Department of Economics Session 2012/2013. EC352 Econometric Methods. Solutions to Exercises from Week (0.052)

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1 Department of Economics Session 2012/2013 University of Essex Spring Term Dr Gordon Kemp EC352 Econometric Methods Solutions to Exercises from Week 10 1 Problem 13.7 This exercise refers back to Equation (13.12) from Wooldridge: log(durat) = (0.031) (0.069) (0.0447) n = 5626, R 2 = af change (0.047) af change highearn, highearn (1) which is part of Example 13.4 Effect of Worker Compensation Laws on Weeks out of Work. This refers to a study by Meyer, Viscusi and Durbin (1995) which examined the impact of a change in the cap on weekly earnings that was covered workers compensation (for injuries etc.). Here the control group is low income workers while the treatment group is high income workers. This is because low income workers had earnings that were below the original cap so raising the cap did not raise the amount of the benefit they could get. In contrast high income workers had earnings that were at or above the cap so raising the the cap did increase the amount of the benefit they could get. Raising the amount of the benefit a worker can get will make it more attractive for the worker to remain on benefit for longer (in the event of an injury). The data used in the example is given in INJURY.DTA and the estimates were obtained using the observations for Kentucky. 1. Using the data in injury.dta for Kentucky, the estimated equation when afchange is dropped from Equation (13.12) is: log(durat) = (0.022) (0.042) n = 5626, R 2 = highearn (0.052) afchange highearn (2) Is it surprising that the estimate on the interaction term is fairly close to that in Equation (13.12)? Explain. The equation of interest is: log(durat) = β 0 + δ 0 afchange + β 1 highearn +δ 1 afchange highearn 1

2 Before After After-Before Control β 0 β 0 + δ 0 δ 0 Treatment β 0 + β 1 β 0 + δ 0 + β 1 + δ 1 δ 0 + δ 1 Treatment - Control β 1 β 1 + δ 1 δ 1 Table 1: Illustration of the Difference-in-Differences Estimator see Equation (13.10). As we can see, the estimated coefficient on afchange in Equation (13.12) is very small in magnitude (and when compared to the estimated coefficients on highearn and on the interaction term afchange highearn) and is statistically very insignificant. Consequently it is not a surprise the results change very little as a result of dropping afchange from the Equation (13.12) while keeping highearn and the interaction term: the change is easily explainable as the result of sampling variability. 2. When afchange is included but highearn is dropped, the result is: log(durat) = (0.023) (0.040) n = 5626, R 2 = af change (0.050) afchange highearn (3) Why is the coefficient on the interaction term now so much large than in Equation (13.12)? Explain [Hint: In Equation (13.10), what is the assumption being made about the treatment and control groups if β 1 = 0?] The coefficient on afchange is the increase in log(duration) for low earners. Raising the cap should have no effect on these workers since their income was below the cap anyway and hence we would assume that δ 0 = 0. This is consistent with the estimates in Equation (13.12) and the results in part 1. In contrast, dropping highearn from Equation (13.12) means that we are assuming that β 1 = 0, i.e., that there was no difference in average pre-policy change log durations between the high and low income groups. This is not very plausible since, for example, the two groups may tend to do different jobs for which the effects of injuries may be quite different. In the results from Equation (13.12) we see that the coefficient on highearn, namely β 1, is strongly significantly different from zero (t-statistic exceeds 5) so we would expect dropping this variable to have quite a strong impact. 2 Exercise C14.7 This question is about analyzing the impact of execution rates and unemployment on murder rates at the US state level using panel data to control for unobserved state specific factors.the data are contained in the file murder.dta and consist of a panel all 50 states plus the District of Columbia with three waves: 1987, 1990, and This gives 153 observations. 2

3 1. Consider the unobserved effects model: mrdrte it = θ t + β 1 exec it + β 2 unem it + a i + u it, i = 1,..., 51 t = 1987, 1990, 1993 where: θ t denotes year specific intercepts; a i denotes the unobserved state specific effects; mrdrte it is the murder rate (per 100,000 population) for state i in year t; exec it is total executions over the past 3 years for state i in year t; and unem it is the unemployment rate for state i in year t. If past executions have a deterrent effect, what should be the sign of β 1? What sign do you think β 2 should have? Explain. β 1 should be negative if, ceteris paribus, past executions have a deterrent effect. β 2 is likely to have a positive effect as it tends to reflect relative social economic deprivation which people seem to think are positively related to crime rates including rates of violent crimes such as murder. 2. Using just the years 1990 and 1993, estimate the equation from part (1) by pooled OLS. Ignore the serial correlation problem in the composite errors. Do you find evidence for a deterrent effect? Running OLS on the data for just the years 1990 and 1993 gives:. regress mrdrte exec unem if year >87 Source SS df MS Number of obs = F( 2, 99) = 5.08 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = mrdrte Coef. Std. Err. t P> t [95% Conf. Interval] exec unem _cons Ignoring the any problems in calculating standard errors resulting from unobserved effects then we don t find a deterrent effect the coefficient on exec is positive (not negative) but is insignificant (p-value of 0.663). 3. Now, using 1990 and 1993, estimate the equation by fixed effects. You may use first differencing since you are only using two years of data. Now, is there any evidence of a deterrent effect? 3

4 The first differencing approach is easy to implement here as the variables have already been created in the data set: cmrdrte is mrdrte minus its first lag (giving the change in the murder rate) cexec is exec minus its first lag (giving the change in the total number of executions) and cunem is unem minus its first lag (giving the change in the unemployment rate) Running the first difference regression using just 1990 and 1993 gives:. regress cmrdrte cexec cunem if year==93 Source SS df MS Number of obs = F( 2, 48) = 2.96 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons There is now evidence of a deterrent effect since the coefficient on cexec is negative every additional execution in a state reduces the murder rate in that state in the next three year period by about 1 per million. Furthermore, this effect is fairly statistically significant (p-value of 0.021). 4. Compute the heteroskedasticity-robust standard error for the estimation in part (3). It will be easiest to use first differencing. Different states are different in many ways and we may worry about heteroskedasticity. Re-running the regression from part (3) but generating heteroskedasticity-robust standard errors gives:. regress cmrdrte cexec cunem if year==93, vce(robust) Linear regression Number of obs = 51 F( 2, 48) = Prob > F = R-squared = Root MSE = Robust cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] 4

5 cexec cunem _cons which indicates that the deterrent effect is now highly statistically significant (p-value is 0 to 3dp). 5. Find the state that has the largest number for the execution variable in How much bigger is this value than the next highest value. Browsing through the data in Stata reveals that the highest observed value for the execution variables is 34 for Texas (id 44) in The next largest value in 1993 was 11 for Virginia (id 47) so the value for Texas is indeed very high. 6. Estimate the equation using first differences, dropping Texas from the analysis. Compute the usual and heteroskedasticity-robust standard errors. Now what do you find? What is going on? Dropping Texas and rerunning the first-difference regression for 1993 with the usual OLS standard errors gives:. regress cmrdrte cexec cunem if year==93 & id!=44 Source SS df MS Number of obs = F( 2, 47) = 0.32 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons so the deterrent effect is rather smaller and is no longer statistically significant (p-value of 0.523). Using heteroskedasticity-robust standard errors gives:. regress cmrdrte cexec cunem if year==93 & id!=44, vce(robust) Linear regression Number of obs = 50 F( 2, 47) = 0.54 Prob > F = R-squared = Root MSE = Robust 5

6 cmrdrte Coef. Std. Err. t P> t [95% Conf. Interval] cexec cunem _cons which implies the deterrent effect is still insignificant (p-value of 0.398). In fact, what we are seeing is that dropping Texas both reduces the magnitude of the estimated deterrent effect quite substantially (the estimate shifts from to ) but also increases the standard errors (from the usual OLS ones increase from to while the heteroskedasticity-robust changes from to 0.079). Clearly including excluding Texas has very dramatic effects on the results and suggests that it is an outlier. 7. Use all three years of data and estimate by fixed effects. Include Texas in the analysis. Discuss the size and statistical significance of the deterrent effect compared with only using 1990 and To run this regression in Stata we use the xtreg command with the fe option. Doing so gives:. xtreg mrdrte exec unem, fe Fixed-effects (within) regression Number of obs = 153 Group variable: id Number of groups = 51 R-sq: within = Obs per group: min = 3 between = avg = 3.0 overall = max = 3 F(2,100) = 0.24 corr(u_i, Xb) = Prob > F = mrdrte Coef. Std. Err. t P> t [95% Conf. Interval] exec unem _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(50, 100) = Prob > F = The deterrent effect is similar to that which we estimated using the first differencing approach with just data from the years 1990 and 1993: however it is no longer statistically significant. We could use standard errors clustered by id:. xtreg mrdrte exec unem, fe vce(cluster id) 6

7 Fixed-effects (within) regression Number of obs = 153 Group variable: id Number of groups = 51 R-sq: within = Obs per group: min = 3 between = avg = 3.0 overall = max = 3 F(2,50) = 1.23 corr(u_i, Xb) = Prob > F = (Std. Err. adjusted for 51 clusters in id) Robust mrdrte Coef. Std. Err. t P> t [95% Conf. Interval] exec unem _cons sigma_u sigma_e rho (fraction of variance due to u_i) Even though the standard errors change quite a lot, the deterrent effect is still insignificant. 7