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I'd also be quite interested. From my own impression at least, "old" birthrates are predictive of "new" birthrates, but directionally not absolutely. My parents were 4 siblings and 7 siblings, respectively, my grandparents afaik also had 6+ siblings for the most part, and my cousins usually were around 3 siblings. These cousins now also seem to mostly have 2-3 kids, with very few childless. So while there is a clear downward-pointing arrow, at any time point we're consistently above the average. At the same time, my childless acquaintances seem to primarily come from academic families that rarely had more than 2 children even in the past. All with considerable variance of course, so I'm open to the possibility that this predictivness is quite weak.