We estimate the effect of BMI on mental health for young adults and elderly individuals using data from the National Longitudinal Study of Adolescent Health and the Health & Retirement Study. To tackle confounding due to unobserved factors, we exploit variation in a polygenic score (PGS) for BMI within two complementary econometric methods that differ in the assumptions they employ. First, we use the BMI PGS as an IV and adjust for PGSs for other factors (depression and educational attainment) that may invalidate this IV. We find a large statistically significant effect of BMI on mental health for the elderly: a 5 kg/m2 increase in BMI (a difference equivalent to moving from overweight to obese) increases the probability of depression by 29%. In contrast, for young adults the IV estimates are statistically and economically insignificant. We show that IV estimates likely have to be interpreted as identifying a weighted average of effects of BMI on mental health mostly for compliers on the upper quantiles of the BMI distribution. Second, we use the BMI PGS as an "imperfect" IV and estimate an upper bound on the average treatment effect for the population. The estimated upper bounds reinforce the conclusions from the IV estimates.