Admissions to T10 econ Phd programs largely hinge on:
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Extremely strong, highly laudatory recommendation letters from Econ Profs whose research is known and respected by the institutions receiving the recommendations. The more well known the recommender, and the more positive the recommendation the better.
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Two Year Pre-doc RA position, preferably at a central Bank (in US: The Federal Reserve) with a laudatory recommendation from a known economist doing respected research.
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Superb GRE scores and undergrad GPA are necessary but not at all sufficient without 1) and 2) at the T10 econ Phd programs, that is particularly so at the T3 (MIT, Harvard, Stanford) where superb stats are a given. Also expected are solid math chops with classes at least through Real Analysis
Undergrad institution matters to the extent that the econ/math departments are staffed by Profs who are doing top level research which will make their recommendations of value to the profs on the admissions committees.