Regression with Probabilitistic Explanatory Variable Let X be a categorical variable.  Instead of knowing for certain whether a particular observation is equal to a given level of X, I have a probability distribution over the possible values of X.  So, for example, X may correspond to racial categories (e.g., white, black, asian, and so forth) where each individual is equal to each racial category with a certain probability.  That is, each individual is equal to each racial category with non-zero probability. For example, individual i may be black with probability 1/3, white with probability 1/3, and so forth.  In other words, I do not know the value of the categorical variable; instead I have a probability distribution over the possible values of the categorical variable.  This distribution might correspond to a system of beliefs.
How should one handle such an explanatory variable?  Specifically, suppose I want to see if there is a racial disparity with respect to a dependent variable Y where X is as described above.  Does it make sense to simply run OLS on a simple linear model: Y = X + u.  How would one interpret the coefficient in this case?  Any suggestions would be greatly appreciated.
 A: I am not sure regressing your outcome on "the probability of being race X" would have a meaningful interpretation.
Given:
$Y=\beta_0+\beta_1X_i+\epsilon$, the standard interpretation of a parameter estimate from OLS is "One unit increase in $X$ corresponds to a $\hat{\beta_1}$ change in $Y$, ceteris paribus."  If you fitted the following model, where $Race_{cat}$ refers to the categories black, white, or other:
$Y=\beta_0+\beta_1Race_{Black}+\beta_2Race_{White}+\beta_3Race_{Other}+\epsilon$
what would the interpretation be? 


*

*$\beta_1Race_{Black}$: One unit increase in the probability of a person being black corresponds to a
$\hat{\beta_1}$ change in $Y$, ceteris paribus.

*$\beta_2Race_{White}$: One unit increase in the probability of a person being white corresponds to a
$\hat{\beta_2}$ change in $Y$, ceteris paribus.

*$\beta_3Race_{Other}$: One unit increase in the probability of a person being other race corresponds to a
$\hat{\beta_3}$ change in $Y$, ceteris paribus.


But, as you say, you do not know the exact race of the individual observations, so if a person has a 1/3 probability of being black, he/she also has a 2/3 probability of not being black.  Therefore, the interpretation is not intuitive.
Perhaps a better argument can be made for including the proportion of black, white, other race people in the individual's "community" of residence.  You mentioned that you have address data, so this may be a feasible solution to your problem.  There is no consensus in the quantitative literature on what constitutes a "community," so researchers in the past have used enumeration areas (equivalent to a census block or a census tract if you are doing research in the U.S.) as the level of aggregation.  Here, the proportions you enter into the model correspond to a physical "reality" (the proportion of Community 1 is 60% black), which may be proxy for some "community-level" constructs, such as community "norms."
See, for example Kravdal (2002), who used average enumeration area level of female education to proxy community-level "norms" for education and "general knowledge" on fertility among women in the community.
References
Kravdal, Ø. (2002). Education and fertility in sub-Saharan Africa: Individual and community effects. Demography, 39(2), 233–250. Retrieved from http://link.springer.com/article/10.1353/dem.2002.0017
