11 Binary dependent variables

Question 1

linear probability model

Answer 1

regression model with a binary dependent variable

Question 2

Disadvantages of the linear probability model

Answer 2

• Predicted probability can be above 1 or below 0!

- Error terms are heteroskedastic

Question 3

nonlinear probability models

Answer 3

Pr(Y = 1) = G(Z)
with Z = β0 + β1X1i + ··· + βkXki
and 0≤G(Z)≤1

Probit: G(Z) = Φ(Z)
Using the cumulative standard normal distribution function Φ(Z )

Logit: G(Z) = 1 / (1 + e^{-Z})
Using the cumulative standard logistic distribution function

Remember:
F(z) = Pr(Z ≤ z)

Question 4

the method used to estimate probit and logit models

Answer 4

Maximum Likelihood Estimation (MLE)

The models are nonlinear in the coefficients, so they can’t be estimated by OLS.

Question 5

likelihood function

Answer 5

The likelihood function is the joint probability distribution of the data, treated as a function of the unknown coefficients.

Question 6

maximum likelihood estimator (MLE)

Answer 6

The maximum likelihood estimator (MLE) are the values of the coefficients that maximize the likelihood function.

MLE’s are the parameter values “most likely” to have produced the data.

Question 7

If Yi is binary, then E(Yi | Xi) =

Answer 7

If Yi is binary, then E(Yi | Xi) = Pr(Yi = 1 | Xi)