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All applications to PhD program must go through official channels. I am not even allowed to give advice or guidance, though I am rarely if ever involved directly in addmissions. I am only expressing my personal opinion here.
For a PhD program like those at Harvard, MIT, or UCLA, the decision problem with applicants from many backwater countries (which includes many EU countries, too!) is that US PhD admissions committees have few benchmarks. An applicant could be the next Einstein or be simply completely unprepared for the US program. And most recommendation letters from unknown references are hard to judge—they are simply credible. They typically make all candidates sound perfect.
A good analogy for the academic competition can be found in sport. There are leagues and then there are leagues. The best players in the Israeli top soccer league are not necessarily capable of playing in the UK Premier Leage. Avi Cohen was the best in Israel, yet barely registered when he tried out in Liverpool. On the other hand, Dirk Nowitzki came out of Wuerzburg=nowhere and turned out to be an MVP even in the NBA.
The top (mostly US) PhD programs are more akin to the UK Premier League or Bundesliga. Many of the top academics from other countries may not be prepared and/or talented enough to compete against the best from all over the world. But there may well be unrecognized Einsteins among them. Most of the time, even the applicants themselves won’t know whether they can “hack it” until they try. Looking back at my own application, I could not have known when I applied, either.
Unfortunately, given that there are some applicants that already do look like the next Einstein (though they never turn out this way) from known channels that admission committees can judge, it is difficult for a school to commit to what is in effect roughly a $500,000 commitment of school resources to an less-known applicant (good or bad) instead of to one of the known quantities.
If there was only one such unknowable case a year, one might take chances. But it is typically about 50 of the about 300 applicants that fall into this “unknowable” category. And top PhD program rarely admit more than 5-10 out of 300 students per year.
What would you do if you were on the admission committee?
But not all is lost. If I were from a “backwater” country and had ambitions to get into a top PHD program, I would first try to enroll in a program that has benchmarks readily understood by such programs. This could be a less-competitive PHD program in the US, UK, or perhaps a top program in a less competitive country (e.g., other EU, Israel). A student who does great in such a program can apply to transfer after a year or two. Or a student could apply to a top US MFE program with the express intent to demonstrate academic excellence and obtain reference letters from some well-known and trustworthy finance professor [i.e., that the committee knows to be reasonably honest].
Incidentally, unless the applicant is truly top-10 of 100 students in such an easier program, and scores at least 95th percentile on the standardized tests in math for a non-native speaker and at least 90th percentile in English for a native speaker, chances of getting into a top PhD program are very low. This is in the applicants’ own interests, too. About 1 in 3 students spends 2-6 years in the PhD program and never succeeds ending up getting a PhD.
There are so many other ventures that are interesting, promising, and better remunerated than finance and economics research for talented applicants, why waste so many years if in the end it comes to nada?