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Conditional Moments and Independence

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  • de Paula, Aureo

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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Suggested Citation

  • de Paula, Aureo, 2008. "Conditional Moments and Independence," The American Statistician, American Statistical Association, vol. 62, pages 219-221, August.
    • Handle: RePEc:bes:amstat:v:62:y:2008:m:august:p:219-221
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    1. Robert S. Pindyck & Julio J. Rotemberg, 1993. "The Comovement of Stock Prices," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(4), pages 1073-1104.
    2. 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 331-349, June.
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    Cited by:

    1. Ebrahimi, Nader & Jalali, Nima Y. & Soofi, Ehsan S., 2014. "Comparison, utility, and partition of dependence under absolutely continuous and singular distributions," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 32-50.
    2. 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.
    3. Mukhopadhyay, Nitis, 2009. "On px1 dependent random variables having each (p-1)x1 sub-vector made up of IID observations with examples," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1585-1589, July.
    4. Matthew Masten & Alexandre Poirier, 2016. "Partial independence in nonseparable models," CeMMAP working papers 26/16, Institute for Fiscal Studies.

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    JEL classification:

    • A2 - General Economics and Teaching - - Economic Education and Teaching of Economics
    • C00 - Mathematical and Quantitative Methods - - General - - - General

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