CAPÍTULO 2 INFORMACIÓN INCOMPLETA E IMPRECISA
2.6 EJEMPLO DE IMPLEMENTACIÓN EN MVL
Given the evidence in Supplementary Table 14 that missing income is endogenous, I employ a second, manual adaptation of an instrumental variables model to test the robustness of my findings. I estimate family income in a first stage regression
Table 6: Instrumental Variable Regression
(1) (2) (3)
Raw Risk Raw Risk Raw Risk
Any Health Shock 540.6 (408.6) 620.7 (637.7) 765.6∗∗ (383.6) Any Health Shock (M) 624.8 (777.8) 706.6 (809.5) 803.4 (790.1) Health -471.2∗ (252.2) -743.2 (1233.8) -484.4∗∗ (230.3) Health t (M) -1556.7 (1395.6) -2411.3 (4093.9) -1487.6 (1354.9) Family Income 3.730∗∗ (1.798) 2.665∗∗ (1.249) 6.095 (4.459) Family Income (M) 1111.8∗∗ (436.2) 1076.9∗∗∗ (415.6) 1578.5∗∗ (787.9) Age -23.06 (39.86) -22.21 (39.20) -11.17 (41.32) Female -13.76 (289.6) 161.1 (302.9) 293.5 (299.5) Education 38.95 (69.13) 68.15 (66.65) 25.38 (74.59) Separated -700.0∗ (370.3) -659.0∗ (372.9) -475.1 (409.3) Widowed -348.7 (1017.3) -249.7 (1029.3) -151.0 (1033.8) Single -1078.1 (826.2) -1037.5 (849.8) -678.8 (942.3) Marital Status (M) -1051.0 (647.0) -1008.1 (659.5) -967.0 (672.4) Risky Med Prob 13.85∗∗∗ (4.885) 14.88∗∗∗ (4.593) 15.29∗∗∗ (4.615) Risky Med Prob (M) 409.8 (786.0) 506.2 (787.9) 615.5 (788.8) Prob Live 20y -1.659 (5.794) 2.141 (11.83) -1.365 (5.487) Prob Live 20y (M) -1523.9 (931.8) -1256.9 (1090.6) -1556.5∗ (900.0) Constant 13794.4∗∗∗ (3010.1) 13745.0∗∗∗ (4131.0) 12489.0∗∗∗ (2847.4)
Observations 1519 1507 1507
R2 0.021 0.025 0.019
Standard errors in parentheses
with local manufacturing jobs and local median income as instruments. The local median income data is from the Census, the year the 13th wave of the survey was sent out (Census 2011). The instruments are jointly significant, with an F-statistic of 49. I use this model to predict family income for all observations in the sample, so there are no missing values. In the second stage, I use these predicted values, and bootstrap the standard errors to correct them. The error terms of the first and second stage regressions are not highly correlated (.0745) and the results are similar to the previous model (see Supplementary Table 15 and column 3 of Table 6), corroborating the evidence that income is exogenous to this model.
Given that the endogeneity tests fail to reject the null hypothesis of exogeneity for both health and income and that the marginal effects of health and income do not substantially change between the OLS and 2SLS regressions, I assume that health and income happen to be exogenous in this model. For the remainder of the paper, I do not use instrumental variables. This result allows me to include nonlinear terms for health and income in the model, which is a key method to test my hypothesis and not possible in a two stage least squares regression.
An OLS regression of Equation 16 is used to test the assumption of constant relative risk aversion with respect to income in the following section and then the effect of a health shock on elicited income risk preference (Theory 2) in Section 8.
7
Eliciting Risk Preferences
To directly compare respondent’s risk preferences to the literature, I elicit relative risk parameters from the premium responses. In order to elicit risk preferences from gamble responses, one has to assume a functional form of the utility function. It is
standard practice to assume a functional form that specifies constant relative risk aversion with respect to income. In the section that follows, I test the strength of that assumption before eliciting risk preferences from gamble responses.
7.1
Testing the Constant Relative Risk Aversion Assump-
tion with Raw Risk Preferences
First, I test the assumption of constant relative risk aversion with respect to income frequently employed in the literature. A respondent’s willingness to pay for health insurance approximates their absolute risk aversion. The marginal effect of income on relative risk aversion can be approximated with non-linear terms. For instance, the effect of a quadratic term for income can be understood as ”the marginal effect of income of risk preferences, relative to one’s level of income”. Given the definition of relative risk aversion from Equation 2 and the constant relative risk aversion specification of the utility function from Equation 3, ∂RRA∂w =α = 0. If the CRRA assumption is true, the marginal effect of non-linear terms for income or health on risk preferences not be statistically significantly different from zero. A significant nonlinear relationship between income and risk preferences is evidence to reject the null hypothesis of constant relative risk aversion with respect to income.
To test this assumption, I estimate respondent’s willingness to pay for health insurance with an ordinary least squares (OLS) regression. Additional higher order terms and interactions are included when theoretically relevant and when significant. The benefit of this parametric model is that the marginal effects of income will be easily interpretable. The drawback is that the relationship between relative risk aversion and income is not specified to be linear. Equation 2 may be
highly nonlinear, and ∂RRA∂w may be non-linear if the CRRA assumption does not hold. With an OLS regression, I can specify nonlinear terms, but the dependent variable is assumed to be a linear function of the covariates. Thus, this model will provide useful evidence as to whether or not the CRRA assumption holds, but will not necessarily provide a correct estimate for the effect of health or income on risk preferences if the marginal effect is nonzero.
Table 7: Constant Relative Risk Aversion Assumption
(1) (2)
Raw Risk Raw Risk Health -450.8∗∗ 15409.5∗∗ (219.2) (7073.5) Family Income 2.605∗∗ 21.74∗∗∗ (1.213) (8.259) Health² -6393.6∗∗ (2745.4) Health³ 788.1∗∗ (331.5) Income x Education -1.067∗∗ (0.434) Observations 1507 1507 R2 0.026 0.052
Standard errors in parentheses
∗p <0.10,∗∗ p <0.05,∗∗∗ p <0.01
Table 7 displays the coefficients on health and income from an OLS regres- sion explaining variation in raw risk preferences. (See the full set of results in Supplementary Table 13). Column 1 shows the results of a linear regression. Col- umn 2 includes theoretically-relevant higher order and interaction terms and those that are statistically significant. There is a linear relationship between raw risk preferences and family income. No higher order terms are significant, nor is a log transformation of income. However, the relationship between income and risk
preferences is heterogenous across individuals with different education: the risk preferences of those that are more educated are less responsive to changes in in- come. With these findings, we fail to reject the null hypothesis of constant relative risk aversion with respect to income. Given this evidence, I procede to use the CRRA assumption to elicit risk preferences in order to have comprable results to the existing literature.
7.2
Eliciting Risk Preferences
In this section, I use the dollar amount that respondents were willing to pay for health insurance (raw risk preferences) and the parameters of the risk preferences question to calculate each individual’s relative income risk aversion, αW.
Assuming that individuals’ utility is well represented by a constant relative risk aversion specification,
U(c) = c
1−ρ
1−ρ, ρ6= 1 (18)
where ρ represents an individual’s risk preferences, then an individual will be willing to pay a premium that satisfies
1∗ (Y −x) 1−ρ 1−ρ ≥psick∗ (Y −L)1−ρ 1−ρ + (1−psick)∗ Y1−ρ 1−ρ. (19)
The maximum premium that an individual will be willing to pay to insure a given risk will satisfy
1∗(Y −x) 1−ρ 1−ρ =psick∗ (Y −L)1−ρ 1−ρ + (1−psick)∗ Y1−ρ 1−ρ. (20)
Given the parameters of the question, respondents faced the problem: 1∗ (100000−x) 1−ρ 1−ρ =.3∗ (60000)1−ρ 1−ρ +.7∗ 1000001−ρ 1−ρ . (21)
We can solve for ρ to get the interval estimates displayed in Table 8. Using the free response data, we can calculate point estimates of ρ. I call these values
elicited risk preferences. Figure 6 visualizes the transformation from raw to elicited risk preferences. The relationship is linear due to the assumed functional form. However, given the assumed CRRA specification of the utility function, Equation 21 can only be approximated when $11,000 ≤ ρ < $15,060. Figure 6 marks in green which observations in the distribution of risk preferences were able to be transformed in to a relative risk parameter. Presumably, past studies have not encountered this issue with the CRRA functional form because they have not allowed such diverse risk preferences.
Table 8: Elicited Risk Preferences: Multiple Choice Maximum Premium Interval
11900 -0.05 ≤ρ <0.00 12000 0.00 ≤ρ <0.02 12050 0.02 ≤ρ <0.07 12150 0.07 ≤ρ <0.14 12300 0.14 ≤ρ <0.24 12500 0.24 ≤ρ <0.47 13000 0.47 ≤ρ
I use point estimates elicted from the raw risk preference variable as the main dependent variable, elicited risk preferences. Summary statistics comparing raw and elicited risk preferences are displayed in Table 9. Positive values of elicited risk preferences indicate risk averse preferences, negative values indicate risk-loving
Figure 6: Comparing Distributions of Raw and Elicited Risk Preferences
preferences, andρ= 0 indicates risk neutral preferences. Table 10 shows a sample of elicited risk aversion estimates from the literature. All studies included assume a CRRA utility function. Our findings fall within a reasonable range.
Table 9: Summary Statistics of Raw and Elicited Risk Preferences
Variable Obs Mean Std. Dev. Min Max
Elicited Risk Preferences 1165 .447 .506 -1.062 1.356 Raw Risk Preferences 1507 12450.52 4895.666 0 40000
Table 10: Elicited Risk Preferences in the Literature
Citation Model Interval
Kimball et al. (2008) 0 < ρ < 3.27 Dave et al. (2010) Eckel-Grossman 0 < ρ < 3.46 Dave et al. (2010) Holt-Laury -1.71 < ρ < 1.37 Harrison and Ng (2016) .3 < ρ < 0.70
Phelps (2019) .8 < ρ < 3.00
A potential limitation to this method of eliciting risk preferences is that indi- viduals may have answered the risk preference questions relative to their current income instead of relative to the hypothetical income of $100,000. Theoretically, we would expect risk aversion to decrease with income, given a concave speci-
fication of the utility function. However, an intital OLS regression on raw risk preferences (see Table 6 Column 1) indicates a significant, positive coefficient on income. Future studies may use individual’s incomes in the elicitation of relative risk parameters to test the robustness of their results.
The elicted risk parameters can be used to model respondent’s utility functions. Given our assumption of a CRRA specification of the utility function and the assumption of the basic insurance model that individuals pay for health insurance to insure against an income loss, the elicited risk values give the curvature of the individual’s utility function with respect to income. Figure 7 shows the most risk- averse (light color) and most risk-loving (dark color) respondent from the multiple choice question (blue) and free response question (red).
Figure 7: Range of Utility Functions
12
16 on elicited risk preferences using the methods employed in Section 6.1. The results suggest that health and income are exogenous to the model, so I do not use instrumental variables. See Supplementary Tables 16 - 18 for detailed results. Note that for estimation of elicited risk preferences, instrumenting income does not make the effect of a health shock and health significant.
Note that, due to the limited range of raw risk values for which I was able to calculate a relative risk parameter, the estimation sample is further reduced to 1,165 observations. However, the sample is not statistically significantly distinct from the raw risk preferences estimation sample. (See Supplementary Table 20 for detailed summary statistics.)
8
Results
With elicited risk preferences, I can directly estimate the relationship between health and relative risk aversion. Table 11 column 1 shows the results of an OLS regression of Equation 16 with only linear terms. Column 2 shows the results of an OLS regression with statistically significant higher order terms and interactions included. The table displays the coefficients on each term, and here I report the marginal effects. Experiencing a health shock does not have a statistically signifi- cant effect on the risk preferences of all respondents: the effect depends on one’s education. Figure 8 shows that the mean effect of a health shock is only statisti- cally significantly different from zero for those that have attended graduate school (22% of the sample). For those with a graduate degree, experiencing a health shock increases their relative risk aversion by .15 (p=.026), or 30% of a standard deviation. These results provide some evidence that income risk preferences may
be health-state dependent, and that income moderates this effect. However, the regression explains only 3% of the variation in risk preferences, which suggests that health is not highly relevant for explaining risk preferences.
Table 11: Health State Dependence of Income Risk Preferences
(1) (2)
Elicited Risk Elicited Risk Any Health Shock 0.0525 (0.0394) -0.445 (0.302) Any Health Shock (M) 0.0103 (0.0893) 0.00976 (0.0885) Age -0.0158∗∗∗ (0.00519) -0.0149∗∗∗ (0.00517) Female -0.0237 (0.0313) -0.0143 (0.0314) Education 0.0119∗ (0.00684) 0.00824 (0.00740) Family Income 0.000203 (0.000134) 0.000178 (0.000133) Family Income (M) -0.0192 (0.0500) -0.0301 (0.0504) Separated -0.0184 (0.0484) -0.0147 (0.0477) Widowed -0.0859 (0.111) -0.0976 (0.112) Single 0.151 (0.105) 0.156 (0.105) Marital Status (M) -0.0210 (0.0950) -0.0112 (0.0950) Risky Med Prob -0.000558 (0.000515) 0.00417∗∗ (0.00186) Risky Med Prob (M) 0.0858 (0.126) 0.150 (0.130) Prob Live 20y -0.000440 (0.000623) -0.000450 (0.000620) Prob Live 20y (M) -0.298∗∗∗ (0.103) -0.271∗∗∗ (0.104)
Risky Med Prob² -0.0000494∗∗∗ (0.0000179)
Health Shock x Education 0.0283∗ (0.0170)
Constant 1.294∗∗∗ (0.350) 1.229∗∗∗ (0.354)
Observations 1165 1165
R2 0.025 0.032
Standard errors in parentheses
∗ p <0.10,∗∗ p <0.05,∗∗∗ p <0.01
Age and time preferences are statistically significant predictors of risk prefer- ences. Older individuals are more risk-loving.13 Figure 9 shows that, on average,
the oldest respondents are only slightly risk averse, while the youngest respondents are most risk averse. This may be due to subjective life expectations changing with
Figure 8: Marginal Effect of a Health Shock, by Years of Education
age. Time preferences have a quadratic relationship with risk preferences, though the curvature is slight. Figure 9 shows that those who value the future the most and those who discount the future the most are more risk-loving. All of these relationships indicate that risk aversion decreases at the end of one’s life, probably due to changes in expectations for the future. A future study may employ joint estimation to illustrate the relationships between time preferences, expectations, and risk preferences.
8.1
Robustness to Health Measurement
In this section, I examine the sensitivity of the primary result, the effect of a health shock on risk preferences, to measurement of health shocks. I estimate elicited risk preferences using the onset of a severe condition, the health shock reported in the traumatic events questionaire, and the decline in self-reported health status. (In previous estimations, I use a composite measure.) Using each health shock measure, I re-estimate the model and allow for higher order terms and interactions, as described earlier in Section 8. No interactions with any measures of health shocks are significant. Table 12 shows all coefficients on health shocks, health, income, and on coefficients that were statistically significant in Table 11, for brevity.
Alternative health measures have different effects. None of the alternative health measures have a statistically significant effect on elicited risk preferences. The sign of the marginal effect differs for men (though not for women): the onset of a severe condition is associated with higher risk aversion, while a self-reported health shock and decline in self-reported health status is associated with risk-loving preferences. Decker and Schmitz (2016) also find that the effect of a health shock on risk preferences is not robust to measurement, and conclude that this is due to the complexity of health: each measurement only captures one aspect of an individual’s health.
Table 12: Robustness to Health Measures
(1) (2) (3)
Elicited Risk Elicited Risk Elicited Risk Condition Onset 0.0766
(0.0544)
Reported Health Shock -0.0137
(0.0778) Drop in SRHS -0.00143 (0.0769) Health -0.0299 -0.0320 -0.0319 (0.0254) (0.0258) (0.0277) Age -0.0150∗∗∗ -0.0152∗∗∗ -0.0152∗∗∗ (0.00517) (0.00520) (0.00518) Education 0.0134∗ 0.0143∗∗ 0.0136∗∗ (0.00690) (0.00687) (0.00691) Family Income 0.000185 0.000185 0.000189 (0.000134) (0.000132) (0.000133) Separated -0.0123 -0.0126 -0.0125 (0.0476) (0.0480) (0.0479) Widowed -0.0901 -0.0862 -0.0854 (0.112) (0.110) (0.110) Single 0.157 0.155 0.156 (0.105) (0.106) (0.106) Risky Med Prob 0.00398∗∗ 0.00413∗∗ 0.00409∗∗
(0.00187) (0.00187) (0.00187) Risky Med Prob² -0.0000482∗∗∗ -0.0000490∗∗∗ -0.0000491∗∗∗
(0.0000180) (0.0000180) (0.0000180) Health Shock x Female -0.161∗
(0.0926)
Observations 1165 1165 1165
R2 0.033 0.032 0.030
Standard errors in parentheses
9
Conclusion
In this paper, I developed two potential theoretical relationships between health and risk preferences. Though I could not empirically evaluateTheory 1, I find evi- dence in the literature that individuals have risk preferences with regard to health that are distinct from their risk preferences with regard to income. Additional research is needed to develop the implications of this finding for models of demand for health insurance.
I evaluated Theory 2 with an ordinary least squares model, and found that income risk preferences are health-state dependent for individuals with a graduate degree. A heterogenous effect across education is not surprising given the persistent strong relationship between health and education (Grossman 2015). However, this direction of the effect is not consistent with Dohmen et al. (2010), which finds that individuals with higher cognitive abilities are less risk-averse. Given the assumption that health shocks are assigned randomly, conditional on gender, and that income and health status are exogenous, this effect can be interpreted as causal. In all models, individuals that experience a health shock are more risk averse. The direction of the health-risk preference relationship is consistent with Sahm (2012), Decker and Schmitz (2016), and Kettlewell (2019). Together, these studies provide conclusive evidence that health has an inverse relationship with risk aversion.
However, there is not strong evidence that health is highly relevant to relative risk aversion. The marginal effect of a health shock is only 30% of a standard de- viation for one-fifth of the sample, and the models explain only 4% of the variation in risk preferences. This finding is consistent with Sahm (2012)’s conclusion that
health is not relevant to risk preferences.
I find evidence that the relevance of health is sensitive to measurement of risk preferences. The linear effects of a health shock and health status are both significant in the two stage least squares regression of raw risk preferences instru- menting for predicted values of income. In this model, a experiencing a health shock increases risk aversion by 16% of a standard deviation. This size of the effect is remarkably similar to that found by Decker and Schmitz (2016) (11% of a standard deviation). This finding may suggest that the effect found by Decker and Schmitz is not robust to risk reference measurement and does not apply to relative risk aversion. However, Schildberg-H¨orisch (2018) finds that the general willingness measure that Decker and Schmitz used is the best predictor of risky be- haviors. Since the measurement of risk preferences substantially impacts results, future studies may explore eliciting risk preferences with alternative functional forms of individual’s utility functions. I find evidence to support the assumption of constant relative risk aversion with respect to income, but the focus on one functional form without substantial exploration of alternatives may be limiting the literature.
All studies of the health-risk preference relationship thus far have only studied the mean effect of risk preferences. Health may have more complex relationship with risk preferences that is obscured by the assumption of a linear functional form. A more flexible parametric model, such as the conditional density estimator (Gilleskie 2004), would allow the effect of the covariates to vary over the distribu- tion of risk preferences and allow a non-linear relationship between the dependent variable and the covariates, while still permitting inference. Figure 10 displays empirical evidence that the effect of a health shock and health vary over the dis-
Figure 10: Evidence of a Nonlinear Health-Risk Preference Relationship
trubution of risk preferences. There is also a theoretical reason to think that health has a nonlinear relationship with risk preferences: relative risk aversion, as defined in Equation 2, is unlikely to have a linear functional form.
10
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