Econometrics

Introduction to Econometrics

Problem Set 4

Due date: October 20th at the beginning of class

You may collaborate with other students. Please hand in your own work. At the top of your answer

sheet, please identify the students with whom you have consulted to complete this problem set.

Question 1

For the following questions test the hypotheses and state your conclusion.

i. Consider the following regression model: Y = �0 + �1X1 + �2X2 + u1

With a sample of 30 people (N=30), which is enough to use a t distribution, you get the

following estimates, the standard errors are in parentheses:

Ŷ = 300 + 10X1 (1.0)

+ 200X2 (25)

(1)

Test the hypothesis that

H0 : �2 = 160

HA : �2 6= 160

at the 5% level of significance.

ii. Consider the following regression model: Y = �0 + �1X1 + �2X2 + �3X3 + u1

With a sample of 33 people (N=33) you get the following estimates, the standard errors

are in parentheses:

Ŷ = 102.19 � 9075X1 (2053)

+ 0.355X2 (0.073)

+ 1.289X3 (0.543)

(2)

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Test the hypothesis that

H0 : �3 = 0

HA : �3 6= 0

at the 1% level of significance.

iii. Consider regression (2). Test the hypothesis that

H0 : �2 > 0

HA : �2  0

at the 5% level of significance.

Question 2

Suppose that you estimate a model of house prices to determine the impact of having beach

frontage on the value of a house. The model looks at the price of house i in thousands of

dollars (PRICEi) as a function of the size of the lot of house i in thousands of square feet

(LOTi), the age of the house i in years (AGEi), the number of bedrooms in the house i

(BEDi), whether house i has a fireplace (FIREi=1 if house i has a fireplace and 0 if not,

this a dummy variable) and whether house i has a beach frontage (BEACHi=1 if house i

has a beach frontage and 0 if not). We have 10,000 observations. For hypothesis tests use the

standard normal distribution.

The regression model you want to estimate is:

PRICEi = �0 + �1LOTi + �2AGEi + �3BEDi + �4FIREi + �5BEACHi + ui

The equation we estimated is below. Standard errors are in parentheses.

dPRICEi = 40 + 35LOTi (5.0)

� 2.0AGEi (1.0)

+ 10.0BEDi (10.0)

� 4.0FIREi (4.0)

+ 100BEACHi (10.0)

(1)

i. You expect the variables LOT, BED and BEACH to have positive coe�cients. Separately

create and test the appropriate hypotheses to evaluate these expectations at the 10%

level for each estimate. What do you conclude about the relationship of LOT, BED and

BEACH to PRICE, respectively?

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ii. You expect AGE to have a negative coe�cient. Create and test the appropriate hypothe-

ses to evaluate these expectations at the 1% level.

iii. You expect BEACH to have a coe�cient di↵erent than zero. Create and test the appro-

priate hypotheses to evaluate these expectations at the 5% level.

iv. Compute and interpret the p-value of part (iii). Interpret your result.

v. Assume your null hypothesis is that the coe�cient on FIRE < 0 and your HA : �4 = 0.5.

Using a test at the 5% level, what would happen to the power of the test, (1 � �), if the sample size increased.

Question 3

Let’s say we wanted to evaluate the e↵ectiveness of a water-cleaning program in Kenya on the

health of the country’s people. The Kenyan government constructed and delivered free clean

piped water to a quarter of the villages in the country. We have the data for a random sample

of individuals in Kenya, and we know if they lived in a village that received free clean water

(CLEANi=1 if the water is provided by the Kenyan government) or not (CLEANi=0). We

also know how many days over the last month each individual in our sample felt sick, Si. The

equation we estimated is below. Standard errors are in parentheses ().

Ŝi = 1.5 (.50)

+ 2.5 (1.25)

CLEANi (2)

where:

Si= sick days in the last month for individual i

CLEANi= indicator equaling 1 if the Kenyan government provided clean piped water to the

village of individual i

i. Interpret �̂0 and �̂1

ii. Can you reject the idea that having clean piped water provided by the Kenyan gov-

ernment has no e↵ect on the number of days a person is sick per month? Use a 95%

confidence interval.

iii. Is the result from ii) combined with the sign of �̂1 unexpected? Why or why not?

iv. Many times when governments of developing countries spend money on providing free

services to a part of their population, they choose to deliver the services to the poorest

villages, as they are typically the ones that are in biggest need of the public programs.

Given this information, what do you think would happen to your original estimate of

the e↵ect of the clean water program on sickness (bigger, smaller, same) from equation

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[2] if you included into equation [2] a variable WEALTH, which represented the wealth

of individual i,? Why?

v. Many times when government’s of developing countries spend money delivering free

services to certain areas of the country they also must engage in other infrastructure

projects (roads, health clinic’s for government workers, etc.) Also, since new and better

infrastructure is now available in these villages, usually non-government organizations

(NGOs) and charities that previously had no access to these areas start to provide their

services for the first time. Given this information, what do you think would happen

to your original estimate of the e↵ect of the clean water program on sickness (bigger,

smaller, same) from equation [2] if you included into equation [2], a variable NGO, which

represents the number of NGOs currently operating in individual i village? Why?

vi. What do you think would happen to your original estimate of the e↵ect of the clean

water program on sickness (bigger, smaller, same) from equation [2] if you included into

equation [2], a variable MONTH (1 to 12), which represents the month individual i was

born? Why?

vii. What is an example of an important variable, excluded from equation [2]? How would

it change your original estimate in equation [2] of the e↵ect of the clean water program

on sickness?

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