Estimating the Extent of Racially Polarized Voting in Multicandidate Contests

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Abstract
We extend the standard ecological regression methodology for analysis of racial bloc voting in single-member districts in which there is one seat to be filled to cover multimember districts without a numbered-place system. In such multimember districts voters may vote for up to k candidates and not all voters need to cast the same number of votes. These complications render the usual single-equation OLS technique inappropriate. The methodology we propose involves a two-step estimating procedure. The variable that is our principal concern, the proportion of white voters who vote for the black (or other minority) candidate, has its value determined by combining information from each of our two separate estimating equations. Thus we cannot directly obtain a confidence limit for our estimated value of this variable from either of the two equations alone. We derive a value of the standard error of our estimate of this variable by combining formula for the variance of the quotients and sums of two variables of known variance and covariance. We verify that our results are robust by comparing our estimates with those generated by the “seemingly unrelated regression” (SURE) model that is designed to cope with correlated errors in regressions that share in the same independent variables. The methodology we develop has an important application in voting rights litigation challenging the constitutionality of the multimember districts and at-large elections, which are the most common form of election mechanism in U.S. cities.