Per the Ray 2009 Homeschool study, the TFR of homeschoolers is apparently likely slightly greater than 3.5.
Why is that?
You can explain some of the deltas based on homeschoolers being more religious. They're also on the average slightly richer and slightly more educated, but these things don't typically result in many more children. They're also far more likely to be married and the wife is usually a stay at home mom, although a fair number also work part-time.
My suspicion is that having a large net disadvantage in the war of indoctrination reduces the fertility of non-homeschooled families. It accomplishes this by making your children less respectful of your authority and thereby making them more annoying and burdensome. Do we have any natural experiments available on this sort of question? Do white families in areas with full-day kindergarten or preschool have a lower TFR?
Did areas where television was late in adoption (e.g. South Africa) have a slower fall in white fertility?
Or do we have to cop out and attribute the homeschool TFR nearly entirely to extra-hardcore religiosity---beyond that which can be easily captured by GSS or similar instruments?
Anecdotally, the homeschool families I know very well personally have this many children:
3*, 3*, 1, 5, 12, 4*, 4 (plus one extra adopted)
where an asterisk indicates that the family is PROBABLY finished having kids, but not a 95%+ certainty.
By the standards of homeschoolers, my wife and I, with likely our 3rd child being born next spring, have a fairly small family. This is probably attributed to the fact that I married pretty late---in my 30s, whereas most of the other homeschool families married in their 20s.