Make scatter plot gdp growth versus population density

Problem 1

Suppose that X and Y are two random variables, and that their joint distribution P(X,Y ) has these moments:

Compute the following quantities: cor(X,Y ); E(3X + 2Y ); var(0.5X + 0.5Y ); cov(2X,X + 4Y ).

COAST65: a measure of a country’s average population density in coastal regions during the 1960’s, with higher values indicating countries with greater population density near the coast of a major body of water.

POPGR6090: average annual rate of population growth from 1960 to 1996.

1. The mean GDP growth rate for all 79 countries (AVERAGE in Excel).

2. The median population growth rate for all 79 countries (MEDIAN in Excel).

Now you will use the GDP-growth data to create some new scatter plots that do not appear in the notes. You will also evaluate some new causal hypotheses. For each part, keep your scatter plot and explanation to one side of piece of paper—for example, scatter plot on the top half of the page, explanation on the bottom half—just as in the notes. (This forces you to be concise!) You will be graded on the quality of your argument, not on whether you got the“right answer,” or the same answer that everyone else got.

Part A: Consider the following claim: “Economic infrastructure simply cannot grow as fast as humans can re-produce or migrate. Therefore, countries with high population growth rates will actually see lower rates of per-capita GDP growth, since the total size of the economy will not be able to grow as fast as the population.”

2. As before, inspect the scatter plot. Does this second causal hypothesis seem to make an accurate predic-tion? If so, name at least one confounder that might complicate matters. If not, explain why you think it doesn’t. Compute any summary statistics that you feel are relevant in making your case.

Part C: Finally, consider the following claim: “Economic growth depends crucially on having major cities near navigable waterways, which will facilitate international trade and allow cheaper access to far-flung domestic markets.”

A famous example of linear regression was carried out in 1885 by Sir Francis Galton, who wanted to see the effect of parents’ height on the height of their children. We will reconsider this problem and take a look at a sample of heights for University of Texas students and their parents. The data is in the file “heights.csv,” and the variables are: SHGT: Student’s height in inches
MHGT: Mother’s height in inches
FHGT: Father’s height in inches.

Part A: Create two scatter plots of child height versus mother’s height, and versus father’s height. Briefly compare the two plots; does either relationship look stronger than the other?

Part D: What is the variance decomposition for each of the regressions in Part C (i.e. what are TV, PV, and UV)? What is R2for each regression?

Part E: For each of the two regressions in Part C, turn in a plot of the residuals versus the x variable. If you see anything in these plots to worry you, comment on it.

Part B: Suppose that you have w dollars to invest in these two stocks. You decide to form a portfolio by placing

your portfolio after 12 months. pw dollars in US Steel, and (1 − p)w dollars in Alleghany, where 0 ≤ p ≤ 1. Let W denote the total value of

5. Suppose that you are risk averse, and that you have a log-utility function for terminal wealth: u(c) = logc. If you start off with w = 1000 dollars to invest in these stocks, what value of p will maximize your expected utility after 12 months? What are the expected return and standard deviation of this portfolio?

6. How does the expected return and standard deviation of the portfolio in (5) compare to those in (3) and (4)? Concisely explain this fact in terms of the concept of risk aversion.