Model Parameters

Range: -50 to 50
Range: -50 to 50
Value for prediction
Set to 0 to ignore
Value for prediction

S-Curve Plot Range

Quick Reference

z = b0 + b1·X1 + b2·X2
Logit: P(y=1) = exp(z) / (1 + exp(z))
Probit: P(y=1) = Φ(z)
Marginal Effects:
Logit: ∂P/∂xj = bj · P · (1 − P)
Probit: ∂P/∂xj = bj · φ(z)
Created by CFA Charterholder • Finance Educator

Results

Predicted Probability
47.50%
Logit Model
Linear Index (z) -0.1000
ME X1 0.1995
OR X1 2.2255

S-Curve: Predicted Probability vs X1

Probability curve over X1 range with b0, b1, b2, and X2 held constant. Dashed lines mark the evaluation point.

Model Assumptions

1 Binary dependent variable — y takes values 0 or 1 only
2 Independent observations — observations are independently sampled
3 No perfect multicollinearity — regressors are not perfectly correlated
4 Correct functional form — the linear index z = b0 + b1X1 + b2X2 is correctly specified
5 Error distribution — Logit assumes standard logistic errors; Probit assumes standard normal errors
6 Exogeneity — regressors should not be correlated with unobserved determinants of the outcome for causal interpretation
7 Non-constant marginal effects — partial effects depend on the evaluation point (they are not constant like OLS)
Note: For binary or dummy regressors, the relevant effect is the discrete probability change ΔP = P(y=1|xj=1) − P(y=1|xj=0), not the calculus-based marginal effect.

For educational purposes. Not financial advice. Market conventions simplified.

Understanding Logit & Probit Models

What Are Logit and Probit Models?

Logit and probit models are regression methods for binary outcomes (y = 0 or 1). Unlike the linear probability model (LPM), which can produce predicted probabilities outside the [0, 1] range, logit and probit use nonlinear link functions that guarantee predictions between 0 and 1.

The logit model uses the logistic CDF: P(y=1|x) = Λ(z) = exp(z)/(1+exp(z)), while the probit model uses the standard normal CDF: P(y=1|x) = Φ(z). Here z = b0 + b1X1 + b2X2 is the linear index.

Logit vs. Probit: When to Use Which

In practice, logit and probit produce very similar predicted probabilities. The choice between them is often driven by convention:

  • Logit is preferred when odds ratios are important for interpretation (e.g., in epidemiology and finance)
  • Probit is preferred in certain fields (e.g., labor economics) and when using the model as part of a larger framework (e.g., bivariate probit)
  • Coefficient magnitudes are not directly comparable between the two models because they assume different latent-error scales. Roughly, logit coefficients ≈ 1.6 × probit coefficients

Understanding Marginal Effects

Unlike OLS, the coefficients in logit/probit do not directly tell you the effect on probability. Instead, you must compute marginal effects, which depend on the evaluation point. The S-curve is steepest near P = 0.5 (where marginal effects are largest) and flattens in the tails (where marginal effects approach zero).

For continuous regressors, the marginal effect is the derivative ∂P/∂xj. For binary or dummy regressors, compute the discrete change in probability instead.

Key Assumptions

  • Binary dependent variable (y = 0 or 1)
  • Independent, randomly sampled observations
  • No perfect multicollinearity among regressors
  • Correct specification of the linear index
  • Logit: logistic error distribution; Probit: normal error distribution
  • Exogeneity of regressors for causal interpretation
McFadden Pseudo R²: Unlike OLS R², the McFadden pseudo R² is based on log-likelihoods and is not directly comparable to OLS R². It is computed as 1 − ln(Lfull)/ln(Lnull) and requires the full dataset, so it is not calculated by this tool.

Frequently Asked Questions

A logit model is a regression model for binary outcomes (y = 0 or 1) that uses the logistic cumulative distribution function to ensure predicted probabilities stay between 0 and 1. The model specifies P(y=1|x) = exp(z)/(1+exp(z)), where z is a linear index of the explanatory variables. Unlike the linear probability model (LPM), logit guarantees predictions within [0, 1].

Both logit and probit are binary response models that constrain predicted probabilities to [0, 1]. Logit uses the logistic CDF, while probit uses the standard normal CDF. In practice, the two models produce very similar predicted probabilities, though their coefficient magnitudes are not directly comparable because the latent-error scales differ. The main practical difference is that logit allows odds ratio interpretation, while probit does not.

Unlike OLS, logit and probit coefficients do not directly give the marginal effect on probability. The coefficient tells you the direction of the effect (positive means the variable increases the probability of y=1), but the magnitude depends on the evaluation point. To find the actual effect on probability, you must compute marginal effects: ∂P/∂xj = g(z) × bj, where g is the PDF of the assumed distribution.

Marginal effects measure how a one-unit change in an explanatory variable affects the predicted probability at a specific evaluation point. For logit: ∂P/∂xj = bj × P × (1−P). For probit: ∂P/∂xj = bj × φ(z). Because the S-curve is steepest near P = 0.5, marginal effects are largest when z is near zero and smallest in the tails. For binary or dummy regressors, the relevant effect is a discrete probability change rather than a derivative.

The odds ratio is specific to the logit model. For coefficient bj, the odds ratio is exp(bj). It represents the multiplicative change in the odds (not the probability) of y=1 from a one-unit increase in xj, holding other regressors fixed. An odds ratio greater than 1 indicates a positive effect on the odds; less than 1 indicates a negative effect. Odds ratios do not apply to probit models because probit does not have a log-odds interpretation.

The linear probability model (LPM) — OLS applied to a binary dependent variable — can produce predicted probabilities outside [0, 1], which is logically impossible. It also suffers from inherent heteroskedasticity since Var(y|x) = P(1−P), which varies with x. Logit and probit models solve both problems: they guarantee predictions in [0, 1] and use maximum likelihood estimation that accounts for the binary nature of the dependent variable.
Disclaimer

This calculator is for educational purposes only. It computes predicted probabilities and marginal effects from user-provided coefficients. It does not estimate coefficients from raw data. Actual model estimation requires maximum likelihood methods and appropriate statistical software. This tool should not be used as a substitute for professional econometric analysis.

Related Calculators

OLS Regression Calculator

Calculate OLS regression coefficients, R-squared, and visualize fitted lines.

Try Calculator →

R-Squared & Adjusted R-Squared Calculator

Compute R-squared and adjusted R-squared for regression models.

Try Calculator →

Regression Hypothesis Testing Calculator: t-Test & F-Test Tool

Test hypotheses about regression coefficients with t-tests and F-tests.

Try Calculator →