Econometrics 2 Pr. Dr. S. Sperlich Assignment 4: Tobit Models ...

Econometrics 2
Pr. Dr. S. Sperlich
Assignment 4: Tobit Models Summer 2008
Exercise 1: We use the dataset car.dta containing the mileage ratings (mpg in miles per gallon)
and weights (weight in pounds) of 74 cars.
(a) Make an ordinary regression between mileage and weight. Discuss the resulting
coefficient.
(b) In our data mpg ranges from 12 to 41. Create a new variable pretending that
the data were censored in the sense that we could not observe a mileage rating
below 17 mpg. Make now a Tobit regression.
(c) What are the marginal effects?
(d) Let us now assume that we do not observe the actual mileage rating of cars
yielding 24 mpg or better. Estimate the model!
(e) Fit a model that is censored from both sides (the so called two–limit Tobit).
Exercise 2: Consider a family saving function for the population of all families in the United
States:
sav = β 0 + β 1 · inc + β 2 · hhsize + β 3 · educ + β 4 · age + u,
where
• inc: household income,
• hhsize: household size,
• educ: years of education of the household head,
• age: age of the household head.
Assume that E(u|inc,hhsize,educ,age) = 0.
(a) Suppose that the sample includes only families whose head is over 25 years
old. If we use OLS on such a sample, do we get unbiased estimators of the β j ?
Explain.
(b) Now suppose our sample includes only married couples without children. Can
we estimate all of the parameters in the saving equation? Which ones can we
estimate?
(c) Suppose we exclude from our sample families that save more than $25,000 per
year. Does OLS produce consistent estimators of the β j ?
Exercise 3: Use the data in fringe.dta for this exercise.
(a) For what percentage of the workers in the sample is pension equal to zero?
What is the range of pension for workers with nonzero pension benefits? Why
is the Tobit model appropriate for modeling pension?
(b) Estimate a Tobit model explaining pension in terms of exper, age, tenure,
educ, depends, married, white and male. Do whites and males have statistically
significant higher expected pension benefits?
1

Econometrics 2
Pr. Dr. S. Sperlich
Assignment 4: Tobit Models Summer 2008
(c) Use the results from (b) to estimate the difference in expected pension benefits
for a white male and a nonwhite female, both of whom are 35 years old, single
with no dependents, have 16 years of education and 10 years of experience.
(d) Add union to the Tobit model and comment on its significance.
(e) Apply the Tobit model from part (d) but with peratio, the pension-earnings
ratio, as the dependent variable. (Notice that is a fraction between zero and one,
but, while it often takes on the value zero, it never gets close to being unity.
Thus, a Tobit model is fine as an approximation.) Does gender or race have an
effect on the pension-earnings ratio?
Exercise 4: Consider a double hurdle model of durable goods expenditure, where the purchase
decision is represented by
Y i1 = 1I Y ∗
i1 >0
but the actual amount purchased is represented by a censored regression model, thereby
avoiding negativity of Y i2 . In this case
Y i2 = Y i1 · max(0,Y ∗
i2).
One interpretation of this model is that observed demand Y i2 is zero either because
an individual genuinely does not consume the good (that is, Yi1 ∗ ≤ 0), or because
for some reason (infrequently of purchase, misreporting, etc.) zero expenditure was
recorded (that is, Yi2 ∗ ≤ 0).
(a) Determine P(Y i2 = 0) when Yi1 ∗ and Y i2 ∗ are independent.
(b) Specify the sample log–likelihood.
(c) What happens when P(Y i1 = 1) = 1 for all sample units?
(d) Which other than the ML–estimator could we use for estimation?
2