## Demaris Logit Model Smsa

Notes: All financial variables are in thousands of dollars. Housing status: Rent (1 if rents; 0 otherwise) Housing status: Own (1 if owns; 0 otherwise) Source: Janet A. Fisher, "An Analysis of Consumer Good Expenditure," The Review of Economics and Statistics, vol. 64, no. 1, Table 1, 1962, p. 67.

Notes: All financial variables are in thousands of dollars. Housing status: Rent (1 if rents; 0 otherwise) Housing status: Own (1 if owns; 0 otherwise) Source: Janet A. Fisher, "An Analysis of Consumer Good Expenditure," The Review of Economics and Statistics, vol. 64, no. 1, Table 1, 1962, p. 67.

a. Comment generally on the fit of the equation.

b. How would you interpret the coefficient of —0.0051 attached to checking account variable? How would you rationalize the negative sign for this variable?

c. What is the rationale behind introducing the age-squared and number of children-squared variables? Why is the sign negative in both cases?

d. Assuming values of zero for all but the income variable, find out the conditional probability of a household whose income is \$20,000 purchasing a durable good.

e. Estimate the conditional probability of owning durable good(s), given: X1 = \$15,000, X3 = \$3000, X4 = \$5000, X6 = 0, X7 = 1, X8 = \$500, X9 = \$300, X10 = 0, X11 = 35, X13 = 1, X14 = 2, X16 = 0.

15.4. The R2 value in the labor-force participation regression given in Table 15.3 is 0.175, which is rather low. Can you test this value for statistical significance? Which test do you use and why? Comment in general on the value of R2 in such models.

15.5. Estimate the probabilities of owning a house at the various income levels underlying the regression (15.7.1). Plot them against income and comment on the resulting relationship.

*"An Analysis of Consumer Good Expenditure," The Review of Economics and Statistics, vol. 64, no. 1, 1962, pp. 64-71.

CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 627

*15.6. In the probit regression given in Table 15.11 show that the intercept is equal to — ¡ix ¡ax and the slope is equal to 1/ax, where ¡ix and ax are the mean and standard deviation of X.

15.7. From data for 54 standard metropolitan statistical areas (SMSA), Demaris estimated the following logit model to explain high murder rate versus low murder rate1':

ln Oi = 1.1387 + 0.0014P, + 0.0561C — 0.4050ft se = (0.0009) (0.0227) (0.1568)

where O = the odds of a high murder rate, P = 1980 population size in thousands, C = population growth rate from 1970 to 1980, R = reading quotient, and the se are the asymptotic standard errors.

a. How would you interpret the various coefficients?

b. Which of the coefficients are individually statistically significant?

c. What is the effect of a unit increase in the reading quotient on the odds of having a higher murder rate?

d. What is the effect of a percentage point increase in the population growth rate on the odds of having a higher murder rate?

15.8. Compare and comment on the OLS and WLS regressions (15.7.3) and (15.7.1).

### Problems

15.9. 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 2820 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.

Li = —2.77231 + 0.347582 ln Income t = (—3.35) (4.05) X2(1 df) = 16.681 (p value = 0.0000)

where Li = estimated logit and where ln Income is the logarithm of income. The X 2 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.

"Optional.

" J. S. Cramer, An Introduction to the Logit Model for Economist, 2d ed., published and distributed by Timberlake Consultants Ltd., 2001, p. 33. These results are reproduced from the statistical package PcGive 10 published by Timberlake Consultants, p. 51.

628 PART THREE: TOPICS IN ECONOMETRICS

15.11. In an important study of college graduation rates of all high school matriculants and Black-only matriculants, Bowen and Bok obtained the results in Table 15.19, based on the logit model.*

TABLE 15.19 LOGISTIC REGRESSION MODEL PREDICTING GRADUATION RATES, 1989 ENTERING COHORT

All matriculants Black only

TABLE 15.19 LOGISTIC REGRESSION MODEL PREDICTING GRADUATION RATES, 1989 ENTERING COHORT

All matriculants Black only