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Pavitra comments on Why are certain trends so precisely exponential? - Less Wrong
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If this were the case, we should expect any noise to accumulate as a random walk. After the Great Depression, the data goes back to where it would have been without the interruption.
GDP trend related to population, not to previous GDP? Over the period of time covered by the above GDP graph, the log graph of US population (seen on http://www.wolframalpha.com/input/?i=united+states+population) looks pretty much like a straight line. GDP per person looks exponential over the last ~60 years (also wolframalpha) [edit] looks linear in constant dollars (from wikipedia)[/edit]
Unless an economic disruption is big enough to greatly change the birth or death rates, the GDP will tend to go back to the straight line, it seems.
If there is easy-to-find GDP data going back to the 1600s, when the log(population) graph is not on the same line as recently, my guess could be tested.
Alternatively, we could examine both datasets closely to see if non-trend-predicted variation in one is reflected in the other.