In this paper we propose a solution to the non-robustness that plagues the estimation of inverted U-shaped relationships using panel data, such as the relation between pollution and income. When dependent and independent variable (like pollution and income) are both time related, separating the effect of the independent variable from time effects brings about a fundamental identification dilemma: the imposition of restrictions on the controls might drive the shape of the relationship between the dependent (pollution) and independent (income) variables. Our solution consists of imposing the very weak constraint that arbitrary cross-sectional units have the same relationship. We apply our methodology to two widely studied cases, namely, SO2 and CO2 emissions. Interestingly, our estimates are insensitive to the required subjective choices, but also strongly differ from the literature so far. We find consistent positive income effects for both cases and time effect estimates with a clear U-shaped trend for SO2-emissions but only slightly so for CO2-emissions.