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Conditional Moments and Independence

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  • Aureo de Paula

    ()
    (Department of Economics, University of Pennsylvania)

Abstract

Consider two random variables X and Y. In initial probability and statistics courses, a discussion of various concepts of dissociation between X and Y is customary. These concepts typically involve independence and uncorrelatedness. An example is shown where E(Y^n|X) = E(Y^n) and E(X^n|Y) = E(X^n) for n = 1, 2,… and yet X and Y are not stochastically independent. The bi-variate distribution is constructed using a well-known example in which the distribution of a random variable is not uniquely determined by its sequence of moments. Other similar families of distributions with identical moments can be used to display such a pair of random variables. It is interesting to note in class that even such a degree of dissociation between the moments of X and Y does not imply stochastic independence. and yet X and Y are not stochastically independent. The bi-variate distribution is constructed using a well-known example in which the distribution of a random variable is not uniquely determined by its sequence of moments. Other similar families of distributions with identical moments can be used to display such a pair of random variables. It is interesting to note in class that even such a degree of dissociation between the moments of X and Y does not imply stochastic independence.

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Paper provided by Penn Institute for Economic Research, Department of Economics, University of Pennsylvania in its series PIER Working Paper Archive with number 08-010.

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Length: 7 pages
Date of creation: 07 Mar 2008
Date of revision:
Handle: RePEc:pen:papers:08-010

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Keywords: indeterminate distributions; moments; independence;

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  1. McDonald, Robert L & Siegel, Daniel R, 1985. "Investment and the Valuation of Firms When There Is an Option to Shut Down," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(2), pages 331-49, June.
  2. Pindyck, Robert S & Rotemberg, Julio J, 1993. "The Comovement of Stock Prices," The Quarterly Journal of Economics, MIT Press, vol. 108(4), pages 1073-1104, November.
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Cited by:
  1. Ebrahimi, Nader & Hamedani, G.G. & Soofi, Ehsan S. & Volkmer, Hans, 2010. "A class of models for uncorrelated random variables," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1859-1871, September.

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