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STATISTICAL MATCHING AND MICROSIMULATION MODELS 75 where (for ease of notation) Y represents Yi, Z represents Z j, and X represents Xk(A) or X k(B). Often, as discussed above, the X(A) variables are selected so that variables in Y and Z will be well explained by X(A). Implicitly it is reasoned that if both ÏYX and ÏZX are close to 1, then the numerator of ÏYZ.X will be close to 0, or, what here amounts to the same thing, ÏYZ will be close to 1. To some extent this reasoning is valid, but it is surprising how variable the correlation between Yi and Zj, ÏYZ, can be even when ÏYX and ÏZX are fairly close to 1. This variability is disturbing since the estimation of these correlations is presumably a major reason the statistical match was performed. The variability of ÏYZ can be seen from the above formula. By setting ÏYZ.X equal to â1 and 1, To take an example from Rodgers (1984), assume that ÏYX equals .8 and ÏZX equals .8. Then ÏYZ ranges from 0.28 to 1.0. More generally, we see that and the correlation between Yi and Zj is completely determined by ÏYX and ÏZX only when at least one of them is essentially 1, or when ÏYZ.X equals 0. Thus, knowledge about the relationships between X(B) and Zj and between X(A) and Yi, from different files, typically is not sufficient to completely inform about the relationships between Yi and Zj. Armstrong (1990:1) points out: Distortion of type (iii) [distortion in the multivariate distribution of X, Y, and Z] is often unavoidable when statistical matching methods are employed. Statistical matching methods involve the assumption that Y and Z are independent conditional on X. When this assumption is violated, type (iii) distortion is inevitable. Moreover, when one of the correlations, ÏYX or ÏZX, is essentially equal to 1, what is the benefit of statistical matching? In that case one could use the linear combination of X(A) as a surrogate for the missing covariate. Paass (1985) thinks that the conditional independence assumption is almost inextricably linked with the distance measure used. This view makes sense because one can make the matches that are consistent with an assumed probabilistic structure more likely through the choice of the distance measure. For example, if one believes that Z and Y are negatively correlated conditioned on X, a distance measure can encourage the joining of records when this obtains. Paass (1985) mentions a variety of ways this can be accomplished, along with some simulation results (see also discussion below). Limitations in Mode ling Even after a statistically matched data set is created, statistical models cannot be