Refer to Section 12.3.1. Using supplementary information improves predictions. Let q_i denote the true proportion of votes for Clinton in state i in the 1992 election, conditional on voting for him or Bush. Consider the model

logit[P(Y_it = 1|u_i)] = logit(q_i) + α + u_i,

where {q_i} are known and {u_i} are independent N(0,σ²). When σ̂ = 0, show π̂_i = q_i exp(α̂)/[1 – q_i + q_i exp(α̂)]. Compared to {q_i}, explain how π̂_i then shifts up or down depending on how the overall Democratic vote compares in the current poll to the previous election (i.e., depending on α̂). When also α̂ = 0, show π̂_i = q_i.