Problems
1. Provide short answers for the following questions:
(a) Read the recent article titled ìDiet Soda Linked To Weight Gainî.
In particular, focus on the main result of the study:
ìResearchers found that the diet soda drinkers had waist circumference
increases of 70 percent greater than those who non-diet soda
drinkers. And people who drank diet soda the most frequently ñ at
least two diet sodas a day ñ had waist circumference increases that
were 500 percent greater than people who didnít drink any diet soda,
the study saidî.
Brieáy explain what may have gone wrong in the study. How would
you design an experiment to see the e§ect of diet sodas on weight
gain?
(b) Explain the di§erence between estimator and estimate. Provide an
example of each.
(c) A population distribution has a mean of 5 and a variance of 12.
Determine the mean and variance of Y from an i.i.d. sample from
this population for
i. n = 20
ii. n = 200
iii. n = 2000
Relate your answer to the Law of Large Numbers.
(d) Use the information in the previous item for n = 2000 and the Central
Limit Theorem to Önd Pr
8:6 Y 11:2

.
2. Suppose the simple linear regression model
Y = 0 + 1X + U;
i.e., K = 1. And suppose we only have 3 observations (n = 1; 2; 3).
(a) Rewrite the model in matrix form. Be explicit about the elements of
the matrices and vectors you deÖne.
(b) Compute explicitly
b =

b
0
b
1
!
= (X0X)
1
X0Y
1
Hint: You need to remember the formula for inverse matrices in the
3×3 case.
(c) Show that the expression for the slope b
1 you obtained in the previous
item equals the formula for the OLS estimator of the slope in the
simple regression case (with n = 3)
b
1 =
Pn
i=1
Xi
X

Yi
Pn
i=1
Xi
X
2
Hint: Use the fact that Pn
i=1 XiYin

XiY i

=
Pn
i=1
Xi
X

Yi = Pn
i=1
Xi
X
Yi

Y

.
3. Consider the simple linear model:
Y = 0 + 1X + U;
where Y is the consumption level of a family, and X is the familyís income
level. Both consumption and income levels are measured in dollars. Take
the OLS estimator for 0 and 1
b
1 =
Pn
i=1
Xi
X

Yi
Pn
i=1
Xi
X
2
;
b
0 = Y
b
1X:
(a) Suppose you estimated
b
1 = 0:514;
b
0 = 25:16:
Provide an economic interpretation of the results.
In the next items, consider the following changes introduced in the
variables of the model. Show mathematically how the original estimators
can be altered by these changes, i.e., calculate the new
least-squares estimators and point out how these estimators can be
obtained from the original estimators.
(b) All observations of X and Y are multiplied by a constant k. Use
your answer here to show how you would interpret b
0 and b
1
if
all observations on consumption and income were measured in cents
instead of dollars.
(c) Only the observations of X are multiplied by a constant k. Use
your answer here to show how you would interpret b
0 and b
1
if
observations on consumption are measured in dollars, but income is
measured in Crowns (1 dollar = 20.21 Crowns).
(d) A constant k is added to each observation of X. Use your answer
here to show how you would interpret b
0 and b
1
if the number 15
(say 15 dollars) is added only to the observations on income.
2
2 Computer Based Problems
1. Production Function. In this exercise you will estimate a Cobb-Douglas
production function:
Yi = AK1
i L
2
i
e
Ui
;
where Y is the production, K is the capital stock, L is the labor force
and U reáects the unobservables that a§ect production that are neither
capital nor labor (some call this term Total Factor Productivity, TFP).
The dataset “production_function.dta” contains the following variables:
output, age, capital, labor. Each observation in the dataset corresponds
to a Örm.
(a) Using a suitable transformation, express the model in ëlinear-in-parametersí
form and give an economic interpretation of the parameters of the
model.
(b) Estimate the parameters of the model by ordinary least squares. Interpret
the estimated coe¢ cients in economic terms. Are they statistically
di§erent from zero at 5% signiÖcance level?
(c) Test the joint signiÖcance of the slope parameters. Comment.
(d) Test the hypothesis that there are constant returns to scale.
(e) Calculate R2
. How much of the variation of output across Örms is
due to the Total Factor Productivity, TFP?
(f) Calculate TFP for each Örm in the data. Then compare its average
and standard deviation for Örms that are less than 3 years old with
those that are more than 14 years old (I am not asking to run any
hypothesis test here, just do simple comparisons). Compare also the
average amount of capital and labor employed by these two groups
of Örms. Comment whether the di§erences across groups make economic
sense. Plot the distribution of TFP for each group.
Hint: To show the distribution, you can use histograms or the command
ëkdensity,íwhich “smooths” the histogram.
(g) Run the regression again, but this time, omit capital. Compare the
results obtained here with item (b). Are the results di§erent for b
2
?
If so, is the di§erence expected?
3