Improving Information for Social Policy Decisions -- The Uses of Microsimulation Modeling: Volume II, Technical Papers (1991)
Chapter: Limitations in Modeling
where (for ease of notation) Y represents Yi, Z represents Zj, and X represents Xk(A) or Xk(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 ρYX and ρZX are close to 1, then the numerator of ρYZ.X will be close to 0, or, what here amounts to the same thing, ρYZ will be close to 1. To some extent this reasoning is valid, but it is surprising how variable the correlation between Yi and Zj, ρYZ, can be even when ρYX and ρZX 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 ρYZ can be seen from the above formula. By setting ρYZ.X equal to −1 and 1,
To take an example from Rodgers (1984), assume that ρYX equals .8 and ρZX equals .8. Then ρYZ ranges from 0.28 to 1.0. More generally, we see that
and the correlation between Yi and Zj is completely determined by ρYX and ρZX only when at least one of them is essentially 1, or when ρYZ.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, ρYX or ρZX, 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 Modeling
Even after a statistically matched data set is created, statistical models cannot be
