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Estimation.

We estimate $\sigma^2$ and $\lambda$ in a preliminary analysis, averaged over as many elections as are available (and for which we have roughly the same information in our explanatory variables). $\sigma$ is estimated from the standard error of the regression produced by regressing $v_i$ on the explanatory variables $x_i$. To estimate $\lambda$, we use the Democratic proportion of the two-party vote in the election following the one used to define $v_i$, if it is available, and regress it on $v_i$ and the original explanatory variables $x_i$. Thus the regression coefficient on $v_i$, our estimate of $\lambda$, represents the variance common to the two election results unexplained by our independent variables.

To evaluate an electoral system, we estimate $\beta$ from the regression in Equation 1, and then calculate the desired summaries and their standard errors from the results of that estimation. For prediction, we estimate $\beta$ from the most recent election for which we have data and apply it, with new values of $x_i$, to make the prediction.

Once $\sigma^2$, $\lambda$, and $\beta$ are estimated, JudgeIt simulates hypothetical election results, $v^{\hyp}$, for the appropriate situation (evaluation, prediction, or counterfactuals), from which we compute all summaries such as seats-votes curves and vote predictions. Estimates and standard errors are computed as described above.