Because this is such a persistent myth, I thought I should briefly report on this massive study that should hopefully put an end to this myth once and for all (I wish! Myths are not so easily squashed.)
This study used data from 377,000 U.S. high school students, and, agreeing with a previous large study, found that first-borns have a one IQ point advantage over later-born siblings, but while statistically significant, this is a difference of no practical significance.
The analysis also found that first-borns tended to be more extroverted, agreeable and conscientious, and had less anxiety than later-borns, — but those differences were “infinitesimally small”, amounting to a correlation of 0.02 (the correlation between birth order and intelligence was .04).
The study controlled for potentially confounding factors, such as a family's economic status, number of children and the relative age of the siblings at the time of the analysis.
A separate analysis of children with exactly two siblings and living with two parents, enabled the finding that there are indeed specific differences between the oldest and a second child, and between second and third children. But the magnitude of the differences was again “minuscule”.
Perhaps it's not fair to say the myth is trounced. Rather, we can say that, yeah, sure, birth order makes a difference — but the difference is so small as not to be meaningful on an individual level.
(2015). The associations of birth order with personality and intelligence in a representative sample of U.S. high school students.
Journal of Research in Personality. 58, 96 - 105.