Honored you would think of me as knowing a good place to point. But alas I do not know. Are you interested in methods people use to do this or?

In some cases the people who do this do know. In other cases they do not.

Most definitely.

So then you measure the trend in the period rate…. Which is clearly just the inverse of the trend in the unidentified time to fill under constant hazard assumption.

But the tension is that, let’s say you just invert that openings per filled opening measure. Then it’s filled openings per openings, a period rate. Fine to measure that since it’s basically just a constant hazard. (Although it assumes constant hazard of course)

I claim without proof (but appealing to what I know about competing risks theory) that it is not. Doesn’t make sense to me that you could measure a trend in a quantity that is statistically unidentified under dependent competing risks.

But my question is whether looking at the trend in either of those (let’s say the second) is valid.

(2) count the active job openings in mid period and divide them by filled job openings (inverse of the period rate). I am sure this also assumes dependence of competing risks.

My experience is that folks tend to either (1) estimate time to fill by averaging the time it took filled jobs to get filled, which is bad for two reasons (ignores censoring and assumes independence of competing risks); or

Indeed!