Probit model

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In statistics, a probit model is a popular specification of a generalized linear model, using the probit link function. Probit models were introduced by Chester Bliss. Because the response is a series of binomial results, the likelihood is often assumed to follow the binomial. Let Y be a binary outcome variable, and let X be a vector of regressors. The probit model assumes that

\Pr(Y=1|X=x) = \Phi(x'\beta),

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 \ \stackrel{\mathrm{def}}{=}\ 1(y^* >0).

Then it is easy to show that

\Pr(Y=1 | X=x) = \Phi(x'\beta).
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