Common mistake 1 with Logit / Probit

Reporting raw logit coefficients as if they were probabilities.

Common mistake 2 with Logit / Probit

Comparing coefficient sizes across logit and probit - the scales differ by roughly 1.6.

Common mistake 3 with Logit / Probit

Bragging about 95% accuracy when 95% of the sample is in one class. Compare against the naive benchmark.

Common mistake 4 with Logit / Probit

Comparing coefficients across logit models with different regressors. The latent scale is fixed by the error variance, so adding a variable rescales every other coefficient even when nothing about their effects changed - which is why average marginal effects are the comparable quantity.

EconStatsCraft › Logit / Probit

Logit / Probit

For a 0/1 outcome. Reports coefficients, average marginal effects (the number you should quote), odds ratios, a classification table and the ROC curve.

What it needs

Binary outcome (0 / 1) - Must take exactly two values.
Independent variables

The intuition

Imagine a latent "propensity" y* = xβ + u that you never see; you only observe whether it crossed zero. Logit assumes u is logistic, probit assumes Normal. The S-shape is not a modelling flourish - it is what a probability must look like if it is to stay inside [0, 1] while the index xβ ranges over the whole real line.

When to use it

The dependent variable takes exactly two values: employed or not, defaulted or not, voted or not.

Where the estimator comes from

With P(y=1\mid x)=F(x'\beta), the log-likelihood is \ell(\beta)=\sum_i\left[y_i\log F(x_i'\beta)+(1-y_i)\log\bigl(1-F(x_i'\beta)\bigr)\right], which is globally concave for both logit and probit - so the maximum is unique and the optimiser cannot get stuck in a local mode. There is no closed form; it is solved numerically.

The model

P(y=1\mid \mathbf{x})=\Lambda(\mathbf{x}^{\prime}\boldsymbol{\beta})=\frac{e^{\mathbf{x}^{\prime}\boldsymbol{\beta}}}{1+e^{\mathbf{x}^{\prime}\boldsymbol{\beta}}}\quad\text{(logit)}

What it assumes

The outcome really is binary.
Observations are independent.
The index xβ is correctly specified - no important omitted variable.
No perfect separation: no regressor that predicts the outcome flawlessly.

Statistical properties

Consistent and asymptotically Normal, and efficient among consistent estimators, IF the link function is correctly specified.
Unlike OLS, a misspecified error distribution makes the coefficients inconsistent, not merely inefficient.
Logit and probit coefficients differ by roughly a factor of 1.6 (the ratio of the logistic and Normal scale parameters); marginal effects are nearly identical.
Omitted variables attenuate coefficients even when they are UNCORRELATED with the included ones - a scaling problem specific to non-linear models and absent in OLS.
Perfect separation makes the likelihood unbounded and the coefficient diverges to infinity.

How to read the output

The raw coefficients are on a latent-index scale. Their SIGN and SIGNIFICANCE are interpretable; their size is not.
Quote the average marginal effect: "a year of extra schooling raises the probability by 2.1 percentage points."
The odds ratio exp(β): above 1 raises the odds, below 1 lowers them.
McFadden's pseudo-R² is not the OLS R². Values of 0.2–0.4 already indicate an excellent fit.
AUC: 0.5 is a coin flip, above 0.8 is strong discrimination.

A worked reading

Insurance status for 2,000 people, 35.3% of them insured. β̂_educ = 0.1129 in the logit, so exp(0.1129) = 1.1195: each extra year of schooling raises the ODDS of being insured by about 12%. The average marginal effect is 0.0245 - two and a half extra percentage points on the PROBABILITY per year - and that is the number to report, because the odds ratio is constant by construction while the effect on the probability is not: it is largest for people near 50% predicted and small at either extreme. Note also McFadden's pseudo-R² of 0.0373. That is not an R² and does not mean the model explains 4% of anything; values around 0.2 already indicate a good fit on this scale, and comparing it to a regression R² is a category error that appears in a great many dissertations.

Symptom → diagnosis → what to do

If you see this	It means	Do this
A coefficient explodes with a huge standard error	Perfect or quasi-complete separation	Drop or combine that regressor; consider penalised (Firth) logit
Accuracy high but AUC near 0.5	The model just predicts the majority class	Compare against the base rate; look at sensitivity, not accuracy
Marginal effects differ a lot from the coefficients	Normal - the coefficients are on the latent index scale	Always quote marginal effects
Predicted probabilities bunch at 0 or 1	Over-fitting, or a near-separating regressor	Simplify, or check for a variable that encodes the outcome
Rare outcome (under ~5%)	Small-sample bias in ML	Use penalised likelihood, or rare-events logit (King & Zeng)

Common mistakes

Reporting raw logit coefficients as if they were probabilities.
Comparing coefficient sizes across logit and probit - the scales differ by roughly 1.6.
Bragging about 95% accuracy when 95% of the sample is in one class. Compare against the naive benchmark.
Comparing coefficients across logit models with different regressors. The latent scale is fixed by the error variance, so adding a variable rescales every other coefficient even when nothing about their effects changed - which is why average marginal effects are the comparable quantity.

Where to read this properly

Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Cengage, chapter 17 (Limited Dependent Variable Models)
Cameron and Trivedi (2005), Microeconometrics: Methods and Applications, Cambridge University Press, chapter 14
Mood (2010) - why logit coefficients cannot be compared across models or groups
King & Zeng (2001) - logistic regression with rare events

Related Regression

Multiple regression (OLS) · Tobit (censored) · Truncated regression · Multinomial logit · Ordered logit / probit · Quantile regression · Poisson / Negative binomial

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Logit / Probit

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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