«School of Economics and Political Science, University of St. Gallen Department of Economics Editor: Martina Flockerzi University of St. Gallen School ...»
–4– the data which have not been satisfactorily solved so far. Section 3 presents a critical survey of the third round papers including some summary statistics. It is shown how simple changes can produce quite opposite results using the same data. In the final Section 4 we draw some conclusions about the usefulness of statistical arguments in policy debates, but also on the moral questions involved in this particular debate. The reason for the latter discussion is that, in contrast to all earlier papers where those authors defending the deterrent effect of capital punishment are keen to mention that this does not imply that this kind of punishment is justified,12) some authors now claim that due to its deterrent effect capital punishment might even be morally required.13) 2 Some Stylised Facts and Easy Estimates  The results of R.D. ADLER and M. SUMMERS are derived from a simple OLS regression.14) Using contemporaneous execution data we get for the period from 1979 to 2004:15) (1) HOMt = 22614 – 71.880 EXECt + ut (33.90) (-4.98)
EXEC number of executions.
The values of the Durbin-Watson test as well as of the Q-statistic indicate that the estimated residuals are highly correlated. Further examination of the residuals indicates second order autocorrelation. Thus, even if the true relation would be a simple bivariate one, the significance of the estimated parameters would be highly dubious. The easiest way to take this into
account is to perform a second order Cochrane-Orcutt transformation. This leads to the following estimates:
(2) HOMt = 22246 – 15.378 EXECt + ut (14.11) (-1.27) ut = 1.459 ut-1 – 0.605 ut-2 + εt, (8.21) (-3.32)
12. See, for example, I. EHRLICH (1975, p. 416), but also J. SHEPHERD (2009)
13. See the discussion between C.S. SUNSTEIN and A. VERMEULE (2005, 2005a), C.S. STREIKER (2005) and D.R. WILLIAMS (2006) mentioned in Section 4 below.
14. Similar regressions are performed by H. DEZHBAKHSH and J.M. SHEPHERD (2006) for the period from 1960 to 2000 and the two sub-periods from 1960 to 1976 and from 1977 to 2000 to get “exploratory evidence” (p. 518).
15. The numbers in parentheses are the estimated t-statistics. D.-W. is the result of the Durbin-Watson test for autocorrelation of the residuals, Q(k) the Box-Ljung Q-statistic with k degrees of freedom, AIC the value of the Akaike criterion. '***', '**', '*' or ('*') denote significance at the 0.1, 1. 5, or 10 percent levels, respectively. The results are derived with EViews, Version 5.1. The sources of the data are given in the Appendix.
R 2 = 0.886, D.-W. = 1.957, Q(2) = 1.762, AIC = 16.782.
According to all statistical criteria, this seems to be a much more reliable equation. However, the estimated coefficient of the number of executions is much smaller and no longer statistically significant at any conventional significance level.
 Another possibility to take the autocorrelation into account is to include lagged dependent variables. This leads to the following result:
Again, according to the statistical criteria, this is a much more reliable result than equation (1). It is even slightly better than equation (2), though the differences are very small and would never prove to be statistically significant if a formal test were to be applied. On the other hand, the estimated long-run effect of the execution variable is now even slightly higher than in equation (1), and both coefficients are significantly different from zero at the five percent level. Thus, it is possible to get very different results for two plausible specifications for the autocorrelation of the error term in model (1) that are very close together according to conventional statistical criteria.
 We get quite similar results if we follow the suggestion of R.D. ADLER and M. SUMMERS to use lagged values of the execution variable in order to take account of the time delay
between cause and effect. Using again OLS we get:
The coefficient of the execution variable and its significance are now somewhat higher, but
there is still considerable autocorrelation in the estimated residuals. Performing a second order Cochrane-Orcutt transformation we get:
Again, according to all statistical criteria, this seems to be a much more reliable equation. But the estimated coefficient of the number of execution is now even positive but, of course, not statistically significant at any conventional significance level. Nevertheless, this estimate would rather support the brutalisation than the deterrence hypothesis.
 As before, we get quite different results if we include lagged dependent variables. Then
In this estimate, the coefficient of the execution variable has still its negative sign but its significance is below the 10 percent level. In the long-run equation, it is, however, still significant at the five percent level. Thus, we find again the situation that, taking the autocorrelation of the residuals into account, we get two rather different results with equally plausible specifications which are hardly distinguishable according to statistical criteria.
 The results depend, however, also on the time period used. As can be seen from Figure 2, before 1940, the overall correlation was positive (ρ = 0.321).16) After 1940, the overall correlation is negative (ρ = -0.635). But there are, nevertheless, considerable sub-periods with a positive correlation. As the following example for the 25 years from 1971 to 1995 shows, it is
even possible to get significant positive results when taking the autocorrelation of the residuals into account:
Taken literally, this equation would imply that every execution leads to 54 additional homicides. If we include lagged dependent variables we get similar results, though the statistical criteria indicate that this estimation is slightly inferior and the execution variable significant
only at the 10 percent level and only in the long-run equation:
which implies that every execution leads to 82 additional murders in the long-run. Thus, even if, given the data from 1940 onwards, most sub-samples show a negative correlation between homicides and executions, it is possible to find sub-periods showing the opposite picture.
Everybody who wants to claim that this negative correlation represents a causal relation has to provide a convincing explanation for the existence of sub-periods showing the opposite (causal) relation.
 Figure 3 shows the t-statistics of the execution variable if we perform rolling regressions over 25 year-periods from 1900 to 2008. If we perform simple OLS estimations, depending on the sub-period we chose, we can get rather negative values, but also high positive ones. If we model the residual process as AR(2), we hardly get any significant results. If we include the lagged endogenous variable up to the second order, we find some significantly negative results. The estimated coefficients vary even more dramatically: between -225 and 132 in the case of the simple OLS regression, between -20 and 54 if we apply the CochraneOrcutt procedure, and between -4241 and 4092 if we include the two lagged endogenous vari
 That most sub-periods of the recent decades show a negative correlation (but some subperiods a positive one) between homicides and executions is just one particular stylised fact of the relation between these two variables. A second one, shown in Figure 4 for the period between 1977 and 2007, i.e. after the suspension by the Supreme Court ended in 1976, is that those states that do not use the death penalty have consistently lower homicide rates than those which execute people. This should be disturbing for anybody defending the deterrence
hypothesis. It might, however, be at least partly due to the fact that reverse causality exists:
those states with higher homicide rates might seem to be more induced to use the death penalty than those with lower rates. Moreover, both series are very highly correlated, with a correlation coefficient of 0.918. In any case, however, if executions have a deterrent effect on murders, this effect should be larger in states with than in states without the death penalty.
 To test this, we used a principal component analysis. The first component represents the common movement of both series, while the second one takes up the difference. If we regress the two series on the first component (PC1), we get the following estimates for the homicide rates in states with (HOMRW) executions for the period from 1977 to 2007, i.e. after the moratorium ended:18)
17. This is the coefficient in the long-run equation corresponding to relation (3a'').
18. If we regress the homicide rate in states without executions on the first component, we get a similar picture, with identical values for the R2 and the t-statistic of the principal component as well as the Durbin-Watson and Q-statistics. To take account of the autocorrelation in the residuals we use for all following models heteroscedasticity and autocorrelation corrected standard errors to calculate the t-statistics.
Figure 4: Homicides rates in U.S. States with and without death penalty, 1977 – 2007 (4a) HOMRWt = 8.034 + 1.202 PC1t + ut (176.71) (48.25) R 2 = 0.959, D.-W. = 0.305, Q(4) = 36.923***, AIC = 0.812.
If executions make a difference between murder rates in the states with and without the death penalty, the second principal component (PC2) that takes account of the differences between
the two series should be influenced by the number of executions. Performing the corresponding regression we get:
(4b) PC2t = – 0.066 + 0.002 EXECt-1 + ut (-0.11) (0.98) R 2 = -0.003, D.-W. = 0.302, Q(4) = 38.550***, AIC = 0.413.
From these estimates it is obvious that the number of executions has no impact whatsoever on the difference in the developments of homicide rates of states with and without executions.
 We get a corresponding result if we regress the difference in the homicide rate between
states with and without death penalty (DIFF) on the (lagged) number of executions:
 A third stylised fact is that homicide rates are consistently lower in Canada compared with the United States.19) Figure 5 shows this for the period from 1977 to 2005. While the last execution was in 1962, death penalty was officially abolished in 1976. Thus, Figure 4 shows the period after Canada abolished the death penalty.
 The correlation between both series is much lower than the one between the different U.S. states. It is, however, still 0.758, indicating that there is considerable common movement
between the developments in the two countries. Thus, we can perform the same test as before.
Estimating principal components and regressing the two homicide series on the first component we get for the U.S. homicide rate (HOMRUS):
 The very high value of the adjusted multiple correlation coefficient indicates that there is a rather strong common development of the homicide rates of the two countries, despite the different position with regard to death penalty. The regression of the second principal component that represents the differences between the two countries on the number of executions in the United States leads to (8b) PC2t = – 0.018 + 0.0005 EXECt-1 + ut (-0.11) (0.20) R 2 = -0.036, D.-W. = 0.937, Q(4) = 7.737, AIC = 1.556.
Thus, similar to the results for the U.S. states, we find that, taking into account the common movements between the two countries, the number of executions in the United States does not have any impact at all on the difference of the development of the murder rates between the two countries.
 Thus, proponents of the death penalty in the United States do have to give convincing
answers to four questions: