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Thus, all of the results obtained in Chapters 1 and 2 may, in principle, be used to study identification of P(y
x = x) when (y, x) realizations are jointly missing. 3) describe the identification region [P(y
x = x)] in principle, they do not provide a transparent description. 1 shows directly that the region has a simple structure. 1: Let P(zy = zx = 1) + P(zy = zx = 0) = 1. 4a) where P(x = x*zyx = 1)P(zyx = 1) r(x)  )))))))))))))))))))))))))) . 4b) a Proof: The Law of Total Probability gives P(y*x = x) = P(y*x = x, zyx = 1)P(zyx = 1
x = x) + P(y*x = x, zyx = 0)P(zyx = 0
x = x).

General Missing-Data Patterns 49 SI[P(y*x = x)] = B [P(y*v = v, x = x)]. 20) vV (b) Let SI[P(y x = x)] be empty. 12) does not hold. 5. General Missing-Data Patterns Consider now a sampling process with a general pattern of missing data in which some realizations of (y, x) may be completely observed, others observed in part, and still others not observed at all. The structure of the problem of inference on P(y*x = x) is displayed by the Law of Total Probability and Bayes Theorem, which give P(y*x = x) = P(x = x*zx = j, zy = k)P(zx = j, zy = k)   P(y*x = x, zx = j, zy = k) )))))))))))))))))))))))))))))) .

A researcher applying assumption MAR must specify the instrumental variable v for which the assumption holds. 1) is the special case in which v has a degenerate distribution. 2 30 2. 4. Statistical Independence Assumption SI has the same identifying power as does observation of data from multiple sampling processes. 4. 2 gives the basic result, and two corollaries flesh it out. 2: (a) Let assumption SI hold. Then the identification region for P(y) is SI[P(y)] = B {P(y
v = v, z = 1)P(z = 1
v = v) + v#P(z = 0
v = v), v  Y}.

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