Logit and Probit Analyses

Deriving the Logit and Probit Models

Perhaps the most straightforward way to derive the logit and probit models is to assume that a political agent's choice is determined by some continuous underlying rating of a political item or a propensity to engage in some behavior known as a latent variable. Each political agent's position on this latent variable (yi∗) is assumed to be a linear function of a set of some observed variables and coefficients (Xiβ) and some unobserved stochastic influences (εi). Thus, a political agent's latent choice preference is given by

If yi∗ was observed, we could simply estimate this equation via linear regression. However, in a binary choice setting, the agent's position on this latent variable is unobserved. Instead, a choice threshold τ divides the latent variable into two categories, and we only observe y, which indicates which choice category the agent selected. We observe yi = 0 for agents who are at or below τ and yi = 1 for agents above the threshold. For instance, a country might have an underlying propensity to fight with its neighbors, but we can only observe this country in one of two discrete choice categories: war or peace.

Since yi∗ is latent, we must make several identifying assumptions to estimate the probability that a political agent makes a particular choice. Although these identifying assumptions are arbitrary, they do not influence the estimate of the relationship between the Xis and choice behavior. The first of these assumptions is to set τ = 0. We make this assumption because we cannot obtain a unique estimate for both τ and the constant term in the model: Since y∗ is unobserved, we could add an arbitrary constant to both τ and the constant term and still generate the same probability of an observed outcome. With this assumption, and noting that yi∗ = Xiβ + εi, the probability that an agent selects the choice category represented by y = 1 given the observed variables Xi is

This equation reveals that the probability that political agents select the choice category denoted by y = 1 will depend in part on the unobserved error term εi. By subtracting Xiβ from both sides of the inequality and assuming that the distribution of εi is symmetric, we can rearrange this equation to read as

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