Economics 704 Spring 2023 Problem Set #1
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Problem Set #1
(Due Sunday, February 19)
Economics 704
Spring 2023
Instructions: Show/prove how you obtained your answers. For the Stata exercise, submit your Stata code, output, and answers to the questions.
1. Consider the following model: yi = xi\ β + ei under the assumptions:
B0) (yi,xi) i.i.d.
B1) E[xixi\ /σ 2 (xi)] is finite and nonsingular
B2) E(ei|xi) = 0
B3) σ 2 (xi) = Var(ei|xi) is finite and well defined for all xi .
Assume that you will observe data (y1 ,x1 ), . . . , (yn,xn) and σ2 ( ·) is known. Define
βˆWLS = (i(z) 
)−1 (i(z) 
)
(a) Show that βˆWLS is consistent for β under Assumptions B0-1-2-3.
(b) Would consistency in part (a) hold if Assumption (B2) is replaced by (B2’) E[xiei] = 0?
(c) What is the asymptotic distribution of ^n(βˆWLS − β) under Assumptions B0-1-2-3?
2. Assume (yi,wi,zi) is an i.i.d. sequence, where wi and zi are scalar. Consider the regression equation yi = wiβ+εi . The joint distribution of (wi,zi) and the conditional mean for εi given (wi,zi) is:
Pr(wi = 1,zi = 1) =
Pr(wi = 2,zi = 1) =
Pr(wi = 2,zi = 2) =
1
6 ,
1
2 ,
1
3 ,
E(εi|wi = 1,zi = 1) = 3
E(εi|wi = 2,zi = 1) = − 1
E(εi|wi = 2,zi = 2) = 0
Definition of the term “plim” (used below): suppose θˆ
θ0 , then we call θ0 the probability limit of θˆ, denoted θ0 = plim(θˆ).
(a) Consider the OLS estimator, where yi is the dependent variable, wi is the independent variable (no constant term). Let βˆOLS denote the resulting scalar estimator. What is plim(βˆOLS )?
(b) Consider the 2SLS estimator, where yi is the dependent variable, wi is the independent variable (no constant term), and zi is the instrumental variable. Let βˆ2SLS denote the resulting scalar estimator. What is plim(βˆ2SLS)?
(c) Consider a weight function h(wi,zi). Suppose we transform the observable data by this weight function:
y˜i = h(wi,zi)yi , 
i = h(wi,zi)wi , 
i = h(wi,zi)zi
and we perform OLS and 2SLS as in parts (a) and (b) using the weighted data ( 
i , 
i , 
i). Is there a choice of weight function so that both OLS and 2SLS on the weighted data are consistent estimators of β? If yes, show such a choice of weight function. If no, show why such a weight function does not exist.
3. Upload the data set CRIME.DTA and the do file hw1 crime.do from the course website. In Stata, run the hw1 crime.do file. This file creates many variables for the crime analysis in Levitt (1997) (and McCrary (2002)).
(a) (R1) Perform OLS regression restricted to year ≥ 73.
Dependent variable = lnviol
Independent variables = constant, llnswornpc, lag llnswornpc, unemprate, lnsta welf, lnsta educ, a15 24, citybla, cityfemh, yy*, cc*, cs*, rr*
and
(R2a) Perform OLS regression restricted to year ≥ 72.
Dependent variable = dlnviol
Independent variables = constant, dllnswornpc, lag dllnswornpc, dunemprate, dlnsta welf, dlnsta educ, da15 24, dcitybla, dcityfemh, yy*, cc*, cs*, rr*
and
(R2b) Perform regression (R2a) again, but this time use the robust command. and
(R2c) Perform regression (R2a) again, but this time use the cluster command and cluster by city.
Focus on the sum of the coefficient estimates for dllnswornpc, lag dllnswornpc in regres- sions R2a, R2b, and R2c. Do the coefficient estimates change? How did the standard error for the sum of the coefficient estimates for dllnswornpc, lag dllnswornpc change? Which method would you choose?
(b) (R3) Use the ivreg command to perform 2SLS restricted to year ≥ 72 (with robust stan- dard errors).
Dependent variable = dlnviol.
Exogenous independent variables = constant, dunemprate, dlnsta welf, dlnsta educ, da15 24, dcitybla, dcityfemh, yy*, cc*, cs*, rr*
Endogenous independent variables = dllnswornpc, lag dllnswornpc
Instrumental variables = elecyear, governor, lagelecyear, laggovernor
Focus on the sum of the coefficient estimates for dllnswornpc, lag dllnswornpc. Explain what Levitt expected to change with these coefficient estimates moving from (R2) to (R3), and compare to your empirical results.
2023-02-16