BMI and Employment: Is There an Overweight Premium?

Abstract

Using pooled data from the Health Survey of England (HSE) and a semi-parametric regression model, this paper aims to estimate the relationship between body weight and employment probability. We show that employment probabilities do not follow a linear relationship and are highest at a body weight over the clinical threshold for overweight. Instead of an “obesity penalty” we find evidence of an “overweight premium”, especially in socially active jobs. These results suggests that there might exists an endogenous social norm governing body weight judgments and influencing employment prospects, which has been recently updated due to an increase in average body weight.

Appendix A: Logistic GAM Estimation

In the logistic GAM, the basic idea is to replace the linear predictor with an additive one. To explain the mechanism let’s focus on a purely additive model of the form:

$$\begin{aligned} g(p(x))= & {} f_0+\sum _{j=1}^pf_j(x_{ij}) \nonumber \\ p(x)= & {} \frac{\exp (f_0+\sum _{j=1}^pf_j(x_{ij}))}{1+\exp (f_0+\sum _{j=1}^pf_j(x_{ij}))} \end{aligned}$$

(5)

where g(.) is a known link function, in our case a logistic, $f_i$ are smooth unknown functions.

Generally let $E(Y|X)=\mu$

$$\begin{aligned} \nu (x)=g(\mu ) \end{aligned}$$

(6)

where $\nu$ is a function of p variables. Assume $Y=\nu (x)+\epsilon$ given some initial estimate of $\nu (x)$, construct the adjusted dependent variable (i.e. pseudo data)

$$\begin{aligned} Z_i=\nu _i+(y_i-\mu _i)\frac{\partial \nu _i}{\partial \mu _i} \end{aligned}$$

(7)

Instead of fitting an additive model to Y, we fit an additive model to the Z’s, treating it as the response variable. Then we apply the iterated reweighed least square (IRLS) (Hastie and Tibshirani 1990; Wood 2006), by initiating $f_o=g(E(Y))$ and $f_1^0= \cdots =f_p^0=0$. Then iterate with $m=m+1$.

From the adjusted dependent variable

$$\begin{aligned} Z_i=\nu ^{m-1}_i+(y_i-\mu ^{m-1}_i)\frac{\partial \nu _i}{\partial \mu _i^{m-1}} \end{aligned}$$

(8)

where

$$\begin{aligned} \nu ^{m-1}= & {} f_0+\sum _{j=1}^pf_j^{m-1}(X_j)\\ \nu ^{m-1}= & {} g(\mu ^{m-1})\\ \mu ^{m-1}= & {} g^{-1}(\nu _i) \end{aligned}$$

Then form the weights $W=\big (\frac{\partial \nu _i^{m-1} }{\partial \mu _i}\big )^2V_i^{-1}$ with $V_i=Var(Y_i)$. Fit an additive model to Z using the backfitting algorithm¹⁷ with weight W, to obtain estimated function $f^m_j$, additive predictor, $\nu _m$ and fitted values $\mu _i^m=p_i$. Repeat the iteration process until the change in deviance is sufficiently small. The last step of this iteration process is simply the additive regression backfitting algorithm.¹⁸ In this semi-parametric setting, the regression in step two is fitted using a smoother, in our case we used penalized cubic regression splines to smooth the estimated residuals of BMI. In the case of additional (linear) covariates, as in the case of (3), the same procedure described above is used, and a linear lest square fit would be used to “smooth” a binary covariate or a continuous covariate for which a linear fit was desired (Wood 2006).

First-stage results of the impact of the instrument on the endogenous variables

	Male			Female
	(1)	(2)	(3)	(4)	(5)	(6)
	All	Social	Non-social	All	Social	Non-social
Social norm (BMI)	0.101 (0.0791)	0.251 (0.144)	0.0276 (0.126)	0.513 (0.0541)	0.474 (0.0989)	0.513 (0.222)
Social norm (obesity)	0.516 (0.0956)	0.652 (0.162)	0.444 (0.188)	0.0935 (0.102)	0.110 (0.106)	0.208 (0.325)
Observations	25,047	11,145	9637	29,287	16,487	7058
F test	59.31	24.65	33.65	99.69	32.00	28.42

1. Standard errors in parentheses
2. Individual level covariates are included for educational attainment, self reported general health, GHQ-12 score, marital status, housing tenure, number infants 0–1 years in household, number children 2–15 years in household, age, ethnic group, year, month of interview and item non-response. Area level covariates are included for rurality, deprivation, respondents used to generate obesity prevalence in health authority, standard deviation of BMI in health authority among respondents, and region. Standard errors are adjusted for health authority level clustering

All the estimations of Eq. (3) were made using the statistical software R. In R the smoothing parameter determining the number of knots is chosen by default via generalized cross-validation (GCV).¹⁹ The idea is simple; let the data speak, and draw a simple smooth curve through the data. The problem is determining goodness-of-fit and error terms for a curve fit by eye. GAMs make this unnecessary and fit the curve algorithmically, using the GCV, in a way that allows error terms to be estimated precisely using either IRLS or back fitting algorithms depending on the type of dependent variable. Some researchers note the fact that such automated smoothing could lead to overfitting as it is not manually controlled by the users. Loader (1996) suggests adjusting the smoothing parameter using visual inspection—i.e. a model displaying implausible wiggles everywhere is highly likely to have been overfitted. The issue of overfitting turns out not to be a concern. Upon request we can provide graphs where we have manually chosen the smoothing parameter by rounding the parameter provided by GCV to the closest integer, from which it is clear that the difference between the automated GCV bandwidth selection and the manual one is negligible. Neither of the two approaches yields any excessive spikes or wiggles. If anything, the GCV algorithm leads to slightly more conservative estimates of the relationship between body weight and employment.