Ecological fallacy and covariates in the estimation of voters   transitions
  We provide a simple formulation of the conditions under which ecological bias should be expected and argue that the bias will affect any method of ecological inference; our claim is supported by formal derivations and several examples where individual data are available. The conditions which we highlight imply that, when they are violated, ecological bias cannot be avoided unless the a suitable model for the effect of specific covariates is incorporated into ecological inference. We also detect situations where the ecological bias cannot be corrected even if the effect of covariates is incorporated into the model. In particular, when the association in the individual data is rather weak and certain transition probabilities are similar functions of a given covariate, ecological inference methods may be unable to disentangle the individual components from their aggregate. In any case, the value of the covariates measured at the level of local units, when these are rather extensive, do not provide enough information on the within unit heterogeneity. Our findings are applied and tested on several data sets, both real and simulated, where individual observations are available in addition to the aggregated ones.,. 
Antonio Forcina, Davide Pellegrino
stat.ME