Omitted-variable bias

In October 2001, Reuters Health reported that

In a study of more than 2,100 secondary school students, researchers found that boys who used computers to do homework, surf the Internet, and communicate with others were more socially and physically active than boys who did not use computers at all.

On the other hand, boys who used computers to play games tended to exercise less, engage in fewer recreational activities, and have less social support than their peers.

The findings, published in the October issue of the Journal of Adolescent Health, suggest that parents should monitor how their sons use their computers—and not just how much time they spend in front of the screen, lead author Dr. Samuel M.Y. Ho from the University of Hong Kong, told Reuters Health.

What problem do you see with this conclusion?

A state study compared the traffic fatality rates (number of fatalities per miles driven) on highways with 55,65, and 75 miles per hour speed limits. They found that highways with a 75 miles per hour speed limit had the lowest fatality rate and highways with a 55 miles per hour speed limit had the highest fatality rate. Traffic fatalities could evidently be reduced by raising speed limits, perhaps because people pay more attention when they are driving fast. What is the biggest statistical problem with this study?

A recent study compared the death rates (number of deaths each year per 1,000 players) of people who play a variety of games, including Bingo (41.3), Beer Pong (1.6), and Mortal Kombat (.04). As a statistician, how would you respond to this conclusion?

The numbers reveal an alarming truth: Bingo may not be as safe as some people have assumed. "This is the first evidence we have seen that violent video games actually reduce the death rate," says PlayGear CEO Pete Elor, "It comes as a blow to the head for people who advocate less violent forms of entertainment."

Lawyer Gerald Hill thinks the church and community need to take action: "When you look at the numbers, there's just no way you can get around it. Bingo is claiming lives right and left. And it doesn't just affect the Bingo community. Friends and families of those we lose to Bingo suffer even a greater loss." We can only hope that this study will cause people to think twice before engaging in such risky behavior.

Before coming to college, a student watched videotapes of a seminar, "Where there's a will, there's an A," which purports to show scientifically supported strategies for guaranteeing A grades. One strategy is to sit in the front of the class. To test this theory, a statistics student asked 50 randomly selected college students to write down their grade point average (GPA) and to indicate where they typically sit in large classrooms.

Here is an excerpt from a debate between two famous economists on the relationship between the growth of the money supply and the rate of inflation. Which economist do you agree with on the question of whether regression implies causation? Steve Hanke: I related the growth rate in the money supply to the inflation rate [in] 147 countries and the money supply measure I used is M2 and what do we end up with? We have almost a perfect correlation. John Cochrane: Correlation is not causation. We've known that forever. Steve Hanke: I also ran a regression and that does imply causation. John Cochrane: Oh, no no. Correlation does not imply causation. Come on now.

In 2005, Granite Construction applied for a permit to build a quarry near Temecula, California. Rocks would be dislodged from a mountain by daily blasts of 10,000 pounds of explosives and then crushed into gravel and sand that are used for concrete, asphalt, and other construction material. Mining and processing would go on 20 hours a day, 6 days a week, with 1,600 trucks entering and leaving the site daily.

The proposed quarry was in a mountain gap where ocean breezes carry cool air to the Temecula Valley. Residents were alarmed about the effect of this quarry on the city's economy and property values. However, an economist hired by Granite argued that in the city of Corona, which is 40 miles away and has had mines for many years, there was a "direct positive correlation" between annual mine production and property values over the past 20 years. Temecula residents evidently should be celebrating that their city was chosen for this quarry-which would have a positive effect on property values.

Provide a clear, persuasive refutation of this statistical argument.

A researcher used annual data for Major League Baseball (MLB) players for 1962 (when the current 162-game schedule was introduced) through 2011 to estimate this equation

where: $\mathrm{Y}=$ average number of home runs per game that season X = average weight (in pounds) of MLB players and the standard errors are in parentheses. The average value of Y is 1.76 and the average value of X is 191.76.

Critically evaluate:

The American Mustache Institute (AMI) is planning a "Million Mustache March" on the nation's capital. Their rallying cry: Pass the "Stimulus to Allow Critical Hair Expenses," or STACHE Act, which would "provide a $\$ 250$ annual tax deduction for expenditures on mustache grooming supplies." According to AMI research, mustached Americans earn 4.3 more money than "clean-shaven Americans" on average per year. Therefore incentivizing mustache growth would boost the economy. > "Given the clear link between the growing and maintenance of mustaches and income, it appears clear that mustache maintenance costs qualify for and should be considered as a deductible expense related to the production of income under Internal Revenue Code Section 212," wrote Dr. John Yeutter, a tax policy professor at Northeastern State University.

A researcher used data for 60 randomly selected colleges: $\mathrm{Y}=$ percent graduation rate (mean 59.65); $\mathrm{X}_{1}=$ students' median math plus reading/ writing SAT score, (mean 1,030.5); and $\mathrm{X}_{2}=$ percent of students with GPAs among the top $10 \%$ at their high school (mean $=46.6$). He found a statistically significant relationship between graduation rate and SAT scores (the t -values are in brackets):

and between the graduation rate and GPA:

However, when he included both SAT scores and GPAs in a multiple regression equation, the estimated effect of GPA on graduation rates was very small and not statistically significant at the $5 \%$ level:

Suppose that a college student's first-year grade point average G can be predicted from the student's high school grade point average H and standardized test scores T: $G=\alpha +\beta_{1} H+\beta_{2} T+\varepsilon$, where $\beta_{1}$ and $\beta_{2}$ are positive, and H and T are positively correlated. If T is omitted from this equation, how will least squares estimate of the coefficient of H be affected? What harm is there if T is omitted?

A student estimated a regression model using annual data for 1990 through 2015,

where: $\mathrm{C}=$ consumption spending $\mathrm{Y}=$ disposable income $\mathrm{W}=$ household wealth, all in billions of dollars The initial estimate of $\beta_{2}$ was not statistically significant at the $5 \%$ level; so the student omitted wealth from the equation, arguing that, "There could have been multicollinearity between income and wealth. Therefore, the regression is more accurate without wealth included." Explain why you either agree or disagree with that reasoning.

A blogger for the statistical software program Minitab answered the question, "How Can I Deal with Multicollinearity?" by writing that "the solution may be relatively simple.... Remove highly correlated explanatory variables from the model.... Because they supply redundant information, removing one of the correlated factors usually doesn't drastically reduce the R-squared." How would you respond?

Suppose that the percentage of the popular vote for US President received by the incumbent party's candidate is related negatively to both the unemployment rate and the rate of inflation. A student learned in a macroeconomics class that there is a negative correlation between unemployment and inflation, and consequently omitted the rate of inflation in order to avoid a multicollinearity problem:

Predict the effect of this omission on the estimated coefficient of the unemployment rate. Explain your reasoning so that a novice will understand your argument.

A researcher used 2011 data for 30 developing countries to estimate a model of real per capita GDP, in US dollars. The six education variables are the percent of people of the appropriate age enrolled in school; for example, the percent of females of primary school age who are enrolled in primary school. The adolescent fertility rate is births per 1,000 women ages 15-19.

The researcher initially estimated Model 1. After noting the high multicollinearity among the explanatory variables, the researcher estimated Model 2.

The author concluded that, "The R-squared decreased from 0.804 to 0.684. However, this model provides more precise estimates of the coefficients of the explanatory variables. The standard errors fall by almost half and the t -values for almost all variables, specifically secondary education for both sexes, are statistically significant, unlike in the previous model." What do you think?

What is the most important statistical problem with each of these studies?