Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-30T16:07:05.823Z Has data issue: false hasContentIssue false
  • English
  • Français

Regression and autoregression with infinite variance

Published online by Cambridge University Press:  01 July 2016

Marek Kanter  and
W. L. Steiger
Marek Kanter
Affiliation:
Sir George Williams University, Montreal
W. L. Steiger
Affiliation:
Université de Montréal
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression Xn = a1Xn-1 + … + akXn-k + Un, the least squares estimates are shown to be consistent as long as the roots of 1 - a1x2 - a2x2 - … - akxk = 0 are outside the complex unit circle, Xn is independent of Un+j,j ≥ 1, and the Un are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.

Keywords

STABLE PROCESSAUTOREGRESSIVE SCHEMEMOVING AVERAGE PROCESSREGRESSION
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 4 , December 1974 , pp. 768 - 783
Copyright
Copyright © Applied Probability Trust 1974 

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] Blattberg, R. and Sargent, T. (1972) Regression with non-Gaussian stable disturbances: some sampling results. Econometrica 39, 501–510.Google Scholar
[2] Box, G. and Jenkins, G. (1970) Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco.Google Scholar
[3] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
[4] Dykstra, R. L. (1970) Establishing the positive definiteness of a sample covariance matrix. Ann. Math. Statist. 41, 2153–2154.Google Scholar
[5] Feller, W. (1971) Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
[6] Granger, C. and Orr, D. (1972) Infinite variance and research strategy in time series. J. Amer. Statist. Assoc. 67, 275–285.Google Scholar
[7] Kanter, M. (1972) Linear sample spaces and stable processes. J. Functional Analysis 9, 441–459.CrossRefGoogle Scholar
[8] Kanter, M. (1973) On the L p norm of sums of translates of a function. Trans. Amer. Math. Soc. 179, 35–47.Google Scholar
[9] Kanter, M. (1975) Stable distributions under change in scale and associated total variation inequalities. To appear.Google Scholar
[10] Kanter, M. and Steiger, W. L. (1975) Sampling properties of some estimates for regression and autoregression with infinite variance. Submitted.Google Scholar
[11] Loève, M. (1963) Probability Theory. Van Nostrand, Princeton, N. J. Google Scholar
[12] Mandelbrot, B. (1963) The variation of certain speculative prices. J. Business 36, 394–419.CrossRefGoogle Scholar
[13] Mandelbrot, B. (1972) Statistical methodology for non-periodic cycles: from the covariance to R/S analysis. Ann. Economic and Social Measurement 1, 259–290.Google Scholar
[14] Searle, S. (1970) Linear Models. Wiley, New York.Google Scholar
[15] Wise, J. (1966) Linear estimators for linear regression systems having infinite variances. Manuscript.Google Scholar