The idea that the Fed can control interest rates, while widely accepted, is not well supported by empirical analysis or theoretical considerations. This is especially true now when an unwieldy balance sheet makes tightening credit an unlikely policy choice.
The Fed itself, in its December (2015) liftoff used an artificial trick to move the repo and funds rates up, but no other rates of importance moved up with them. In fact, rates fell across the board after the famous Fed liftoff.
What if? Maybe zero rates are mostly market determined and the Fed is irrelevant. We have been assuming that inflation is 1 -2 percent, but what if that isn't right. Measuring inflation is a notoriously difficult project. Maybe inflation is not 1-2 percent, but is actually minus 1-2 percent. If so, zero rates would make perfect sense. The real rate, actual rates minus expected inflation, would then be slightly positive.
If inflation is actually negative, not positive, then the a low real rate of interest would be consistent with sub par economic growth. It would explain a lot. However, it suggests that risk assets may be over-extended since the risk premium is unlikely to be much higher than four or five percent at best.
This means the stock market is not your grandfather's stock market. With a zero risk free rate, risk assets will only earn a risk premium, which, it seems, has been fallling -- maybe one percent, maybe less, but certainly less than the real growth rate of GDP.
This suggest that while recessions are likely, major recessions are unlikely. What is most likely is that economic growth will simply grind to a halt. The Fed, meanwhile will, at some point, push repo rates and funds rates higher, but other rates will remain on the floor (until the reality dawns that sovereign debt is probably not a good bet).
So, what zero rates are telling us is not that the Fed has its thumb on rates. Instead zero rates are consistent with mild deflation and a declining risk premium. This means unexpectedly low, though still positive, expected stock market returns, and a stagnant economy.