Assignment
For the following questions test the hypotheses and state your conclusion.
i. Consider the following regression model: Y = β0 + β1 X1 + β2 X2 + 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:
Y^ = 300 + 10X1 + 200X2 (1)
(1.0) (25)
Test the hypothesis that
H0 : β2 = 160
HA : β2 ≠ 160
at the 5% level of significance.
ii. Consider the following regression model: Y = β0 + β1 X1 + β2 X2 + β3 X3 + u1
With a sample of 33 people (N=33) you get the following estimates, the standard errors are in parentheses:
Y^ = 102.19 - 9075X1 + 0.355X2 + 1.289X3 (2)
(2053) (0.073) (0.543)
Test the hypothesis that
H0 : β3 = 0
HA : β3 ≠ 0
at the 1% level of significance.
iii. Consider regression (2). Test the hypothesis that
H0 : β2 > 0
HA : β2 ≤ 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 = β0 + β1 LOTi + β2 AGEi + β3 BEDi + β4 FIREi + β5 BEACHi + ui
The equation we estimated is below. Standard errors are in parentheses.
PRIC^ Ei = 40 + 35LOTi - 2.0AGEi + 10.0BEDi - 4.0FIREi + 100BEACHi (1)
(5.0) (1.0) (10.0) (4.0) (10.0)
i. You expect the variables LOT, BED and BEACH to have positive coefficients. 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?
ii. You expect AGE to have a negative coefficient. Create and test the appropriate hypothe- ses to evaluate these expectations at the 1% level.
iii. You expect BEACH to have a coefficient different than zero. Create and test the appropriate 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 coefficient on FIRE < 0 and your HA : β4 = 0.5. Using a test at the 5% level, what would happen to the power of the test, (1 - β), if the sample size increased.
Question 3
Let's say we wanted to evaluate the effectiveness 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 ().
S^ i = 1.5 + 2.5 CLEANi (2)
(.50) (1.25)
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 β^ 0 and β^ 1
ii. Can you reject the idea that having clean piped water provided by the Kenyan government has no effect 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 β^ 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 effect of the clean water program on sickness (bigger, smaller, same) from equation [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 effect 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 effect 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 effect of the clean water program on sickness?