Part 1 was a couple of weeks ago, when I only looked at one pair of years. I have now done 5 pairs of years. Low correlations indicate that pitchers tend not to be clutch one year and again clutch the next year.
I looked at all pitchers that had at least 250 plate appearances against opposing hitters in both of five consecutive two-year periods both with and without runners on base. There were around 70 such pitchers in each of the five cases. Data from David Pinto's "Day by Day Database" which is based on Retrosheet.
I found the differential between the batting average they allowed with runners on base (ROB) and the batting average they allowed with no runners on (NONE). I did the same for Slug%. So if a pitcher allowed a .240 BA with ROB and a .260 BA with NONE, his differential was -.020. That means he did better in the clutch than otherwise.
Did pitchers maintain about the same clutch performance in each year? Probably not. The table below shows the correlation between the first year's differential and the second year's differential for both BA and Slug%. They all tend to be pretty low or negative. If a pitchers in general tended to have about the same differential in each year, the correlation would be much higher. That is what I would expect if they really had a constant ability (or inability) with runners on base. But what this means is that many pitchers have a good differential one year and a bad or mediocre one the next year.
In both cases, with runners on and with none on, there is a large number of PAs. So if there really is some clutch ability here, we should see it. Also, pitchers, unlike hitters, can actually bear down and throw a little harder with runners on (or try harder to break off good curve balls). A batter really can't swing harder, for example, depending on the situation. But pitchers could save a little extra for when they needed it with runners on. Batters probably don't save a little extra.
So the one group that could theoretically have some control over the clutch don't seem to have it even when looking at a fairly large number of observations.