The Linear Probability Model: When the Dependent Variable Is Binary

Published: March 19, 2026

Article by Ryan O'Connell, CFA, FRM

The linear probability model is the simplest econometric approach to modeling a binary outcome. When a credit analyst needs to estimate the probability that a borrower will default, the most direct method is to apply ordinary least squares (OLS) to a dependent variable coded as 0 (no default) or 1 (default). The result is a model whose fitted values can be read as predicted probabilities and whose coefficients have an unusually clean interpretation. The LPM has real advantages — and well-known limitations that every practitioner should understand. For the nonlinear alternatives that address those limitations, see our guide on logit and probit models.

What Is the Linear Probability Model?

The linear probability model applies OLS regression to a binary (0/1) dependent variable. Because Y can only be 0 or 1, the conditional expectation E(Y | X) equals the probability that Y = 1 given X. The LPM models this probability as a linear function of the regressors.

Where:

• E(Y | X) = P(Y = 1 | X) — the probability that the outcome equals 1, given the values of X
• Y — binary dependent variable (0 or 1)
• β₀ — intercept
• β₁, …, β_k — slope coefficients, each measuring the effect of a one-unit change in the corresponding X on P(Y = 1)
• X₁, …, X_k — independent variables (regressors)

The LPM is a special case of multiple regression where the dependent variable happens to be binary. Note that having a binary dependent variable is distinct from using dummy variables as independent regressors — in the LPM, it is the outcome itself that takes only two values.

Interpreting Linear Probability Model Coefficients

LPM coefficients have a direct, intuitive interpretation that makes the model attractive for applied work.

Continuous regressors. The coefficient β_j represents the change in P(Y = 1) for a one-unit increase in X_j, holding all other variables constant. This is a percentage-point change, not a percent change. For example, if β = −0.0008 on credit score in a loan default model, a 1-point increase in credit score reduces default probability by 0.08 percentage points. A 100-point improvement (say, from 650 to 750) reduces default probability by 8.0 percentage points.

Binary regressors. When X_j is a 0/1 dummy variable, the coefficient represents the ceteris paribus difference in predicted probability between the two groups. For instance, a “secured loan” dummy with coefficient −0.05 means secured loans have a predicted default probability 5 percentage points lower than unsecured loans, holding borrower characteristics constant.

Constant marginal effects. The LPM imposes the same absolute effect regardless of the baseline probability. A 100-point credit score improvement has the same 8.0-percentage-point effect whether the borrower starts at 5% or 50% default risk. This linearity is a simplification — near the boundaries of 0 and 1, marginal effects should logically compress toward zero. For models with diminishing marginal effects near the boundaries, see logit and probit models.

Advantages of the Linear Probability Model

Despite its limitations, the LPM remains widely used in applied econometrics for several good reasons:

• Direct coefficient interpretation. Coefficients are marginal effects. There is no need to compute average marginal effects (AME) or marginal effects at the mean (MEM) as with logit or probit. What you estimate is what you interpret.
• Computational simplicity. OLS has a closed-form solution — no iterative maximum likelihood estimation is required. R² remains valid, and joint hypothesis tests can be conducted using heteroskedasticity-robust Wald statistics available in all standard software.
• Works reasonably well in the interior. When predicted probabilities cluster roughly between 0.2 and 0.8 (a rough heuristic, not a hard rule), LPM marginal effects often approximate logit and probit average partial effects closely.
• Compatible with causal inference designs. The LPM is standard in difference-in-differences and instrumental variable settings because adding a nonlinear link function can complicate the identification argument without meaningfully changing the estimated effects.

It is worth noting that the main concerns with the LPM are functional form and inference — not coefficient bias. Under correct exogeneity assumptions (Wooldridge’s MLR.1–MLR.4), LPM coefficients remain unbiased for the linear projection, regardless of the functional form limitations described below.

Limitations of the Linear Probability Model

The LPM has four well-known problems, all stemming from applying a linear function to a probability that is inherently bounded between 0 and 1.

1. Predicted probabilities outside [0, 1]. Nothing constrains the linear function to produce values between 0 and 1. With extreme regressor values — a very high credit score combined with low debt — the model can predict −0.05 or 1.12. These have no meaning as probabilities.

2. Inherent heteroskedasticity. Because the dependent variable is binary, the conditional variance of Y given X takes a specific form that depends on the regressors:

Because p(X) varies with X by construction, the error variance is non-constant — the model is heteroskedastic. This means standard OLS standard errors are invalid for inference. For the general treatment of heteroskedasticity and its consequences, see our guide on heteroskedasticity.

3. Non-normal errors. The error term in the LPM can take only two values: (1 − p(X)) when Y = 1 and −p(X) when Y = 0. This is clearly not normally distributed. However, with large samples, the central limit theorem ensures asymptotic normality of the OLS estimator, so this is primarily a small-sample concern.

4. Constant marginal effects. The LPM forces each regressor to have the same absolute effect on the probability regardless of the baseline. A credit score improvement should logically have a smaller effect on default probability for a borrower already at 2% risk than for one at 40% risk. The LPM cannot capture this diminishing-returns pattern.

When to Use the LPM vs. Logit or Probit

The choice between the LPM and nonlinear alternatives depends on the research goal (causal effect estimation vs. probability prediction), the distribution of predicted probabilities, and interpretive needs.

In practice, all three models often produce similar average partial effects when predicted probabilities cluster in the interior (roughly 0.2 to 0.8 — a heuristic, not a hard rule). The choice matters most when modeling outcomes near the probability boundaries. Important: raw logit and probit coefficients are not directly comparable to LPM coefficients. To compare results across models, compute marginal effects. For full coverage of logit and probit estimation, see our guide on logit and probit models.

Robust Standard Errors for the Linear Probability Model

Because the LPM is inherently heteroskedastic — Var(Y | X) = p(X)[1 − p(X)] depends on X by construction — standard OLS standard errors are invalid. The t-statistics, p-values, and confidence intervals produced by default OLS output cannot be trusted.

The solution: heteroskedasticity-robust (White) standard errors. These adjust the variance-covariance matrix using observation-specific squared residuals rather than a common σ². Robust standard errors are the best-practice default for LPM inference because heteroskedasticity is guaranteed with binary outcomes. Their justification is asymptotic — they rely on large-sample properties, so they are most reliable with moderate to large sample sizes.

WLS as a theoretical alternative. In principle, one could apply weighted least squares by weighting each observation by 1/√[p̂(1 − p̂)], where p̂ is the fitted probability. In practice, when any fitted probability falls outside [0, 1], the quantity p̂(1 − p̂) becomes nonpositive, making the implied WLS weights invalid. This practical complication makes WLS unreliable for many LPM applications.

LPM Example: Loan Default Prediction

To see how the linear probability model works in practice, consider a bank that estimates the probability of loan default using data from 12,500 consumer loans.

For the broader credit risk framework — including probability of default within regulatory and Basel contexts — see our guide on credit risk and probability of default.

Common Mistakes

These are the most frequent errors practitioners make when working with the linear probability model.

1. Using standard OLS standard errors. Heteroskedasticity is built into any LPM with nonzero slope coefficients. Reporting standard OLS standard errors produces invalid t-statistics and p-values. Always use heteroskedasticity-robust standard errors.

2. Ignoring out-of-range predicted probabilities. When a nontrivial share of observations have fitted values below 0 or above 1, the linear specification is poorly matched to the data. Always report the fraction of out-of-range predictions. If the share is substantial, consider logit or probit.

3. Using the LPM when bounded probabilities or calibration quality is the objective. The LPM estimates marginal effects well, but it is generally not the best choice when the goal is to produce well-calibrated probability scores for individual observations. Logit and probit produce predictions that are bounded in (0, 1) and are typically better calibrated, especially in the tails.

4. Assuming constant marginal effects are always a problem. For many causal-inference questions — such as whether a regulatory change affects corporate bond default rates — the average partial effect is the quantity of interest. The LPM’s constant-effect assumption approximates this well in many settings. The linearity becomes most problematic when modeling outcomes near the 0 or 1 boundaries.

5. Confusing percentage points with percent changes. An LPM coefficient of 0.0045 means a 0.45 percentage-point increase in default probability, not a 0.45% increase. At a 10% baseline default rate, 0.45 percentage points represents a 4.5% relative increase — these are very different numbers. Always specify percentage points when reporting LPM results.