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The logit model is one of the most widely used models in econometrics to analyze situations where the dependent variable is binary, meaning it can only take two possible outcomes such as success or failure, yes or no, employed or unemployed, and so on. The underlying idea behind the logit model is that the probability of the dependent variable taking a certain value can be expressed as a non-linear function of a linear combination of explanatory variables. Ordinary linear regression is not appropriate for such problems because it can predict probabilities outside the \[0,1] interval and because the variance of the error terms is heteroskedastic. The logit model resolves this by using a logistic cumulative distribution function to link the explanatory variables to the probability outcome, ensuring that the predicted values always lie within the unit interval. The model begins with the specification of a latent variable framework. Suppose we have an unobservable continuous latent variable $y_i^*$ that depends linearly on a set of covariates: $$ y_i^* = X_i \beta + u_i, $$ where $X_i$ is the vector of explanatory variables, $\beta$ is the parameter vector, and $u_i$ is a random error term. What we observe is not $y_i^*$, but rather a binary outcome variable $y_i$, defined as: $$ y_i = \begin{cases} 1 & \text{if } y_i^* > 0, \\ 0 & \text{otherwise}. \end{cases} $$ Thus, the probability that $y_i = 1$ is equal to the probability that the latent variable is greater than zero. If we assume that the distribution of $u_i$ follows a logistic distribution, then the probability that $y_i = 1$ can be expressed as: $$ P(y_i = 1|X_i) = \frac{e^{X_i \beta}}{1 _____ ___ ____ _____ _______ _____ ________ ________ ___ ___.
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