problem 1 consider a linear model to explain

Question

Problem 1: 

Consider a linear model to explain monthly beer consumption

Write the transformed equation that has a homoskedastic error term, and explain how you would estimate this equation.

Problem 2: 

Consider the following model of birth weight, bweight, as a function of the number of prenatal doctor visits, pnvisits,

Estimation by OLS provides the following estimates, which are all highly statistically significant, β₀= 8.0, β₁ = 0.02, and β₂ = -0.0004.

a) What is the effect on birth weight of increasing the number of prenatal visits from 3 to 4?

b) What number of prenatal visits maximizes log(bweight)?

c) What does the negative coefficient on the quadratic term imply about the relationship between birth weight and prenatal visits? Does the sign make sense? Explain.

d) Suppose we believe the effect of prenatal visits is different for whites and blacks. Write down a new model incorporating this hypothesis. Clearly define any new variables required.

e) Now suppose that we want to test the hypothesis that both the effect of prenatal visits and the intercept differ for whites and blacks. Write down this model and explain in detail how to test this hypothesis.

Problem 3:

This problem uses data on air quality. Download the file AIRQ from the course website under Resources > Air Quality Data. AIRQ contains observations for 30 standard metropolitan statistical areas (SMSAs) for California in 1972.  The data set contains he following variables: airq (air quality - lower number is better), vala (value added of companies in $1,000 US), rain (amount of rain in inches), coas (indicator variable equal to 1 if the area is at the coast and 0 otherwise), dens (population density), and medi (average income per person in US$).

a) Estimate a linear regression model that explains air quality from the other variables using OLS.

b) Perform a Breusch-Pagan test for heteroskedasticity related to all 5 explanatory variables using auxiliary regressions.

c) Perform a White test for heteroskedasticity using auxiliary regressions. Attach your results, and write out the form of the test as well as the value of the statistic and its interpretation.

d) What might be a problem with the White test in this application? What is a possible solution?

e) Perform a Goldfeld-Quandt test for heteroskedasticity under the assumption the variance of the error increases with population density. Attach your results, and write out the form of the test as well as the value of the statistic and its interpretation.