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Use the data in ELEM94 95 to answer this question. See also Computer Exercise $\mathrm{C} 10$ in Chapter $4 .$(1) Using all of the data, run the regression lavgsal on bs, tenrol, lstaft, and lunch. Report thecoefficient on $b s$ along with its usual and heteroskedasticity-robust standard errors. What do youconclude about the economic and statistical significance of $\hat{\beta}_{b s} ?$(ii) Now drop the four observations with $b s>.5,$ that is, where average benefits are (supposedly)more than 50$\%$ of average salary. What is the coefficient on $b s ?$ Is it statistically significantusing the heteroskedasticity-robust standard error?(iii) Verify that the four observations with $b s>.5$ are $68,1,127,1,508,$ and $1,670 .$ Define fourdummy variables for each of these observations. (You might call them $d 68, d 1127, d 1508$ , and $d 1670 .$ Add these to the regression from part (i) and verify that the OLS coefficientsand standard errors on the other variables are identical to those in part (ii). Which of the fourdummies has a $t$ statistic statistically different from zero at the 5$\%$ level?(iv) Verify that, in this data set, the data point with the largest studentized residual (largest $t$ statistic on the dummy variable) in part (iii) has a large influence on the OLS estimates. (That is, run OLS using all observations except the one with the large studentized residual.) Does dropping,in turn, each of the other observations with $b s>.5$ have important effects?(v) What do you conclude about the sensitivity of OLS to a single observation, even with a largesample size?(vi) Verify that the LAD estimator is not sensitive to the inclusion of the observation identified inpart (iii).

(i) see video (ii) evidence of heteroskedasticity (iii) see video (iv) robust SEs differ getting either higher or lower but slightly (v) WLS is more precise

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Chapter 8

Heteroskedasticity

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but one this is the estimated equation. The usual S standard errors are in green and in round brackets. The hetero skid elasticity, robust standard Iran's are in blue and in square bracket. As you can see the robot standard Iran's are somehow larger in all cases on variable lock of E. X. B pp. Yeah the robust T statistic is 1.5 not too high. So we are not very convinced if performance its link to spending hard to. The f statistic has a value of 132.7 and a p value almost zero. Therefore there is the strong evidence of heterocyclic elasticity for three and four. This is the same equation estimated by weighted least square usual. Standard herons are in round and green brackets and weighted least square standard Iran's are in blue square brackets. We don't care about it. Their standard barrel over there. Constant term the intercept. So I don't report the centered errol oh, weighted least square for it. Yeah. Oh sorry the round one hour the usual wait at least square and the square one is robust. Type all part three. Mhm. We have to compare the estimates from weighted least square approach to L. S. Approach. Yeah. And we see that the old L S n W L. S coefficients on lunch or the same. But for other variables they differ. The waiting list square estimate on lock of spending is much larger than the old L. S coefficient. Based on this estimate a 10% increase in spending. Let me write that down. That is equivalent to an increase of 0.1 in lot of spending. Let's do you a pulling 65 percentage point increase in the math past rate the weighted least square T statistic is much larger too. Before we get 1.5 now we get 3.8. More than double past four. We compare the robot standard Iran's with the usual W. L. S. Standard Iran's and we find that for the key variable lack of spending, the robust standard error is somewhat larger than the usual one robust scented errol, our lock of enrollment also somewhat higher and for lunch, the robust version is slightly lower than the usual one, part five, the weighted least square estimated is more precise and by precise, I mean, robust standard errol is smaller than the robust version of the old LS estimate. For the first one we got 1.82 and for the second one two point 35

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