Mixed spline-Fourier-kernel estimation for longitudinal data: carbon emissions in Latin America

[Objetive] The objective was to develop a mixed nonparametric estimator that combines truncated splines, Fourier series and kernel smoothing for longitudinal data, and to describe how carbon dioxide emissions per person relate to income in Latin America and the Caribbean. [Methodology] The estimator was placed in a mixed-effects model with a random effect for each unit. Its kernel part removes the predicted unit effects before smoothing, its spline and Fourier coefficients are estimated by generalized least squares, and its tuning parameters are chosen by generalized cross-validation on whitened residuals. It was evaluated in eighteen simulation scenarios and applied to 163 countries observed from 1990 to 2022, including 25 Latin American and Caribbean countries. [Results] The proposed estimator reduced the error of the classical mixed estimator by 10% to 59%, more so under strong within-unit correlation, although additive mixed models were slightly more accurate for smooth functions. Worldwide, the within-country income elasticity of emissions fell from 0.71 at low incomes to 0.32 at middle and high incomes and was indistinguishable from zero above about 33,000 dollars. Latin American and Caribbean countries lie below that level and their elasticity showed no decline. Costa Rica emitted far less than its income predicts and, since 2010, about 7% less than its own expected path. [Conclusions] The estimator makes mixed nonparametric regression reliable for longitudinal data while keeping interpretable change points. Regional emissions still rise with income, with Costa Rica as a notable exception.