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Homework1-AK

# Homework1-AK - Homework 1 ECO 7427 ANSWER KEY Prof Sarah...

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Homework # 1 ECO 7427 ANSWER KEY Prof. Sarah Hamersma 1. This was a hard question. I gave substantial partial credit for wrong answers that were well thought-out. But please do make sure you read this so you know the right answer. Outline of answer: a) clustering standard errors is still needed, even with dummies b) explanation of what dummies can and cannot successfully fix c) explanation of why clustering will make SEs bigger even if it’s unneeded John Lott’s analysis uses county-level data from several states and looks at the impact of state-level treatments. Note that he does not use individual data at all – the unit of observation is the county. This means when he refers to using county fixed effects, this is equivalent to an “individual” fixed effect from the perspective of his sample where each observation is a county. He argues that including county fixed effects implicitly includes state fixed effects. This argument is correct. However, this only moves us one step closer to the real question: Does including state fixed-effects mean you don’t need clustering at the state level? The answer is that you still may need clustering. State fixed effects are an important component of an analysis that uses state-level treatments. As noted in the example above, there may be correlated outcomes Y within a state that are not picked up by observable X’s. This can be thought of as an omitted variables (endogeneity) problem – so if this is the case, and we do not include state fixed effects, our estimates of the treatment effect will be biased and inconsistent (not to mention the standard errors!). Including a state fixed effect allows us to explain some of this variation. Econometrically, it will force the expected value of the residuals within each state to be zero (if they averaged something else, this would have been incorporated into the estimate of the fixed effect by construction). Suppose that these state fixed effects properly fix the point estimates (i.e. there is no longer an omitted variables problem). What does the error structure look like now? Well, within each state there are several counties. We can estimate a regression and look at the residuals within each state – they will average zero (as noted above) but depending on the state they might be spread widely or distributed narrowly around zero. This is a heteroskedasticity problem – solve it with the “robust” function to fix your standard errors. Where does the clustering come in? It is worth noting that the clustering problem would have been HUGE if we ignored the fixed effects to start with, and so including them does make the problem smaller (which is why some of our intuition suggested that it could fix the problem). However, it may still remain. The issue is that we have controlled only for a very specific form of correlation among observations within a state – we have controlled for a form of correlation in which every observation in the state has a common (state-level) component of 1

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