ECON 141 UCB OLS Estimator of Minimum Wage Effect Exercise

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Question 3 

Let Yi be the log unemployment rate in city i; let MWi be a “minimum wage indicator” i.e., 1 if city i increased the minimum wage and 0 otherwise. An economist is interesting in studying the effect of an increase in the minimum wage on the unemployment rate. The true model of the world is given by 

Yi = ?0 + ?1MWi ? Edi + Ui
where Edi be “high education indicator”, i.e., 1 is city i has a high proportion of highly educated people 

and 0 otherwise. Moreover, 

MWi = 1{?0 + ?1Edi ? ?i} where ?i ? U(0,1) IID and ?1 < 0. Also 

Ui = ?(Edi ? ?Ed) + ?i where ?i ? N(0,1) IID and ?Ed = E[Ed]. 

An economist decides to run OLS on the following regression: Yi = ?0 + ?1MWi + Vi 

(a)  Derive the probability limit of the OLS estimator of the slope (?1). 

(b)  Based on your answer in (a). Is the OLS estimator of the slope a consistent estimator of ?1?   Suppose the true effect is zero, i.e., ?1 = 0. Will the economist be concluding that the minimum wage increases or decreases unemployment (on average)? 

An economics, Mr. Paredes, proposes an alternative estimation technique based on comparing cities that increased the minimum wage to those that did not, but controlling for the education level. That is, E[Y|MW =1,Ed=e]?E[Y|MW =0,Ed=e]wheree?{0,1}.