This study by Bush et al. has shown that rare birds are not the most traded birds. My master thesis was on the international bird trade and I found that, although there may not be much trading in rare birds, the scarcity of a bird species does drive the price up. Additionally, poachers may have up to two years warning before a species is up-listed to a level of higher protection by CITES, causing poachers to fill their boots before trade in a particular species becomes illegal.

Unfortunately, the same study found that the demand for reptiles and mammals was related to their rarity.