Lose Weight for Money Only if Over-Weight: Marginal Integration for Semi-Linear Panel Models

Body mass index (BMI), weight(kg)/ height(m)2, is a widely used measure of obesity in medical science. In economics, there appeared studies showing that BMI has a negative (or no) effect on wage. But BMI is a tightly specified function of weight and height, and there is no a priori reason to believe why this particular function is the best to combine weight and height. In this paper, we address the question of weight effect on wage, employing two-wave panel data for white females (and white males); the same panel data with more waves were used originally in Cawley (2004). We posit a semi-linear model consisting of a nonparametric function of height and weight and a linear function of the other regressors. The model is differenced to get rid of the unit specific effect, which results in a difference of two nonparametric functions with the same shape. We estimate each nonparametric function with a ‘marginal integration method’, and then combine the two estimated functions using the same shape restriction. We find that there is no weight effect on wage up to the average weight, beyond which a large negative effect kicks in. This negative effect’s magnitude is greater than that in Cawley (2004) who used a linear BMI model. The linear model gives the incorrect impression of the presence of a ‘wage gain’ by becoming slimmer than the average and of a ‘wage loss’ that is less than what it actually is when going above the average.