In this exercise, we extend Exercise 15.29 by also considering the possibility that the probability of arrest is jointly determined with the crime rate and the number of police per capita. The idea is that when the crime rate is high, the police may intensify their efforts to reduce crime by increasing the arrest rate. Consider the same model as in Exercise 15.29.

a. It is possible that the crime rate and police per capita are jointly determined and that \(L P O L P C\) and LPRBARR might be endogenous. Hence we consider estimating the model by 2SLS. As instruments we use the \(\log\) of tax revenue per capita (LTAXPC) and the \(\log\) of the ratio of face-to-face crimes relative to other types of crimes (LMIX). Estimate the first-stage regression of LPOLPC on the other variables, except \(L C R M R T E\), and the two instruments. Test the joint significance of the IV. Can we reject the null hypothesis that the IV are weak? Estimate the first-stage regression of LPRBARR on the other variables, except LCRMRTE, and the two instruments. Test the joint significance of the IV. Can we reject the null hypothesis that the IV are weak?

b. Using the instruments in (a), estimate the model, treating both LPOLPC and LPRBARR as endogenous, by 2 SLS. Are the deterrence variables significant?

c. Test for the endogeneity of LPOLPC and LPRBARR using the regression-based Hausman test. What do you conclude in each case?

d. The estimation in (b) ignores unobserved county heterogeneity. For each variable, except the time-invariant variables WEST and URBAN, obtain the variables in the deviation about the county mean form, that is, apply the within transformation to each variable. Estimate the first-stage model for both LPOLPC and LPRBARR with the variables in deviation from the mean form. Test the joint significance of the two transformed instruments.

e. Using the transformed instruments and other variables, estimate the model, treating both LPOLPC and LPRBARR as endogenous, by 2SLS. What differences do you observe between these estimates and those in part (b)? Recall that you must adjust the standard errors for the correct degrees of freedom, as in Example 15.5. (Note: You may investigate whether your software has an automatic command to do 2SLS with panel data as a check.)

f. Test for the endogeneity of \(L P O L P C\) and \(L P R B A R R\) using the regression-based Hausman test. What do you conclude in each case?

Data From Exercise 15.29:-

In this exercise, we re-examine the data in Exercise 15.22, a panel of data from North Carolina. Consider a model in which the log of crime rate (LCRMRTE) is a function of the log of police per capita (LPOLPC), the log of the probability of arrest (LPRBARR), the log of the probability of conviction (LPRBCONV), the log of average prison sentence (LAVGSEN), and the log of average weekly wage in the manufacturing sector \((L W M F G)\) and indicator variables for the western region (WEST) and urban counties (URBAN).

a. It is possible that the crime rate and police per capita are jointly determined and that LPOLPC might be endogenous. Hence we consider estimating the model by 2SLS. As instruments we use the log of tax revenue per capita (LTAXPC) and the log of the ratio of face-to-face crimes relative to other types of crimes (LMIX). Estimate the first-stage regression of LPOLPC on the other variables, except LCRMRTE, and the two instruments. Test the joint significance of the IV. Can we reject the null hypothesis that the IV are weak?

b. Using the instruments in (a), estimate the model by 2SLS. Are the deterrence variables significant?

c. Test for the endogeneity of LPOLPC and test the validity of the surplus instrument. What do you conclude in each case?

d. The estimation in (b) ignores unobserved county heterogeneity. For each variable, except the time-invariant variables WEST and URBAN, obtain the variables in the deviation about the county mean form, that is, apply the within transformation to each variable. Estimate the first-stage model with the variables in deviation from the mean form. Test the joint significance of the two transformed instruments.

e. Using the transformed instruments and other variables, estimate the model by 2SLS. What differences do you observe between these estimates and those in part (b)? Recall that you must adjust the standard errors for the correct degrees of freedom.

f. Using the transformed instruments and other variables, test for the endogeneity of LPOLPC and test the validity of the surplus instrument. What do you conclude in each case?

Data From Exercise 15.22:-

What is the relationship between crime and punishment? This important question has been examined by Cornwell and Trumbull \({ }^{18}\) using a panel of data from North Carolina. The cross sections are 90 counties, and the data are annual for the years 1981-1987. The data are in the data file crime. In these models, the crime rate is explained by variables describing the deterrence effect of the legal system, wages in the private sector (which represents returns to legal activities), socioeconomic conditions such as population density and the percentage of young males in the population, and annual dummy variables to control for time effects. The authors argue that there may be heterogeneity across counties (unobservable county-specific characteristics).

a. What do you expect will happen to the crime rate if (i) deterrence increases, (ii) wages in the private sector increase, (iii) population density increases, and (iv) the percentage of young males increases?

b. Consider a model in which the log of crime rate (LCRMRTE) is a function of the log of the probability of arrest (LPRBARR), the log of probability of conviction ( \(\angle P R B C O N V)\), the log of the probability of a prison sentence (LPRBPRIS), the log of average prison sentence (LAVGSEN), and the log of average weekly wage in the manufacturing sector (LWMFG). Estimate this model by OLS. (i) Discuss the signs of the estimated coefficients and their significance. Are they as you expected? (ii) Interpret the coefficient on LPRBARR.

c. Estimate the model in (b) using a fixed effects estimator. (i) Discuss the signs of the estimated coefficients and their significance. Are they as you expected? (ii) Interpret the coefficient on LPRBARR and compare it to the estimate in (b). What do you conclude about the deterrent effect of the probability of arrest? (iii) Interpret the coefficient on LAVGSEN. What do you conclude about the severity of punishment as a deterrent?

d. In the fixed effects estimation from part (c), test whether the county level effects are all equal.

e. Based on these results, what public policies would you advocate to deal with crime in the community?