If the specialist teachers were significant factor, shouldn't the private sector teacher wage in those statistics be much higher, considering that the proportion of private sector for 4 year college education is much higher than the proportion of private sector for K-12?
By their own admission they drew their conclusion based on inadequate data:
Information on the pay of private school teachers by metro area has a fair amount of missing data since there may have been insufficient samples in many areas. For those metro areas for which we have data, San Antonio, San Francisco, and Minneapolis paid their public school teachers more than twice as much as private school teachers. In the Phoenix, Houston, and Raleigh metro areas, private school teachers earned more than public school teachers.
The point I made is that these numbers don't control for numerous variables.
Look at
the BLS data yourself. It is broken down by numerous categories (experience, skill and other factors). The data is broken down by grade level and university specialty. And the margin of error for each line of data differs from <1 to >15.
Do the authors describe how they controlled for all these variables? Do they mention what weight they gave to data where the data differed in certain categories? Do they explain what accounts for these differences?
Private and public university teachers make about the same on average. Then you look at public teachers vs private for non-university positions and the difference is abnormally large, ($33.70 vs $20.74). Something is accounting for this difference besides just unions or public school positions.
If you look closer there are curious patterns. Public university teachers earn more at lower skill/tenure levels and less at higher levels. It averages out to similar pay overall. The range is narrower for public and wider for private teachers at the university level. But we don't know if a few specialty programs skew the private university results or many universities average out the public university teacher results.
Non-university pay for public school teachers is markedly higher for prekindergarten/kindergarten teachers ($30.35) and lower for private prekindergarten/kindergarten teachers ($12.71). What's going on there? It affects the averages. Was it given equal weight in the average as a category or a proportional weight based on the number of workers in that category? It looks like it is the weight by category, not number of workers in the category. That could skew the average.
Private school teachers other than university is the same at level 5 as public then public teachers jump ahead at level 6. Both are equal again at level 11. This irregularity suggests missing data might be at work. At level 4 the public school data is missing and at level 12 the private school data is missing. Adding in a level 12, the highest, and subtracting out a level 4 from the public school wages shifts the whole average up.
The data used to make these analyses sounds good but only presents a tentative conclusion which the authors themselves admit. This is not some clear proof that public teachers are overpaid, and there is nothing in this data that suggests what a fair wage is, which was the premise of claiming teachers were 2/3 up the scale of white collar workers.