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# Risk Preferences in the PSID: Individual Imputations and Family Covariation

^{}Corresponding author.

Unobserved heterogeneity greatly complicates empirical analysis in economics. Unobserved heterogeneity in preferences is particularly troublesome because there are so few theoretical restrictions on the distribution of preference parameters in the population. Therefore, despite potential pitfalls, we have developed direct survey measures of preference parameters based on hypothetical choices and appropriate econometric techniques for dealing with the inevitable measurement error in any such measures. Our work on survey measures of preference parameters focuses on risk tolerance (Robert B. Barsky, F. Thomas Juster, Kimball, and Shapiro 1997 (BJKS hereafter) and Kimball, Claudia R. Sahm, and Shapiro 2008 (KSS hereafter)), time preference and the elasticity of intertemporal substitution (BJKS), and labor supply elasticities (Kimball and Shapiro 2008).

Risk tolerance is central to portfolio choice and many other economic decisions, such as choices about insurance and career choices. In this paper, we discuss how to go from categorical survey responses to imputed values of preference parameters. The procedure takes into account measurement error from survey response, and has implications for the appropriate use of imputed preference parameters in econometric analysis. We present the risk tolerance imputations for the survey responses in the Panel Study of Income Dynamics (PSID). We also present quantitative evidence on the covariation in risk preferences within families.

## I. Survey Measures of Risk Preferences

Numerous surveys have fielded measures of an individual’s willingness to take risk, including the Health and Retirement Study (HRS), which pioneered the use of hypothetical gambles in a large survey to measure the economic preference parameter of risk tolerance (BJKS 1997). In this paper, we analyze the gambles fielded in the 1996 PSID that ask respondents the following:^{1}

Suppose you had a job that guaranteed you income for life equal to your current, total income. And that job was (your/your family’s) only source of income. Then you are given the opportunity to take a new, and equally good, job with a 50–50 chance that it will double your income and spending power. But there is a 50–50 chance that it will cut your income and spending power by a third. Would you take the new job?

Individuals who answered that they would take this risky job were then asked about a riskier job:

Now, suppose the chances were 50–50 that the new job would double your (family) income, and 50–50 that it would cut it in half. Would you still take the new job?

In contrast, individuals who would not take the initial risky job were asked about a less risky job:

Now, suppose the chances were 50–50 that the new job would double your (family) income, and 50–50 that it would cut it by 20 percent. Then, would you take the new job?

Conditional on their first two responses, individuals are asked to consider a risky job with either a 75 percent downside risk or a 10 percent downside risk. These responses allow us to order individuals into six categories. Unlike the HRS, the PSID asked these questions only of working family respondents and did not ask them of other household members. The targeting of the questions to workers in the PSID particularly affects the selection of the youngest and oldest respondents, so we have limited our analysis to respondents between the ages of 20 and 69.

After collecting over ten years of gamble responses in the HRS and similar questions in surveys like the PSID and the National Longitudinal Survey, a number of lessons on measuring risk preferences have emerged. First, the gamble responses are subject to considerable measurement error. KSS report a rank correlation in the gamble response categories of 0.27 across individuals’ responses in the first two waves of the HRS. In addition, Sahm (2007) shows that much of the transitory variation in the gamble responses remains unexplained, even after including a rich set of individual and household covariates. To address the problem of measurement error, BJKS (1997), KSS (2008), Sahm (2007), and this paper use multiple responses from some individuals to isolate the variance owing to measurement error from the variance in true risk preference.

Second, extraneous details in the description of the gambles can affect the measurement of risk preferences. For example, in the original HRS version and the PSID version of the question, the risky job is described as a new job. This frame has the potential to induce status quo bias in which individuals are averse to taking the new job independent from its income risk. Starting in the 1998 wave, the HRS addressed this potential problem by using a scenario in which the individual has to move for health reasons and is given a choice between two new jobs. This variation in the question wording in the HRS also allows us to estimate the degree of status quo bias in the original version and to correct the estimates of risk tolerance from the PSID. Finally, the interpretation of a job-related gamble may vary across workers who are at different stages of their career. In designing the question, a choice of jobs was used to create a large shock to lifetime resources. The fraction of lifetime income associated with labor income, however, likely declines with age. The job gamble may be particularly hard for retirees and other nonworkers to interpret. The HRS now uses an investment gamble related to an unexpected inheritance for respondents age 65 and older, and gives the job-gamble question only to those under age 65. Similarly, the PSID targeted its job-related question only to workers.

## II. Individual Imputations

The responses to hypothetical gambles in the PSID suggest that most individuals have a low tolerance for risk, though there is substantial heterogeneity. The first column of Table 1 shows that 31 percent of the respondents rejected all of the risky jobs, but almost 7 percent accepted all the risky jobs. An advantage of the hypothetical gambles relative to qualitative measures of risk tolerance is that one can use them to quantify the degree of risk tolerance and its dispersion across individuals. As in BJKS and KSS, we assume that individuals have constant relative risk aversion utility and will reject the risky job when its expected utility is less than that of the safe job. Along with the risks specified in the questions, these assumptions allow us to assign a range for the coefficient of relative risk tolerance to each gamble response category. Previous analysis from the panel of gambles in the HRS suggests that these questions provide a noisy signal of risk tolerance reflecting both status quo bias and classical measurement error. Therefore, we estimate a model of noisy log risk tolerance, *ξ* = log *θ* + *e*, where log *θ* is distributed
$N(\mu ,{\sigma}_{x}^{2})$, and the *e* is classical measurement error distributed
$N(0,{\sigma}_{e}^{2})$. The KSS estimation procedure is an ordered probit with known cutoffs based on which gambles respondents accept or reject.

With only a single response from each PSID respondent, it is not possible to identify separately the variance of true log risk tolerance and the variance of the response error. The PSID responses identify the mean and the total variance of the noisy signal *ξ*. We impose the estimate of the variance of true log tolerance
${\sigma}_{x}^{2}$ from the HRS to divide the PSID total variance into variance of true preferences and variance of error. For the PSID, the estimates of the parameters are *μ* = −1.05,
${\sigma}_{x}^{2}=0.76$, and
${\sigma}_{e}^{2}=1.69$.^{2} Using these distributional parameters, we can impute individual-level estimates of preference parameters based on the conditional expectation of the true parameter given the individuals’ survey responses. Table 1 provides the individual imputations for the PSID. The conditional expectation of each preference parameter is computed using the moment generating function, so, as shown in the last two columns, the reciprocal of imputed risk tolerance is not equal to imputed risk aversion. Researchers interested in studying differences in risky behavior can use these individual imputations as a covariate. The imputations offer advantages relative to categorical controls for gamble responses. First, the imputations summarize the sequence of gamble responses in a single cardinal measure of preferences that can be used to assess the quantitative predictions of behavioral models. Second, our estimation procedure accounts for the measurement error in the survey responses, so the imputations are the conditional expectations of the individual’s true preferences. The use of the imputed values in regression analysis substantially reduces the attenuation bias arising from survey response error when the imputed values of risk tolerance are used as explanatory variables. Nonetheless, these imputations conditioning only on individual’s gamble responses understate the true variation in preferences, so they may not capture all of the relevant differences in risk attitudes across individuals and would only partially control for risk tolerance in OLS estimation.

The application in KSS that uses individual imputations (as in Table 1) to study stock ownership makes these points more concrete. Using categorical controls for the gamble response category or imputations that do not account for response error leads to an attenuation bias that can substantially understate the responsiveness of behavior to risk tolerance. Even with imputations that address response error, standard multivariate estimators may not be consistent due to a nonstandard errors-invariables problem.^{3} The main issue is that the imputations based on gamble responses do not capture all the differences in true risk tolerance. To the extent that other covariates are correlated with the unmeasured part of risk tolerance, they will be correlated with the error term in the OLS regression that includes the imputations. Thus, the estimated coefficients on the other covariates would also include the indirect effects of risk tolerance. To address this issue, KSS provide a consistent GMM estimator using the imputations that scales up the covariance between imputed risk tolerance and other covariates.^{4} As an example of the difference this correction makes, compared to OLS, the estimated difference in stock ownership rates between men and women is 40 percent lower with the GMM estimator and is no longer statistically different from zero at the 5 percent level.

The PSID illustrates how preference parameters can differ according to values of covariates. In particular, there are important differences in measured risk preference by age. For example, 61 percent of the individuals in their sixties reject all of the risky jobs versus only 23 percent of individuals in their twenties. As Table 2 shows, this pattern holds across all six gamble response categories with older individuals more concentrated in lower, less risk tolerant categories. The interpretation of such age effects remains open.^{5} One possibility is that risk tolerance, in terms of the curvature of the utility function, diminishes with age. Alternatively, consumption commitments or habits may increase with age and make individuals less willing to risk a loss in income. Finally, the interpretation of the job-related gamble may simply vary with age in a way that is unrelated to true risk preferences. In any case, when the individual imputations also condition on age, the differences are sizeable. For example, conditioning on age in addition to the gamble response category, a 30-year-old in the least risk tolerant category has an imputed risk tolerance of 0.25, whereas a 50-year-old with the same gamble responses has an imputed risk tolerance of 0.16. One option would be to impute risk tolerance to individuals based on both their gamble response category and their age. This method constrains researchers who want to use the imputation as a covariate in behavioral studies. The specification of age effects in the behavioral model has to match those in the risk tolerance estimation, or a spurious correlation between imputed risk tolerance and the behavior under study could arise. In the application in the next section, we use a rough control for differences in ages.

In summary, Table 1 provides imputed values of risk preference parameters based on responses to a hypothetical gamble about lifetime income in the PSID. The imputations control for survey response error. Neglecting this response error will substantially understate the correlation of survey measures of risk preferences with other variables. KSS show how to use such imputed values in multiple regressions—either by imputing the preference parameters based on multiple covariates or by using a GMM procedure that adjusts for the fact that the imputed values do not capture all the cross-sectional variation in the true preferences.

## III. Family Covariation

We now apply the methodology sketched in Section II to study the covariation in preferences among family members. The PSID has risk preference responses from members of different generations of the same families, and the HRS has responses from both spouses.

We use our maximum-likelihood approach to quantify the covariation in family members’ preferences. Consider the correlation in risk tolerance between a father *f* and his adult child *c*. Because of the differences across age documented in Table 2, we allow the mean and variance of noisy log risk tolerance *ξ* to differ across fathers and children, such that
${\xi}_{f}\sim N({\mu}_{f},{\sigma}_{f}^{2})$ and
${\xi}_{c}\sim N({\mu}_{c},{\sigma}_{c}^{2})$, where, as above, the variances are sums of the variance of the true parameter log *θ* and of the response error *e*.^{6} Because the response errors *e* are uncorrelated across family members, we can estimate
$\text{Cov}({\xi}_{f},{\xi}_{c})={\sigma}_{fc}^{2}$.

The numbers below the diagonal in Table 3 present the variance-covariance matrix of log risk tolerance for various family members (standard errors in parentheses). The numbers above the diagonal (in **bold**) are correlation coefficients. The main diagonal is the variance of true log risk tolerance (from estimates presented in Section II). We find a positive association between parents and their adult children. The correspondence between fathers and their children is relatively weak, though positive. The mother-child covariance is over 60 percent larger than the father-child covariance and is statistically different from zero at the 10 percent level.^{7} The mother-child covariance is over one-fifth of the within-person variance. We do not find a stronger correlation between parents and children of the same gender. The correlations are noteworthy given the fact that parents and children with an average age difference of over 20 years are at very different life stages and in most cases have not resided together for some time. The role of the family in shaping risk preferences is even more apparent in the gamble responses of siblings. Again, each adult sibling in the pair is either the head or spouse in an independent family when answering the gambles. The covariance in risk tolerance among siblings is more than twice the size of the mother-child covariance and is almost 50 percent of the within-person variance. The average age difference between the siblings is only five years, which likely makes their interpretation of the gambles more comparable. Clearly, there are a number of factors that could lead siblings to form similar risk preferences: transmission from common parents, shared experiences within the family, and similar peer and social environments.^{8}

The HRS offers one more dimension of within-family variation, since it poses the gamble to both the husband and wife in a household.^{9} We find a covariance between spouses that is similar to the covariance between siblings and is about 40 percent of the within-person variation. Both assortative mating and common experiences in the marriage could help account for the correlation.

The substantial covariance within families is also important for interpreting the variance of risk tolerance on the main diagonal in Table 3. The estimated variance from the HRS uses the persistent component of individuals’ gamble responses over time to identify risk preference. It is possible that a repeated misinterpretation of the question could lead to persistent measurement error, which then would bias upward the estimated variance of true risk tolerance. The size of the sibling and spousal covariances makes it unlikely that the true variance of risk tolerance on the main diagonal is much smaller. In other words, the size of the sibling and spousal covariances leaves little room for a large variance of persistent idiosyncratic response error. This finding is important because there are few other ways to get a handle on the variance of persistent idiosyncratic response error.

Our results showing a correlation in risk preferences among family members are largely consistent with related studies. Using a subset of the PSID gamble responses in their study on the intergenerational transmission of wealth, Kerwin K. Charles and Erik Hurst (2003) find a strong correspondence between parent and child risk tolerance, particularly at the tails of the distribution. They make sample restrictions that result in a more homogeneous group of parent and child households which leads to a stronger parent-child correlation than we find in the full sample. Nonetheless, the basic finding of inter-generational transmission in risk preferences is similar.^{10} Thomas J. Dohmen et al. (2008) use experimentally validated qualitative measures of willingness to take risk in the German Socioeconomic Panel to also show that parents and children, as well as married couples, have similar attitudes toward risk.

## IV. Conclusion

In this paper, we apply a survey-based method for imputing individual risk preferences to respondents in the PSID. These procedures draw on estimates and previous lessons from analysis of the HRS gamble responses. We provide individual estimates of risk preferences based on the gamble response categories that can be used in other behavioral studies—both to study the effects of risk tolerance and to control for risk tolerance when looking at other effects. We use the gamble responses to document a substantial covariance in risk preferences among family members. In addition to its intrinsic interest, this covariance in risk preferences across family members helps validate these risk tolerance measures by putting an upper bound on the variance of idiosyncratic response error.

## Acknowledgments

Kimball and Shapiro gratefully acknowledge the support of National Institute on Aging grant 2-P01-AG10179.

## Footnotes

^{1}This question was included in the 1996 PSID (both original respondents and offsprings in split-off households).

^{2}The PSID responses are also adjusted by −0.21 for status quo bias (again using the parameter estimated in the HRS). See the online Appendix, available at http://www.aeaweb.org/articles.php?doi=10.1257/aer.99.2.363, for details of the estimation and imputation procedures.

^{3}In univariate analysis, the estimated effect of risk tolerance on the behavior of interest would be consistent, but the *R*^{2} would be underestimated.

^{4}The scaling factor is the variance of true risk tolerance divided by the variance of imputed risk tolerance. This true-to-proxy variance ratio is 6.3 in the HRS and 4.6 in the PSID.

^{5}With a cross section of responses, the distribution of gamble responses by age may also incorporate differences in risk tolerance across birth cohorts. Ulrike Malmendier and Stefan Nagel (2008) find an association between individuals’ current willingness to take financial risks and the path of aggregate stock market returns experienced over their lifetimes.

^{6}An alternative parameterization would be to allow the differences of means of log risk tolerance *μ* to be functions of the differences in ages in the family pairs.

^{7}The standard deviation of the common mother-child component in log risk tolerance is statistically different from zero at the 5 percent level. (The 95 percent confidence interval of the standard deviation and the 95 percent confidence interval of the variance cover nonequivalent regions of the parameter space.)

^{8}Looking for some evidence on these factors, we tested for a difference in the covariance for siblings who share both parents as opposed to those who share only one parent. The difference was statistically insignificant.

^{9}One complication is that a spouse is sometimes present during the HRS interview, which might bias an individual’s response and lead to a spurious correlation in gamble responses. We limit our analysis to pairs of responses that were given in separate interviews in 1992. This may understate the true correlation if spouses with similar preferences choose to be together during interviews more than those with dissimilar preferences.

^{10}Charles and Hurst (2003) use a different method for assessing the covariation in preferences across parents and children. Applying our maximum likelihood procedure to their restricted sample of parent-child pairs yields a covariance of 0.25 (standard error of 0.12). The point estimate from their parent-child sample is higher than our parent-child covariance estimate of 0.15 (standard error of 0.08), though the difference is not statistically significant.

The views presented are solely those of the authors and do not necessarily represent those of the Federal Reserve Board or its staff.

## Contributor Information

Miles S. Kimball, Department of Economics, University of Michigan, Ann Arbor, MI 48109–1220.

Claudia R. Sahm, Federal Reserve Board, 20th Street and Constitution Ave., NW, Washington, DC 20551.

Matthew D. Shapiro, Department of Economics, University of Michigan, Ann Arbor, MI 48109–1220.

## References

- Barsky Robert B, Juster F Thomas, Kimball Miles S, Shapiro Matthew D. Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Study. Quarterly Journal of Economics. 1997;112(2):537–79.
- Charles Kerwin K, Hurst Erik. Intergenerational Wealth Correlations. Journal of Political Economy. 2003;111(6):1155–82.
- Dohmen Thomas J, Falk Armin, Huffman David, Sunde Uwe. The Intergenerational Transmission of Risk and Trust Attitudes. IZA Discussion Paper 2380 2008
- Kimball Miles S, Sahm Claudia R, Shapiro Matthew D. Imputing Risk Tolerance from Survey Responses. Journal of the American Statistical Association. 2008;103(483):1028–38. [PMC free article] [PubMed]
- Kimball Miles S, Shapiro Matthew D. Labor Supply: Are Income and Substitution Effects Both Large or Both Small? National Bureau of Economic Research Working Paper 14208 2008
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- Sahm Claudia R. Board of Governors of the Federal Reserve System Finance and Economics Discussion Series 2007–66. 2007. How Much Does Risk Tolerance Change?

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- Risk Preferences in the PSID: Individual Imputations and Family CovariationRisk Preferences in the PSID: Individual Imputations and Family CovariationNIHPA Author Manuscripts. May 2009; 99(2)363

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