| 概要 |
Conventional cross-tabulation of two variables assumes that the data are available for the same individuals, households, or firms. This pre-requisite is not always met. One way to proceed in that case is to cross-tabulate exact observations with imputed values. Estimates will now also be subject to imputation error, in addition to the sampling error, which increases the standard errors and may introduce a bias. The objective of this paper is to provide analytical approximations for both the bias (that may be adopted for a bias correction) and the standard errors, so that estimates can be obtained without the use of bootstrapping. We include Monte-Carlo simulation results that show that: (i) bias-corrected estimates are more accurate, (ii) the contribution of the imputation error to total variance is not negligible, and (iii) standard errors obtained by analytical approximation almost perfectly coincide with estimates based on bootstrapping. An empirical example is also provided. |