We show that errors in the measurement of police are a primary impediment to the accurate estimation of the effect of police on crime.  Collecting multiple measures of the number of police for approximately 250 U.S. cities from 1960-2010, we provide evidence that, when measured in growth rates, police are measured poorly in the data that are commonly used in the prevailing literature.  Under the classical measurement error model, in which the measurement errors are assumed to be additive and random, the classic result is that least squares estimates of the police elasticity will be substantially underestimated. 

Using our multiple measures of police, we suggest an omnibus test of the classical measurement error model.  This test is motivated by the fact that under classical measurement errors, a consistent estimate of the effect of police on crime can be obtained using an instrumental variables framework in which one noisy measure of police is used as an instrument for a second noisy measure of police.  The crux of this approach is to isolate “good” variation in police that is common to both measures.  We note that with two measures of police, there are two possible configurations of this IV regression: one in which the first measure of police is the endogenous covariate and the second measure of police is the instrument and an alternative configuration in which the second measure of police is the endogenous covariate and the first measure of police is the instrument.  Under the classical measurement error model, both configurations should lead to the same consistent estimate of the police elasticity.  This intuition leads to an overidentification test of the equality of the estimated police elasticities under the two different IV configurations.  Applying this procedure, we fail to find evidence against the null hypothesis of classical measurement errors.  As a result, consistent estimates of the police elasticity can be estimated straightforwardly via instrumental variables.

Correcting for measurement errors in police and conditioning on an unrestricted set of state-by-year effects, we estimate elasticities of crime with respect to police of roughly -0.35 for violent crime and -0.15 for property crime.  Elasticities are largest for murder (-0.58), robbery (-0.57), motor vehicle theft (-0.33) and burglary (-0.20).  Notably, our estimates are similar in magnitude to those in the literature that use instrumental variables such as mayoral elections and firefighters to isolate plausibly exogenous variation in police.  The similarity between our measurement error-robust estimates of the police elasticity and those in the prevailing IV literature suggests that a leading explanation for the discrepancy between least squares and IV estimates of the police elasticity may be measurement errors instead of simultaneity between police and crime.  We note that a considerable advantage of our approach is that the elasticities we report are estimated with extraordinary precision in comparison to those in the prevailing literature.

Finally, leveraging our resulting police elasticities and using estimates of the cost of crime to victims, we calculate that every dollar spent on policing delivers approximately $1.50 in social benefits, though there is considerable heterogeneity among cities.