The simplest version of ListCorrelate, ListCorrelate[ker,list], only runs the kernel with respect to the list across the range where there is no overlap with the end points. But there are a variety of additional argument forms that allow additional behaviors.

As you read though the documentation you will come across this form:

ListCorrelate[y2, y2, {-1, 1}, 0]

this allows maximal overhang at both ends of the kernel (y2 in your case) with the list (y2 as well) and pads the ends past the overhang with 0s. This gives

{8, 23, 44, 70, 100, 133, 168, 204, 168, 133, 100, 70, 44, 23, 8}