# Why are the fitted probabilities for the linear probability model and the probit model identical?

I estimated a linear probability model (LPM) $P(y=1|x_1) = b_0 +b_1x_1 + u$ and a probit model $P(y=1|x_1) = \Phi(b_0 +b_1x_1 + u)$, where $\Phi()$ denotes the cumulative normal distribution. The regressor $x_1$ is the same binary variable.

I observed that the predicted probabilities for both models $\hat{y} =\hat{b_0} +\hat{b_1}x_1$ and $\hat{y} =\Phi(\hat{b_0} + \hat{b_1}x_1)$ are identical across both models. Why is this the case from a theoretical point of view?

• Assuming $y$ is binary, $y = b_0 +b_1x_1 + u$ is not a linear probability model (LPM), just a usual linear regression model. $P(y=1\mid x_1) = b_0 +b_1x_1 + u$ is the specification for the LPM. Also $y = \Phi(b_0 +b_1x_1 + u)$ is not a probit model - $P(y=1\mid x_1) = \Phi_u(b_0 +b_1x_1)$ is. Are these just typos of the question, or you actually estimated the specifications that you write? – Alecos Papadopoulos Dec 18 '13 at 19:52

Because the model's saturated: with a binary response & a single binary predictor you get a perfect fit no matter what link function you use, with the estimated probability being the observed probability for each value of $x_1$. Think of fitting a line to two points: it doesn't matter whether it's straight or curvy; if it has two parameters it'll go through them.