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Shape Matters: Evidence from Machine Learning on Body Shape-Income Relationship

We study the association between physical appearance and family income using a novel data which has 3-dimensional body scans to mitigate the issue of reporting errors and measurement errors observed in most previous studies. We apply machine learning to obtain intrinsic features consisting of human body and take into account a possible issue of endogenous body shapes. The estimation results show that there is a significant relationship between physical appearance and family income and the associations are different across the gender. This supports the hypothesis on the physical attractiveness premium and its heterogeneity across the gender.

Disease Control and Prevention (CDC), the standard weight status categories associated with BMI ranges for adults are as follows: below 18.5 (underweight), 18.5-24.9 (normal or healthy weight), 25.0-29.9 (overweight), 30.0 and above (obese). BMI is calculated as weight in kilograms divided by height in meters squared. We refer reported BMI (measured BMI) to the one based on reported height and weight (measured height and weight). In the tables, height, weight and BMI are those measured by professional tailors at the survey sites. For both genders, reported BMI is slightly larger than measured BMI on average.
In addition to the bio-metric measurements, the data contains other variables for individual characteristics and socio-economic backgrounds. Education grouped into nine categories is 16.29 years for males and 15.75 years for females on average. Experience is calculated as potential experience = age − education − 6 and its mean is 17.54 years for males and 18.62 years for females. Fitness is defined as exercise hours per week. Its mean and median are 4.24 hours and 2.5 hours, respectively, for males. For females, its mean and median are 3.74 hours and 2.5 hours, respectively.
The data also include the number of children. Marital status is classified as three groups: single, married, divorced/widowed. Occupation consists of white collar, management, blue collar, and service. Race has four groups, including White, Hispanic, Black, and Asian. Birth region is grouped into five groups, including Midwest, Northeast, South, West, and Foreign. The majority in the dataset are white collar married white males and females born in the Midwest. As we will discuss later, the data also contains 40 body measures that include height and weight. The list of body measures are provided in Table S.3.

Reporting Errors in Height and Weight
The following equation estimates the personal background that explains reporting error in height and weight: Reporting Error H i = πX i + µHeight i + ε i , (S.1) where X i is a set of covariates, including family income, age, age squared, occupation, education, marital status, fitness, race, and birth region. Height i is the true height in millimeters, and Weight i is the true weight in kilograms. We found dependence between reporting errors and some covariates. Table S. 4 reports the estimation results. The standard errors are estimated by bootstrapping and are reported inside the parentheses. In equation (S.1), the coefficient of the true height is not statistically significant for both genders. We observe different results across genders. For males, family income is negatively correlated with the reporting error in height at the 1% significance level. Age squared is positively correlated with the reporting error. Hispanic males are more likely to under-report their height than White males. Males who were born in the Northeast are more likely to over-report their height than those from the Midwest. On the other hand, the coefficient of family income is not statistically significant for females. Older females are more likely to under-report their height. The estimation results are summarized in Fig S.1.
In equation (S.2), true weight is negatively correlated with the reporting error in weight (at the 1% significance level) for both genders: heavier people have a tendency to under-report their weight. For females, interestingly the coefficient of fitness is statistically significant at the 5% significance level and it is negatively correlated with the reporting-error in weight. Thus, females who spend more time on exercise have a tendency to under-report their weight. However, we find little evidence that other  Conditional quantile function is a useful tool to estimate heterogeneity in a conditional distribution. It also measures the proportion of reporting errors that are positive or negative. Fig S.3-S.4 present the estimation of the conditional quantiles of the reporting errors in height and weight conditional on the true measures, namely, Q τ [Reporting Error H | Height] and Q τ [Reporting Error W | Weight] for τ ∈ (0, 1), with their 95% confidence bands, respectively. We estimate the conditional quantiles using nonlinear polynomial regression. The figures display the median and the 10%, 25%, 75%, and 90% quantiles across genders. In Fig S.3, the results show that there is heterogeneity in the conditional distribution of the reporting error in height for both genders. Over-reporting of height is more pronounced for males than females. We notice that more than 75% of the sample of the males over-report their height. Interestingly, the median regression lines in both genders are approximately parallel with the horizontal line, which implies that the conditional median of the reporting error is independent of the true height. Bollinger also found that median regressions for earnings will be more robust to the reporting error than mean regressions [1]. Thus, it would be more natural to impose a restriction on the conditional quantile of the reporting error of  height than the conditional mean (e.g., Hu and Schennach [2] and Song [3]). Fig S.4 also displays apparent heterogeneity in the conditional distribution for both genders. It shows that, for heavier than average people, the under-reporting of weight is more pronounced for females than males. We notice that within this group, almost 75% of the females under-report their weight. The median regressions in both genders are dependent on the level of the measured weight. This indicates that there are substantial nonclassical errors in the reported weight; therefore, a restriction on the conditional quantile may not be valid.

Height, Weight, and Reporting Errors
Most papers in the literature estimate the relationship between body shapes and family income by replacing body shapes with their observed proxies, such as height or weight. However, these measurements are hardly accurate to fully describe body shapes. Furthermore, including only height or weight without controlling for the other, as in the literature, could create an omitted variable problem. For instance, consider two people who have the same height but different weight. Comparing only height will not identify the difference in their body shapes. Thus, we consider the following two regression July 9, 2021 Supplementary Information 4 equations: where X i is a set of controls, including experience, squared experience, race, occupation, education, marital status, and number of children. We can test the importance of controlling for weight by comparing the estimated coefficients from the two equations. In addition, as mentioned, the data contains measurements on height and weight, both reported by subjects, and measured by on-site measurers. Therefore, by comparing estimates of measured ones with their reported counterparts, we can see the effect of reporting errors on the estimation results.  Table S.5 does not include reported weight. The column for males shows that education is statistically significant in the income equation. The coefficient of the reported height is positive and statistically significant at the 10% significance level. The column for females is somewhat different than that for males: the coefficient of experience, experience 2 , and education are statistically significant. In addition, the coefficient on the reported height is positive and statistically significant at the 5% significance level. In equation (S.4), we add the reported weight to the set of regressors. The column for males shows that the coefficient of the reported height becomes statistically insignificant, and the coefficient of the reported weight is also insignificant. However, in the column for females, the coefficient of the reported height is still positive and statistically significant, but the coefficient on the reported weight is insignificant.
In Table S.6, we use the measured height and weight to estimate the income equation. Interestingly, the coefficients on the height for males in both equations are positive and statistically significant at the 1% significance level. Their magnitudes are larger than those from Table S.5. When the measured weight is added, its coefficient is still insignificant for males. For females, the coefficients on the measured height are statistically significant in both equations, and their magnitudes are larger than those from Table S.5. When the measured weight is added, its coefficient becomes significant at the 10% significance level, which shows a negative association between family income and weight. Thus, we confirm apparent reporting errors in height and weight. Particularly, the impacts of the reporting errors on the estimation results are more severe in males than females. These reporting errors introduce attenuation bias into the estimates. Furthermore, the estimation results from two equations (S.3-S.4) are different. It shows that using height only as a proxy to body shapes might be too simple to describe the delicate figures of the physical appearance. We refer to the main text for more results based on BMI.
To further investigate the role of the measurement errors on the body types, we run the following regression equation: where Body i is a set of body measurements that include 40 measurements of various parts of the body. A full list of the measurements is provided in Table S.3. As these are more sophisticated than simple measurements of height and BMI, it is less likely that the measurement errors on body type are prevalent.  Moreover, interactions between different body measurements are possible, as they have close relationships in constructing a body shape. Therefore, we consider the original regressors (X i and Body i in equation (S.5)) and interaction terms of Body i as a set of regressors. This gives 797 covariates, which makes the OLS regression inconsistent. We note that OLS is consistent under some regularity conditions only if the number of observations is larger than the number of covariates. To mitigate the issue of high-dimensional data, we use the following Lasso regression, which is valid under a sparsity assumption: where Z is a vector of covariates, including the interaction terms with size p = 797 and λ ∈ [0, 1) is a regularization parameter. We construct the lasso fit using 10-fold cross-validation.

Graphical Autoencoder
Mathematically, human body shapes can be represented as curved surfaces, or more formally manifolds M (i) embedded in R 3 , where i is an index identifying each individual. A manifold is a topological space that locally looks like Euclidean space near each point. The statistical models that this study considers are, in a generic form, the regression of an economic variable Y with respect to a manifold-structured regressor M and other covariates X: where φ is a known function up to unknown parameter θ and is an error term. Here, a problem rises regarding the manifold regressor M, as the regressor M is an abstract, geometric object and not a usual vector variable, as in other typical economic and statistical models. In other words, there is no statistical model that naturally accepts the manifold regressor M, unless M is somehow converted into a vector form.
Owing to the above bottleneck, one may consider measuring a few geometric dimensions, such as lengths and girths, and use those measurements to encode body shapes. However, as our study shows, such simplistic measurements are not an accurate characterization of complex geometric objects, such as human body shapes. Instead, data driven parameterizations such as in Wang, [7] Baek and Lee, [4] and Pishchulin et al. [8] provide more comprehensive and reliable codification of body shapes; however, many of these works assume that the human body shape distribution is linear, leading to inaccurate encoding of body shapes [5,6].
In this study, we employ a data-driven, nonlinear parameterization of body shapes achieved via a graphical autoencoder. An autoencoder is a certain type of artificial neural network that possesses an hourglass shaped network architecture. An autoencoder can be thought of as two multilayer perceptron (MLP) models cascaded sequentially, where the first MLP codifies a high-dimensional input into a lower dimensional embedding (encoder) and the second MLP reconstructs the original input back from the encoded embedding (decoder). Because of the dimensional bottleneck created in the middle, the neural network is promoted to search for the most effective way of compressing the high dimensional input into the lower dimensional embedding.
Our proposed concept of a graphical autoencoder is an extension of such notion of autoencoders to manifold-structured data (see Fig 4 for a schematic overview). Similar to many geometric data analysis applications, we discretize a manifold M to a triangular mesh, achieving piece-wise linear approximation of the original surface. A triangular mesh is a graph G = {V, E, F}, where V is a set of vertices/nodes, E are edges interconnecting the vertices, and F are triangular facets. We equip the meshes G 1,··· ,N with a semantic correspondence structure, such that the graph elements of the same index correspond to the same body part location across all meshes G 1,··· ,N . This process is commonly called "registration" or "correspondence matching" in computer graphics and geometry processing literature and can be achieved via methods such as Zuffi and Black [9]; Wei et al. [10]; and Sun et al. [11,12].
In this setting, the graphical autoencoder is defined as follows: Here, each of the layers f 1 · · · f m and g 1 · · · g m are modeled as a simple perceptron: where W i are neural weights and b i are bias. σ is the activation function, where we empirically decide to be rectified linear unit (ReLU) activation for f 1 · · · f m−1 and g 1 · · · g m−1 . We set linear activation for the terminal layers f m and g m (i.e. no rectification). Finally, we train the graphical autoencoder to minimize the mean square error between the original mesh and the reconstructed mesh: where p = f (V ) by definition, θ f and θ g are the model parameters of f and g respectively, and V is the list of vertex coordinates of a graphical model. Note that the mean square error is not an ideal metric for shape dissimilarity. For example, it is widely known that the mean square error is not invariant to posture changes or other isometric transformations (e.g. rotations, parallel translations, etc.), implying that the two models with the exact same body shape but different postures or orientation may be deemed as different under the mean square error metric. However, instead of seeking an alternative metric, we employ the following provisional steps to maintain a reasonable computational load: First, all scan data were normalized to have the same position, orientation, and posture. The centroid of each individual scan was set to (0,0,0). The anterior-posterior (AP) axis and the height (inferior-superior) axis were aligned with the z-axis and the y-axis, respectively. The body postures (joint angles) were adjusted to a standard "A" posture. The skeletal landmarks available in the CAESAR dataset were used for such joint angle alignment. Second, the geometric autoencoder was trained with randomly corrupted inputs, similar to the denoising autoencoders [13]. The random corruption (noise) added to the 3D scan data allows the network to develop more robust features. In a similar spirit, we also added dropout layers [14], with the dropout rate of 30%. Such stochastic perturbations help the autoencoder develop strong and robust shape descriptors. Lastly, we regularized the network weights with the L 2 regularizer to promote generalizable shape descriptors.
Empirically, the above provisions (i.e. data normalization, random corruption, and weight regularization) were effective in overcoming the limitations of the mean squared error. We tested the network through multiple training sessions with different initial network parameters. For each training session, we not only monitored the trend of the loss function but also inspected the graphical rendering of reconstructed 3D models. The training results were consistent both quantitatively and qualitatively from such observations.

Graphical Autoencoder on CAESAR Dataset
The first experiment was conducted to test the ability of the graphical autoencoder in embedding the geometric information underlying in the data. To achieve this, we applied the aforementioned graphical encoder to the entire CAESAR dataset, with varying embedding dimension d from 1 to 20, as reported in Fig S.7. The embedding accuracy was below 3e −4 m 2 in most cases. Particularly, when the dimension d was 3, it showed the lowest MSE, in both training and validation losses, which justifies estimating d = 3 as the intrinsic dimension. As shown in Fig S.8, when the sample was divided into males and females, we found d = 2 for males and d = 3 for females.
For the meaning of the embedded parameters in the third dimension, the first component, P 1 , discerned to be related to height of a person and P 2 to the body volume (obesity/leanness). Interestingly, as P 3 increases the body shape became more feminine, (namely, more prominent chest and hip-to-waist ratio) and, conversely, as it decreases the body shape became more masculine with less prominent chest and curves (see

Endogenous Obesity
One hurdle in estimating the relationship between physical appearance and income is the endogeneity concern associated with the body types, such as obesity and stature. To resolve the issue of endogenous obesity, we use the proxy variables approach, in which observed proxies to the unobserved common determinants of obesity and income control for the possible endogeneity in obesity. A set of the observed proxies includes fitness, car size, and birth state. Fitness is measured as hours of exercise per week. Car size is classified as two groups: Sedan (compact, economy, intermediate, full size, luxury, sports car) and Non-sedan (SUV, minivan, station wagon, truck, van). Birth state is classified as five groups: Foreign, West, Midwest, South, and Northeast. We choose these variables as relevant proxies to the unobserved common determinants of obesity, as fitness and car size could be related to personality and individual preference to body types, and birth state could reflect local and childhood nutrition environments that are correlated with individual obesity. To control for endogeneity in obesity, we assume the conditional independence of obesity and unobserved determinants of family income, conditional on the observed proxies, as commonly used in the literature on treatment effects.
We estimate equation (7) in the main text by controlling for the various subsets of the proxy variables. For comparison, we use measured Height, BMI, and hip-to-waist Ratio in place of P 1 , P 2 , and P 3 , respectively. The estimation results are reported in Table S.11. The estimated coefficients of height are similar to those in Table S.8 for both genders (i.e., columns for equation (4) in the main text). In particular, the coefficient of height is still positive in the equation for both genders. On the other hand, the estimated coefficient of BMI for females becomes negative and significant in the female subsample. Table S.12 reports the estimation results for the main text equation (7) with P 1 , P 2 , and P 3 instead. Notably, fitness and car size are not statistically significant; however, birth state (Northeast) is statistically significant. When all proxy variables are included, the estimated coefficient of P 1 is 0.054 for males. It is statistically significant at the 1% significance level. Thus, taller males tend to have higher family income. However, we do not find a statistically meaningful relationship between male obesity and the family income. For females, the P 2 measurement is negatively associated with the family income, and its coefficient is statistically significant at the 1% significance level. Thus, we find that female obesity matters for the family income in a negative sense; however their stature and hip-to-waist ratio are not associated with family income. The results are qualitatively similar to those from Table S.10, where obesity is assumed to be exogenous.

Endogenous Stature
Here, we conjecture that P 1 or stature would possibly be the endogenous regressor. To address this issue, we look for a set of valid instrumental variables (IV). Our identification strategy is to assume that unobserved determinants of Shoe size, Pants size, and Jacket Size for males (or Blouse Size for females) are uncorrelated with individual cognitive and noncognitive abilities. Given the condition, we adopt these ). Complete circles denote observed variables and dashed circles denote unobserved determinants. Arrows denote direct causal relations. A line without arrows denotes dependence between two variables. The primary parameter of interest is the association between body type and family income. However, body type (stature) is endogenous because of the dependence of body type determinants and cognitive and noncognitive abilities. Shoe size, jacket (blouse) size, and pants size are determined by body type as well as unobserved determinants, such as personal size preference. We assume that such unobserved size determinants are uncorrelated with ability and that they are excluded from family income. Given these conditions, they can serve as legitimate IVs to identify the causal relation. Indeed, it is plausible to assume that these variables are legitimate IVs. First, they are unlikely to be correlated with the unobserved determinants of family income, such as cognitive and noncognitive abilities. For instance, it is natural to assume that highly capable people do not necessarily wear bigger shoes or pants given their foot length or waist circumference. Furthermore, it is unlikely that these IVs directly cause the family income. Second, as shown in the estimation results below, they are also strongly correlated with the body type.
As each size determinants or variations are unobserved, we estimate them from the projection of the observed size on the most relevant body part. In practice, we consider the projection of the reported shoe size, pants size and jacket (or blouse size) on the measured foot length, waist circumference, and chest circumference, respectively. The residuals from each projection are used as the estimated size determinants or variations (analogous to the 'residuals as instruments' in Hausman and Taylor [17]).
We consider the following first-step reduced form equation: where Shoe Size Determinants i is the estimated individual i's variation or determinants in shoe size and Jacket Size/Blouse Size Determinants i is the estimated variation in jacket size for males (or blouse size for females), and Pants Size Determinants i is variation in pants size, and where X i are a set of exogenous regressors and ν i is idiosyncratic shocks to the P 1i . By construction, ν i is the component that generates the endogeneity. From the reduced-from equations (S.11), we estimate the control function ν i . In the second-step, we then estimate the income equation by adding the control function as follows: Asν corrects for the sources of the endogeneity, we can consistently estimate the parameters associated with the physical appearances. Another advantage of the control functions approach is that we can test whether the physical appearances are endogenous by checking if π = 0. Table S.14 reports estimation results for the equations (S.11-S.12). In the columns for the equation (S.11), all IVs are statistically significant and positively correlated with stature in both genders. Experience is also positively associated with individual's stature and the relation is nonlinear. People who were born in foreign countries or the Northeast are likely to be shorter than those born in the Midwest. For males, education is negatively correlated with stature. Asian people are less likely to be taller than White people. For females, Hispanic and Asian people are less likely to be taller than White people.
In the columns for the equation (S.12), the estimated coefficient of P 1 is 0.097 for males, which is larger than that in Table S.12. It is statistically significant at the 5% significance level. Thus, taller males tend to have higher family income. Interestingly, we do not find a statistically significant relationship between male obesity and family income. We estimate that one standard deviation increase in P 1 measurement is associated with $0.097 × 70, 000 = $6, 790 increase in the family income for a male who earns $70, 000 of median family income. This is equivalent to $ 0.097 6.8 × 70, 000 = $998.5 increase in family income per centimeter. Note that for males, one standard deviation in P 1 is equivalent to 6.8 centimeters in height and one standard deviation in P 2 is equivalent to 4.07kg/m 2 in BMI.
The estimation results for the covariates resemble those in previous tables. As shown in the literature on the returns to education, education has a positive impact on family income. Its estimated coefficient is 0.046 and it is statistically significant at the 1% significance level. Males born in the Northeast tend to have higher family income than those born in the Midwest. Interestingly, the estimated coefficient ofν 1 is statistically insignificant. Thus, we find no strong evidence that P 1 is a endogenous regressor.
For females, the estimated coefficient of P 1 is negative and the estimated coefficient of P 3 is positive, but they are not statistically significant. The P 2 measurement is negatively associated with family income. Its coefficient is −0.069, and it is statistically significant at the 1% significance level. Thus, we find that female obesity is negatively correlated with her family income, but her stature or hip-to-waist ratio is not. One standard deviation decrease in P 2 measurement is associated with $0.069 × 70, 000 = $4, 830 increase in family income for a female who earns $70, 000 family income. This can be interpreted as $ 0.069 5.17 × 70, 000 = $934.2 increase per one unit of BMI. Note that for females one standard deviation in P 1 is equivalent to 6.8 centimeters in height and one standard deviation in P 2 is equivalent to 5.17kg/m 2 in BMI.
For females, experience is positively correlated with family income. As commonly reported in the literature on the wage equation, the experience displays a quadratic functional form. Education has a positive impact on family income and its coefficient is statistically significant at the 1% significance level, which is similar to the finding for males. Similarly, females born in the Northeast tend to have higher family income than those born in the Midwest. The estimated coefficient ofν is statistically significant. Thus, we find substantial evidence that stature is endogenous in the female's income equation.
For comparison, we apply the control functions approach to income equations, where height, BMI, and hip-to-waist ratio are used in place of the extracted body features. The corresponding reduced-form and structural equations are the same as the equations (S.11-S.12). Table S.13 reports the estimation results. For males, we estimate that one centimeter increase in height is associated with $0.01 × 70, 000 = $700 increase in family income for a male who earns $70, 000 of median family income. The estimated relationship is smaller than that when the extracted features from the deep learning are used. For females, pants size is not associated with height in the first step regression. The estimated coefficient ofν is significant in the female sample, which implies evidence of endogenous female height. For females, interestingly, coefficients of height, BMI, and hip-to-waist ratio are all insignificant. Thus, we do not find strong evidence of the association between body shape and family income when using height, BMI, and hip-to-waist ratio. Consequently, we observe that the estimation results with the classical measurements are volatile across different regression models-OLS, proxy variable approach, and control functions approach. However, those with the deep-learned body parameters are very stable across different models, and interestingly, capture gender differentials in the impact of body types on income.      Table S.14. The association between body-type parameters and family income -Control function approach.