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