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## Problem 1: Suppose we want to estimate the effect of income on smoking,smoke.- = ?g + ,31income; + ?

Problem 1: Suppose we want to estimate the effect of income on smoking,smoke.- = ?g + ,31income; + ?zege; + 6;. We have data on lottery winnings that we want to use as an instrument for income. (i) Describe how to estimate til using two stage least squares.(ii) Suppose instead that we want to estimate smoke,- = ?g + .51 log(incomei) + ?zage; + 5;. Describe how to estimate ?1 using two stage least squares.(iii) Instead of what you describe in (ii), consider estimating ,61 by 1. Regress income on lotto and age. Compute the fitted values, imcf. 2. Regress smoke on log(income;] and age._Call this estimate 1″â”””â. Argue that s{Â°ââ”ââ is inconsistent. [toss 0L5 residuals havesome important properties, the difference log(income) â log[income) is not an 0L8 resid-ual]. Problem 2: A researcher is interested in the effect of military service on earnings. He collects datafrom a random sample of 4000 workers aged 40 and runs the DIS regression, is = ?n + ?iVETt + 6;, where yg is log annual earnings and VET; equals 1 if the person served in the military and 0otherwise. (i) Use selection bias to explain why the OLS estimates are likely to be unreliable. What is thetreated group? What is the untreated group? What is the hypothetical, unobserved groupto which we would like to compare the treated group to estimate the causal effect of beinga veteran? (ii) Use omitted variables bias to explain why the 01.8 estimates are likely to be unreliable.Give one example of an important omitted variable. Based on this omitted variable, inwhich direction do you expect 01.5 to be biased? (iii) In the US, during the Vietnam war, there was a draft (i.e. mandatory military service forsome people) in which priority for the draft was determined by a national lottery. (Ihedays of the year were randomly reordered 1 through 365. Those with birthdates orderedfirst were drafted before those with birthdates ordered second, and so forth.) What mustwe assume about the lottery to use it as an instrument to estimate the effect of militaryservice on earnings? (iv) Angrist, Chen, and Song {2011) estimate a model similar to the above. Read this paper (itâsonly five pages}. What are the estimateis) of ii reported in this paper? What are the mainfindings of this paper?