I do have income data for individuals from different families over three generations (y, yparents, ygrandparents). To determine the two-generational social mobility, I run a simple linear regression:
log(y) = beta * log(yparents) + e,
which I can interpret as the intergenerational income elasticity.
In order to determine the effect of the grandparental generation, I ran the following regression:
log(y) = beta1 * log(yparents) + beta2 * log(ygrandparents) + e
Interestingly, the coefficient of the grandparental generation is more or less equal to the squared coefficient of the parental generation (which is what we would expect if the intergenerational transmission of status follows an AR(1) process. Furthermore, the coefficient of the parental generation does hardly change once we include the grandparental generation in the regression model.
Is there any problem related to this second regression (caused by endogeneity, etc.) or is this a valid approach? And if there is a problem: How can I address it, without having more information about either one of the generations?