 econometrics

| April 27, 2015

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the homoskedasticity p-value is larger than the significance level (implying a failure to reject). In this case one may elect to use the heteroskedasticity robust version of the test as this test is robust to all forms of heteroskedasticity, as well as when the errors are in fact homoskedastic. Q.5. (5) A key conclusion of Brown et al. (2014) is that university endowment payouts respond asymmetrically to positive and negative shocks. In response to positive shocks, universities tend to leave current payouts unchanged while following negative shocks, universities actively reduce payout rates. Comment on the asymmetry in payout responses uncovered in Model (2) over that in Model (1). From an economic standpoint we see that when the return to an endowment is positive, the increase in the payout is roughly 6 times smaller than the decrease in the payout when the return to the endowment is negative. Statistically, we see that there is no change to the payout when returns are positive, but there is a statistically significant decrease in the payout when returns are negative. Q.6. (5) The heteroskedasticity robust p-values for statistical significance for β1 and β2 in Model (2) are 0.104 and 0.000, respectively. If you were conducting these tests at the 5% level what would your conclusions be regarding these asymmetric effects? In this case we would fail to reject the null hypothesis that β1 = 0 and we would reject the null hypothesis that β2 =. Thus, individually, the asymmetric effects of negative and positive returns hold at the 5% level. Q.7. (10) Detail why the two, single hypothesis tests you just investigated are inappropriate to test for the asymmetric effect of endowment returns. Setup a proper hypothesis that would allow you to test for the asymmetric effect suggested by Brown et al. (2014). These two single tests are inappropriate because each one ignores the other hypothesis. A true test for asymmetric effects would need to hold jointly and would require some form of an F-test. In our setting the appropriate null hypothesis is H0 :β1 = 0; β2 6= 0. However, this is a complicated null hypothesis to test. Mathematical: Please answer the following questions being as statistically precise as possible. 5 M.1. (10) You have the following regression model, yi = β0 + β1xi + β2x 2 i + εi where V ar(ε|xi) = σ 2x 2 i . Write out the feasible GLS regression that would produce OLS parameter estimators that were BLUE. yi/xi = β0 (1/xi) + β1 + β2xi + εi/xi M.2. (15) Consider a binary dependent variable y. Let ¯y represent the proportion of ones in the sample. Let ˆq0 = m0/n0 represent the % correctly predicted for the outcome y = 0 via the linear probability model (i.e. a fitted probability less than 0.5) and ˆq1 = m1/n1 represent the % correctly predicted for the outcome y = 1 via the linear probability model (i.e. a fitted probability greater than 0.5) where n0 (n1) is the total number of observations where yi = 0 (yi = 1) and m0(m1) is the total number of observations where ˆyi = 0 (ˆyi = 1). If ˆp is the overall % of outcomes that are correctly predicted (ˆp = (m0 + m1)/n), show that pˆ = (1 − y¯)ˆq0 + ¯yqˆ1. (3) Let n equal the total number of observations, n0 the number of observations where yi = 0 and n1 the number of observations where yi = 1. Then n = n0 + n1. Further, note that y¯ = n −1 Pn i=1 = n1/n. This implies that 1 − y¯ = n0/n. Now, by definition we have pˆ = m0 + m1 n . (4) Now, note that qˆ1 is the proportion of observations for which yi = 1 that the model correctly predicts, which we can quantify as qˆ1 = m1/n1. Similarly, qˆ0 is the proportion of observations for which yi = 0 that the model correctly predicts, which we can quantify as qˆ0 = m0/n0. Finally, we have pˆ = n0(m0/n0) + n1(m1/n1) n = n0qˆ0 + n1qˆ1 n = (n0/n)ˆq0 + (n1/n)ˆq1 = (1 − y¯)ˆq0 + ¯yqˆ1. (5) Bonus: Please answer the following question using proper spelling and grammar. B.1. (5) What does the fox say?

andpoint? Hint: Think about degrees of freedom. Q.20. (10) Detail the two main conditions that settler mortality must satisfy for it to be considered a valid instrument. Be precise. Q.21. (5 each) Please interpret both coefficient estimates in model (2) in Table 3. Q.22. (10) Can we say that the estimates of β1 in Table 3 are closer to the true value of β3 than those in either Table 1 or 2? Why or why not? Q.23. (15) Note that in models (1)-(3) in Table 1, the estimates of β1 are larger than their counterparts in Table 2. Comment on the theoretical implications that these larger estimates carry. Hint: Read Acemoglu, Johnson and Robinson. Q.24. (15) For models (1)-(3) in Table 1, the standard errors for βbIV 1 are larger than their counterparts in Table 2. Detail why this is not that surprising to you. Q.25. (10) The estimates of β2 which appear in Table 3 now have the wrong sign and are statistically insignificant with p-values larger than 0.15. Comment on the implication of this as it pertains to correlation between institutional quality and settler mortality. 6 Q.26. (10) If you disagreed with Acemoglu et al.’s (2001) assertion that early settler mortality was a valid instrument for current institutional quality, explain how you are hamstrung from a statistical standpoint. Q.27. (10) McArthur and Sachs (2001) suggest that the ‘disease environment’ and health characteristics of country belong in the Acemoglu et al. (2001) model. If disease environment was positively correlated with institutional quality and had a negative impact on growth, comment on the likely bias of Acemoglu et al.’s (2001) estimates? Further, discuss, given the magnitude found, why this may not be a valid concern pertaining to ruling out institutions as a driver of cross-country economic growth. Q.28. (10) Having considered the estimates appearing in Tables 1 through 3, detail to the best of your ability the likely impact that institutional quality has on economic growth. Be careful not to overstep your bounds, but to also say something with economic and statistical substance. 