Empirical Exercise 12: Death Penalty

Question, Hypothesis, Theory

1. Question: Does punishing criminals with the death penalty have a deterrent effect on crime, particularly homicide

2. Hypothesis:  The death penalty deters homicide, i.e., the existence or application of death penalty laws is associated with a lower murder rate.

3. Theory: 

   1. Potential criminals assess costs and benefits. The death penalty raises the cost of crime (through the possibility of execution), thereby reducing the incentive to commit murder

   2. Counter Arguments

      1. Criminals may not behave rationally or perform cost-benefit analyses.

      2. The perceived probability of execution may be too low to affect behavior.

      3. The death penalty may have cultural consequences (e.g., "cheapening life") and even lead to more violence.

      4. The costs of implementing the death penalty may outweigh its benefits eg maintaining an inmate on death row 

Furmam decision rule: how many states had death penalty legal

-between 1972-1977 supreme court deemed executions unconstitutional , then after 1977 some bring it back

-states that in 1972 were using death penalty cannot use any more 

-states that did not have death penalty not affected 

-crimes will go up in the first group of states which are no longer able to use death penalty

-can implement differences in differences empirical strategy to measure time series variation: executions go down and crime goes up

• 44 bring it back, 5 states keep it illegal in 1977: almost all states had time variation in the death penalty law

  • Only 5 states do not change their death penalty laws at all throughout the sample period

type of dataset, variables, no of observations

-panel dataset of states and years to estimate existence of death penalty law effect on homicide rates 

• independent variable: existence of death penalty in the statute book 

  • Whether by law of that state it is possible to execute someone

• dependent variable: crime rate

• 1962-2000, observe 49 states over 41 years

• 2009 observations: balances panel as we observe every single one of 49 states over 41 yrs = 49 x 41= 2009

• Every observation is weighted by the population of the state

  • states such as California or Texas receive a higher weight than North Dakota or Maine

  • The standard errors are heteroskedasticity robust, but not clustered within states

-what is the effect of death penalty being legal on crime

• State fixed effects and decade fixed effects

  • Absorb(state)

  • One dummy variable for each decade Idecade

murderpc: murder per 100k people

-0.95: when death penalty is legal, one fewer murder among 100k: so there is a deterrent effect

-Highly statistically significant: reduces crime, having death penalty reduces crime

decade vs year fixed effects

-controlling only for decade effects assumes that crime levels are identical on average for different years belonging to the same decade/changes in death penalty laws are equally likely to take place for different years within the same decade. 

• We know later you are in the 70s, the more likely death penalty is likely to be legal, as you could only bring it back after 1977 so have to use year fixed effects

-not controlling for year effects would therefore introduce a negative bias in our estimates: -0.95 to -0.47: year fixed effects reduce the estimate 

-When you use correct SE and use year fixed effects, result become not significant [p value is 0.524] so no effect of death penalty on murder rates

population model

-independent variable legalit takes value 1 if it is legal to execute someone in that state

-X’it: vector of additional control variables

-individual state fixed effects: lambda i

-One dummy variable for each decade Idecade eg D60: dummy for the decade 60s

robust vs clustered standard errors criticism

• Coefficient is still -0.95: estimated coefficient does not change

• 0.2 standard error when using only robust, standard error increase to 0.57 when you use clustered SEs

  • SE changed so p value changed to 0.101

  • Coefficient is no longer statistically significant of death penalty being legal on murder rate: was significant before with robust SEs

  • Clustering the standard errors at the state level increases them significantly: estimate is no longer significant at the 5% level

changing the independent variable from legal to execs

-right independent variable should not be the legal presence of the death penalty in the state but whether how often the death penalty is applied eg no of executions by state

-legal requires criminals to be aware of state laws and know for their specific crime they will be executed

-effect of executions on murder rates: better measure of deterrence 

-Every extra execution decreases the murder rate by .13 murders per 100,000 residents

-small effect on the estimate but a massive effect on the standard error: from 0.012 to 0.069

-Texas is not an outlier in the classical sense, since including/excluding it does not affect the estimate

-Texas has a much larger number of executions than other states so excluding Texas dramatically reduces the variation in the data

-from the formula for the variance of an estimate, reducing variation will increase the standard error.

-an execution in California (many murderers) deters less than an execution in Maine (few murderers) because signal about chances of being executed is weaker

-number of executions per 100k residents: executions per capita rather than total no of executions like above execs

-not significant effect

-Number of executions per 1000 prisoners

what are the effects of changing the variable from legal

-number of executions per homicide

-using these variables as the right-hand side variables on the regressions shows no effect of executions on the murder rate: not significant p value 

Conclusion of the Dezhbakhsh and Shepherd study

• Deterrence effect depends on details of regression 

  • whether cluster or not 

  • whether decade or year fixed effects 

  • whether Texas in or out 

  • whether number of executions, or executions per capita 

-Deterrence effect cannot be said to be very robust

what were the issues with the previous study

-harsher policy implemented by public policy eg longer jail sentences, harsher prison conditions, increased resources + increased use of death penalty will have direct impact on homicide rate

-direct effect from police resources to homicide rates 

-reverse causality from homicide rates to executions as public will support use of death penalty when homicide rate is out of control so higher murder rate may  increase likelihood that you bring in death penalty

dataset from Dezhbakhsh, Rubin and Shepherd (2003) study + population model

-panel of US counties from 1977-1996: how murder rates depend on 3 separate deterrence variables + controls 

-larger dataset: counties instead of states

• Likelihood of being arrested conditional on committing a homicide

• likelihood of being given a death sentence conditional on being arrested

• likelihood of execution conditional on being given a death sentence

• X’it it includes controls such as the assault rate, the robbery rate, income, race, age

• variables probarrit, probsenit and probexecit are calculated only at the state level: take the same value for all the counties in the same state/year

• 3 variables are endogenous so at least 3 instruments to see if factors that affect them are uncorrelated with murder rates

instruments to measure the endogenous variables

• Expenditure in police payroll [at state level]

• Expenditures in judicial/legal payroll ie criminal justice expenditure [at state level]

• No of prison admissions [at state level]

• Republican vote shares in the previous election [at county level]

  • Counties that are more republican voting have attitudes that are more tough on crime 

hoq to run instrumental regression

• 3 first stage equation for probarr, probsen, probexec eg 

• If an observation is from the year ’77, then rep76it is equal to the value of the Republican vote share in that county for the Presidential election of ’76. The other variables (rep80it, etc) take 0 value

• Vote shares enter as diff variables for diff elections: allow effect of previous presidential election to be diff on probarr, depending on whether election took place in 76.80 etc so rep76 diff to rep80

• Can collapse into single variable repit where vote share is in most recent prior election + effect of repub voting on probarr etc assumed to be same regardless of election year it came from 

run 3 2nd stage regression taking fitted values for 3 variables and regressing them against murderrate

-We can see from the regression that the three variables are associated with less crime

-probsen not significant -probability of an execution, in particular, reduces crime significantly

-enter the Republican share of the vote as a single variable rather than as six separate variables as in above regression like with repit

-single variable: replast
-estimated deterrence effects on crime are now positive for two of the three variables

-increases in the likelihood of being sentenced to death and of being executed are associated  with a higher murder rate

-drop if state==48 (5,589 observations deleted)

-run 3 first stage regressions and get fitted values for second stage

-just dropping Texas massively affects the estimates, flipping their sign for two out of three cases: for probsen and probexec

-drop if state==6 (1,277 observations deleted)

-run 3 first stage regressions and get fitted values for second stage

-Just dropping California flips the sign of the effect of the execution rate

are our estimates robust

-robustness is when estimates are more/less same following small changes to regression

• So our estimates are not robust to minor change in regression or dropping one state eg texas, california

running a placebo experiment on minor crimes

1. death penalty is not applied for crimes such as aggravated assault, robbery, rape, auto theft, larceny and burglary

• if instruments are truly exogenous we will find that the predicted probability of being executed is associated with a lower homicide rate, but has no causal effect on more minor crimes

• Need y variable similar to exiting y variable: robberypc [has same things in the error term to murderpc]

• conversely, if we find that the predictability of being executed ‘decreases’ (in our dataset/sample) the other crimes, effect cannot be a causal, must be due to a correlation between instruments and the other crime dependent variables

  • would cast doubt on the exogeneity of the instruments

-done for other minor crimes as well

-likelihood of being executed increases aggravated assault, robbery, rape, burglary and larceny, and it decreases auto theft

-Instruments are not exogenous: would give true effect of 0

placebo experiment 2 on states that don’t have death penalty

1.  number of county/year observations belonging to states that do not have a death penalty law in place at that moment ie 5 states that never made it legal again

   1. since there are no executions, predicted no of executions [fitted value of regression of probexecit on instruments] cannot have deterrence effect on crime

   2. If you find an effect of the predicted number of executions in states with no death penalty law, we know that this effect cannot be causal, but will instead be driven by the correlation between the instruments and the murder rate

      1. Instruments are not valid: correlated with other unobserved factors affecting homicide which is causing an ‘effect’

running regression on states with no death penalty

1. Effect should be 0 here as legal=0 but it is massively significant

over identifying restrictions

1. 9 instruments and 3 independent variables 

   1. Splitting instruments we should obtain approximately the same estimates regardless of whether we use one set of instruments or the other 

      1. sample size is predictably much smaller as we are only focusing on states without the death penalty

      2. even for the states without the death penalty, increase in the predicted no of executions (they would have if the death penalty was in place, which it is not, so exec=0) ‘deters crime’

         1. effect is large and statistically significant. This finding casts a lot of doubt on the suitability of this empirical design

splitting up instruments

-only regress for the following six instrumental variables; rep76,80,84 etc  

-only regress for the following three instrumental variables: policexp, judicalxp, prisonerspc

-diff results depending on which instruments you use

impact of instruments used

• Our prediction is that instruments: expenditures in the policing and judicial systems, the number of prison admissions, and the Republican vote shares all increase the likelihood of being arrested, sentenced to death, and being executed (but have no direct effect on murder rates)

however the effects are completely inconsistent both across and within regressions

• Eg judicial expenditures and the Republican vote sharers lead to a decrease in the likelihood of being arrested. 

• Police expenditures and the number of prison admissions lead to a decrease in the likelihood of being sentenced to death. 

• Police expenditures, judicial expenditures and one of the Republican vote shares are associated with a lower likelihood of being executed. 

  • This is contrary to our prediction and again suggests that these instruments are unlikely to be exogenous

conclusion of this study

-it is only by cherry picking the instruments and the sample that Dezhbakhsh, Rubin and Shepherd manage to find a deterrence effect of the death penalty as:

• instruments are bad 

• robustness is low 

• placebos show that instruments have direct effect on dependent variable

internal validity

• Use of panel data and fixed effects to control for unobserved state specific characteristics

• DID to help isolate effect of death penalty by comparing states with death penalty and states without

• IVs to tackle endogeneity of deterrence variables

• OVB: Even after controlling for observable factors, other unobserved time-varying factors (e.g., societal attitudes, other criminal justice reforms) may bias results.

• Reverse Causality: Higher murder rates may lead to tougher criminal policies, including the reinstatement of the death penalty, rather than the death penalty reducing crime.

• Weak or Invalid Instruments: The instruments used (like political variables and expenditures) may themselves be correlated with unobserved determinants of crime. 

  • Placebo tests showed that these instruments often correlate with other crimes they shouldn’t affect, undermining their validity.

• Serial Correlation: Original standard errors were not clustered, leading to underestimation of uncertainty. 

  • When clustering is introduced, statistical significance often disappears.

external validity

• Specific to the U.S. Context: The study uses U.S. state-level data, may not generalize to countries with different legal systems, cultures, or governance structures.

• Time Period Specific: The empirical window (mostly post-1970s) may not reflect longer-term or future deterrence dynamics.

• Heterogeneity Across States: States differ significantly in enforcement, crime levels, judicial efficiency, and public attitudes, making generalization within the U.S. itself difficult.