Repetitive jQuery click functions for changing multiple elements
As far as I can tell, the code simplifies as follows :

$("#timeframe_pills_container li a").click(function(e) {
  var $li = $(this).closest("li");
  var index = $li.index();
  var $dropdown = $("#timeframe_dropdown_container");
  $li.addClass("active").siblings().removeClass("active");
  $dropdown.find("li").eq(index).addClass("active").siblings().removeClass("active");
  $dropdown.find(".dropdown-toggle p").text($(this).text());
}).eq(0).click();

$("#timeframe_dropdown_container li a").click(function(e) {
  var index = $(this).closest("li").index();
  $("#timeframe_pills_container li a").eq(index).click();
});


.eq(0).click() establishes the intial state, so you don't need to set class="active" (twice) or the button text, in HTML.