Probit Model - Introduction

Introduction

Suppose response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes form

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

where Pr denotes probability, and Φ is the Cumulative Distribution Function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood.

It is also possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable

where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive:

Y = \mathbf{1}_{\{Y^\ast>0\}} = \begin{cases} 1 & \text{if }Y^\ast > 0 \ \text{ i.e. } - \varepsilon < X'\beta, \\ 0 &\text{otherwise.} \end{cases}

The use of the standard normal distribution causes no loss of generality compared with using an arbitrary mean and standard deviation because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount.

To see that the two models are equivalent, note that

\begin{align} \Pr(Y = 1 \mid X) &= \Pr(Y^\ast > 0) = \Pr(X'\beta + \varepsilon > 0) \\ &= \Pr(\varepsilon > -X'\beta) \\ &= \Pr(\varepsilon < X'\beta) \quad \text{(by symmetry of the normal dist)}\\ &= \Phi(X'\beta) \end{align}

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