Conditional and Unconditional Statistical Independence
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.
|Date of creation:||1987|
|Date of revision:||Dec 1987|
|Publication status:||Published in Journal of Econometrics (1988), 75(2): 341-348|
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- Phillips, Peter C B, 1984.
"The Exact Distribution of LIML: I,"
International Economic Review,
Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 249-61, February.
- Peter C.B. Phillips, 1983. "The Exact Distribution of LIML: II," Cowles Foundation Discussion Papers 663, Cowles Foundation for Research in Economics, Yale University.
- Peter C.B. Phillips, 1982. "The Exact Distribution of LIML: I," Cowles Foundation Discussion Papers 658, Cowles Foundation for Research in Economics, Yale University.
- Peter C.B. Phillips & Vassilis A. Hajivassiliou, 1987. "Bimodal t-Ratios," Cowles Foundation Discussion Papers 842, Cowles Foundation for Research in Economics, Yale University.
- Peter C.B. Phillips & Joon Y. Park, 1986.
"Statistical Inference in Regressions with Integrated Processes: Part 2,"
Cowles Foundation Discussion Papers
819R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1987.
- 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.
- 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.
- Dawid, A. P., 1985. "Invariance and independence in multivariate distribution theory," Journal of Multivariate Analysis, Elsevier, vol. 17(3), pages 304-315, December.
- 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.
- 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.
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