Here in Phoenix most work this way, but when a road crosses central avenue the walk counter hits 0 and then there is about a 10 second delay before the traffic lights change.

The problem is, people have realized this. So peds continue to begin crossing as the counter gets low (central has a lot of ped crossing because the light rail runs down the middle of the road).

They really ought to reverse that. Make the light shorter for cars and drivers can no longer use the countdown clocks as a signal. They'd have to actually look at their own light. Combine that with starting the pedestrians early and you make it much harder for drivers to be jerks.

I've seen both. On wide multilane roads I've seen the coundown hit zero and the "don't walk" signal come on before the traffic light changes to yellow. On narrower crossings the yellow tends to appear as the counter hits zero or very shortly after.