CFA Level II Exam · Extensions of Multiple Regression

Qualitative Dependent Variables: Probit, Logit and Discriminant Analysis

Updated 7 October 2026 · Fact-checked

A qualitative dependent variable takes a limited set of values, such as default or no default. Linear regression can give probabilities outside 0 to 1, so you use probit (normal CDF), logit (logistic CDF) or discriminant analysis (a score that classifies). On the exam you interpret the output; you do not estimate it.

Understand Qualitative Dependent Variables: Probit, Logit, Discriminant

Most regressions you have seen have a continuous dependent variable, such as a return. Sometimes the outcome is a category: a bond defaults or it does not, a firm is acquired or it is not. You code this as a binary variable, 1 or 0. Variables like this are called qualitative dependent variables.

Why not run ordinary least squares on it? That is a linear probability model. It has three problems. Fitted values can fall below 0 or above 1, which makes no sense as a probability. The errors cannot be normal, because the outcome only takes two values. And the errors are heteroskedastic, because their variance depends on the predictors.

Probit and logit fix this by passing a linear index through an S-shaped function that stays between 0 and 1. Probit uses the cumulative standard normal distribution. Logit uses the cumulative logistic distribution. Both are estimated by maximum likelihood, not OLS. The logit model is built on the log odds: ln[p ÷ (1 − p)] = b0 + b1X1 + ... + bkXk.

Because the link is nonlinear, a coefficient is not the change in probability for a one-unit change in X. The sign tells you the direction: a positive coefficient raises the probability of the outcome. The size of the probability change depends on where you start on the curve. In a logit model, the slope coefficient is the change in log odds. The two models usually give similar results. The logistic distribution has fatter tails.

Discriminant analysis is a different approach. It uses the predictors to build a linear function that produces a score, often called a discriminant function. You compare the score with a cutoff to place an observation into a group, such as bankrupt or not. It classifies, and it assumes the predictors follow certain distributional conditions. The classic example is the Altman Z-score for bankruptcy. Probit and logit give a probability. Discriminant analysis gives a score and a classification.

Key formulas to remember

Linear probability model: Y = b0 + b1X1 + ... + bkXk + ε, where Y is 0 or 1; Fitted values can fall outside 0 to 1. Errors are heteroskedastic.
Logit model: p = 1 ÷ (1 + e^−(b0 + b1X1 + ... + bkXk)); Uses the logistic CDF. The probability always lies between 0 and 1.
Log odds: ln[p ÷ (1 − p)] = b0 + b1X1 + ... + bkXk; Each slope is the change in log odds for a one-unit change in that X, holding others constant.
Odds: odds = p ÷ (1 − p); p = odds ÷ (1 + odds); Use to convert between probability and odds.
Probit model: p = N(b0 + b1X1 + ... + bkXk), where N is the standard normal CDF; Uses the cumulative normal. The coefficient sign shows direction only.
Discriminant function: Score = a0 + a1X1 + ... + akXk; classify by comparing the score with a cutoff; Produces a classification, not a probability.

How to solve Qualitative Dependent Variables: Probit, Logit, Discriminant questions

Use this method for any item-set question on binary outcomes.

1Identify the dependent variable. If it is a yes/no outcome coded 1 or 0, you are in qualitative dependent variable territory.
2Find the model named in the vignette: linear probability, probit, logit or discriminant analysis. Check the exhibit title and the notes.
3If the question asks why OLS is a poor choice, cite fitted values outside 0 to 1, non-normal errors and heteroskedasticity.
4For probit or logit, read the sign of each coefficient first. Positive means a higher probability of the outcome. Do not read the size as a change in probability.
5If asked for a probability under logit, compute the linear index, then apply p = 1 ÷ (1 + e^−index). With log odds, convert via odds = e^index.
6Check the estimation method. Probit and logit use maximum likelihood. Do not apply R-squared or the OLS F-test logic to them.
7For discriminant analysis, compute the score and compare it with the cutoff given. State the group it falls in.
8Check that your answer is in the right form: probability between 0 and 1, odds, log odds or a class label.

Quickest way: Three-check shortcut

When to use it: Use when time is short and the question is conceptual or asks you to read a coefficient.

Is the outcome 0 or 1? Then OLS is inappropriate; think probit, logit or discriminant.
Does the question ask for probability? Probit and logit give it. Discriminant analysis gives a score and a group.
Is it a coefficient question? Use the sign for direction. For logit, the coefficient is a change in log odds. For a probability, plug in and convert.

Common mistakes in Qualitative Dependent Variables: Probit, Logit, Discriminant

Reading a logit or probit slope as the change in probability.
It is how you read an OLS slope.
Fix: The relationship is nonlinear. A logit slope is the change in log odds. Only the sign and direction carry over directly.
Treating log odds as a probability.
The output of the linear index looks like a number you can use directly.
Fix: Convert: odds = e^index, then p = odds ÷ (1 + odds).
Saying probit and logit are estimated by OLS.
Both are called regressions.
Fix: They use maximum likelihood estimation.
Saying discriminant analysis gives the probability of default.
Both approaches classify observations.
Fix: Discriminant analysis produces a score used to assign a group. Probit and logit give probabilities.
Confusing which distribution goes with which model.
The names are similar and easy to swap under pressure.
Fix: Probit is the normal CDF. Logit is the logistic CDF.
Forgetting the linear probability model's flaws.
Students remember only that probabilities can be out of range.
Fix: Also recall non-normal errors and heteroskedasticity.

Worked examples

Example 1

An analyst models whether a firm defaults within a year (1 = default, 0 = no default) using a logit model. The estimated equation is: ln[p ÷ (1 − p)] = −3.0 + 2.0 × (Debt/Equity) − 1.0 × (Interest coverage). A firm has Debt/Equity of 1.5 and interest coverage of 2.0. Q1: What are the log odds of default? Q2: What is the probability of default, to two decimals? Q3: What does the coefficient of −1.0 on coverage tell you?

Show the solution

Q1: Log odds = −3.0 + 2.0 × 1.5 − 1.0 × 2.0 = −3.0 + 3.0 − 2.0 = −2.0.
Q2: Odds = e^−2.0 = 0.1353.
Probability = 0.1353 ÷ (1 + 0.1353) = 0.1353 ÷ 1.1353 = 0.1192, about 0.12.
Q3: The negative sign means higher interest coverage lowers the log odds, so it lowers the probability of default, holding Debt/Equity constant. The size is a change in log odds, not in probability.

Answer: Q1: −2.0. Q2: about 0.12, or 12%. Q3: higher coverage reduces the probability of default; the coefficient is a change in log odds per unit.

Example 2

A researcher wants to predict whether a company will be acquired (1) or not (0). A colleague proposes an OLS regression with this 0/1 variable. The OLS model gives a fitted value of 1.18 for one company. The colleague suggests a probit model instead. Q1: What is wrong with the OLS result? Q2: How does probit address this? Q3: How does discriminant analysis differ from probit?

Show the solution

Q1: A fitted value of 1.18 as a probability is above 1, which is impossible. This is a known flaw of the linear probability model. Its errors are also non-normal and heteroskedastic.
Q2: Probit applies the cumulative standard normal distribution to the linear index, so the fitted probability always lies between 0 and 1. It is estimated by maximum likelihood.
Q3: Discriminant analysis builds a linear score from the predictors and compares it with a cutoff to classify the company as acquired or not. It gives a classification rather than a probability.

Answer: Q1: A probability of 1.18 is impossible. Q2: Probit keeps probabilities within 0 and 1 using the normal CDF. Q3: Discriminant analysis gives a score and a group, not a probability.

Exam tips

Expect conceptual questions on why OLS fails for a binary outcome. Memorise the three problems.
If the vignette gives a logit equation and asks for a probability, do the two-step conversion: log odds to odds to probability. Check that your final answer lies between 0 and 1.
Match the distribution to the model: probit is normal, logit is logistic. Wrong options often swap them.
Do not read slope size as a probability change. A positive or negative sign is usually what the question wants.
Note whether the question asks for a probability or a classification. That points you to logit or probit versus discriminant analysis.

Qualitative Dependent Variables: Probit, Logit, Discriminant: frequently asked questions

What is the difference between probit and logit?

Probit uses the cumulative normal distribution and logit uses the cumulative logistic distribution. The logistic has fatter tails. Both are estimated by maximum likelihood and usually give similar conclusions.

Why not use linear regression for a binary dependent variable?

Fitted values can fall outside 0 and 1, the errors cannot be normal, and the errors are heteroskedastic. Probit and logit keep probabilities inside 0 to 1.

What is discriminant analysis in CFA Level II?

It builds a linear function of the predictors that gives a score. You compare the score with a cutoff to assign an observation to a group, such as bankrupt or not bankrupt. The Altman Z-score is a well-known example.

How do I interpret a logit coefficient?

It is the change in log odds of the outcome for a one-unit rise in that variable, holding others constant. A positive sign raises the probability. To get a probability, you must convert the full index.