What is the underlying idea behind the probit model? Explain how parameters are estimated in the probit model.

The underlying idea behind the probit model is to model the probability of a binary outcome as a function of explanatory variables using a cumulative normal distribution function. In many cases, particularly in economics and social sciences, the dependent variable is binary, taking values such as success/failure, employed/unemployed, or yes/no. Unlike the linear probability model (LPM), which directly estimates probabilities using ordinary least squares (OLS), the probit model ensures that predicted probabilities remain between 0 and 1 by applying a nonlinear transformation.  

### Theoretical Foundation  

In the probit model, the latent variable approach is used to explain the relationship between the binary dependent variable and independent variables. Suppose there exists an unobserved continuous variable \( Y^* \) such that  

\[
Y^* = X\beta + \epsilon
\]  

where:  
- \( Y^* \) is a latent (unobserved) variable,  
- \( X \) is a vector of explanatory variables,  
- \( \beta \) is a vector of coefficients,  
- \( \epsilon \sim N(0,1) \) is a normally distributed error term.  

The observed binary outcome \( Y \) is determined as follows:  

\[
Y =
\begin{cases} 
1, & \text{if } Y^* > 0 \\
0, & \text{if } Y^* \leq 0
\end{cases}
\]  

This framework suggests that the probability of observing \( Y = 1 \) is given by:  

\[
P(Y = 1 | X) = P(Y^* > 0 | X) = P(X\beta + \epsilon > 0)
\]  

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