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Linear probability model

The linear probability model (LPM) is a regression technique in econometrics that estimates the probability of a binary outcome—such as success or failure, participation or non-participation—as a linear function of explanatory variables, using ordinary least squares (OLS) estimation.^[1] Formally specified as P(Y=1 \mid X) = \beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k, where Y is the binary dependent variable and X represents covariates, the model treats the outcome probability directly as the conditional expectation E(Y \mid X).^[1] The resulting coefficients \beta_j provide straightforward interpretations as the marginal change in probability associated with a unit increase in X_j, holding other factors constant, making it particularly appealing for average partial effects in empirical analysis.^[2] Despite its simplicity and low computational demands, the LPM has notable drawbacks that stem from its linear assumption.^[1] Predicted probabilities can fall outside the valid [0,1] range, especially for covariate values far from the mean, leading to potential nonsensical forecasts.^[2] Additionally, the error term exhibits heteroskedasticity because the conditional variance \text{Var}(Y \mid X) = P(Y=1 \mid X) \cdot (1 - P(Y=1 \mid X)) varies with the predicted probability, necessitating robust standard errors for valid inference.^[1] These issues have historically prompted the development of nonlinear alternatives like logit and probit models, which enforce the [0,1] bound through functional forms such as the logistic or cumulative normal distribution.^[2] The LPM remains a popular choice in modern econometrics, especially for causal inference with binary treatments or outcomes, as advocated by influential works emphasizing its robustness when interest lies in average effects rather than precise probability forecasts. For instance, under conditions like symmetric covariate distributions or exhaustive binary indicators, OLS estimates from the LPM can closely approximate average partial effects from nonlinear models without significant bias.^[2] Its ease of implementation in software and compatibility with panel data or instrumental variables further enhances its utility in applied research, though users must verify the fraction of in-range predictions and address heteroskedasticity to ensure reliability.

Overview

Definition and Purpose

The linear probability model (LPM) is a regression technique used to estimate the probability of a binary outcome occurring as a linear function of one or more explanatory variables, where the dependent variable takes values of 0 or 1.^[3] In this framework, the model directly specifies the conditional expectation of the binary variable, which equals the probability that the outcome equals 1 given the covariates.^[4] The general form of the LPM is given by

P(Y=1 \mid X) = \beta_0 + \beta_1 X_1 + \cdots + \beta_k X_k,

where Y is the binary dependent variable, X = (X_1, \dots, X_k) are the explanatory variables, and the \beta coefficients represent the change in the probability of Y=1 associated with a one-unit change in the corresponding X_j, holding other variables constant.^[3] This direct interpretation of coefficients as marginal effects makes the LPM particularly appealing for modeling dichotomous outcomes, such as whether an individual participates in a government program or experiences a specific event.^[5] In econometrics and statistics, the LPM serves primarily to provide straightforward estimates of how covariates influence the likelihood of binary events, facilitating policy analysis and causal inference in settings with limited dependent variables.^[5] To ensure consistent estimation, the model relies on key assumptions inherited from ordinary least squares regression: linearity in the parameters, no perfect multicollinearity among the regressors, and exogeneity (i.e., the conditional mean of the error term given the covariates is zero).^[3]

Historical Development

The linear probability model (LPM) emerged in the mid-20th century within econometrics as a straightforward application of ordinary least squares regression to binary dependent variables, addressing qualitative choice problems where outcomes were limited to 0 or 1.^[6] Its roots lie in earlier efforts to adapt linear regression techniques to non-continuous data in social sciences, with pre-1970s applications including estimates of labor force participation.^[6] The model's appeal stemmed from its computational ease, allowing economists to estimate marginal effects directly without complex nonlinear optimization, which was particularly valuable in an era of limited computing resources.^[6] By the 1970s, the LPM was popularized in pedagogical texts, notably Damodar Gujarati's Basic Econometrics (first edition, 1978), which introduced it to students and practitioners as an accessible entry point for binary regression analysis in applied fields like labor economics and policy evaluation. This period marked its establishment as a baseline tool, often contrasted with emerging nonlinear methods but favored for interpretability in empirical work. The model's evolution reflected a tension between practicality and theoretical rigor; it served as a precursor to more sophisticated alternatives until the late 20th century, with Takeshi Amemiya's Advanced Econometrics (1985) providing a seminal theoretical treatment, including derivations of estimation properties and bias analyses that highlighted its role in advanced binary modeling. Throughout, the LPM's persistence in applied social sciences underscored a preference for simplicity in contexts where exact probability bounds were secondary to causal inference.^[6]

Model Specification

Basic Linear Form

The basic linear form of the linear probability model (LPM) specifies the conditional probability of a binary outcome directly as a linear function of the covariates. For an observation i, let Y_i be a binary dependent variable taking values 0 or 1, and let X_i be a $1 \times K vector of explanatory variables including a constant term. The model is given by

P(Y_i = 1 \mid X_i) = X_i \beta,

where \beta is a K \times 1 vector of parameters.^[3] This formulation treats the probability itself as the response variable in a linear regression setup, without invoking an underlying continuous process. The coefficients in the LPM have a direct and intuitive interpretation in terms of probabilities. Specifically, the coefficient \beta_j on covariate X_{ij} measures the change in the probability P(Y_i = 1 \mid X_i) associated with a one-unit increase in X_{ij}, holding all other covariates constant.^[3] Unlike in nonlinear models such as logit or probit, these marginal effects are constant across all values of the covariates, simplifying the analysis of how changes in predictors affect the outcome probability. The LPM can be rewritten in a regression form that reveals its implicit error structure: Y_i = X_i \beta + \varepsilon_i, where the error term is \varepsilon_i = Y_i - X_i \beta. Under the model, the errors satisfy E(\varepsilon_i \mid X_i) = 0, ensuring the conditional mean of Y_i given X_i is correctly specified as X_i \beta.^[3] However, because Y_i is binary, the conditional variance of the errors is \text{Var}(\varepsilon_i \mid X_i) = p_i (1 - p_i), where p_i = X_i \beta, which varies with X_i and induces heteroskedasticity.^[3] Key assumptions for the basic linear form include the strict exogeneity condition, E(\varepsilon_i \mid X_i) = 0, which underpins the unbiasedness of the parameters, along with no perfect multicollinearity among the covariates in X_i.^[3] Notably, normality of the errors is not required for the ordinary least squares estimator to be consistent, as consistency relies on the first two moments rather than the full distribution.^[7] This direct probability specification provides a straightforward foundation, which can alternatively be interpreted through a latent variable lens in related formulations.

Latent Variable Formulation

The linear probability model arises from a latent variable framework in which an unobserved continuous variable determines the observed binary outcome through a threshold crossing. Specifically, assume a latent variable Y_i^* = X_i \beta + u_i, where X_i is a vector of covariates, \beta is a parameter vector, and u_i is a mean-zero error term; the observed binary outcome is then Y_i = 1 if Y_i^* > 0 and Y_i = 0 otherwise.^[8]^[1] The connection to the observed binary variable follows from the probability P(Y_i = 1 | X_i) = P(u_i > -X_i \beta). Under standardization where X_i \beta lies between 0 and 1, this equals X_i \beta if the distribution of u_i implies a linear cumulative distribution function for -u_i in the relevant range, yielding the linear form of the model.^[1]^[2] Linearity in probabilities requires u_i to follow a specific distribution, such as uniform on [0, 1] (with Y_i^* = X_i \beta - u_i), which produces exact linearity within bounds; alternatively, truncated logistic or normal distributions can approximate linearity over limited ranges of X_i \beta, though the uniform case provides a precise justification.^[2]^[9] This formulation connects the linear probability model to foundational ideas in early probit analysis, which employed a similar latent threshold structure but with normally distributed errors to model S-shaped probabilities, whereas the linear probability model simplifies by assuming a uniform error distribution and a fixed threshold at zero.^[10]^[8]

Estimation Methods

Ordinary Least Squares

The linear probability model (LPM) is typically estimated using ordinary least squares (OLS), a method that minimizes the sum of squared residuals defined as \sum_{i=1}^n (Y_i - \mathbf{X}_i \boldsymbol{\beta})^2, where Y_i is the binary outcome, \mathbf{X}_i includes the covariates and an intercept, and \boldsymbol{\beta} are the parameters.^[5] This objective yields a closed-form solution for the parameter estimates: \hat{\boldsymbol{\beta}} = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{Y}, assuming \mathbf{X}^\top \mathbf{X} is invertible, which requires no perfect multicollinearity among the regressors. Under the LPM assumptions of strict exogeneity—where E[Y_i | \mathbf{X}_i] = \mathbf{X}_i \boldsymbol{\beta}—and no perfect multicollinearity, OLS produces consistent estimates of \boldsymbol{\beta}, even with a binary dependent variable Y_i.^[2] This consistency arises because the LPM specifies a population linear projection of Y onto \mathbf{X}, ensuring that as the sample size grows, \hat{\boldsymbol{\beta}} converges in probability to the true \boldsymbol{\beta}.^[11] The resulting predicted values, \hat{Y}_i = \mathbf{X}_i \hat{\boldsymbol{\beta}}, are interpreted directly as estimates of the probabilities P(Y_i = 1 | \mathbf{X}_i), providing straightforward marginal effects equal to the coefficients themselves. However, these predictions may lie outside the [0,1] interval, particularly for covariate values far from the sample means.^[6] In finite samples, OLS estimates in the LPM are unbiased under the core assumptions, despite the model's binary nature inherently implying heteroskedasticity in the residuals. Regardless, consistency ensures reliable large-sample inference without the need for maximum likelihood estimation, as the linearity simplifies the procedure compared to nonlinear binary choice models.^[2]

Adjustments for Heteroskedasticity

In the linear probability model (LPM), heteroskedasticity arises inherently from the binary nature of the dependent variable. Specifically, the conditional variance of the outcome Y_i given covariates X_i is \operatorname{Var}(Y_i \mid X_i) = p_i (1 - p_i), where p_i = X_i \beta represents the predicted probability. This variance is maximized at p_i = 0.5 (value 0.25) and approaches zero as p_i nears 0 or 1, resulting in non-constant error variance across observations. Ordinary least squares (OLS) estimation of the LPM ignores this heteroskedasticity, assuming homoscedastic errors, which invalidates conventional standard errors (SEs). These biased SEs typically understate uncertainty, particularly for predictions near 0 or 1 where the true variance is small, leading to overly narrow confidence intervals and inflated t-statistics for hypothesis testing. To address this, heteroskedasticity-robust SEs, based on White's covariance matrix estimator, provide consistent inference without assuming a specific form of heteroskedasticity. The robust variance-covariance matrix is given by

(X'X)^{-1} X' \hat{\Omega} X (X'X)^{-1},

where \hat{\Omega} is a diagonal matrix with elements \hat{u}_i^2 (squared OLS residuals) or, exploiting the known LPM form, \hat{p}_i (1 - \hat{p}_i) using fitted probabilities \hat{p}_i = X_i \hat{\beta}. This "sandwich" estimator ensures valid inference even under the LPM's heteroskedasticity. Alternatively, weighted least squares (WLS) achieves efficiency by weighting observations inversely to their conditional variances, using weights $1 / \sqrt{p_i (1 - p_i)}. Since p_i depends on the unknown \beta, estimation proceeds iteratively: start with OLS to obtain initial \hat{p}_i, compute weights, re-estimate via WLS, and repeat until convergence. This feasible generalized least squares approach yields asymptotically efficient estimates under the LPM assumptions.

Properties and Interpretation

Advantages

The linear probability model (LPM) is prized for its ease of estimation, as it employs ordinary least squares (OLS), a standard method available in virtually all econometric software packages, eliminating the need for iterative algorithms required in maximum likelihood estimation for nonlinear models like logit or probit.^[5] This approach ensures reliable convergence even in challenging cases, such as when covariates perfectly predict the binary outcome, where nonlinear models may fail.^[5] Consequently, the LPM is particularly accessible for researchers without advanced computational resources. A key strength lies in its interpretability, where the estimated coefficients directly represent the average marginal effects of covariates on the probability of the binary outcome, expressed in straightforward probability units.^[12] For instance, a coefficient of 0.05 indicates a 5 percentage point increase in the outcome probability for a one-unit change in the predictor, holding other factors constant, which facilitates clear communication in policy evaluations and economic analyses.^[2] The LPM also excels in computational speed, leveraging the efficiency of OLS to handle large datasets rapidly, making it ideal for exploratory analyses or scenarios demanding quick approximations.^[13] This efficiency is especially beneficial in big data contexts or repeated estimations, where nonlinear alternatives can be prohibitively time-intensive.^[14] Furthermore, the model's flexibility allows seamless incorporation of interaction terms, dummy variables, or higher-order polynomials within the linear framework, preserving the same OLS estimation procedure and enabling the capture of nuanced relationships without methodological overhaul.^[12] This adaptability supports its widespread use in empirical applications, from causal inference to panel data settings.

Limitations and Biases

One primary limitation of the linear probability model (LPM) is the boundary problem, where predicted probabilities can fall outside the [0, 1] interval, resulting in nonsensical interpretations such as negative or greater-than-one probabilities.^[5] This occurs because the linear functional form imposes no constraints on the range of predictions, unlike nonlinear alternatives.^[15] For instance, when explanatory variables take extreme values, the model may forecast probabilities exceeding unity, undermining its reliability for probability estimation.^[16] The LPM also suffers from inherent heteroskedasticity due to the binary nature of the dependent variable, where the error variance equals p(1-p) and varies with the predicted probability p.^[17] Consequently, ordinary least squares (OLS) estimation remains consistent but becomes inefficient, as the standard errors are incorrect and the estimator does not achieve the minimum variance among unbiased estimators.^[18] This non-constant variance can lead to unreliable inference, particularly in hypothesis testing, although adjustments like heteroskedasticity-robust standard errors can mitigate the issue for practical use.^[19] In small samples, the LPM is prone to bias and poor model fit, especially when true probabilities approach the boundaries of 0 or 1.^[20] Such bias arises from the misspecification of the linear form relative to the underlying nonlinear probability relationship, exacerbating finite-sample distortions in coefficient estimates.^[21] These problems are more pronounced in datasets with limited observations or extreme event probabilities, reducing the model's accuracy for causal inference.^[2] Finally, the LPM lacks a strong theoretical foundation, particularly in economic choice models, as it does not derive from random utility maximization principles that underpin models like probit and logit.^[6] Instead, it serves as an ad hoc linear approximation to binary outcomes, without a direct microeconomic justification for assuming a linear probability response.^[22] This absence of behavioral grounding limits its applicability in contexts requiring derivations from individual decision-making processes.^[23]

Comparisons and Alternatives

Versus Logit and Probit Models

The linear probability model (LPM) posits a linear relationship between the predictors and the probability of the outcome, expressed as P(Y=1|X) = \beta_0 + X\beta, which can yield predicted probabilities outside the [0,1] interval. In contrast, the logit model employs the logistic cumulative distribution function (CDF), P(Y=1|X) = \frac{1}{1 + \exp(-X\beta)}, and the probit model uses the standard normal CDF, P(Y=1|X) = \Phi(X\beta), both producing an S-shaped curve that inherently bounds predictions between 0 and 1. This nonlinearity in logit and probit better accommodates the bounded nature of binary outcomes, avoiding the implausible extrapolations possible in LPM, particularly when predictors push the linear index far from the mean.^[24]^[4] Marginal effects in the LPM are constant and directly given by the coefficients \beta, representing the uniform change in probability for a unit change in each predictor. Logit and probit models, however, feature marginal effects that vary with the values of the predictors, as the slope of the S-curve flattens at extreme probabilities; these are typically computed as average marginal effects across the sample or evaluated at the means of the covariates. For instance, in applications like mortgage denial predictions, the LPM marginal effect for a debt-to-income ratio might be fixed at 0.061, whereas probit effects range from 0.03 to 0.09 depending on other variables. This variability in logit and probit provides a more nuanced depiction of how effects diminish at the tails but requires additional computation for interpretation.^[4]^[25] Estimation in the LPM relies on ordinary least squares (OLS), which minimizes squared residuals and yields consistent estimates under mild conditions, though standard errors must be adjusted for heteroskedasticity inherent in binary data. Logit and probit models are estimated via maximum likelihood estimation (MLE), maximizing the log-likelihood function; for logit, this is \ell(\beta) = \sum_i \left[ y_i (X_i \beta) - \log(1 + \exp(X_i \beta)) \right], which accounts for the binary distribution and produces efficient estimates but involves numerical optimization. While OLS is computationally straightforward and scalable to large datasets, MLE for logit and probit is more demanding, especially with many parameters.^[24]^[25] In terms of performance, the LPM offers simplicity and ease of interpretation, with coefficients directly approximating average partial effects under symmetric covariate distributions, but it is less statistically efficient and prone to bias in average partial effects when covariates are asymmetric, particularly at extreme probabilities where predictions may fall outside [0,1]. Logit and probit models generally provide better efficiency and accuracy for binary outcomes, with superior handling of tail probabilities— for example, simulations show that probit and logit quasi-MLE reduce bias in partial effects compared to OLS LPM in asymmetric settings—though their nonlinear nature complicates direct coefficient interpretation. Overall, logit and probit yield similar predictions to LPM in the central range but diverge in the tails, where their bounded S-shapes offer advantages for realistic probability modeling.^[2]^[4]

Selection Criteria for Use

The linear probability model (LPM) is particularly suitable for empirical research when the primary objective is to estimate marginal effects that are constant across the distribution of covariates, as the model's coefficients directly represent changes in the probability of the outcome occurring. This interpretability makes it preferable in large samples, where asymptotic properties ensure reliable inference, or during exploratory analysis to identify key predictors without the complexity of nonlinear functional forms. The LPM performs well when predicted probabilities are bounded away from the extremes of 0 and 1, generally when the proportion of positive outcomes (i.e., the fraction of 1s in the binary dependent variable) falls between approximately 0.2 and 0.8, as this range minimizes the risk of implausible predictions outside [0,1] and aligns closely with logistic approximations.^[13] Key trade-offs in selecting the LPM involve prioritizing ease of interpretation and computational simplicity over potential gains in efficiency from nonlinear alternatives like logit or probit models. Researchers often opt for the LPM when the loss in efficiency is minimal relative to the benefits of straightforward coefficient estimates, especially in settings where marginal effects are of central interest rather than the full probability curve. It is especially advantageous in panel data applications incorporating fixed effects, where nonlinear models can suffer from the incidental parameters problem, leading to biased estimates, whereas the LPM integrates seamlessly with within-group transformations for causal identification.^[26] Empirical guidelines for employing the LPM include evaluating the baseline proportion of successes: if it exceeds 10-20%, the model tends to yield reasonable approximations, but performance deteriorates with very rare or common events due to heteroskedasticity and boundary issues. To assess suitability, compare the LPM's ordinary R-squared to the McFadden pseudo-R² from fitted logit or probit models; a comparable or superior fit in the LPM, combined with similar average partial effects, supports its use, particularly when sample sizes are large enough to mitigate heteroskedasticity via robust standard errors. In contemporary empirical work, the LPM continues to play a prominent role in causal inference frameworks, such as differences-in-differences designs, where it facilitates estimation of average treatment effects on binary outcomes without requiring nonlinear adjustments, even as alternatives like logit exist for distributional analysis.^[27]^[26]

References

[1]
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Linear Probability Model (LPM). 1. So far we assume p is constant since we flip the same coin again and again. In reality, we ask different persons whether ...
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It is essential to use robust standard errors since the ui u i in a linear probability model are always heteroskedastic. Linear probability models are easily ...
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The linear probability model is a linear regression model for binary variables, where the predicted value is a probability. It models Pr(Y=1|X) as a linear ...
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The main reason that the LPM works so well to estimate experi- mental impacts is that treatment status is a binary variable (not a continuous variable, which ...
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The initial modelling suggestions of the Cowles Commission have been extended and new departures have emerged (e.g. see Fair, 1984; Griliches and Intriligator, ...Missing: origins | Show results with:origins
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Wooldridge (2010) treats it as an ordinary regression with inconvenient flaws. ... Why do credible revolutionaries prefer the discredited linear probability ' ...
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At the Cowles Commission, Koopmans continued his study of the transportation problem that he had initiated in 1942. By the end of 1946 he realized that his ...
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In this lecture we discuss under which assumptions the OLS (Ordinary Least Squares) estimator has desirable statistical properties such as consistency and ...
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### Historical Notes and General Setup for LPM and Latent Variable Formulation
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Latent response variable formulation defines a threshold τ on a continuous u* variable so that u = 1 is observed when u* exceeds τ while otherwise u = 0 is ...
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The Method of Probits. C. I. BlissAuthors Info & Affiliations. Science. 12 Jan 1934. Vol 79, Issue 2037. pp. 38-39. DOI: 10.1126/science.79.2037.38 · PREVIOUS ...Missing: latent | Show results with:latent
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[17]
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[18]
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[20]
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You get to the same model but the latent interpretation has a bunch of applications ins economics (for example, random utility models) and psychometrics ...
[21]
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Limitations of the Linear Probability Model. • Even though the parameters of the linear model are easily interpreted, there are limitations. • A major problem ...
[22]
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The higher correlation, heteroscedasticity becomes more important (b is more inefficient). • There are several theoretical reasons why the σ may be related to.
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### Summary of Comparisons: LPM, Logit, and Probit
[28]
Mostly Harmless Econometrics
### Summary of Mostly Harmless Econometrics on Specified Topics
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9 Difference-in-Differences - Causal Inference The Mixtape
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