Linear Probability Model

The Linear Probability Model (LPM) is OLS applied directly to a binary outcome $y_i \in \{0, 1\}$. Under the zero-conditional-mean assumption, $\mathbb{E}[y_i \mid x_i] = \Pr(y_i = 1 \mid x_i) = \beta_0 + \beta_1 x_i$, so each $\hat\beta_j$ is a change in probability of the outcome (in percentage points). It is the simplest possible binary-response model and the coefficients are directly interpretable as marginal effects.

When to use

Reach for LPM as the first pass whenever the outcome is binary — it produces directly interpretable coefficients and lets you focus on the design rather than the link function. Two unavoidable drawbacks: (i) predicted probabilities can fall outside [0, 1] because nothing in OLS constrains them — the Andersen ultimatum-game lecture hits 1.19 — which motivates logit / probit; (ii) the residual variance $p(x)(1 - p(x))$ depends on $x$, so Heteroskedasticity is built in by construction — always report robust standard errors.