Does moderate weight loss affect subjective health perception in obese individuals? Evidence from field experimental data

The data used in the present analysis originate from a field experiment that was conducted by RWI—Leibniz-Institut für Wirtschaftsforschung. Its prime objective was to test whether monetary incentives are an effective instrument for assisting obese individuals in losing body weight. Four medical rehabilitation clinics operated by the German Pension Insurance of the federal state of Baden–Württemberg and the association of pharmacists of Baden–Württemberg cooperated with RWI in this project. The Pakt für Forschung und Innovation, which is part of the excellence in research initiative of the German federal government, provided funding. The study protocol of the project was approved by the ethics commission of the Chamber of Medical Doctors of Baden–Württemberg. See Augurzky et al. (2018) and Augurzky et al. (2014) for a more detailed discussion of the project.

Upon admission to one of the four involved clinics, 695⁴ obese individuals were recruited for participation in the experiment between March 2011 and August 2012. The medical staff in charge was advised to approach any new patient whose BMI exceeded 30⁵ and to invite him or her to take part in the experiment. Yet, participation was entirely voluntary and had no consequence for any treatment or advice the patient received over their rehab stay, which usually takes 3  weeks. The prime objective of rehab stays in these clinics is to preserve, or to restore, patients’ workableness. Our study population is hence biased toward the working population, which is, however, no challenge to the internal validity of our analysis. For the vast majority of participants, obesity was not the prime reason for being sent to rehabilitation. Yet, many suffered from health problems related to overweight such as chronical back pain. Hence, all obese patients, irrespective of participation in the experiment, were advised to reduce their body weight.

At rehab discharge, participants’ body weight was measured again and participants were set an individual weight-loss target by the physician in charge, which they were prompted to realize within 4  months. Physicians were asked to choose a weight-loss target of about 6–8% of current body weight. Yet they were in principal free to deviate from this guideline. Near the end of the rehab, the participants received a questionnaire, which they were prompted to answer. The questionnaire covered a wide range of questions regarding socio-economic characteristics and weight-related behavior, such as exercising or eating habits. Most importantly, participants were also asked about their current health status. Two health questions addressed self-rated health and physical well-being, in a standard fashion. The questionnaire was collected (in a sealed envelope) at the appointment with the physician in which the weight-loss target was fixed.

Right after rehab discharge, the participants were randomly assigned to one control and two treatment/incentive groups, and subsequently informed about the result of the randomization by regular mail (intervention). While in this letter all participants were prompted to realize their weight-loss target, treatment group members were informed about the monetary reward they could earn by being successful in losing weight.⁶ For one treatment group, the maximum reward was € 150; for the other, it was € 300. If participants failed to realize at least 50% of the contractual weight loss, they did not receive any money. If they were partially successfully, i.e., they lost more than 50% but less than 100%, they were rewarded proportionally to the degree of target achievement.⁷

By the end of the 4 month weight loss period, all participants received another letter, by which they were prompted to visit a specified pharmacy in a specific week for a weigh-in. Body weight measured in the pharmacy served as basis for the cash out of rewards. Upon attending the weigh-in all participants, irrespective of the experimental group they were assigned to, received an expense allowance of € 25. Each letter was accompanied by a questionnaire, which included the same set of questions as the questionnaire the participants had answered by the end of the rehab stay. In particular, the questions addressing subjective health were exactly the same and made no reference to information the patients had provided earlier.

The experiment included two further phases: a 6-month weight-maintenance phase, which directly followed the weight reduction phase, and a subsequent 12-month follow-up phase. In the weight-maintenance phase, participants who were at least partially successful in meeting their weight-loss target were offered another monetary reward for not exceeding their target weight. In the follow-up phase, participants were not exposed to any monetary incentives for weight loss. In both phases, the weigh-in procedure was the same as for the weight-loss phase. The present analysis only uses information up to the end of the weight reduction phase. The reason for this is that in the weight reduction phase the exogenous source of weight variation, i.e., being member of the control or the treatment arm of the experiment, is clearly random by the design of the experiment. This applies less to the subsequent weight-maintenance phase, since the second randomization was conditional on success in the previous phase.

The econometric analysis rests on information which was collected at rehab discharge and by the end of the weight-loss phase. While the information regarding body weight is complete for the first time of measurement, this does not hold for the second, since roughly one-fourth of the participants did not attend the weigh-in by the end of the weight-loss phase. In consequence, weight-change information is available for only 517 participants. Augurzky et al. (2018) comprehensively discuss the issue of experiment drop-out and its possible implications. Using a battery of different econometric techniques, they find that the results are rather robust to correcting for selective drop-out. Unlike body weight, which was measured in the clinic or the pharmacy, the information regarding self-rated health and physical well-being was collected through a written questionnaire. This renders item non-response an issue, which further reduced the size of the estimation sample to 485 individuals in the self-rated health estimation and 468 in the physical well-being estimation, for which weight and health information is available for either time of measurement.

We employ two variables to measure the outcome subjective health perception: (i) self-rated health (SRH) and (ii) physical well-being (PWB). Self-rated health is measured by asking the respondents “how would you describe your current health status?” and allowing for five possible answers: “excellent”, “good”, “satisfactory”, “poor” and “bad”.⁸ Physical well-being is measured by asking the respondents “how would you describe your current physical well-being?”, allowing for the same five possible answers.

While either variable measures subjective health perception, they potentially capture different aspects of it. PWB emphasizes subjectiveness in health perception even stronger, while SRH leaves more room for objectifying the reported health status. For instance, an obese individual without any health impairments might rate her physical well-being as excellent. At the same time, she is probably aware that her excess weight is a risk for her health. Although feeling healthy she might therefore report a relatively poor SRH, to account for potential health risks.

While any questionnaire the participants were asked to fill in included questions about SRH and PWB, the present empirical analysis focusses on SRH and PWB that was reported by the end of the 4-month weight reduction phase. These variables, denoted as \({\textit{SRH}}_1\) and \({\textit{PWB}}_1\) enter the econometric model at the left-hand side.⁹ The analysis also makes use of self-rated health and physical well-being reported at rehab discharge, i.e., at the outset of the weight reduction phase. As single item measures that do not refer to any objective health indicator but are purely subjective in nature, SRH and PWB are well suited for analyzing self-perceived rather than objectively measured health effects.

Table 1 displays the (joint and marginal) sample distribution of SRH for both considered times of measurement. Not surprisingly—all respondents underwent medical rehabilitation for some reason—the share of individuals who regarded themselves in excellent or good health is smaller than in general population surveys such as the German Socioeconomic Panel (SOEP). Nevertheless, SRH exhibits substantial heterogeneity between individuals. From Table 1, it also becomes obvious that self-rated health considerably varies at the individual level over the observation period.¹⁰ For 54% of the participants, we observe a change in SRH (off-diagonal elements in Table 1), while 46% report the same category of SRH at the beginning and by the end of the weight reduction phase (values highlighted bold in Table 1). 60% of all changes are improvements in SRH (cells above the principal diagonal). Among the participants who reported SRH changes, 81% report a change to an adjacent category. Yet, some rather drastic shifts in SRH, e.g., from ‘excellent’ to ‘poor’ or the other way round, are observed.

The corresponding (joint and marginal) sample distribution of PWB at rehab discharge (\({\textit{PWB}}_0\)) and at the end of the weight-loss phase (\({\textit{PWB}}_1\)) is displayed in Table 2.¹¹ Comparable to SRH, physical well-being exhibits substantial heterogeneity between individuals and varies at the individual level over time. For 60% of the participants, we observe a change in PWB (off-diagonal elements in Table 2), while 40% report the same category of PWB at the beginning and by the end of the weight reduction phase (values highlighted bold in Table 2). 60% of all changes are improvements in PWB (cells above the principal diagonal). Among the participants for which reported PWB changes, 79% report a change to an adjacent category.

Self-rated health and physical well-being are obviously closely related measures and are strongly correlated in the sample. However, as their correlation is far from perfect, the two variables seem to capture different aspects of subjective health perception. Table 3 displays the (joint and marginal) sample distribution of SRH and PWB at the end of the weight-loss phase. Most respondents report the identical answer category for both variables (61%). However, 25% of the respondents reported better SRH, while 14% of the respondents reported better PWB.¹² Only 1% of the respondents deviated by more than two answer categories (bad SRH and excellent PWB).¹³

Body weight, which is the key explanatory variable in the present analysis, is measured in terms of the body mass index.¹⁴ Rather than its level, we consider the absolute change (\({\textit{BMI loss}}\equiv {\textit{BMI}}_0-{\textit{BMI}}_1\)) between rehab discharge and the end of the weight-reduction phase as regressor. By this choice, we emphasize that the focus of the analysis is on the effects of within-individual weight loss rather than between-individual heterogeneity in the level of BMI.¹⁵

The variation of weight change in the sample is quite substantial. 81% of the participants lost weight. Mean weight change is 1.56 BMI units. The median of the weight loss distribution (1.49) is close to the mean. The 95% quantile is 4.71, indicating that a substantial share of participants managed to materially reduce body weight over the 4 month weight-loss phase. Yet, the 5% quantile is \(-1.37\), pointing to substantial weight gain being not a rare phenomenon in the sample; see Fig. 1 for sample distribution of BMI loss. Figure 1 also illustrates that members of the incentive groups were on average clearly more successful in reducing body weight (cf. Augurzky et al. 2018). While the median weight loss is 1.82 BMI units for the former, it is only 0.85 BMI units for the latter. Yet, it also becomes visible that the weight loss variation in the respective group is substantial and exceeds the variation between the groups.

From the first panel of Table 4, one can see that—in a descriptive sense—participants who lost weight are more likely to report good health. Among this group of the participants around 40% reported good or excellent health, while this only holds for around 19% of the participants who gained weight. 38% of the latter reported poor or bad health. In contrast, the corresponding share of participants who lost weight is only 19%. According to a Wilcoxon rank-sum test, the distribution of \({\textit{SRH}}_1\) clearly differs (p-value 0.000) between individuals who lost weight and individuals who did not. The estimated probability for an individual from the former group to be in better health than an individual from the latter is 0.65. These descriptive findings line up with the general pattern of results found in the literature that less body weight is associated with better self-ratings of health. Considering physical well-being instead of self-rated health yields a very similar picture. Again, according to a Wilcoxon rank-sum test, the distribution of \({\textit{PWB}}_1\) clearly differs (p-value 0.000) between individuals who lost weight and individuals who did not.

If the same descriptive analysis is applied to self-rated health measured at the beginning of the weight-loss phase, i.e., to \({\textit{SRH}}_0\) instead of \({\textit{SRH}}_1\), we still find a significant (p-value 0.044), though less distinct, deviation in the distribution of self-rated health. At the one hand, this suggests that weight loss might be endogenous and, in turn, calls for an empirical approach that does not interpret the mere correlation as causal effect. On the other hand, this pattern suggests analyzing the effect of BMI loss on SRH conditionally on its initial level \({\textit{SRH}}_{0}\) in order to account for persistent unobserved heterogeneity and to eliminate variation in the dependent variable that cannot be explained by a change in BMI. For this reason, \({\textit{SRH}}_0\) enters the econometric analysis as control variable.

If the same analysis is applied to physical well-being measured at the beginning of the weight-loss phase (\({\textit{PWB}}_0\)), we do not find a clearly significant difference (p-value 0.159) in the distribution of physical well-being. Yet, the share of respondents who rate their physical well-being at least satisfactory is still higher for respondents who lost weight. Hence, analogously to the regression explaining self-rated health, we control for \({\textit{PWB}}_0\) when our outcome variable is \({\textit{PWB}}_1\) in order to account for persistent unobserved heterogeneity.¹⁶

As another approach to account for unobserved heterogeneity, we also control for initial body mass index \({\textit{BMI}}_0\). Though all participants were obese at the time of recruitment, \({\textit{BMI}}_0\) exhibits pronounced heterogeneity ranging from 28 up to 60.¹⁷ The average of the initial BMI is 37.26, while the median value of 36.03 is somewhat smaller, indicating that distribution of initial BMI is skewed to the right.

Due to the relatively small estimation sample, we abstain from specifying a rich regression model with a large number of controls. As basic socioeconomic characteristics we only control for age and gender.¹⁸

As discussed in Sect. 1, we use exposition to monetary weight loss incentives as instrument for weight change. Though random assignment to the experimental groups is a very strong argument for the instrument being exogenous, direct effects of the group assignment on subjective health might still be a challenge for exogeneity. One such channel is anxiety of not earning the reward because of insufficient weight loss, which may negatively impact on health.¹⁹ We cannot rule out that this channel plays some role. However, this effect should downward bias the estimated effect since only the members of the incentive groups are subject to such adverse effects of the incentives. This possible direct effect should hence not generate a spurious regression result in the IV estimation. Moreover, to dig deeper into this issue, we stratified the analysis of weight loss effects with respect to: (i) the weight-loss target \([\text {kg}]\) (sample split at the median) and (ii) the size of the reward (€ 150 and € 300). One may hypothesize that a more ambitious target and a higher amount of money at stake are more prone to elicit anxiety. Yet, regarding the effect of weight loss, we see no significant differences between the respective groups.²⁰ We take this as indication that possible anxiety of not earning the reward does not generate a major endogeneity problem. Another possible channel is that not weight loss itself, but the measures taken to achieve the reduction in body weight, affect subjective health. Though it is almost impossible to disentangle these two channels, the results of Augurzky et al. (2018), who find a much stronger effect of weight loss incentives on weight loss than on weight reducing activities such as doing sports and health eating, argue in favor of weight loss being the prime channel through which the incentives operate.

Though the experiment involved two treatment groups which were offered incentives of different size, in the regression analysis we use a simple dummy that indicates random assignment to one of the treatment groups. Pooling the treatment groups is in line with the finding of Augurzky et al. (2018) that the size of offered monetary reward proved to be immaterial for realized weight loss. Descriptive statistics for all variables that enter the preferred regression model are provided in Table 5.

In this section, we present and discuss results for the regression models we introduced in the previous section. Columns one and two in Table 6 display the estimation results of the naïve model that does not take possible endogeneity into account.

The results are in line with those of the majority of the related literature. In both specifications we find a statistically significant association between weight change and subjective health perception, where weight loss is positively associated with the inclination to report a better status of subjective health. In terms of magnitude, the estimated coefficient is similar for both measures of subjective health perception.

In quantitative terms the point estimate of 0.15 (self-rated health) translates into an average increase in the probability of rating one’s health ’satisfactory’ or better of 3.7 percentage points if one reduces her or his body weight by one BMI unit. For physical well-being the average marginal effect is of similar magnitude. A reduction of body weight by one BMI unit is associated with an increase in the probability of rating one’s physical well-being ‘satisfactory’ or better by 4.7 percentage points.

Turning to the coefficients of the control variables, \({\textit{BMI}}_{0}\) is not significantly associated with subjective health perception. Yet, not surprisingly, the coefficients estimated for initial subjective health perception (\({\textit{SRH}}_{0}\) and \({\textit{PWB}}_{0}\)) are positive and highly significant, revealing pronounced persistence in subjective health perception, which has already been observed in Tables 1 and 2. The simple ordered probit regressions indicate a gender differential in subjective health perception, with women exhibiting a less favorable subjective health rating both in terms of SRH and PWB. The regression analysis does not yield a significant association between age and subjective health perception²⁴. We also find no significant influence of age on physical well-being.

As discussed above, the results presented in columns one and two of Table 6 might suffer from endogeneity bias regarding the coefficient attached to BMI loss. In this subsection, we discuss estimates that address this issue by the use of an instrumental variable.²⁵ Columns three and four of Table 6 display coefficients for the model that relies on weight variation induced by randomly assigned weight loss incentives for identifying the coefficient of prime interest. Besides the coefficients of the equation of prime interest (upper panel), average marginal effects of BMI loss as well as estimates for the auxiliary equation explaining BMI loss (lower panel) are also displayed.

Starting with the instrumental equation, estimation results indicate that cash incentives have a substantial effect on achieved weight loss, see second panel of Table 6 columns three and four. This result, which has already been established in the literature (e.g., Augurzky et al. 2018; Volpp et al. 2008; John et al. 2011; Cawley and Price 2013; Paloyo et al. 2014), is important for the present analysis, as it points to the experiment generating exogenous variation in BMI that can be used for identification. Indeed, the indicator \({\textit{incentive}}\) proved to be a rather strong instrument for BMI loss. The relevant F-statistic²⁶ is 27.72 and 28.54, respectively,²⁷ which clearly exceed the conventional threshold value of 10 (Stock et al. 2002). Besides this key result regarding instrument relevance, the estimates for the auxiliary equation indicate that those who start with a high initial BMI are more likely to lose weight. Yet it is worth mentioning that including controls in the instrumental variable equation is of minor importance since the randomization balances the covariates between the groups.²⁸ Dropping the controls from the auxiliary equation thus has very little effect on the results.

Turning to the equation of main interest, we see almost no change in the coefficients of the control variables as compared to simple ordered probit. Yet, with respect to the effect of weight change on subjective health perception, we find a pattern of results that in some respect deviates from its counterpart from naïve estimation. The point estimates are smaller and the coefficients of BMI loss turn statistically insignificant. However, the estimated BMI loss coefficients are still positive. For the model explaining self-rated health, the deviation in the point estimates and the corresponding marginal effects is very small. Yet, due to the large standard errors the 95-% confidence interval around the IV-estimates is wide, more precisely \([-0.092, 0.345]\). For the model explaining physical well-being the deviation from the IV estimate and its counterpart from simple ordered probit is more pronounced. But still, the associated confidence interval, which is \([-0.144, 0.287]\), includes the coefficient from the naïve approach. The same line of argument applies to the corresponding marginal effects. The 95-% confidence intervals around the estimated mean effects are \([-0.022,0.085]\) and \([-0.042,0.086]\), respectively. They clearly include the relatively precisely estimated effects from simple ordered probit estimation. Yet, they are that wide that one cannot reject effects which are much smaller (even zero effects and effects to the opposite direction) or much bigger than those one obtains from the naïve estimation approach.

The lack of statistical significance of the estimated effects in the IV approach, thus, seems first of all to be a standard error issue, and can hardly be interpreted as evidence for the absence of a weight loss effect on subjective health. Evidently—although the instrument is not weak—augmenting the naïve model by the instrumental equation inflates the noisiness of the estimates substantially. The reason for this is that the instrument, despite the large F-statistic, still explains a relatively small fraction of the variation in the endogenous variable BMI loss. The partial R-squared is, indeed, just 0.051 despite the relatively large F-value of 27.72; compare Fig. 1.²⁹

The finding that instrumenting BMI loss does not fundamentally change the estimated key coefficients in our main specifications is mirrored by the estimates of the cross-equation error correlation. The estimate is positive but of moderate or even negligible magnitude and statistically insignificant. Though the positive sign argues in favor of unobserved confounders may play some role in the correlation of subjective health perception and BMI, the estimates still provide little evidence for endogeneity of BMI loss being a major issue. Moreover, based on linear specifications (see next subsection) that employ OLS and 2SLS rather than ordered probit and IV ordered probit, Hausman–Wu tests do not yield evidence for any systematic deviation between instrumental variables and naïve estimation.³⁰ One possible reason for this pattern is that relying on short-term, within-individual variation cuts or at least weakens already several channels—for instance, health-conscious attitudes, educational and family background, and certain genetic endowments—that are likely to be major sources of endogeneity in cross-sectional data based analyses.

To sum up this discussion, though the instrumental variables estimation does not yield a clear cut result regarding the effect of weight change on subjective health perception, the pattern of results is still telling. While, the point estimates argue for the naïve empirical approach suffering from some upward bias, the rather noisy IV estimates can hardly put the general finding of weight-gain in obese individuals affecting subjective health perception detrimentally into question. This in particular holds since the IV approach reveals little evidence for the naïve estimates suffering from a severe endogeneity bias. In qualitative terms our results, hence, do not conflict with the bulk of the literature. They also do not contradict those of Cullinan and Gillespie (2016)³¹, who carefully designed their analysis to allow for a causal interpretation of the link between body weight and self-rated health. This appears to be an interesting finding, given that the present analysis exploits a source of variation for identification that is very different from what is used in Cullinan and Gillespie (2016) and that, in consequence, the nature of the estimated effect also differs. While Cullinan and Gillespie (2016) rest on genetics as a persistent and long-term determinant of body weight, the present analysis uses short-term extrinsic incentives. Thus, the result of the former can be interpreted such that permanently reducing the BMI of an obese individual to a normal level will improve her SRH substantially. In contrast, our results—at least in terms of the point estimates—suggest that also a small reduction in body weight will make an obese individual instantaneously feel healthier. This distinction is important for the question of how to motivate obese individuals to lose weight. Even if substantial weight loss is known to pay-off in the long-run for sure, obese individuals may still need some instantaneous improvement in subjectively perceived health in order to keep the discipline to continue their weight loss efforts.

Comparing our results in quantitative terms to those of Cullinan and Gillespie (2016) is not straight forward as their analysis relies on a very different source of weight variation and since they consider a categorical measure of the weight status (healthy weight, overweight, obese grade I, obese grade II) as key explanatory variable, rather than an continuous measure of weight change as we do in our analysis. Nevertheless we use a simple simulation to translate our results into figures that allow for being compared to the results of Cullinan and Gillespie (2016). More specifically, based on the point estimates from the IV model (Table 6, column 3), for each participant we calculate the change in the probability of reporting to be in excellent health—this is the probability Cullinan and Gillespie (2016) focus on when discussing their results—that would occur if she reduced her body weight to a BMI of 25. We hence consider a shift from obesity to a healthy body weight as Cullinan and Gillespie (2016) do. Moreover, following the line of how they present their results, we examine the mean change in this probability separately for men and women and for grade II obese (\(\hbox {BMI} > 35\)) and grade I obese individuals.³² At least for grade I obese individuals our results are surprisingly similar to those of Cullinan and Gillespie (2016). For women we calculate a mean change of 7.0 percentage points while the corresponding value in Cullinan and Gillespie (2016, IV with controls) is 9.1. For grade I obese men our approach yields a mean change of 9.7 percentage points while the corresponding value in Cullinan and Gillespie (2016) is 11.6. For the grade II obese the results are less well aligned, but the overall pattern is still similar. Cullinan and Gillespie (2016) report effects of 17.1 (women) and 20.1 (men) percentage points on the probability to be in excellent health, while we calculate mean effects of 31.7 (women) and 40.0 (men) percentage points, respectively.³³

We ran several robustness checks in order to test how sensitive the results are to changes of the model specification. (i) We estimated all discussed model specifications with additional controls for education, employment, and income measured at rehab discharge; see Tables 12 and 13 in Appendix. This does not change the overall pattern of results. The coefficients of the naïve model are hardly affected. Their counterparts from instrumental variables estimation remain inconclusive, as they do not change consistently in the same direction. For \({\textit{SRH}}\) the point estimate of the BMI loss coefficient gets smaller, while it gets bigger for \({\textit{PWB}}\). (ii) We reduced the number of health categories to just three merging ‘good’ with ‘excellent’ and ‘bad’ with ‘poor’. This just marginally affects the coefficient estimates; see Table 11 first panel in Appendix. As another robustness check, (iii) we excluded individuals with extreme changes in BMI, in order to check for few extraordinary cases possibly driving the empirical results. We considered two definitions of extreme, (\({\textit{BMI loss}} > 5\)) and (\({\textit{BMI loss}} < -2 ~|~{\textit{BMI loss}} > 5\)). For both, the pattern of results remains largely unchanged; see Table 17 and 18 in Appendix.

In order not to rely exclusively on fully parametric model, (iv) we also ran two-stage least squares (2SLS) regression. To avoid interpreting SRH as being measured on an interval scale, we recoded the dependent variable to have just two categories and, in consequence, we estimated linear probability models by 2SLS and as reference also by OLS. Since transforming five-category SRH to a binary health indicator involves the somewhat arbitrary choice of a cutoff category, we tried all four possible variants; see Table 11 second to fifth panel and Figs. 2–7 in Appendix. If the left-hand side variable is specified to indicate one of the two extreme categories (‘better than good’ or ‘bad’) the effect of BMI loss gets very small or even vanishes. Interestingly, this holds for both OLS and 2SLS. One possible explanation is that these categories are rather rare in the sample, hampering the identification of effects on these extreme categories; see Table 1. An alternative, less technical, explanation is that relatively small changes in body weight will rarely be the reason for a shift in subjective health perception to an extreme. If an interior category is chosen as cutoff, the linear model insofar mirrors the results of ordered probit estimation as the naïve estimator yields a significant and favorable effect of weight loss on SRH, while IV does not. For the variant with the left-hand side variable indicating poor or bad SRH, the 2SLS coefficient even turns negative. This does not apply to the variant with an indicator for ‘SRH neither good nor excellent’ serving as dependent variable. There, the 2SLS coefficients are similar or larger than their OLS counterparts.

Since due to drop-out the estimation sample is considerably smaller than the initial 695 recruited participants, non-random sample attrition might bias our results. (v) To address the concern, we estimated model specifications that include a third equation explaining experiment attrition that is jointly estimated with the remaining two. As an ‘instrument’ for attrition this additional equation includes a dummy variable ‘pharmacy in town’ which indicates whether the assigned pharmacy for the weigh-in lies in the same zip-code as the respondent’s place of residence. The results from these specifications suggest, that non-random sample attrition is likely to be no issue; see Table 15 and 16 in Appendix.³⁴

As displayed in Table 5, initial body weight varies substantially in our sample. To check whether this heterogeneity is mirrored by heterogeneity in the effect of weight loss on self-perceived health, (vi) we stratified the analysis by initial BMI. Figure 8 in Appendix displays the respective estimated coefficients³⁵ of BMI loss, which do not reveal a striking pattern of heterogeneity with respect to \({\textit{BMI}}_{0}\).

Since earlier studies found gender differences in the relationship between BMI and SRH (Imai et al. 2008), besides the pooled model, we also conduct separate regression analyses for males and females in the specification of reference as well as in the robustness checks. The naïve regression models yield results that are rather similar to those from the pooled model. However, for the stratified instrumental variable analysis the results deviate more from the naïve estimation compared to the pooled model: In the male sample, the estimated effect of BMI loss on PWB gets smaller, which is similar in the female sample for the effect of BMI loss on SRH. However, neither the cross-equation error correlation in the male nor the cross-equation error correlation in the female sample is significant, which is in line with our main results, stating that endogeneity of BMI loss does not seem to be a major issue. Due to the rather small sample sizes and the instrument becoming relatively weak in these sub-samples the instrumental variable analysis seems not to be overly informative and has to be taken with a grain of salt.