6.18 Suppose that, given sets of covariates X1 and X2 (possibly overlapping, that is, not disjoint), yi1 and yi2 are bivariate normal with means xi1????1 and xi2????2, variances ????2 1 and ????2 2 = 1, and correlation ????. The data consist of a random sample of units i with xi1 and xi2 always observed, yi2 always missing and yi1 missing if and only if yi2 >0. Let mi1 be the missingness indicator for yi1. Show that Pr(mi1 = 1 ∣ yi1, xi1, xi2, ????, ????) = 1−Φ

(

−xi2????2 − (????∕????1)( yi1 − xi1????1) √

1 − ????2

)

, 150 6 Theory of Inference Based on the Likelihood Function whereΦis the standard normal cumulative distribution function.Hence, give conditions on the parameters under which the data are MAR and under which the missingness mechanism is ignorable for likelihoodbased inference. (This model is considered in detail in Example 15.5.)