Simple Regression

For the two most recently completed major league baseball seasons, identify those players with at least 502 official times at bat in each season. Use the simple regression model to see how well each player's batting average in the most recent season is predicted by his batting average the preceding season. Do these results exhibit regression toward the mean?

Imagine that you are a US government statistician in the 1860s. Use the census data for the years 1790-1860 to predict the US population in 1930. To do so, estimate the equation $Y=\alpha +\beta X+\varepsilon$, where $Y$ is the population and X is the year (1790, 1800, and so on). Repeat with Y equal to the natural logarithm of population and compare your results.

Has there been any long-term trend in voter turnout in US presidential elections? Does the Democratic or Republican presidential candidate tend to do better when there is a heavy turnout?

Select a US President who ran for office twice; for example, Bill Clinton, George Bush, Barack Obama, or Donald Trump. For each of the 50 states, calculate this person's percentage of the total votes cast for the Democratic and Republican presidential candidates in each year; do not include the votes cast for other candidates. Is there a statistical relationship between these two sets of data? Are there any apparent outliers or anomalies?

An old Wall Street saying is, "As January goes, so goes the year." Use the simple regression model to see whether the February-throughDecember percentage change in the Dow Jones industrial average of stock prices is well predicted by the percentage change in January.

Use the simple regression model to see whether the annual percent-age-point change in the interest rate on 10-year Treasury bonds is well predicted by the change the preceding year.

Use the simple regression model to see whether the annual percentage change in the Dow Jones industrial average of stock prices is well predicted by the percentage change the preceding year.

It has been argued that interest rates move up and down in lockstep with the rate of inflation. Test this theory by using annual data to estimate the model, $\mathrm{R}=\alpha +\beta \mathrm{P}+\varepsilon$, where R is the interest rate on 1-year Treasury bonds and P is the annual rate of inflation.

Are future interest rates better predicted by current interest rates or by the current rate of inflation? Use annual data to estimate these two models, $\mathrm{R}=\alpha_{1}+\beta_{1} \mathrm{~ S}+\varepsilon_{1}$, and $\mathrm{R}=\alpha_{2}+\beta_{2} \mathrm{P}+\varepsilon_{2}$, where R is the interest rate on 1-year Treasury bonds, $S$ is interest rate the previous year, and P is the annual rate of inflation the previous year.

Find the heights and weights of the quarterbacks, running backs, and wide receivers who have been inducted into professional football's hall of fame. For each of these three positions, estimate a linear regression model with height as the dependent variable and time as the explanatory variable and another linear regression model with weight as the dependent variable and time as the explanatory variable. In each case, let the time variable equal the year in which the player first played professional football.

Pick a date and approximate time of day (e.g., 10:00 in the morning on April 1) for booking nonstop flights from an airport near you to at least a dozen large US cities. Determine the cost of a coach seat on each of these flights and the distance covered by each flight. Use your data to estimate a simple regression model with ticket cost the dependent variable and distance the explanatory variable. Are there any outliers?

Collect data for at least 50 years on the cost of attending your school (or another school that interests you). Taking into account the increase in the overall price level during these years, use the simple regression model to see whether there has been a trend in the real cost of attending this school.

Investigate how well mutual fund performance is predicted by past performance.

Use data for several presidential elections to estimate a simple regression model, where $\mathrm{Y}=$ percent of two-party vote for US President received by the incumbent party's candidate and $X=$ percent change in real disposable income in the last year of the incumbent party's term. Is there a statistically persuasive and substantial effect? Which elections appear to be outliers or anomalies?

This website https://www.spotrac.com/mlb/rankings/_/year/2024 has MLB player salaries. This site https://www.espn.com/mlb/war/leaders/_/type/seasonal/year/2024 has data for player wins above replacement. Looking at the $30-50$ highest-paid players for the most recently completed season, how strong is the relationship between pay and performance?

Looking at the most recently completed MLB season, how strong is the relationship between team payroll and regular-season team performance?

Looking at the most recently completed National Basketball Association (NBA) season, how strong is the relationship between team payroll and regular-season team performance?

Looking at the most recently completed English Premier League (EPL) season, how strong is the relationship between club payroll and regular-season club performance?

This website https://www.multpl.com/shiller-pe/table/by-year has data on the Shiller PE ratio (CAPE) on January 1 of every year going back to 1872. This website https://www.multpl.com/s-p-500-historical-prices/table/by-year has data on the SP 500 on January 1 of every year going back to 1871. Starting with January 1, 1872, how well does the Shiller P/E ratio at the start of each year predict the percentage change in the SP 500 that year?

Ask a random sample of US citizens to think of the last two digits of their Social Security number. Then show them a picture of a bottle of wine (or some other item of uncertain value) and ask them to guess the price of the wine. Do people with higher Social Security numbers tend to guess higher prices?

One measure of the relative importance of skill and luck in competitions is the consistency of the outcomes. Calculate the correlation between the winning percentages of Major League Baseball (MLB) teams before and after the All-Star Break. Compare this to the correlation for National Basketball Association (NBA) teams before and after their All-Star Break.

Follow the instructions for the previous question, comparing the correlations for National Football League (NFL) teams and English Premier League (EPL) teams for the first-half and second-half of their seasons. You can use winning percentages for NFL teams but may want to use points for EPL teams since many games are drawn.