Probit model

The probit model is a popular specification of a binary regression model. Let $Y$ be a binary outcome variable, and let $X$ be a vector of regressors. The probit model assumes that

P r (Y = 1 | X = x) = Φ(x'β)

where $Φ$ is the cumulative distribution function of the standard normal distribution. The parameters $β$ are typically estimated by maximum likelihood.

The probit model can be obtained from a simple latent variable model. Suppose that

y * = x'β + ε,

where $\epsilon | x \sim N(0,1)$ , and suppose that $Y$ is an indicator for whether the latent variable $y *$ is positive:

$Y \equiv 1(y^* >0).$

Then it is easy to show that

P r (Y = 1 | X = x) = Φ(x'β).

Categories: Econometrics | Statistics