Actually, there is one, but from the late 90’s. It is telling that it hasn’t been re-done or publicized. Link below. Two paragraphs of note: [emphasis mine]
As can be seen from Table 2 and Table 3, the states with high levels of joint custody had significantly lower divorce rates four years later. States with higher levels of joint custody had an average four-year decline in the divorce rate approximately double that for states with medium levels of joint custody. On a percentage basis, between 1989 and 1994 the rate in the High joint custody group declined by 8%, in the Medium group by 4%, and in the Low group by less than 1%….
If the greater decline in divorce rates for High joint custody states results from declining marriage rates in previous years, then we would expect marriage rates for these states to show larger decreases in the early 1980s than the Low joint custody states. As can be seen from Table 4, the reverse is true. The low joint custody states actually had greater declines in marriage rates during the early 1980s.
Full study is here: http://proactivechange.com/divorce/research-rates.htm
