Logit — Binary logistic regression

Logit models the probability of a binary dependent variable ($Y \in \{0,1\}$) through the logistic link. It is the standard model when the outcome is "yes/no" (default, participation, pass/fail…).

When to use

Use Logit when $Y$ is binary. Logit coefficients are interpreted via the odds ratio $e^{\beta}$; to interpret directly in probability terms, read the marginal effects.

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Model specification

$$P(Y_i = 1 \mid X_i) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 X_{1i} + \dots + \beta_k X_{ki})}}$$

Estimated by Maximum Likelihood (MLE). For coefficient $\beta_j$: $e^{\beta_j}$ is the odds ratio; marginal effects depend on the value of $X$.

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Interpretation & diagnostics

• Odds ratio $e^{\beta_j} > 1$ ⇒ increases odds; $< 1$ ⇒ decreases.
• Marginal effects (average AME or at-the-mean MEM) give the change in probability.
• Fit: Pseudo-$R^2$ (McFadden), classification accuracy, ROC/AUC.
• Check: perfect separation, multicollinearity.

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Running in EcoLab

1. Modeling module → Limited dependent variable family → Logit.
2. Select the binary $Y$ and the $X$ variables.
3. Run; read coefficients, odds ratios, marginal effects, AUC; export the replication code.

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Input / output example

Input (illustrative): default (0/1) on leverage, roa, size.

Output (format, illustrative figures — not real results):

Variable	Coefficient	Odds ratio	AME	p-value
leverage	1.20	3.32	0.18	0.000
roa	−2.10	0.12	−0.31	0.001

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Replication code

Stata

* ===== Logit — Binary logistic regression =====
* Estimate the logit model
logit vo_no don_bay roa quy_mo

* Odds ratios
logit vo_no don_bay roa quy_mo, or

* Average marginal effects (AME)
margins, dydx(*)

* Classification table (sensitivity, specificity)
estat classification

* ROC curve and AUC
lroc

R

# ===== Logit — Binary logistic regression =====
# Estimate the logit model
model <- glm(vo_no ~ don_bay + roa + quy_mo,
             data   = df,
             family = binomial(link = "logit"))

summary(model)

# Odds ratios
exp(coef(model))
exp(confint(model))

# Average marginal effects (AME)
library(margins)
mfx <- margins(model)
summary(mfx)

# Classification & AUC
library(pROC)
pred  <- predict(model, type = "response")
roc_obj <- roc(df$vo_no, pred)
auc(roc_obj)
plot(roc_obj, main = "ROC Curve — Logit")

Python

# ===== Logit — Binary logistic regression =====
import statsmodels.api as sm
from sklearn.metrics import roc_auc_score, classification_report

# Prepare data
X = sm.add_constant(df[["don_bay", "roa", "quy_mo"]])
y = df["vo_no"]

# Estimate the logit model
model = sm.Logit(y, X).fit()
print(model.summary())

# Odds ratios
import numpy as np
print("Odds Ratios:")
print(np.exp(model.params))

# Average marginal effects (AME)
mfx = model.get_margeff()
print(mfx.summary())

# Classification & AUC
y_pred = model.predict(X)
print("AUC:", roc_auc_score(y, y_pred))
print(classification_report(y, (y_pred > 0.5).astype(int)))

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Limitations

• Raw coefficients are not probability effects — use marginal effects.
• Assumes the logistic form; for the normal-CDF link see Probit.

Video tutorial

Video Tutorial: Guide to running Logit (logistic regression) in EcoLab

See also

• Probit · Tobit · Catalog