Well, whatever correlation they noticed back in 1996 obviously didn't hold up just a few years later:
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And again, poverty rates fell in the 60s and 70s, while crime rose.
On a state level, there are certainly some outliers that don't make sense. E.g. Maine has one of the lowest homicide rates, but is in the middle of the rankings for poverty rates; Kentucky has a high poverty rate, but is in the middle for homicide rates.
Poverty rates are slightly higher in rural areas than urban ones (except in the US south, where rural poverty is much higher than urban), but for decades rural crime rates were significantly below urban. How does
that work?
And of course, we can't forget the impact of structural racism (and classism) on all of this, as police have deliberately targeted poverty-stricken urban minority neighborhoods for overpolicing, which is obviously going to damage the social fabric of those communities.
(And doesn't it follow that if poverty causes crime, then why shouldn't we just throw money at poor people to reduce crime rates, instead of spending hundreds of billions on law enforcement and jails?)
The bottom line is that
nobody really knows what causes the changes in crime rates that we've seen over the past few decades. Criminologists and other researchers have spent decades banging their heads against that brick wall, and have had no success in finding any simple answers.
It's frustrating, but it also means that the usual pat answers -- poverty, unemployment, Twitter, whatever -- are probably wrong, or at best a tiny piece of a complex puzzle.