1.   Ambient air pollution is a major public health issue in large cities around the world. Recent evidence also points toward the health risks from forest fires. In particular, exposure to air pollution during pregnancy affects the child’s cognitive development.

(a)       Collecting data from 2100 individuals, you estimate the following relationship

Ravens = 0.12 - 0.063×Pollution, R2 = 0.12

(0.8)  (0.017)

where Ravens is the score on the Raven’s intelligence test. Pollution is a continuous variable of air pollution during pregnancy.

What is the effect on exposure to an additional unit of air pollution during pregnancy on the Ravens intelligence test?

Is the relationship between Pollution and Ravens statistically significant?

Calculate the 95% confidence interval for the coefficient Pollution

The researchers are interested in determining the causal relationship between exposure to air pollution during pregnancy and later life intelligence. They are however concerned about reverse causality. Suggest a reason why they should worry about reverse causality in this particular case.

After conducting the Ravens intelligence tests, the researchers learn that 10% of the scores were measured with error. The computer system randomly added or withdrew one point for 10% of the subjects. Should the researchers worry about the inference they are drawing from the regression model? If so, why and how?

2.   Collecting data from 435 individuals, you estimate the following relationship using Linear Probability Model

Scholarship = 0.12 + 0.013× Grades,

(0.14)    (0.008)

where Scholarship is a binary variable that takes a value of 1 if the person received a scholarship to go to university, and 0 otherwise. Grades is a continuous variable that captures grades in high school. It varies between 0 and 100.

What is the effect of a one unit increase in Grades?

The model was estimated using Lineal Probability Model with normal standard errors. Shall we trust the inference from this model. If so, why/why not?

10% of students have an average grade of 89 or above. What is the predicted likelihood of receiving the scholarship for an individual with Grades = 89? Is Linear Probability Model the right model? If not, why?

3.   Women generally earn less than men. Consider the two potential earning structures depicted below.

Write the population regression functions used to estimate model (a) and model (b)

In the graphs below, mark the intercepts and slopes for the two graphs using the elements specified in the population regression models.

4.   Interpret the coefficients

(a) Y = β0 + β1ln (X) + u

(b) ln (Y) = β0 + β1ln (X) + u