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Conditional and Unconditional Statistical Independence

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Conditional independence almost everywhere in the space of the conditioning variates does not imply unconditional independence, although it may well imply unconditional independence of certain functions of the variables. An example that is important in linear regression theory is discussed in detail. This involves orthogonal projections on random linear manifolds, which are conditionally independent but not unconditionally independent under normality. Necessary and sufficient conditions are obtained under which conditional independence does imply unconditional independence.

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File URL: http://cowles.econ.yale.edu/P/cd/d08a/d0824-r.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 824R.

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Length: 13 pages
Date of creation: 1987
Date of revision: Dec 1987
Publication status: Published in Journal of Econometrics (1988), 75(2): 341-348
Handle: RePEc:cwl:cwldpp:824r

Note: CFP 705.
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  1. Ullah, Aman & Zinde-Walsh, Victoria, 1984. "On the Robustness of LM, LR, and W Tests in Regression Models," Econometrica, Econometric Society, vol. 52(4), pages 1055-66, July.
  2. Peter C.B. Phillips & Vassilis A. Hajivassiliou, 1987. "Bimodal t-Ratios," Cowles Foundation Discussion Papers 842, Cowles Foundation for Research in Economics, Yale University.
  3. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
  4. Hillier, Grant H., 1985. "On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation," Econometric Theory, Cambridge University Press, vol. 1(01), pages 53-72, April.
  5. Dawid, A. P., 1985. "Invariance and independence in multivariate distribution theory," Journal of Multivariate Analysis, Elsevier, vol. 17(3), pages 304-315, December.
  6. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
  7. Chib, Siddhartha & Tiwari, Ram C. & Jammalamadaka, S. Rao, 1988. "Bayes prediction in regressions with elliptical errors," Journal of Econometrics, Elsevier, vol. 38(3), pages 349-360, July.
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Cited by:
  1. Sokbae Lee & Oliver Linton & Yoon-Jae Whang, 2009. "Testing for Stochastic Monotonicity," Econometrica, Econometric Society, vol. 77(2), pages 585-602, 03.
  2. Kyungchul Song, 2007. "Testing Conditional Independence via Rosenblatt Transforms," PIER Working Paper Archive 07-026, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  3. Oliver Linton & Pedro Gozalo, 1996. "Conditional Independence Restrictions: Testing and Estimation," Cowles Foundation Discussion Papers 1140, Cowles Foundation for Research in Economics, Yale University.

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