Logit & Probit Models: Binary Dependent Variables in Regression

Published: March 19, 2026

Article by Ryan O'Connell, CFA, FRM

When a bank evaluates a loan application, the outcome is binary: default or no default. An analyst might be tempted to use ordinary least squares to predict this 0/1 outcome, but OLS can produce predicted “probabilities” below zero or above one — nonsensical results for a probability model. The logit model solves this problem by mapping the linear predictor through the logistic cumulative distribution function (CDF), constraining all predicted probabilities to the interval (0, 1). The closely related probit model uses the standard normal CDF instead. Both are estimated by maximum likelihood and have become the standard tools for modeling binary dependent variables in finance and economics. This guide covers the logit and probit specifications, maximum likelihood estimation, how to interpret coefficients through marginal effects and odds ratios, and when to choose logit versus probit versus the linear probability model.

The Problem with OLS for Binary Outcomes

When the dependent variable takes only the values 0 and 1, applying OLS produces the linear probability model (LPM). The LPM interprets the fitted values as probabilities: P̂(Y = 1 | X) = β̂₀ + β̂₁X₁ + … + β̂_kX_k. While simple and easy to interpret, the LPM has three fundamental limitations for binary outcomes:

• Predicted probabilities outside [0, 1]. Nothing prevents the linear function from producing fitted values of −0.15 or 1.30 — values that have no meaning as probabilities. In credit scoring with extreme borrower profiles, this occurs routinely.
• Inherent heteroskedasticity. Because Var(Y | X) = P(X)[1 − P(X)], the error variance depends on X by construction whenever the slope coefficients are nonzero. Standard errors require robust corrections at minimum.
• Constant marginal effects are unrealistic. The LPM assumes a 1-unit increase in credit score has the same effect on default probability whether the borrower starts at 5% or 50% risk. In reality, marginal effects should diminish near the boundaries.

The Logit Model (Logistic Regression)

Where:

• P(Y = 1 | X) — the predicted probability that the binary outcome equals 1, given the values of the independent variables
• Y — the binary dependent variable (0 or 1)
• β₀ — the intercept
• β₁ … β_k — the slope coefficients for each independent variable
• X₁ … X_k — the independent (explanatory) variables
• exp(·) — the exponential function

The logistic function is symmetric around zero: when the linear index equals zero, the predicted probability is exactly 0.5. As the index increases toward positive infinity, the probability approaches 1; as it decreases toward negative infinity, the probability approaches 0. This S-shaped curve naturally captures diminishing marginal effects near the boundaries.

The logit model can be equivalently expressed through the log-odds (logit) transformation, which linearizes the relationship:

Where:

• P — the predicted probability P(Y = 1 | X)
• P / (1 − P) — the odds of Y = 1 (the ratio of the probability of the event to the probability of the non-event)
• log[P / (1 − P)] — the log-odds (the natural logarithm of the odds), also called the logit

This leads to a powerful interpretation: exponentiating a coefficient gives the odds ratio. If β₁ = 0.045, then exp(0.045) = 1.046, meaning a one-unit increase in X₁ multiplies the odds of Y = 1 by 1.046 (a 4.6% increase in the odds), holding all other variables constant.

The logit model also has a latent variable interpretation: an unobserved continuous variable y* = β₀ + βX + e determines the binary outcome, with Y = 1 when y* > 0. In the logit model, the error term e follows a logistic distribution (mean 0, variance π²/3).

The Probit Model

The probit model replaces the logistic CDF with the standard normal CDF, denoted Φ(·):

Where:

• Φ(·) — the cumulative distribution function (CDF) of the standard normal distribution, which maps any real number to a probability between 0 and 1

In the latent variable framework, the probit model assumes the error follows a standard normal distribution (mean 0, variance 1) rather than a logistic distribution. Because the standard normal has a smaller variance than the logistic distribution (π²/3 ≈ 3.29), probit coefficients are systematically smaller: logit coefficients are approximately 1.6 times probit coefficients. This scaling difference is mechanical — it does not reflect a substantive disagreement between the models.

In practice, logit and probit produce nearly identical predicted probabilities and marginal effects. The logit model has slightly heavier tails, but the difference is rarely meaningful in applied work.

Maximum Likelihood Estimation

Because the logit and probit models are nonlinear in the parameters, OLS cannot be used. Instead, both models are estimated by maximum likelihood estimation (MLE), which finds the parameter values that make the observed data most probable.

For each observation i, the contribution to the likelihood is P(X_i)^y_i × [1 − P(X_i)]^{(1 − y_i)}. Taking the logarithm and summing across all n observations gives the log-likelihood:

Where:

• ln L — the log-likelihood (the natural logarithm of the total likelihood of the observed data)
• y_i — the observed binary outcome for observation i (0 or 1)
• P(X_i) — the predicted probability of Y = 1 for observation i, given its covariate values
• n — the number of observations in the sample

There is no closed-form solution — the computer uses iterative numerical methods to find the maximum. Under standard regularity conditions, the MLE is consistent, asymptotically normal, and asymptotically efficient. Standard errors are computed from the inverse of the Hessian matrix (the matrix of second derivatives of the log-likelihood).

For testing multiple restrictions simultaneously (the logit/probit analog of the F-test in OLS), the Likelihood Ratio (LR) test compares the log-likelihoods of the restricted and unrestricted models: LR = −2(ln L_r − ln L_ur) ~ χ²_q, where q is the number of restrictions. For the OLS version of multiple hypothesis testing, see our guide on hypothesis testing in regression.

Model fit is assessed with McFadden’s pseudo R-squared: 1 − (ln L_full / ln L_null), where the null model includes only an intercept. As a rough heuristic, pseudo R-squared values of 0.2 to 0.4 are often seen in well-fitting binary models, though the appropriate range depends on the application and data. The percent correctly predicted — the share of observations where the predicted probability correctly classifies Y at a 0.5 threshold — provides an intuitive but crude alternative. However, this metric depends heavily on the chosen cutoff and can be misleading with imbalanced samples: in a loan portfolio where only 5% of borrowers default, a model that predicts “no default” for everyone achieves 95% accuracy while being completely uninformative.

Interpreting Logit and Probit Coefficients

The partial effect (marginal effect) of a continuous variable x_j on the predicted probability is:

Where:

• ∂P / ∂x_j — the marginal effect of variable x_j on the predicted probability
• g(·) — the density function of the link distribution: the logistic density g(z) = P(1 − P) for logit, or the standard normal density φ(z) for probit
• β₀ + βX — the linear index (the linear combination of all regressors and their coefficients)
• β_j — the estimated coefficient on variable x_j

The scale factor g(z) is always positive, so the marginal effect has the same sign as β_j. Because the marginal effect varies with X, two standard summary measures are used:

• Marginal Effect at the Mean (MEM): Evaluate the partial effect at the sample mean values of all regressors. Simple to compute, but the “average borrower” may not be representative of any actual borrower.
• Average Marginal Effect (AME): Compute the partial effect for every observation in the sample, then average. This uses the full distribution of X values and is generally preferred by applied researchers.

For logit, the maximum scale factor is 0.25 (when the predicted probability equals 0.5), so the largest possible marginal effect for any variable is 0.25 × β_j. For probit, the maximum scale factor is approximately 0.40 (at z = 0).

Logit vs. Probit vs. Linear Probability Model

In practice, all three models produce similar marginal effects when predicted probabilities are well within the (0.2, 0.8) range. The choice matters most at the tails — when modeling events with very high or very low base rates (such as loan defaults, which typically occur in 1–5% of a portfolio), logit and probit provide more reliable probability estimates than the LPM. When comparing logit and probit results, always compare average marginal effects (AMEs), not raw coefficients — the raw coefficients differ by a factor of approximately 1.6 due to the different scale of the underlying distributions, but the AMEs will be nearly identical.

Applications in Finance

Logit and probit models are widely used across financial applications wherever the outcome is naturally binary:

• Credit scoring and probability of default. Banks estimate the probability that a borrower will default on a loan using financial ratios, credit history, and macroeconomic variables as predictors. Logit is the dominant model in the industry due to its odds ratio interpretation. For the full probability of default framework including Basel regulatory requirements, see our guide on credit risk and probability of default.
• Customer churn prediction. Financial services firms model whether a client will close an account (Y = 1) or remain (Y = 0) based on account activity, fee sensitivity, service interactions, and competitor pricing.
• Merger and acquisition prediction. Researchers model whether a target firm will receive a takeover bid as a function of firm size, Tobin’s Q, leverage, free cash flow, and industry concentration.
• Buy/sell signal models. Quantitative analysts predict whether a stock will outperform a benchmark over the next period (Y = 1) or not (Y = 0), using fundamental ratios, technical indicators, and momentum variables as inputs.

Common Mistakes

1. Interpreting logit or probit coefficients as marginal effects. A logit coefficient of −0.008 does not mean the probability drops by 0.008. The marginal effect depends on the values of all regressors through the density function g(Xβ). Always report average marginal effects (AME) or marginal effects at the mean (MEM) alongside raw coefficients to convey the economic magnitude of the relationship.

2. Comparing pseudo R-squared to OLS R-squared. McFadden’s pseudo R-squared typically ranges from 0.2 to 0.4 for well-fitting binary models. A pseudo R-squared of 0.25 does not mean the model “only explains 25% of the variation” — it is a fundamentally different metric measured on a different scale. Comparing pseudo R-squared values across logit models is reasonable; comparing them to OLS R-squared values is not.

3. Not reporting marginal effects. Publishing only raw logit coefficients or odds ratios without marginal effects makes it impossible for readers to assess the economic magnitude of each predictor. Marginal effects translate results into probability-scale changes that can be compared across studies and models.

4. Ignoring finite-sample bias and separation. These are related but distinct problems. Finite-sample bias: MLE is a large-sample estimator, so with fewer than roughly 10 events (Y = 1) per estimated parameter, logit and probit coefficients can be substantially biased and standard errors inflated. A common rule of thumb is at least 10 events per predictor. Separation is a different issue: it occurs when a predictor (or combination of predictors) perfectly predicts the outcome for a subset of observations, causing one or more coefficient estimates to diverge toward ±∞. Separation can arise even in moderately large samples if a category has zero events — for example, if no borrower with collateral ever defaults in the sample.

5. Comparing raw coefficients across logit and probit models. Logit coefficients are approximately 1.6 times probit coefficients because the logistic distribution has a larger variance (π²/3 ≈ 3.29) than the standard normal (variance = 1). Comparing magnitudes directly without adjusting for this scale difference is meaningless. Compare marginal effects instead — they will be nearly identical across the two models.