Britain sealed a top-20 ranking in the medal standings, equalling their previous best finish at the inaugural Winter Games in 1924.
"I am proud of the team's achievements, we have competed to the utmost of our ability," Hay told BBC Sport.
"I'm delighted with everybody, not just with the medals, but with the number of finals and top half finishes we have had.
"We have won four medals which sets a new standard for us. We have come a long way from Vancouver 2010 where we won just the one medal. The team has a young profile and I believe we can do even better in Pyeongchang in 2018."
The Russian Federation finished first in the standings, with 33 medals, 13 of them being gold.