Linear Probability Model

The linear probability model, ctd.
When Y is binary, the linear regression model
Yi = β0 + β1Xi + ui is called the linear probability model.
• The predicted value is a probability:
• E(Y|X=x) = Pr(Y=1|X=x) = prob. that Y = 1 given x
• Yˆ = the predicted probability that Yi = 1, given X
• β1 = change in probability that Y = 1 for a given ∆x:
Pr(Y = 1 | X = x + ∆x ) − Pr(Y = 1 | X = x ) β1 =
∆x
5

Example: linear probability model,
HMDA data
Mortgage denial v. ratio of debt payments to income
(P/I ratio) in the HMDA data set (subset)

6

1

Linear probability model: HMDA data, ctd. n = -.080 + .604P/I ratio deny (.032) (.098)

(n = 2380)

• What is the predicted value for P/I ratio = .3? n Pr( deny = 1| P / Iratio = .3) = -.080 + .604×.3 = .151
• Calculating “effects:” increase P/I ratio from .3 to .4: n Pr( deny = 1| P / Iratio = .4) = -.080 + .604×.4 = .212
The effect on the probability of denial of an increase in P/I ratio from .3 to .4 is to increase the probability by .061, that is, by 6.1 percentage points.

7

Linear probability model: HMDA data, ctd
Next include black as a regressor: n = -.091 + .559P/I ratio + .177black deny (.032) (.098)
(.025)
Predicted probability of denial:
• for black applicant with P/I ratio = .3: n Pr( deny = 1) = -.091 + .559×.3 + .177×1 = .254
• for white applicant, P/I ratio = .3: n Pr( deny = 1) = -.091 + .559×.3 + .177×0 = .077
• difference = .177 = 17.7 percentage points
• Coefficient on black is significant at the 5% level
• Still plenty of room for omitted variable bias…
8

2

The linear probability model:
Summary
• Models Pr(Y=1|X) as a linear function of X
• Advantages:
• simple to estimate and to interpret
• inference is the same as for multiple regression (need heteroskedasticity-robust standard errors)
• Disadvantages:
• Does it make sense that the probability should be linear in X?
• Predicted probabilities can be <0 or >1!
• These disadvantages can be solved by using a nonlinear probability model: probit and logit