## 1. INTRODUCTION

In this paper we investigate the role of body mass index (BMI)1 in the production of log wages. The simplest approach to studying this would proceed by running a regression of the outcome, log wages, on the BMI variable and potentially a variety of controls. The resulting coefficient on the BMI variable, however, could not be convincingly argued to capture a causal impact, since it fails to account for the potential endogeneity of BMI. By endogeneity we mean that there could be factors unobserved by the econometrician which simultaneously correlate with BMI and log wages. One such confounding variable could be preferences for long-term investments, which we mean to represent characteristics that simultaneously impact decisions affecting both health and human capital accumulation. The existence of such unobserved confounding variables produces a bias and inconsistency in the simple regression-based estimator described above, requiring us to employ a more elaborate procedure that accounts for potential confounding on unobserved characteristics. To this end, we adopt in this paper a standard two-equation triangular treatment–response model in which the outcome of interest is log wages and the endogenous treatment variable is BMI.

The primary methodological innovation of our approach is that we permit the BMI variable to enter the wage equation nonparametrically. The default specification in applied treatment–response modeling assumes a linear relationship between the treatment and the outcome, and defines the slope of this line as the causal effect of interest. This linearity assumption, however, is not credible in all situations. In our study, for example, it is plausible that wages are relatively unresponsive to marginal changes in BMI in the ‘underweight’ or ‘normal’ ranges, and relatively more responsive to marginal changes in BMI in the ‘overweight’ or ‘obese’ ranges. Alternatively, underweight individuals may experience a wage penalty similar to overweight or obese individuals, producing an inverted U-shaped relationship between BMI and log wages. Standard linear treatment–response models cannot capture these features of the data if they are present. Although a linear relationship might be found *ex post* by testing down from more flexible specifications, it does not seem desirable to impose these restrictions *ex ante*. To our knowledge, no applied studies in this area have investigated this possibility extensively. Therefore, we hope that use of a more flexible empirical specification which can either refute or provide support for the specifications used in past work offers a useful contribution to this literature.

A second, yet more minor, methodological contribution is that we introduce a model capable of accommodating skew in the distribution of the continuous endogenous treatment variable. This particular modeling assumption is made with an eye toward our empirical application, where the unconditional and conditional BMI distributions have a pronounced right skew. Our approach for handling this skew is to generalize the error distribution associated with the treatment variable to the class of skew-normal type distributions (e.g., Azzalini and Dalla Valle, 1996; Branco and Dey, 2002). We find that use of the skew-normal distribution provides an adequate fit to our data, outperforms the standard semi-log specification, and is parsimonious as it introduces only one additional parameter. In addition, the methods described here can be easily generalized to handle cases where there is skew in the outcome variable, skew in both the outcome and the endogenous treatment variable, and when the variables being modeled are censored, binary, or ordered.2 In all cases, however, careful diagnostic checking should be performed to assess whether the maintained distributional assumptions are sufficiently flexible. We fit the model from a Bayesian perspective and present a computationally attractive posterior simulator which handles the endogeneity problem, skew, and nonparametric component simultaneously, and involves only standard Gibbs steps.

Studies of the effect of weight-for-height3 on economic outcomes must contend with observational data. One approach that has been used to identify a causal effect with such data is differencing, within twins or siblings (e.g., Averett and Korenman, 1996; Behrman and Rosenzweig, 2001; Baum and Ford, 2004), or over time with panel data (e.g., Baum and Ford, 2004; Cawley, 2004), to purge the model of unobserved confounding variables. These approaches require within-sibling variation in weight-for-height or temporal variation in weight-for-height to identify the effects of interest. Since such variation may not be substantial, particularly in longitudinal data studies with a short time dimension, it may often prove difficult to precisely estimate effects of interest with such an approach. Alternatively, some studies have made use of instrumental variables (e.g., Behrman and Rosenzweig, 2001; Cawley, 2004) to identify a causal effect.

We follow the instrumental variables approach and use parent BMI as a source of exogenous variation. The validity of this IV strategy hinges on the assumptions that parent BMI is conditionally correlated with the child's adult BMI, and that parent BMI can be excluded from the child's adult log wage equation. The first assumption is not controversial and is empirically testable, but the second assumption may be met with more skepticism. In particular, it is possible that the correlation between parent BMI and child BMI is due partly to learned preferences for long-term investments in health capital, which may be correlated with learned preferences for investments in human capital, which in turn are known to have a direct effect on earnings. In this case, parental BMI may embody unobserved factors that have a structural effect on the child's adult earnings, potentially undermining the validity of our identification strategy.

We offer three reasons why such concerns should not necessarily invalidate the use of parental BMI as valid instruments. First, and perhaps most importantly, the validity of the instruments depends on their *conditional* uncorrelation with the errors, and thus the inclusion of a rich set of proxies for family preferences and background characteristics will surely bolster the case for the use of parental BMI in practice. Specifically, the set of variables we include in the wage equation should reduce the problem a ‘shared family environment’ mechanism may present, as they serve as proxies for family preferences, even if this problem is likely to exist unconditionally (i.e., without any controls). To this end, we include in the wage equation covariates capturing aspects of environment and background including indicators for parental education, parental occupation, and parental income when the worker was 10 years old.

Second, the health literature suggests that the ‘shared family environment’ mechanism in the production of adult BMI outcomes is relatively weak. Stunkard *et al.* (1986), for example, obtain a sample of information from Danish adoptees, and find a strong correlation between the weight class of adoptive children and their biological parents, yet find no correlation between the weight class of adoptive children and their adoptive parents. In a review of several studies of the determinants of BMI, Maes *et al.* (1997) similarly conclude that ‘it is unlikely that environmental factors shared with family members contribute substantially to variance in BMI’ (pp. 329–330). In another review, Grilo and Pogue-Geile (1991) write: ‘this suggests little environmental effect of parental weight on offspring weight through modeling’ (p. 534). These studies provide suggestive evidence that the correlation between parent and child BMI has little to do with shared environmental factors, and, therefore, parental BMI can potentially be excluded from the wage equation, especially given adequate controls for family characteristics and background characteristics of the child.

Finally, since we have two instruments available4 we can conduct the standard exercise of assuming that one parent's BMI is a valid instrument, including the other in the log wage equation, and calculating the Bayes factor in favor of the hypothesis that the other parent's BMI coefficient in the log wage equation is equal to zero. From a theoretical perspective, however, it seems plausible that the BMIs of the parents are either jointly valid as instruments, or jointly invalid, thus calling into question what is actually learned from this procedure. On the empirical side, however, the correlation between parental BMI was found to be reasonably small (around 0.16), suggesting that something can be learned from this exercise, and that its implementation is not obviously redundant or ‘circular’. For both men and women, these Bayes factors are found to support the restricted model, providing evidence that parent BMI can be excluded from the log wage equation, and informal evidence of the validity our identification strategy.

Using data from the 1970 British Cohort Study, we apply our estimation algorithm and find strong evidence that BMI affects log wages. We find that log wages are decreasing throughout the BMI support, and that this result holds for both men and women. For men, the wage penalty to marginal increases in BMI is modest provided the individual is in the ‘normal’ BMI range, whereas penalties are comparably large for overweight and obese men. For women the results are essentially reversed. The largest penalties for a marginal increase in a woman's BMI are found over the ‘normal’ BMI range. In addition to these nonlinearities within gender groups, we find evidence of differential BMI wage penalties across these groups. We find some evidence that for individuals with BMIs in the alternatively defined normal range of 20–25 the wage penalty for a marginal increase in BMI is smaller for men than it is for women. Overweight and obese men, however, receive substantially larger wage penalties to marginal increases in BMI than comparably overweight and obese women. To our knowledge, these results have not been documented before in the literature. These findings also raise questions about the credibility of the assumption of linearity between log wages and BMI that has been made in past work.

The outline of this paper is as follows. In the following section we describe our model, strategy for handling the nonparametric component, skew, and endogeneity issues, and the associated Bayesian posterior simulator. Section 3 describes our data, while Section 4 presents our empirical results. The paper concludes with a summary in Section 5.