Identification Problems in the Social Sciences by Charles F. Manski

Author:Charles F. Manski [Manski, Charles F.]
Language: eng
Format: epub
Tags: Business & Economics, Econometrics, Social Science, Research, Statistics
ISBN: 9780674442832
Google: awGDAAAAIAAJ
Publisher: HarvardUP
Published: 1995-09-15T20:46:18+00:00

(4.18) P(γ | x) = P(γ | x, s = 1) P(s = 1 | x) + P(γ | x, s = 0) P(s = 0 | x)

The sampling process identifies P(γ | x, s = 1) and P(γ | x, s = 0). The sampling process does not restrict the stratum distribution P(s | x), because inference on P(s | x) is precisely the response-based sampling problem, with s here in the role of γ earlier. So sampling from both strata reveals P(γ | x) to be an unrestricted mixture of the two distributions P(γ | x, s = 1) and P(γ | x, s = 0).

How informative is the sampling process about P(γ | x)? The answer depends on how closely related the outcomes γ and strata s are to one another, conditional on x. At one extreme, suppose that γ and s are statistically independent conditional on x; that is, P(γ | x, s = 1) = P(γ | x, s = 0). Then equation (4.18) reduces to

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