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Key Terms 457

differences in mortgage denial rates for otherwise similar black applicants and
white applicants.

Binary dependent variables are the most common example of limited dependent
variables, which are dependent variables with a limited range. The final quarter of the
twentieth century saw important advances in econometric methods for analyzing
other limited dependent variables (see the Nobel Laureates box). Some of these
methods are reviewed in Appendix 11.3.

Summary

1. When Y is a binary variable, the population regression function shows the
probability that Y = 1 given the value of the regressors, X1, X2, c, Xk.

2. The linear multiple regression model is called the linear probability model
when Y is a binary variable because the probability that Y = 1 is a linear
function of the regressors.

3. Probit and logit regression models are nonlinear regression models used
when Y is a binary variable. Unlike the linear probability model, probit and
logit regressions ensure that the predicted probability that Y = 1 is between
0 and 1 for all values of X.

4. Probit regression uses the standard normal cumulative distribution function.
Logit regression uses the logistic cumulative distribution function. Logit and
probit coefficients are estimated by maximum likelihood.

5. The values of coefficients in probit and logit regressions are not easy to
interpret. Changes in the probability that Y = 1 associated with changes in
one or more of the X’s can be calculated using the general procedure for
nonlinear models outlined in Key Concept 8.1.

6. Hypothesis tests on coefficients in the linear probability, logit, and probit
models are performed using the usual t- and F-statistics.

Key Terms likelihood function (447)
maximum likelihood estimator
limited dependent variable (432)
linear probability model (434) (MLE) (447)
probit (437) fraction correctly predicted (447)
logit (437) pseudo-R2 (448)
logistic regression (437)