Why does a linear probability model fitted with OLS fail on a binary outcome?

## What the linear probability model is Fit ordinary least squares with a 0/1 response and read the fitted value as a probability: ``` Y = b0 + b1*x1 + b2*x2 + error, Y in {0, 1} E[Y | X] = P(Y = 1 | X) ``` Because the expected value of a 0/1 variable *is* the probability that it equals 1, this is not nonsense on its face — the regression really is estimating a conditional probability, and `b1` reads as "a one-unit increase in x1 raises the probability of the event by b1", straight in percentage points. The problems are with the functional form and the error structure. ## Problem 1: unbounded predictions A plane has no ceiling. Fit a churn model on tenure and monthly spend and you will find real rows with fitted values like -0.12 and 1.30. These are not merely inelegant: - They cannot be used in an expected-value calculation. A -0.12 probability of churn multiplied by a customer's lifetime value produces a negative expected loss. - Clipping to [0, 1] afterwards is a patch, not a fix. It leaves the coefficients that produced the impossible values untouched and creates a discontinuity in the fitted function. - The failure concentrates precisely where the decisions matter: at the extreme, highest-risk and lowest-risk accounts. How often this bites depends on the data. If every fitted value sits between 0.2 and 0.8, the linear approximation is nearly harmless; if the base rate is 2% or the predictors are strong, it is everywhere. ## Problem 2: heteroscedasticity by construction For a Bernoulli outcome with success probability `p`, the conditional variance is ``` Var(Y | X) = p * (1 - p) ``` Since `p` varies with the predictors, so does the error variance — it is maximised at `p = 0.5` (variance 0.25) and shrinks toward 0 at both extremes. Homoscedasticity is a core assumption behind the usual OLS standard-error formula, and here it is violated *by the definition of the outcome*, not by accident of the data. The consequence is specific and worth stating precisely: the coefficient estimates remain unbiased for the linear approximation, but the default standard errors, t-statistics and p-values are wrong. Heteroscedasticity-robust standard errors repair the inference without repairing the functional form. A related, more cosmetic point: the residuals can only take two values at any given `X` (namely `1 - fitted` and `-fitted`), so they are nowhere near normal. This matters far less than it is usually claimed to — OLS does not require a normal outcome — so a candidate who names non-normality as the *main* defect has the priorities backwards. ## Problem 3: a constant slope is the wrong shape The linear model asserts that a one-unit change in a predictor moves the probability by the same amount everywhere. Substantively that is rarely credible. Moving an account from a 45% to a 55% churn risk is one thing; the same nominal change applied to an account already at 95% would take it to 105%. Real effects taper as the outcome approaches certainty in either direction. The logistic form encodes exactly that taper: its marginal effect is `b * p * (1 - p)`, largest when `p` is near 0.5 and vanishing at the extremes. ## What the logit link changes Modelling `log(p / (1 - p))` as linear makes the fitted probability `1 / (1 + exp(-(b0 + b1*x1 + ...)))`. That curve is bounded in (0, 1) with asymptotes it never crosses, it saturates smoothly, and the model is fitted by maximum likelihood using the Bernoulli distribution, so the variance structure is part of the model rather than an assumption it violates. The price is interpretability: coefficients now speak in log-odds and need exponentiating into odds ratios or converting into marginal effects before anyone outside the modelling team can act on them. ## Why practitioners still fit the linear model anyway Honesty here impresses interviewers. In applied economics the linear probability model is common and defensible, for concrete reasons: - **Direct interpretation.** The coefficient is already an average partial effect in percentage points, with no conversion step. - **Close agreement in the middle.** When fitted probabilities stay roughly within 0.2 to 0.8, its coefficients approximate the logistic model's average marginal effects well. - **Fixed effects and instruments.** Linear estimators handle high-dimensional fixed effects and instrumental-variable designs cleanly, where the nonlinear analogues are harder. The defensible version always comes with robust standard errors, a check on how many fitted values fall outside [0, 1], and an acknowledgment that the model is an approximation to a conditional probability rather than a description of one. The indefensible version is fitting OLS to a 0/1 column, reading the default p-values, and reporting a negative probability with a straight face.