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Use the data in ELEM94 $_{-} 95$ to answer this question. The data are on elementary schools inMichigan. In this exercise, we view the data as a cluster sample, where each school is part of adistrict cluster.(i) What are the smallest and largest number of schools in a district? What is the average numberof schools per district?(ii) Using pooled OLS (that is, pooling across all $1,848$ schools), estimate a model relating lavgsalto bs, lenrol, lstaff, and lunch; see also Computer Exercise 11 from Chapter $9 .$ What are thecoefficient and standard error on $b s ?$(iii) Obtain the standard errors that are robust to cluster correlation within district (and alsoheteroskedasticity). What happens to the $t$ statistic for bs?(iv) Still using pooled OLS, drop the four observations with $b s>.5$ and obtain $\hat{\beta}_{b s}$ and its cluster-robust standard error. Now is there much evidence for a salary-benefits tradeoff?(v) Estimate the equation by fixed effects, allowing for a common district for schools withina district. Again drop the observations with $b s>.5 .$ Now what do you conclude about thesalary-benefits tradeoff?(vi) In light of your estimates from parts (iv) and (v), discuss the importance of allowing teachercompensation to vary systematically across districts via a district fixed effect.

(i) 1 and 162, average 22.3 (ii) -.516 (.110) (iii) bs is marginally significant (iv) t stat drops to under 1 (v) fe estimate is -.523 after dropping (vi) see video

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Chapter 14

Advanced Panel Data Methods

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part one. The smallest number of schools is one you may find that 271 out of 537 districts have only one elementary school in the sample. Yeah, the largest number of schools in the district is 162. Yeah, the average number of schools is about 22.3 mhm or to the old. L s estimate of beta B s is about minus 0.516 and the usual s standard payroll is 0.11 for three. The standard barrel on B s robust. You within district cluster correlation and heterocyclic elasticity is about point 25 three. This gives a T statistic on Bs of minus two point oh four, so T statistic has reduced, and now the variable bs is only marginally significant. At four, you may recall from computer exercise 12 in chapter nine, the coefficient on B s is very sensitive to the inclusion of the observation with the highest PS. When the four observations with Bs greater than 40.5 are dropped, our estimate on Bs becomes minus point 186 with a robust standard error of 0.273 and the T statistic, which is the ratio of the estimate data on the standard. Errol is minus 0.68 and given this t start, we are unable to reject the non hypothesis that beta bs is zero. In other words, there is little evidence of the trade off between salary and benefit. Thank you. First five, when we use all the observations, the fixed effects estimate of beta bs is minus one point oh five mhm. And the standard error is point 09 six. And this value is very close to the theoretical value of minus one. Yeah, and for sure, the 95% confidence interval contains minus one. If we drop the four observations with B s greater than 40.5, the fixed effects estimate of beta Bs becomes minus 0.5 chu three and the standard error is point 147 And now we can reject the null hypothesis that beta bs is zero. So, in other words, there exist evidence that there is a trade off between benefits and salary. However, you may find that the 95% confidence interval construct from mhm this beta had value and the standard error does not contain minus one. This is equivalent to saying we reject the null hypothesis that Beta had beta B s equals minus one. Okay against the two sided alternative. So again, there is little evidence that the trade off between benefits and salary is 1 to 1. Part six. We get such a result in part five and four because we have the district fixed effects. The district effect captures the fact that some districts pay higher salaries and higher benefits for reasons not captured by our assistant controls. And once we control for the systematic differences across districts, we find a trade off. The fixed effects estimate is based on the variation in salary benefits packages across schools within districts. Okay, One reason different compensation packages may exist within districts is because of different teacher age distributions with red within districts. Yeah,

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