Measuring tail thickness to estimate the stable index α: A critique

Journal of Business and Economic Statistics

Volumn 15, Issue 1, 1997, Pages 74-81

  McCulloch, J Huston ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Bond returns; Foreign exchange returns; Generalized Pareto distribution; Hill estimator; Infinite variance; Monte Carlo distribution of stable likelihood ratio statistic; Pareto distribution; Stable distributions; Stock returns; Tail index

Indexed keywords

EID: 0002485569     PISSN: 07350015     EISSN: 15372707     Source Type: Journal    
DOI: 10.1080/07350015.1997.10524689     Document Type: Article

References (49)

1. Akgiray, V., and Booth, G. G. (1988), “The Stable-Law Model of Stock Returns,” Journal of Business & Economic Statistics, 6, 51-57.
2. Bergstrøm, H. (1952), “On Some Expansions of Stable Distribution Functions,” Arkiv fur Mathematik, 2, 375-378.
3. Blattberg, R. C., and Gonedes, N. J. (1974), “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices,” Journal of Business, 47, 244-280.
4. Bollerslev, T., Chou, R. Y., and Kroner, K. F. (1992), “ARCH Modeling in Finance,” Journal of Econometrics, 52, 5-60.
5. Boothe, P., and Glassman, D. (1987), “The Statistical Distribution of Exchange Rates,” Journal of International Economics, 22, 297-319.
6. Brorsen, B. W., and Yang, S. R. (1990), “Maximum Likelihood Estimates of Symmetric Stable Distribution Parameters,” Communications in Statistics—Simulation and Computation, 19, 1459-1464.
7. Buckle, D. J. (1995), “Bayesian Inference for Stable Distributions,” Journal of the American Statistical Association, 90, 605-613.
8. Chambers, J. M., Mallows, C. L., and Stuck, B. W. (1976), “A Method for Simulating Stable Random Variables,” Journal of the American Statistical Association, 70, 340-344.
9. Csorgo, S. (1984), “Adaptive Estimation of the Parameters of Stable Laws,” in Limit Theorems in Probability and Statistics (Bolyai Colloquia, 36), ed. P. Revesz, Amsterdam: North-Holland, pp. 305-367.
10. Csorgo, S. (1987), “Testing for Stability,” in Goodness of Fit (Bolyai Colloquia, 45), eds. P. Revesz, K. Sardoki, and P. K. Sen, Amsterdam: North-Holland, pp. 101-132.
11. David, H. A., Hartley, H. O., and Pearson, E. S. (1954), “The Distribution of the Ratio, in a Single Normal Sample, of Range to Standard Deviation,” Biometrika, 41, 482-493.
12. Davidson, R., and MacKinnon, J. G. (1993), Estimation and Inference in Econometrics, New York: Oxford University Press.
13. Dewachter, H., and Gielens, G. (1994), “A Note on the Sum-Stability of Speculative Returns,” Economic Notes by Monte Dei Paschi di Siena, 23, 116-124.
14. Diebold, F. X. (1993), Comment on “Modeling Asset Returns With Alternative Stable Distributions,” by S. Mittnik and S.T. Rachev, Econometric Reviews, 12, 339-342.
15. DuMouchel, W. H. (1973), “On the Asymptotic Normality of the Maximum-Likelihood Estimate When Sampling From a Stable Distribution,” The Annals of Statistics, 1, 948-957.
16. DuMouchel, W. H. (1975), “Stable Distributions in Statistical Inference: 2. Information from Stably Distributed Samples,” Journal of the American Statistical Association, 70, 386-393.
17. DuMouchel, W. H. (1983), “Estimating the Stable Index a in Order to Measure Tail Thickness: A Critique,” The Annals of Statistics, 11, 1019-1031.
18. Feuerverger, A., and McDunnough, P. (1981), “On Efficient Inference in Symmetric Stable Laws and Processes,” in Statistics and Related Topics, eds. M. Csorgo et al., Amsterdam: North-Holland, pp. 108-122.
19. Gibbons, M., and Hess, P. (1981), “Day of the Week Effects and Asset Returns,” Journal of Business, 54, 579-596.
20. Hill, B. M. (1975), “A Simple General Approach to Inference About the Tail of a Distribution,” The Annals of Statistics, 3, 1163-1174.
21. Holt, D., and Crow, E. L. (1973), “Tables and Graphs of the Stable Probability Density Functions,” Journal of Research of the National Bureau of Standards, 77B, 143-198.
22. Hols, M. C. A. B., and de Vries, C. G. (1991), “The Limiting Distribution of Extremal Exchange Rate Returns,” Journal of Applied Econometrics, 6, 287-302.
23. Jansen, D. W., and de Vries, C. G. (1991), “On the Frequency of Large Stock Returns: Putting Booms and Busts Into Perspective,” Review of Economics and Statistics, 73, 18-24.
24. Janicki, A., and Weron, A. (1994), Simulation and Chaotic Behavior of a-Stable Processes. New York: Marcel Deklcer.
25. Koedijk, K. G., Schafgans, M. M. A., and de Vries, C. G. (1990), “The Tail Index of Exchange Rate Returns,” Journal of International Economics, 29, 93-108.
26. Koutrouvelis, I. A. (1980), “Regression-Type Estimation of the Parameters of Stable Laws,” Journal of the American Statistical Association, 75, 918-928.
27. Lau, A. H., Lau, H.-S., and Wingender, J. R. (1990), “The Distribution of Stock Returns: New Evidence Against the Stable Model,” Journal of Business & Economic Statistics, 8, 217-233.
28. Lau, H.-S., and Lau, A. H. (1993), “The Reliability of the Stability-Under Addition Test for the Stable-Paretian Hypothesis,” Journal of Statistical Computation and Simulation, 48, 67-80.
29. Lee, J. H., and Brorsen, B. W. (1995), “A Cox-Type Non-Nested Test for Time Series Models,” working paper, Oklahoma State University, Economics Dept.
30. Liu, S. M., and Brorsen, B. W. (1995), “Maximum Likelihood Estimation of a GARCH-Stable Model,” Journal of Applied Econometrics, 10, 273-285.
31. Loretan, M., and Phillips, P. C. B. (1994), “Testing the Covariance Sta-tionarity of Heavy-Tailed Time Series,” Journal of Empirical Finance, 1, 211-248.
32. Mandelbrot, B. (1963), “The Variation of Certain Speculative Prices,” Journal of Business, 36, 394-419.
33. McCulloch, J. H. (1979), “Linear Regression With Symmetric Stable Disturbances,” Working Paper 63, Ohio State University, Economics Dept.
34. McCulloch, J. H. (1985), “Interest-Risk Sensitive Deposit Insurance Premia: Stable ACH Estimates,” Journal of Banking and Finance, 9, 137-156.
35. McCulloch, J. H. (1986), “Simple Consistent Estimators of Stable Distribution Parameters,” Communications in Statistics—Simulation and Computation, 15, 1109-1136.
36. McCulloch, J. H. (1996), “Financial Applications of Stable Distributions,” in Statistical Methods in Finance (Handbook of Statistics, 14), eds. G. S. Maddala and C. R. Rao, Amsterdam: Elsevier, pp. 393-425.
37. McCulloch, J. H. (in press a), “On the Parameterization of the Afocal Stable Distributions,” Bulletin of the London Mathematical Society, 28.
38. McCulloch, J. H. (in press b), “Numerical Approximation of the Symmetric Stable Distribution and Density,” in A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions, eds. R. Adler, R. Feldman, and M. S. Taqqu, Boston: Birkhauser.
39. McFarland, J. W., Pettit, R. R., and Sung, S. K. (1982), “The Distribution of Foreign Exchange Price Changes: Trading Day Effects and Risk Measurement,” Journal of Finance, 37, 693-715.
40. Moran, P. A. P. (1971a), “The Uniform Consistency of Maximum-Likelihood Estimators,” Proceedings of the Cambridge Philosophical Society, 17, 435-439.
41. Moran, P. A. P. (1971b), “Maximum-Likelihood Estimation in Non-Standard Conditions,” Proceedings of the Cambridge Philosophical Society, 17, 441-450.
42. Oh, C. S. (1994), “Estimation of Time Varying Term Premia of U.S. Treasury Securities: Using a STARCH Model With Stable Distributions,” unpublished Ph.D. dissertation, Ohio State University, Economics Dept.
43. Paulson, A. S., Holcomb, W. E., and Leitch, R. A. (1975), “The Estimation of the Parameters of the Stable Laws,” Biometrika, 62, 163-170.
44. Samorodnitsky, G., and Taqqu, M. S. (1994), Stable Non-Gaussian Random Processes, New York: Chapman and Hall.
45. So, J. C. (1987), “The Sub-Gaussian Distribution of Currency Futures: Stable Paretian or Nonstationary?” Review of Economics and Statistics, 69, 100-107.
46. Stuck, B. W. (1976), “Distinguishing Stable Probability Measures, Part I: Discrete Time,” Bell System Technical Journal, 55, 11-82.
47. Tsionas, E. G. (in press), “Bayesian Inference in Econometric Models With Symmetric Stable Paretian Disturbances,” Journal of Econometrics.
48. Young, M. S., and Graff, R. A. (1995), “Real Estate Is Not Normal: A Fresh Look at Real Estate Return Distributions,” Journal of Real Estate Finance and Economics, 10, 225-259.
49. Zolotarev, V. (1986), One-Dimensional Stable Laws, Providence, RI: American Mathematical Society.