Exercise 5. Stock and Watson (2019) Exercise E7.2 In the empirical exercises on earning and height in previous chapters, you estimated a relatively large and statistically significant effect of a workers height on his or her earnings. One explanation for this result is omitted variable bias: Height is correlated with an omitted factor that affects earnings. For example, Case and Paxson (2008) suggest that cognitive ability (or intelligence) is the omitted factor. The mechanism they describe is straightforward: Poor nutrition and other harmful environmental factors in utero and in early childhood have, on average, deleterious effects on both cognitive and physical development. Cognitive ability affects earnings later in life and thus is an omitted variable in the regression.

1. Suppose that the mechanism described above is correct. Explain how this leads to omitted variable bias in the OLS regression of Earnings on Height. Does the bias lead the estimated slope to be too large or too small? If the mechanism described above is correct, the estimated effect of height on earnings should disappear if a variable measuring cognitive ability is included in the regression. Unfortunately, there isnt a direct measure of cognitive ability in the data set, but the data set does include years of education for each individual. Because students with higher cognitive ability are more likely to attend school longer, years of education might serve as a control variable for cognitive ability: in this case, including education in the regression will eliminate, or at least attenuate, the omitted variable bias problem. Use the years of education variable (educ) to construct four indicator variables for whether a worker has less than a high school diploma (LTHS = 1 if educ < 12, 0 otherwise), a high school diploma (HS = 1 if educ = 12, 0 otherwise), some college (SomeCol = 1 if 12 < educ < 16, 0 otherwise), or a bachelors degree or higher (College = 1 if educ 16, 0 otherwise).

2. Focusing first on women only, run a regression of (1) Earnings on Height and (2) Earnings on Height, including LTHS, HS, and SomeCol as control variables. (a) Compare the estimated coefficient on Height in regressions (1) and (2). Is there a large change in the coefficient? Has it changed in a way consistent with the cognitive ability explanation? Explain. (b) The regression omits the control variable College. Why? (c) Test the joint null hypothesis that the coefficients on the education variables are equal to 0. (d) Discuss the values of the estimated coefficients on LTHS, HS, and SomeCol. (Each of the estimated coefficients is negative, and the coefficient on LTHS is more negative than the coefficient on HS, which in turn is more negative than the coefficient on SomeCol. Why? What do the coefficients measure?)

3. Repeat part 2, using data for men.