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The Gelman-King Model

JudgeIt is used to model two-party electoral systems; choose one party to identify as ``Party 1''. (All results for Party 2 are clearly the opposite for those of Party 1.)

In any particular election year, let $v_i$ be the share of the two-party vote received by Party 1 in district $i$ . We model the resulting vote share as

$$v_i = X_i'\beta + \gamma_i + \varepsilon_i$$

where $X_i'$ is a vector of predictor variables with coefficient $\beta$ , and $\gamma_i \sim N(0,\lambda \sigma^2)$ and $\varepsilon_i \sim N(0,(1-\lambda) \sigma^2)$ are the systematic and random error terms. In this presentation, $\sigma^2$ is the total error variance, and $\lambda$ is the share attributed to the systematic error component. The error terms in each district are independent of each other and of those in each other district in the system.

The standard approach to estimating the unknown quantities is to model each year under the Bayesian framework, with noninformative priors on the $\beta$ and $\sigma^2$ parameters in each year. To estimate $\sigma^2$ , take the total variance estimate in each year and pool those estimates together; then use the mean of the pooled estimates as the value of $\sigma^2$ as the value for each election.

To estimate $\lambda$ for the electoral system, note that the systematic component of the error is proportional to the votes received in each district in two subsequent elections, yielding

$$v_{i,t+1} = \lambda v_{i,t} + X_i'\beta + \gamma_i + \varepsilon_i,$$

where $X_i'$ may include as many variables as are available from elections $t$ and $t+1$ . The value of $\lambda$ used is the mean of these estimated values. Note that estimates of $\lambda$ can only be obtained when two subsequent elections use the same electoral map, i.e. no redistricting has taken place.