a) The nonlinear Logistic regression model in lecture 1-2 specifies shooting probability pi ≡
P r(Shootingi = 1) by
log pi
1 − pi
= X0
iβ(⇔ pi = (1 + exp(−X0
iβ))−1
),
where Xi are dependent variables and β is the unknown coefficients we try to identify and estimate.
Fryer’s racial bias study uses the above pi specification because it guarantees 0 ≤ pi ≤ 1. But
the above pi specification is just one of the many possible specifications with the same property
0 ≤ pi ≤ 1. Provide an alternative expression of p
∗
i
in terms of β and Xi that also guarantees
0 ≤ p
∗
i ≤ 1.
b) How would you like to estimate β with the alternative p
∗
i you come up with in (a)? Define your
estimate of β as the solution to an optimization problem. Hint: Mimic the method we use to
estimate logistic regression.
c) Between the logistic method in (a) and your method in (b), which is a better way to estimate racial
bias? Why?
Sample Solution