Economics 571 Problem Set #6 Solutions Wooldridge, 3.9 (i) I would ...

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Economics 571. Problem Set #6 Solutions. Wooldridge, 3.9. (i) I would expect β1 to be negative and β2 to be positive. The more pollution in a given community ...
Economics 571 Problem Set #6 Solutions Wooldridge, 3.9 (i) I would expect β1 to be negative and β2 to be positive. The more pollution in a given community, the less someone would be willing to pay for a home in that community (all else equal), so that β1 should be negative. In addition, we would expect β2 to be positive since rooms should be positively correlated with the size of the home, and larger homes will command higher prices. (ii) One possible explanation for the negative correlation is that pollution [and thus log(nox)] will tend to increase as one lives closer to the inner city or closer to a large metropolitan area. In addition, homes in such areas tend to be smaller on average (and homes in rural areas or suburbs are comparably larger ) so that we would see a negative correlation between log(nox) and rooms. (iii) To answer this question, we make use of our omitted variables bias formula discussed in class. Let β1 denote the coefficient on log(nox) from the short regression (not including rooms) and θ1 be the coefficient on log(nox) from the long regression (which includes rooms). Then, we obtain Cov(log(nox), rooms) p βˆ1 → θ1 + θ2 , Var(log(nox)) with θ2 denoting the coefficient on rooms from the long regression. ”Applying” this result to our two regressions, we obtain −1.043 ≈ −.718 + .306

Cov(log(nox), rooms) . Var(log(nox))

Based on our results in (ii), we expect a negative correlation between log(nox) and rooms. This is consistent with the above omitted variables bias formula, while a positive (or zero) covariance would not have been consistent. Wooldridge C3.4 Computer output and code for this regression is provided as an attachment. (ii) Regression results are provided in the attachment. First, note that the intercept has no useful meaning here, as it corresponds to the predicted attendance rate when ACT =0 and priGPA = 0. The minimum values in sample are not even close to zero. (iii) Somewhat surprisingly, the coefficient on ACT is negative, indicating that as ACT scores increase by one point, the predicted attendance rate decrases by about 1.7 percent.

2 For priGPA, every additional unit increase in cumulative GPA leads to about 17.2 percent more classes attended. (iv) d = 75.7 + 17.26(3.65) − 1.71(20) = 104.5. atndrte

There are some people in the sample with these types of values, although 3.65 is relatively high on the priGPA scale. (There appears to be one observation in the data set with exactly these values of the covariates). Note that this is something of a silly prediction since atndrte can not exceed 100, and there is nothing about our model that constrains our predictions in this way. The data just say that such an individual would likely attend all of his or her classes. (v) For person A d = 75.7 + 17.26(3.1) − 1.71(21) = 93.3. atndrte

and for person B: d = 75.7 + 17.26(2.1) − 1.71(26) = 67.49. atndrte

So, the predicted difference in attendance rates is 93.3-67.49 =25.81. Wooldridge, 4.5

(i) The Normal approximation to the 95 percent confidence interval (which is appropriate in suitably large samples) is .412 ± 1.96(.094) = [.228.596].

(ii) We form our test statistic under this null hypothesis: (.412 − .4)/.094 = .128. Thus, we can not reject this null hypothesis at the 5 percent level. (iii) We again form the test statistic (.412 − .1)/.094 = −6.26. Thus, we clearly reject this null at the 5 percent level (since the critical value from the Normal tables is 1.96). Wooldridge, C4.5

3 (i) Stata output is provided as a separate attachment. As the attachment shows, when rbisyr is dropped as a covariate, the coefficient on hrunsyr becomes statistically significant at the 1 percent level (when formerly it was not significant at the 10 percent level). In addition, the coefficient has increased significantly in magnitude. This should not be regarded as surprising - home runs and rbis are strongly positively correlated and capture a similar aspect of offensive production. When including them both, neither seems individually significant (though jointly they are), and when dropping one, the remaining covariate becomes statistically significant. (ii) When adding these three new variables, the only one that is significant at conventional levels is runsyr - in fact sbasesyr has the opposite sign of what we might expect.