  This paper investigates the problem of making inference about a parametric
model for the regression of an outcome variable $Y$ on covariates $(V,L)$ when
data are fused from two separate sources, one which contains information only
on $(V, Y)$ while the other contains information only on covariates. This data
fusion setting may be viewed as an extreme form of missing data in which the
probability of observing complete data $(V,L,Y)$ on any given subject is zero.
We have developed a large class of semiparametric estimators, which includes
doubly robust estimators, of the regression coefficients in fused data. The
proposed method is DR in that it is consistent and asymptotically normal if, in
addition to the model of interest, we correctly specify a model for either the
data source process under an ignorability assumption, or the distribution of
unobserved covariates. We evaluate the performance of our various estimators
via an extensive simulation study, and apply the proposed methods to
investigate the relationship between net asset value and total expenditure
among U.S. households in 1998, while controlling for potential confounders
including income and other demographic variables.
