PPHA 311: Statistics for Data Analysis II

PPHA311

ProblemSet2

Winter2024

1 Crime and economic conditions

In this question, you will work with state-level data on murder rates, economic conditions, and capital punishment in 2018.

You should use R to answer this question and include code and output in your submission.

Refer to the datafile cp.csv available on Canvas, which reports data for 2018. This dataset has the following variables:

1. Create a scatter plot between the unemployment rate and the murder rate. Please put the unemployment rate as X axis and the murder rate as Y axis. (1 point)

2. Estimate the effect of unemp on mrate using OLS.

mrate = β₀ + β₁unemp + u

Interpret β^ˆ₀ and β^ˆ₁. Your answer should also reference the statistical significance of these estimates. (2 points)

3. Calculate the predicted values of the murder rate for a state with unemployment rates of 2.5 percent and 5 percent, respectively. You can calculate the predicted values by hand (show your work) or using RStudio (show your code and output). (2 points)

4. Do you think this regression analysis is suitable for estimating the causal effect of the unemployment rate on the murder rate? Please explain your reasoning. (2 points)

5. Capital punishment is a highly controversial policy issue today. Proponents argue that capital punishment has a strong deterrent effect.

Add the capital punishment rate (execr) to the model in (2) and reestimate the following regression (i.e., the following regression equation).

mrate = β₀ + β₁ · unemp + β₂ · execr + ϵ

Report the estimated coefficient and intercept. Interpret the coefficient on execr (including the statistical significance).

Does the estimated equation suggest a deterrent effect of capital punishment? Based on your answers for (2), discuss whether the data support that capital punishment discourages murders. (2 points)

6. Compare the model fit in (2) and (5) using adjusted R squared. How did the model fit change from (2)? Discuss. (2 points)

7. Construct a 95 percent confidence interval for the effect of a 1 percentage point increase in the execution rate on the murder rate. Based on the confidence intervals, discuss whether you can reject a large deterrent effect. (2 points)

2 The Returns to Schooling

Many countries allocate a large budget to educational expenditures. Are these budgets justified? In this question, you will work with data on wages and education.

You should use R to answer this question and include your code and output in your submission. Refer to the datafile dwage.csv available on Canvas. This dataset has the following variables:

1. Estimate the following bivariate regression model via OLS and interpret the size and statistical significance of your estimates (i.e.,both β^ˆ₀ and β^ˆ₁). (2 points)

wage = β₀ + β₁educ + u

2. Calculate the residuals (ˆu_i) for your OLS regression from Part (1) above.

Using R, calculate: (a) the mean of ˆu_i, (b) the covariance between ˆu_i and educ. Report your result to 6 decimal places.

Calculate the mean residual for each year of education and plot the mean residual on the y-axis for each year of education on the x-axis. Describe the plot. Which years of education show the largest residuals? (2 points)

3. Now, estimate the following bivariate model via OLS. Interpret the ˆγ₁ coefficient. Using the regression result, discuss whether the model estimated in question (1) is likely to be an unbiased estimate of the return to education. Why or why not? (2 points)

educ = γ₀ + γ₁exper + v

4. Now, estimate the following multivariate regression model via OLS. Has your estimate of β^ˆ₁ changed from what you found in question (1)? (2 points)

wage = β₀ + β₁educ + β₂exper + β₃SMSA + u

5. Recalculate the residuals (ˆu_i) for your OLS regression from Part (4). Calculate the covariance between ˆu_i and each of the X’s: educ, experience, and SMSA. Report your results to 6 decimal places. Based on these covariances, should we conclude that the regression is unbiased and identifies the causal effect of education on earnings? (2 points)