Does the Link Matter? This is based on Santos Silva (1999). Consider a binary random variable Yi such that P(Yi = 1|x) = F(0 +

Does the Link Matter? This is based on Santos Silva (1999). Consider a binary random variable Yi such that P(Yi = 1|x) = F(β0 + β1xi), i= 1, . . . , n, where the link F(·) is a continuous distribution function.

(a) Write down the log-likelihood function and the first-order conditions of maximization with respect to β0 and β1.

(b) Consider the case where xi only assumes two different values, without loss of generality, let it be 0 and 1. Show that  F(1) =



xi=1 yi/n1, where n1 is the number of observations for which xi = 1. Also, show that  F(0) =



xi=0 yi/(n − n1).

(c) What are the maximum likelihood estimates of β0 and β1?

(d) Show that the value of the log-likelihood function evaluated at the maximum likelihood estimates of β0 and β1 is the same, independent of the form of the link function.