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2011 Final - ECON 241B Final Exam 1 Winter 2011(20 points...

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ECON 241B, Final Exam Winter, 2011 1 (20 points) You wish to estimate the effect of unemployment on criminal behavior in Gotham City. In your model, Crime t , is a function of the unemployment rate, Unemp t , and the presence of Batman, Batman t . You hy- pothesize that unemployment causes an increase in criminal behavior, as represented by β 1 in the following model (assume that all variables are expressed as deviations from means): Crime t = β 1 Unemp t + β 2 Batman t + ε t (1) You make the following assumptions: E [ Unemp t * ε t ] = E [ Batman t * ε t ] = 0 E [ Unemp t * Batman t ] > 0 (When unemployment is low, business is good for Bruce Wayne and he is more often traveling for business) You gather data on crime rates and unemployment rates in Gotham City, but the presence of Batman is unobservable, so you proceed with the following model: Crime t = β 1 Unemp t + v t (2) (a) (4pts) Derive the OLS estimator of the unemployment effect from (2), ˆ β OLS 1 , and show whether or not the estimator is consistent for the true parameter from (1), β 1 . (b) (2pts) If inconsistent, will your estimator be too large or too small? Explain. You also are able to gather data on mass layoffs, Mass t , that happen in Gotham City throughout the time period of interest. These mass layoffs are not seasonal fluctuations and can be thought of as random shocks. Batman’s presence is not contemporaneously affected by mass layoffs. (c) (4pts) List all required conditions for Mass t to be a valid instrument for the endogenous regressor. (d) (5pts) Assuming these conditions hold. Derive the 2SLS estimate of the unemployment effect, ˆ β 2 SLS 1 and show whether your estimator is consistent. Write out the equations for both stages. How can we test one of the requirements for a valid instrument using the first-stage results? You can write the second stage of the 2SLS strategy as: Crime t = β 1 ˆ Unemp t +( v t + β 1 ( Unemp t - ˆ Unemp t )) (3) Suppose you estimate eq. (3) by OLS and use OLS standard errors for inference, ignoring the fact that ˆ Unemp t is estimated in the first stage. Assuming conditional homokedasticity, the asymptotic variance of your estimated standard errors when using OLS on the second stage can be written as: AVAR ( ˆ β OLS , 2 ndStage 1 ) = VAR ( v t + β 1 ( Unemp t - ˆ Unemp t )) E [ ˆ Unemp 2 t ] (4) Assume that v t and ( Unemp t - ˆ Unemp t ) are independent. For accurate inference, we should use an estimate of the following asymptotic variance (this would be the asymptotic variance of the IV estimator): AVAR ( ˆ β IV 1 ) = VAR ( v t ) E [ ˆ Unemp 2 t ] (5) 1
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ECON 241B, Final Exam Winter, 2011 (e) (2pts) If we mistakenly use an estimate of (4), would you expect your estimated standard errors to be too big or too small?
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