From the household budget survey of 1980 of the Dutch Central Bureau of Statistics, J. S. Cramer obtained the following logit model based on a sample of 2,820 households. (The results given here are based on the method of maximum likelihood and are after the third iteration.) The purpose of the logit model was to determine car ownership as a function of (logarithm of) income. Car ownership was a binary variable: Y = 1 if a household owns a car, zero otherwise.

L̂_i = −2.77231 + 0.347582 ln Income

t = (−3.35) (4.05)

χ²(1 df ) = 16.681 ( p value = 0.0000)

where L̂_i = estimated logit and where ln Income is the logarithm of income. The χ² measures the goodness of fit of the model.

a. Interpret the estimated logit model.

b. From the estimated logit model, how would you obtain the expression for the probability of car ownership?

c. What is the probability that a household with an income of $20,000 will own a car? And at an income level of $25,000? What is the rate of change of probability at the income level of $20,000?

d. Comment on the statistical significance of the estimated logit model.