Today, the Center on Budget and Policy Priorities posted the above graphic on Twitter (here) suggesting that the huge rise in the Incarceration rate (yellow line) had little impact on either the Violent (blue line) or Property crime rates (gray line). My colleague, Riccardo Fiorito, posted a reply on Twitter (here) suggesting (well more than suggesting, he actually offered an elasticity coefficient) that maybe there is some small effect. Wisely or not, I also replied suggesting that a time series model could provide a test of the idea.
I was able to find the data on which the CPB graph was based and started developing a state space model. My first inclination was to include both total crimes (adding violent and property crimes together) and the total number of prisoners both as dependent variables. That is, crimes and incarcerations form a system: crimes generate some incarcerations for those caught, tried and convicted and incarceration rates must send some message to criminals (imagine if no one was caught, tried and convicted). I also tested a model where total crimes was the single dependent variable and incarcerations was the single independent variable. And, I tested two other models controlling for World and US economic conditions. Without entering the debate about the role of economic conditions, if there is some relationship between poor economic performance and incarcerations, I wanted to control for the effect. Finally, I estimated total crimes and total incarcerations rather than rates as presented in the CPB graph. I was not sure what the "rate" represented (per 100,000 population, per 100,000 adult male population, etc.) so I used the raw numbers (a rate model could be estimated later if anyone is still interested).
The best model was chosen using the lowest AIC (Akaike Information Criterion) statistic. The models were all estimated in R using the dse package (I can make the models available if anyone is interested). The best model was the systems model (total crimes and incarcerations as the output variables) controlling for economic conditions in the World System. The US is a globalized country and controlling for conditions in the World economy is a bit more general than just controlling for US economic conditions.
The best way to understand the estimation is from the Impulse Response graph (above). The two plots on the upper part of the figure show the impact of a one-time increase in crimes on both crimes and incarcerations (controlling for World economic conditions). What is interesting is that it takes the law enforcement system about four years to respond to a one-time shock in crime with increased incarcerations. You can also see that incarcerations increase disproportionately at a five-to-one ratio (an increase in one crime creates five more incarcerations fours years in the future, Riccardo thought the lag length might be two years). The lower panel shows the effect of an increase in incarcerations on the total crimes. Incarcerations do decrease the crimes but the effect is very small (and non-significant using bootstrap t-statistics).
So, in summary, incarcerations increased so dramatically because the criminal justice system responded disproportionately to increase in the crime. The effect on criminal activity was slight, possibly because it takes the criminal justice system so long to respond positively (a four year lag seems to insure that the reaction is quite divorced from the cause).
All this might be moot as can be guessed from the CPB graph. My forecast for the future is that both criminal activity and incarcerations will drop to quite low levels (but notice the upper 98% bootstrap prediction interval, the dashed green line) by 2040.
