Is there a rule of thumb for absolute purchasing power parity to hold?

We find an example where real exchange rate (RER) is stationary and the nominal exchange rate and the price levels are cointegrated but purchasing power parity (PPP) does not hold, which reveals a fault of the unit root and cointegration tests in this use. We argue that the distribution of an RER misalignment can be used in testing absolute PPP. Then we apply this new test and the coefficient restriction test to study the validity of absolute PPP in 40 main countries and areas (versus the US) in light of the Harrod-Balassa-Samuelson effect. The econometric proofs show that absolute PPP holds or closely holds in most countries when their averaged relative GDP per capita (GDPPs, against the US with the US = 1) are greater than 0.7. And it does not hold in almost all countries when their averaged GDPPs are smaller than 0.7. Thus, a rule of thumb for the theory to hold is that the GDPP should be above 0.7.