if they want to identify the causal effect of sports participation on wages where is individual i’s wage in 1994 and is an indicator variable for whether the individual played a sport in high school; and are a set of variables that control for individuals’ demographic characteristics, their family backgrounds, and the characteristics of their schools; and are measures of the individual’s conventional and unconventional abilities; and, is a random error term, which is uncorrelated with the explanatory variables.

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

Stevenson (2010) estimates the effect of participating in high school sports on adult wages using samples drawn from the 1979 Longitudinal Survey of Youth (NLSY). NLSY randomly selected teenagers and young adults in 1979 and surveyed them annually until 1994 when they were adults. Stevenson argues that economists need to estimate

(1)

How does Stevenson conceptually define conventional and unconventional ability?

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

Stevenson’s category of “unconventional skills,” are “less easily measured” than conventional academic abilities, and include attributes such as “the ability to communicate, the ability to work well with others, competitiveness, assertiveness, and discipline” (Stevenson, 2010, p. 286).

(1)

In table 1 of her paper, Stevenson presents estimates of using a variety of empirical specifications. The measures of ability she uses at this point in the paper are principally measures of conventional ability, not unconventional ability. In particular, she presents estimates of using the following two specifications:

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝜇𝑖(2)

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝜏 𝑖(3)

What is the interpretation of for girls using specification (2)?

(1)

In table 1 of her paper, Stevenson presents estimates of using a variety of empirical specifications. The measures of ability she uses at this point in the paper are principally measures of conventional ability, not unconventional ability. In particular, she presents estimates of using the following two specifications:

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝜇𝑖(2)

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝜏 𝑖(3)

What is the interpretation of for girls using specification (2)?

in the regression for girls that controls for demographics, family characteristics and school characteristics. It is statistically significant since is more than twice its standard error (0.039). The estimate implies that students who play high school sports earn, on average, 10.6 percent more as adults, holding their demographics, family characteristics and school characteristics constant.

(1)

What is the interpretation of using specification (3)?

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝜇𝑖(2)

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝜏 𝑖(3)

(1)

The estimate of decreases to when measures of conventional ability are added to the regression, and are slightly less significant, now only at the 10 percent level. The estimate implies that students who play high school sports earn, on average, 7.2 percent more as adults, holding their demographics, family characteristics and school characteristics constant.

Explain the difference in the two estimates using the concept of an omitted variable.

The first estimate of is probably upward biased due to omitting ability from the regression. In the regression that omits conventional ability (CA),

𝐸 ( �̂�1)≈ 𝛽1+𝛽5 ∙𝑐𝑜𝑣 (𝑆𝑝𝑜𝑟𝑡𝑠 ,𝐶𝐴)𝑣𝑎𝑟 (𝑆𝑝𝑜𝑟𝑡𝑠)

(+)

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝛽6𝑈𝐴𝑖+𝜀𝑖

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝜇𝑖(2)

ln (𝑤𝑎𝑔𝑒𝑖 )=𝛽0+𝛽1𝑆𝑝𝑜𝑟𝑡𝑠𝑖+𝛽2𝐷 𝑖+𝛽3 𝐹 𝑖+𝛽4𝑆𝑖+𝛽5𝐶𝐴𝑖+𝜏 𝑖(3)

(1)

(+)

and are also more likely to earn higher wages as adults. Hence, overestimates the importance of playing sports because part of the reason that people who played sports earn higher wages is because they have greater conventional ability, on average, not because they played sports.

𝐸 ( �̂�1)≈ 𝛽1+𝛽5 ∙𝑐𝑜𝑣 (𝑆𝑝𝑜𝑟𝑡𝑠 ,𝐶𝐴)𝑣𝑎𝑟 (𝑆𝑝𝑜𝑟𝑡𝑠)

(+)(+)

Virtually everyone believes that people with greater academic ability have higher earnings, on average, holding other factors constant, i.e., almost everyone believes that is positive. It also seems reasonable that is positive, for example because people with greater academic ability face lower opportunity cost of playing sports (e.g., foregone grades) and are more likely to be eligible to play. If these assumptions are correct then omitting conventional ability from the first regression causes an upward bias in the estimate of the impact of playing sports on earnings because people with greater conventional ability are more likely to play sports

What general point is Stevenson making by presenting these estimates of based on the NLSY?

Stevenson (2010) compares the estimates of from different specifications to demonstrate that there is “substantial self-selection into athletics and, without adequate controls, estimates of the impact of athletics on education will be upwardly biased” (p. 286). I’m sure she believes that the estimates in table 1 are biased because they do not control adequately for unconventional abilities that are important in being successful at sports and in the labor market.

What Stevenson wants to do is take high school girls who prior to Title IX would have been doing something other than sports, e.g., watching television, games, reading books, or talking with their friends, and randomly assign them to either not play sports or play sports. She can’t truly randomly assign girls but she can do something akin to it. What does she find that is akin to random assignment.

Population of high school girls who prior to Title IX were not doing sports

Treatment Group: Play Sports

Control Group: Don’t Play Sports

Stevenson (2010) exploits the variation in the participation of high school boys across states prior to Title IX to estimate the effect of playing sports on the educational and labor market outcomes of women. It is a natural experiment because girls are more likely to play sports in some states than others and the choice of where to live is sort of random, i.e., parents are not likely to move to a state because girls are more likely to play sports there.

Stevenson (2010) presents persuasive evidence that Title IX increased the number of years of education acquired by women. Since people with more education earn higher wages, on average, we would expect that Title IX would have increased the wages of women. But, it also increased the supply of people with more education, potentially leading to a decrease in the wages. Stevenson summarizes this “difficulty,” saying that “generated a large labor supply shock, which may mitigate against any human capital gain that might increase wages (p. 297).