I’ve recently come across a 2011 post by Matt Yglesias (via, via) in which he presents the following little theory of population density and crime:
higher density helps reduce street crime in an urban environment in two ways. One is that in a higher density city, any given street is less likely to be empty of passersby at any given time. The other is that if a given patch of land has more citizens, that means it can also support a larger base of police officers. And for policing efficacy both the ratio of cops to citzens and of cops to land matters. Therefore, all else being equal a denser city will be a better policed city.

While plausible, this is also somewhat surprising because in the past people have come up with ideas on how density might *increase* crime. Which is not too surprising given that there is a positive correlation between density and crime (denser cities have higher crime rates). A while back, I half-heartedly reviewed the literature on this; people seem to have come to the conclusion that there’s not a lot to it. But that would suggest the effect is (close to) zero rather than negative.

As it happens, I have a dataset for 125 U.S. cities sitting on my hard drive. So let’s run some quick regressions. All are weighted by a variable that divides 1990 population size by the unweighted sample mean for 1990 population size. That means that each city is given a weight proportional to its size while the sample size stays the same; as a consequence, each crime has the same influence on the results irrespective of whether it happens in a small or a large city. While Yglesias writes in the context of having been assaulted, I will not use data on assault, which seems not to be particularly valid, but rather robbery, as official robbery rates appear to correlate highly with the true rates and robbery is the prototypical street crime. I use 1990-2000 changes in density per square mile and changes in robberies per 100,000 population known to the police as the variables of interest. The use of change data takes care of stable differences between cities that may contaminate the results. The estimation method is linear WLS of changes in (untransformed) rates.

I am not going to go through the trouble of embedding tables in blogger, but simply report results for the variable of interest in the text. Bivariate regression: B = -.25 (p < .001), meaning that an increase in density of 1 person per square mile is associated with a decrease of .25 robberies per 100,000 persons.

Next, let’s worry about immigration. It is, unsurprisingly, correlated positively with density and there are some students of crime who think that immigration decreased crime rates in the 1990s U.S. While I don’t necessarily agree with this, let’s control for changes in the percentage of the population that is foreign-born anyway. This makes next to no difference: B = -.27 (p < .001).

This may mean that density reduces robbery, robbery reduces density, there’s a bunch of unmeasured variables that influence both, or a combination of the above. I am not going to solve that problem here. But what I will do is control for some initial conditions (i.e., 1990 levels of variables) that may influence both of our variables of interest. First, the robbery rate is particularly likely to decline where it is high, so let’s control for 1990 levels of robbery rates. Also, better economic conditions will tend to attract people (and hence increase density) and perhaps also foster future decreases in crime. So let’s throw in 1990 values for poverty and unemployment rates, as well as the median of 1989 household income. This leads to a substantial reduction in the coefficient: -.14 (p < 0.001).

Is that a lot? The mean of changes in population density is 579 per square mile with a standard deviation of 842; for changes in the robbery rate, those values are 366 and 345, respectively. If we were to interpret the coefficient of the last regression as causal, this would mean that, in the sample as a whole, increases in density averted 579*(-.14) = 81 robberies per 100,000 population, meaning that changes in density would be responsible for about a seventh of the observed decline in homicides. That’s a lot.

Of course, you shouldn’t take these little analyses all that seriously. I haven’t worried about functional form or heteroskedasticity and the equation isn’t all that convincing as a causal model.

Still. File under “suggestive”.