Endogenous regressor and instrumental variable In this exercise we consider the following model: Yi = Bo + Bifil + ui (1) where y; is a variable reflecting the crime in city i (in deviation from its mean) and where rij is a variable reflecting the size of the police force in city i (in deviation from its mean). Further, we observe the variable zil reflecting whether there are elections for the city council: 1 if there are elections for the city council in city i, 2,1 0 if there are no elections for the city council in city i. cross section of n = 1500 observations (a simulated dataset). In this exercise we assume homoskedasticity. (2a) The variable til may be an endogenous regressor in (1). For what reasons may this be the case? Explain why zil could be a good instrument for til. Duchin-Wu-Hausman test to chool encogen (20) Show that the IV estimator B1.IV = :1(211 )(yi ) !=(zil - z)(x1 - 71) is equal to (71-70)/(21 - 2o), where y and j denote the sample means of y and cover election cities (with zil = 1) and where yo and To denote the sample means of y and r over non-election cities (with zi1 = 0). Give also an intuitive motivation for this estimator of B1. Hint: you can use that = and 1 i To, where ni and no are the numbers of election cities and non-election cities. n! Endogenous regressor and instrumental variable In this exercise we consider the following model: Yi = Bo + Bifil + ui (1) where y; is a variable reflecting the crime in city i (in deviation from its mean) and where rij is a variable reflecting the size of the police force in city i (in deviation from its mean). Further, we observe the variable zil reflecting whether there are elections for the city council: 1 if there are elections for the city council in city i, 2,1 0 if there are no elections for the city council in city i. cross section of n = 1500 observations (a simulated dataset). In this exercise we assume homoskedasticity. (2a) The variable til may be an endogenous regressor in (1). For what reasons may this be the case? Explain why zil could be a good instrument for til. Duchin-Wu-Hausman test to chool encogen (20) Show that the IV estimator B1.IV = :1(211 )(yi ) !=(zil - z)(x1 - 71) is equal to (71-70)/(21 - 2o), where y and j denote the sample means of y and cover election cities (with zil = 1) and where yo and To denote the sample means of y and r over non-election cities (with zi1 = 0). Give also an intuitive motivation for this estimator of B1. Hint: you can use that = and 1 i To, where ni and no are the numbers of election cities and non-election cities. n