ECMT6002 Chap.11 Binary Choice Models
Binary Choice Models
Binary outcomes require a model for probabilities rather than an unrestricted conditional mean. The linear probability model is simple but can predict outside the unit interval and has heteroskedastic disturbances. Probit and logit pass an index through a cumulative distribution function, keeping fitted probabilities valid.
Their coefficients describe index changes, so substantive interpretation usually requires a marginal effect, discrete probability change or odds statement evaluated at specified covariates. Because a binary conditional mean equals the probability of observing one, an LPM slope has an immediate percentage-point reading.
Its constant slope and unbounded fitted line can be implausible at extreme covariate values, and the Bernoulli variance makes robust inference necessary. Logit uses the logistic cumulative distribution and probit uses the standard normal cumulative distribution. Both impose an S-shaped probability response and can be motivated through an unobserved latent index crossing a threshold.
Maximum likelihood assigns each observed one its fitted probability and each zero one minus that probability. Raw nonlinear coefficients operate on the index; in logit they also shift log odds. A continuous marginal effect multiplies the coefficient by the link density at a stated covariate profile, while a binary regressor calls for a discrete probability difference. Classification adds a threshold chosen for a decision task.
Accuracy alone may hide severe false-negative or false-positive performance, particularly when one outcome is rare, so probability calibration and the full confusion matrix remain important.
What this chapter covers
- 01
Linear probability model
- 02
Latent index representation
- 03
Probit probability
- 04
Logit probability and odds
- 05
Maximum likelihood estimation
- 06
Marginal effects and classification
Worked example · free
Converting logit index to probability
- +1Exponentiate the index: exp(log 3) = 3.
- +1In logit, this exponential is the predicted odds p divided by 1 minus p.
- +1Convert odds to probability using 3 divided by 1 plus 3.
- +1The probability is 0.75 and the odds are 3 to 1.
Key terms
- Linear probability model
- An OLS model with a binary dependent variable whose conditional mean is interpreted as a probability.
- Probit
- A binary response model using the standard normal cumulative distribution function.
- Logit
- A binary response model using the logistic cumulative distribution function.
- Odds ratio
- A ratio comparing the odds associated with different regressor values, often obtained by exponentiating a logit coefficient.
Binary Choice Models FAQ
Why not interpret a logit coefficient as a probability change?
The coefficient changes the linear index and log odds. The derivative of probability multiplies that coefficient by the logistic density, so the probability effect varies with the covariate values.
When is the linear probability model useful?
It gives directly interpretable average probability slopes and is easy to estimate with fixed effects or instruments. Robust inference is needed, and fitted values outside the probability range can limit prediction quality.
How should probit and logit be compared?
They use different link functions and coefficient scales, so raw coefficient magnitudes are not directly comparable. Compare predicted probabilities, marginal effects or fit on the same sample and target task.
Why can accuracy be misleading?
When one outcome dominates, a rule that always predicts the majority class can achieve high accuracy while never finding the rare cases. Open the result into true positives, false positives, false negatives and true negatives. Choose a threshold with the application costs in view and evaluate probabilities separately from classification labels.
Exam move
Move repeatedly among four representations: latent index, probability, odds and marginal effect. Before interpreting a number, label which representation it belongs to. Practise computing predicted probabilities at two covariate values and reporting the discrete difference. Use classification accuracy only with a declared threshold and alongside a probability based measure.
Begin with an LPM coefficient and translate it into percentage points, then list the range and heteroskedasticity limitations separately. For logit, practise moving from an index to odds and probability in both directions. Evaluate the same continuous marginal effect at low, middle and high predicted probabilities to see why it is not constant.
For a dummy regressor, calculate two complete predicted probabilities rather than applying a derivative. Write the Bernoulli log-likelihood contribution for y equal to one and zero before combining observations, which prevents the common missing complement error. Build confusion matrices from small prediction lists and compute sensitivity, specificity and accuracy.
Compare each classifier with the majority-class benchmark and state the cost of each error type. When comparing probit with logit, use probabilities or marginal effects rather than raw coefficients. Finish every interpretation by naming the evaluation point, held-fixed covariates and whether the reported quantity is an index, odds change, probability change or classification result.
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