Questions: The identity link is the standard one with normal responses but is not often used with binary or count responses. Why do you think this is?
My idea: The range for a linear predictor, and the range of the identity link applied to a binomial probability or to a Poisson mean. I am kinda confused
If you use the identity link for the binary data, then your regression equation would be $$ {\rm Pr}(Y_i|x_i)\equiv p_i = \beta_0 + \beta_1 x_i. $$ Without any restriction on the coefficients, RHS can range in real-line (for example, a negative value). Since this does not make any sense for probability $p_i$, we transform it so that $g(p_i)$ takes its value in $[-\infty, \infty]$.
