EMET8002 Chap.4 Binary Outcomes With Logit and Probit
Binary Outcomes With Logit and Probit
From straight lines to curves
When the outcome is a yes or no, OLS gives the linear probability model, which is easy to read but can predict probabilities below 0 or above 1. Logit and probit models pass the linear index through the logistic function or the standard normal distribution, keeping predictions between 0 and 1, and are estimated by maximum likelihood.
Reading the results
Coefficients give sign and significance only.
The chapter shows how to compute predicted probabilities, marginal effects that depend on the starting point, and the risk and odds ratios that compare two groups, and when to prefer logit or probit.
Data work before a grade-repetition logit
The Week 4 lab models whether a PISA student has repeated a grade.
It rewards careful data work before estimation: dropping missing outcomes, recoding dummies stored as 2 and 1, and explaining why one dummy describes a two-category variable. After estimation, marks follow from clustered standard errors by school, joint tests on mother's education, odds ratios, marginal effects for father's education and a comparison of logit and probit predictions.
What this chapter covers
- 01
Dummy regressors and the dummy variable trap
- 02
The linear probability model and its limits
- 03
Logistic and standard normal link functions
- 04
Maximum likelihood estimation
- 05
Predicted probabilities
- 06
Marginal effects at stated values
- 07
Risk ratios and odds ratios
- 08
Choosing between logit and probit
Worked example · free
A marginal effect on a probit curve
- 1At 0.3 the index is −2.5 + 0.96 = −1.54, and the normal table gives 0.062.
- 1At 0.4 the index is −2.5 + 1.28 = −1.22, which gives 0.111.
- 1The marginal effect is 0.111 minus 0.062, or 4.9 percentage points.
Key terms
- Linear Probability Model
- An OLS regression on a binary outcome whose coefficients are changes in probability.
- Logit Model
- A binary outcome model that uses the logistic function to map the index into a probability.
- Probit Model
- A binary outcome model that uses the standard normal distribution function as its link.
- Risk Ratio
- The predicted probability of an event in one group divided by that in another group.
- Link Function
- The function in a generalised linear model that connects the linear index to the outcome.
- Dummy Variable Trap
- The perfect multicollinearity created by including a dummy for every category.
Binary Outcomes With Logit and Probit FAQ
Why not just use OLS for a binary outcome?
OLS gives the linear probability model, which is easy to interpret but has heteroskedastic, non-normal errors and can predict probabilities outside the range from 0 to 1. Logit and probit avoid impossible predictions by using a curved link function.
How do you interpret a probit coefficient?
It is the change in the z-value from a one-unit change in the regressor. Its sign and significance can be read directly, but the size of the effect requires computing predicted probabilities and the difference between them.
What is the difference between a risk ratio and an odds ratio?
A risk ratio divides two predicted probabilities, while an odds ratio divides two odds, each odds being a probability over one minus that probability. In a logit, the odds ratio for a dummy equals e raised to its coefficient at any starting values.
Should I choose logit or probit?
In most cases either works. Logit is faster, has a closed form and gives odds ratios directly, and it is slightly more robust to outliers. Probit fits better in some settings such as random effects models, and some selection models require it, so let your paper's choice and your software guide you.
Why do marginal effects change with the starting value?
Logit and probit curves are steeper in the middle than in the tails. The same change in a regressor therefore moves the predicted probability more when the starting probability is moderate than when it is close to 0 or 1, which is why a marginal effect must be quoted with its starting point.
Assessment move
Practise the four-step marginal effect method on a table of normal probabilities until it is automatic, and always write down the starting values you chose. Then compute a risk ratio and an odds ratio from the same two probabilities and explain why they differ. Finally, rewrite one logit result three ways, as a predicted probability, a marginal effect and an odds ratio, and say which a policy reader would find most useful.
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