Course-taking beyond the level of advanced algebra (what is normally called "Algebra 2" in the United States) has been strongly linked to future academic and economic attainment; for example, a U.S. Department of Education study (Adelmann, 1999) found that students who complete mathematics coursework beyond the level of a course in advanced algebra are more than twice as likely to pursue and complete a postsecondary degree. Similarly, Horn and Nunez (2000) report that students whose parents never attended college are more than twice as likely to enroll in four-year colleges if they take courses beyond advanced algebra; and Carnevale & Desrochers (2003) found that “Algebra II is the threshold mathematics course taken by people who eventually get good jobs in the top half of the earnings distribution”, despite the fact that relatively few actually make use of that mathematical knowledge on the job.
(I realize that these studies focus on economic outcomes, rather than "understanding, analytic ability etc" as you asked, but I think this is the closest you are likely to get in terms of measurable benefit to students.)
However -- and this is a huge caveat -- these results show a correlation, not causation, and the question you ask is probably impossible to answer. That is because course-taking beyond the basic level is a consequence of many factors, like intelligence, socioeconomic status, etc., and so it is unsurprising that the students who take those advanced courses go on to have better outcomes -- they were already having better outcomes before they took the courses, which is why they took them in the first place!
To actually demonstrate that learning higher-level math has a beneficial outcome you would need to do some kind of controlled experiment in which kids were randomly assigned to one of two educational tracks, and see what happens. Research like that would be (obviously) unethical and hence cannot be done.
On the other hand, there is a kind of naturally-occurring version of this experiment unfolding in the US right now. Several states have enacted new, tougher high school graduation requirements in which all students must take and pass Algebra 2 in order to graduate. In most states, there is no such requirement. So in principle one could study the outcomes in those states and try to see if those new policies lead to discernible effects in their graduates, as contrasted not only with other states but also with the status quo ante. (Algebra 2 is perhaps not as high-level as you have in mind, but I don't think there would be much support for requiring all students to take Calculus in order to graduate high school.)
Edited to add:
Afterthought: The hypothetical study I described in my last paragraph would probably not show much if any effect, for the following reason: Faced with a new obligation to get all students into, and successfully through, a course that large numbers of them would not have taken in previous years could reasonably be expected to have the unintended consequence of the course being somewhat watered-down. There's no evidence yet that this has happened, but it's not implausible to expect the system to accommodate the new requirements in ways that the designers of those requirements were not hoping for.
