1. The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.
| Student |
GPA |
ACT |
| 1 |
2.8 |
21 |
| 2 |
3.4 |
24 |
| 3 |
3 |
26 |
| 4 |
3.5 |
27 |
| 5 |
3.6 |
29 |
| 6 |
3 |
25 |
| 7 |
2.7 |
25 |
| 8 |
3.7 |
30 |
(a) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation
GP^A = β^o + β^1ACT
Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points?
[b] Compute the residuals for each observation and verify that the residuals (approximately) sum to zero.
[c] What is the predicted value of GPA when ACT = 20?
[d] How much of the variation in GPA for these eight students is explained by ACT? Explain.
2. Consider data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs).
The following simple regression was estimated using data on n = 1,388 births:
bwˆght = 119.77 - 0.514ciˆgs
[a] What is the predicted birth weight when cigs = 0? What about when cigs = 20 (one pack per day)? Comment on the difference.
[b] Does this simple regression necessarily capture a causal relationship between the child's birth weight and the mother's smoking habits? Explain.
[c] To predict a birth weight of 125 ounces, what would cigs have to be? Comment.
[d] The proportion of women in the sample who do not smoke while pregnant is about 0.85. Does this help reconcile your finding from part (c)?
[3] Suppose that the Mankiw-Romer-Weil model describes the economy. However, human capital now accumulates according to the following equation:
dht/dt = shYtKtγδht
with γ > 1. So, human capital accumulates at more than proportionate way: every unit of output invested in education produces more human capital the more physical capital there is in the economy.
[a] Write down the equations of motion for human capital, physical capital and the equation for Y. Population and technology are constant. What can the fact that human capital accumulation depends on physical capital as described above represent?
[b] Write the two equations of motion in terms of only ht and Kt (hint: express them as percentage changes, for example (dht/dt)/(ht).
[c] As you can see from the equations above, these two equations describe the dynamic evolution of the economy. The state of the economy at time t is completely captured by the vector (Kt; ht). If we know this, we will also know (from the previous equations) towards what direction the economy will be moving. For example, for some values of (Kt; ht) we might have h? > 0 and K? < 0, i.e. the economy will evolve with a decrease in physical capital but an increase in human capital. For a certain set of states of the economy (Kt; ht), capital does not change, even though h may still be changing.
i. Find the set of states of the economy (Kt; ht) for which physical capital doesn't change. (Just write down the equation, no need to explicitly solve for the variables).
ii. Find the set of states of the economy (Kt; ht) for which human capital doesn't change. (Just write down the equation, no need to explicitly solve for the variables).
[d] When the economy belongs to both sets of points, neither its capital nor its human capital can change over time. Therefore, these are the steady states of the economy. Find them. How do they depend on γ (assume γ < (1-α-β)/β).
[e] What happens if instead γ ≥ (1-α-β)/β ? Why?