1-4899-2937-2CrossRefGoogle Scholar
12. Fang, K-T. & Zhang, Y-T.. Generalized Multivariate Analysis. New York: Springer-Verlag, 1990.Google Scholar
13.Fraser, D.A.S. & Ng, K.W. Multivariate regression with spherical error. In Krishnaiah, P.R., (ed.), Multivariate Analysis V. New York: North-Holland, 1980.Google Scholar
14. Hardle, W. Applied Non-parametric Regression. Cambridge: Cambridge University Press, 1990.10.1017/CCOL0521382483CrossRefGoogle Scholar
15. Judge, G.G., Griffiths, W.E., Carter, R.C., Lutkepohl, H. & Lee, T.-C.. The Theory and Practice of Econometrics, 2nd ed., New York: Wiley, 1985.Google Scholar
16. Kawani, N.C. The unbiasedness of Zellner's seemingly unrelated regression equations estimators. Journal of the American Statistical Association 62 (1967): 141–142.10.1080/01621459.1967.10482895CrossRefGoogle Scholar
17. Kelejian, H.H. & Prucha, I.R.. Independent or uncorrelated disturbances in linear regression. Economic Letters 19 (1985): 35–38.CrossRefGoogle Scholar
18. Kelker, D. Distribution theory of spherical distribution and a location scale parameter generalization. Sankhya A 43 (1970): 419–430.Google Scholar
19. McGuirk, A., Robertson, J. & Spanos, A.. Modeling exchange rate dynamics: Non-linear dependence and thick tails. Econometric Reviews 12 (1993): 33–63.CrossRefGoogle Scholar
20. McGuirk, A. & Spanos, A.. Modeling stock returns: The Student's t model with dynamic heteroskedasticity. (Mimeo), VPI & State University, 1992.Google Scholar
21. Muirhead, R.J. Aspects of Multivariate Statistical Theory. New York: Wiley, 1982.10.1002/9780470316559CrossRefGoogle Scholar
22. Phillips, P.C.B. Time series regression with a unit root. Econometrica 55 (1987): 277–301.10.2307/1913237CrossRefGoogle Scholar
23. Phillips, P.C.B. & Loretan, M.. Testing covariance stationarity under moment condition failure: An application to common stock returns. Cowles Foundation Discussion Paper No. 947, 1990.Google Scholar
24. PraagVan, M.S. Van, M.S. & Wesselman, B.M.. Elliptical multivariate analysis. Journal of Econometrics 41 (1989): 189–203.CrossRefGoogle Scholar
25. Praetz, P.D. The distribution of share price changes. Journal of Business 45 (1972): 49–55.10.1086/295425CrossRefGoogle Scholar
26. Prais, S.J. A note on heteroskedastic errors in regression analysis. Review of the International Statistical Institute 21 (1953): 28–29.10.2307/1401514CrossRefGoogle Scholar
27. Prucha, I.R. & Kelejian, H.H.. The structure of simultaneous equations equation estimators: A generalization towards nonnormal disturbances. Econometrica 52 (1984): 721–736.10.2307/1913473CrossRefGoogle Scholar
28. Robertson, J. & Spanos, A.. Modeling the volatility in U.S. interest rates. Mimeo, VPI & State University, 1991.Google Scholar
29. Rogalski, R. & Vinso, J.D.. Empirical properties of foreign exchange rates. Journal of International Business Studies 9 (1978): 69–79.10.1057/palgrave.jibs.8490663CrossRefGoogle Scholar
30. Sansone, G. Orthogonal Functions. New York: Wiley, 1959.Google Scholar
31. Silverman, B.W. Density Estimation. London: Chapman and Hall, 1986.Google Scholar
32. Spanos, A. Statistical Foundations of Econometric Modelling. Cambridge: Cambridge University Press, 1986.CrossRefGoogle Scholar
33. Spanos, A. On re-reading Haavelmo: A retrospective view of econometric modeling. Econometric Theory 5 (1989): 405–429.10.1017/S0266466600012597CrossRefGoogle Scholar
34. Spanos, A. On non-linear and heteroskedastic models: The exponential and related regression models. Mimeo, VPI & State University, 1992.Google Scholar
35. Spanos, A. The Student's t autoregressive model with dynamic heteroskedasticity. Mimeograph, University of Cyprus, 1993.Google Scholar
36. Spanos, A. On theory testing in econometrics: The case of the Efficient Market hypothesis. Journal of Econometrics, forthcoming, 1995.10.1016/0304-4076(94)01633-BCrossRefGoogle Scholar
37. Spanos, A. On normality and the linear regression model: Karl Pearson's argument revisited. Econometric Reviews, forthcoming.Google Scholar
38. Theil, H. Estimates and their sampling variance of parameters of certain heteroskedastic distributions. Review of the International Statistical Institute 19 (1951): 141–147.10.2307/1401241CrossRefGoogle Scholar
39. White, H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48 (1980): 817–838.10.2307/1912934CrossRefGoogle Scholar
40. Zellner, A. An Introduction to Bayesian Inference in Econometrics. New York: Wiley, 1971.Google Scholar
41. Zellner, A. Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. Journal of the American Statistical Association 71 (1976): 400–405.Google Scholar