Using Survey-Based
Risk Tolerance
Miles Kimball, Claudia Sahm and
Matthew Shapiro
Global Theme
• Theoretical concepts are often measured
imperfectly.
• When using an imperfect measure, it is essential
to correct for measurement error. Researchers
usually fail to do this and make improper
inferences.
• Example from a different research project:
controlling for past religiosity when looking to see
whether college major affect religiosity. It leads
to misleading results (exaggerating the effect of
college major) to just add the best available
measure of past religiosity to OLS regression.
Risk Aversion and Risk Tolerance
• Think in terms of a von NeumannMorgenstern expected utility function
E (1-γ)-1 W1-γ
where W is wealth, γ is (constant) relative
risk aversion (RRA) and its reciprocal
θ=γ-1 is relative risk tolerance (RRT).
• Sometimes it is most convenient to think in
terms of RRA γ and sometimes it is most
convenient to think in terms of RRT θ.
The Need for a Measure of
Risk Preferences
• Unobserved heterogeneity in preferences
can lead to biased coefficients, destroying
identification.
• E.g., think of regressing, with constant δ
E Δ C = a + b r + (1+δ) E (Δ C)2/2 + e
(Dynan JPE 1993) when the true equation is
E Δ C = a + b r + (1+γ) E (Δ C)2/2 + ε
With γ negatively correlated with E (ΔC)2/2.
The Bias
• e = ε + [γ-δ] E (Δ C)2.
• Since relative risk aversion γ is negatively
correlated with E (Δ C)2/2, e will be
negatively correlated with E (Δ C)2/2.
• Therefore, the estimate of δ in the first
equation will be biased downward.
• Indeed, Dynan (1993) found an estimate
implying a zero coefficient for E (Δ C)2/2.
Evidence that the Bias is Large
• With no direct control for risk preferences,
“Precautionary Saving and SelfSelection—Evidence from the German
Reunification ‘Experiment’” by Matthias
Schuendeln and Nicola Fuchs-Schuendeln
QJE 2005 finds that former East Germans
–whose risk preferences would have had
less effect on their occupations--exhibited
a strong precautionary saving effect, while
those who grew up in W. Germany did not.
Controlling for Risk Preferences
• One of the big motivations for measuring
RRT is to “control for risk tolerance” in
estimation.
• Clearly, one could control for RRT if one
had a perfect measure of θ. Not a likely
event!
• We show how to control for risk tolerance
and obtain consistent estimates even
with an imperfect measure.
We also show how to estimate:
• average risk tolerance
• the difference in average risk tolerance
between groups.
• the overall variance in risk tolerance
across the population.
• the effects of risk tolerance on behavior.
Obstacles to be Confronted
• Data that only give a range for risk
tolerance.
• Response error.
• Data that limit the number of respondents
with multiple observations on risk
tolerance.
-HRS Wave I 100% sample
-HRS Wave II 20% sample
-PSID: one observation per respondent.
The HRS Wave I and II Risk
Tolerance Question
• “Suppose that you are the only income
earner in the family, and you have a good
job guaranteed to give you your current
(family) income every year for life. You are
given the opportunity to take a new and
equally good job, with a 50-50 chance it
will double your (family) income and a 5050 chance it will cut your (family) income
by a third. Would you take the new job?
Follow-up Questions
• If the respondent would take the new job:
“Suppose the chances were 50-50 that it would
double your “family” income and 50-50 that it
would cut it in half. Would you still take the new
job?
• If the respondent would not take the new job:
“Suppose the chances were 50-50 that it would
double your (family) income and 50-50 that it
would cut it by 20 percent? Would you then take
the new job?
Advantages of a Quantitative
Measure
• Makes it possible to construct a sensible
one-dimensional cardinal proxy.
• Linked to economic theory:
Sharerisky = θ E[Retrisky-r]/Var(Retrisky).
• Enables us to be serious about
measurement error.
Contrast with the Qualitative
Questions on the SCF
“Which of the statements comes closest to the
amount of financial risk that you and your
(spouse/partner) are willing to take when you
save or make investments?”
1. take substantial financial risks expecting to
earn substantial returns.
2. take above financial risks expecting to earn
above average returns.
3. take average financial risks expecting to earn
average returns.
4. not willing to take any financial risk.
Problems with the SCF Questions
1. Because they are not quantitative, it is
hard to know how they map into the
economic theory.
2. Different respondents may interpret
words like “substantial” and “not willing to
take any” differently.
3. It is clear these questions have
measurement error, but it is not easy to
tell how much.
A More Subtle Problem with the
SCF Risk Preference Questions
4. They are awfully close to asking people what
kind of portfolio they actually have.
• Thus, there is a serious danger the answers
could be influenced by the respondent’s actual
portfolio, including those aspects of the actual
portfolio that are a historical accident rather
than a function of risk preferences.
• Formally, the measurement error for these
questions may be correlated with the behavior
we want to explain.
Constructing a Cardinal Proxy for
Risk Tolerance
• Find the range of risk tolerance values
consistent with a set of responses in the
absence of response error.
• Use Maximum Likelihood Estimation (MLE) to
estimate a lognormal distribution for risk
tolerance in the absence of response error to
develop intuition.
• Use repeated observations to estimate the
variance of response error and to adjust other
estimates for response error.
• Construct a cardinal proxy as E(θ|c) where c is
the answers to the hypothetical questions above.
Bounding Risk Tolerance in the
Absence of Response Error
• Cutpoint for “double or cut by a third”:
.5[2W]1-γ/(1-γ) +.5 [.667W]1-γ/(1-γ)=W1-γ/(1-γ)
• Dividing through by W1-γ/(1-γ):
.5 [21-γ + .6671-γ] = 1.
• γ=2 solves the equation: .5 [.5 + 1.5] = 1.
• Therefore, in the absence of response
error, taking the risky job means γ≤2 and
θ≥.5; staying safe means γ≥2 and θ ≤.5.
Measurement Error
• High cognitive demands lead to survey
response error.
• The fact that we only have ranges
generates additional measurement error.
• Uncorrected, this leads to errors-invariables problems
– coefficient of RRT likely to be too small
– coefficients of other variables are biased as
they pick up some of the effects of RRT.
Modeling Response Error
• x = ln(θ)~N(μ,σx2)
• xw= x + εw ~ N(μ,σx2 + σε2) ~ N(μ,σ2)
• εw ~N(μ,σε2) is the transitory response error
– transitory fluctuation in perception of own RT
– transitory error in calculation of the bounds
• ε is assumed not to affect real-life
decisions.
– This assumption (made below) could be contested.
– It would matter if exp(x+ε) (instead of exp(x)=θ) governs
behavior, since even if an entirely different realization of ε
governs actual behavior, the effect of ε on exp(x+ε) is nonlinear
and raises exp(x+ε) on average.
Probabilities of Falling into a
Category Given Response Error
Likelihood Function
E(θ|c)
Advantages of the Cardinal Proxy
h=E(θ|c)
• One-dimensional, quantitative measure that
flexibly summarizes the details of the information
we have about risk tolerance
– ranges
– response error
– multiple observations for some respondents,
single observations for others
• Univariate OLS regressions with the cardinal
proxy as an independent variable are unbiased.
Limitations of the Cardinal Proxy
h=E(θ|c)
• Captures only about 20% of the total
variance of underlying risk tolerance.
λ = Var(θ)/Var(h) ≈ 5
• OLS regressions of h on demographic
variables underestimate group differences
by roughly this factor of 5.
• Multivariate OLS regressions with h as an
independent variable are biased and
underestimate the contribution of θ to R2.
Because λ is known, we can still
get consistent estimates.
• People often talk as if having an
imperfect measure inevitably
leads to inconsistent estimates.
• But it is possible to correct for
measurement error, if it has
known characteristics.
A Nonstandard
Errors-in-Variables Problem
• θ = h+u = E(θ|c)+u
• h┴u
[E(u|h)=0]
• Contrast with classical errors-in-variables
problem:
– Note that h = θ – u.
– In a classical errors-in-variables problem
θ ┴ -u, or equivalently θ ┴ u .
– But in this case Cov(θ,u) = Var(u) > 0.
The Underlying Structural Model
•
•
•
•
y = θδθ + zδz + ν (everything demeaned)
E(v|θ,z,ε) = 0
y = risky asset share
z = sex, education, age, race, log(income),
log(net worth)
• Note the assumption that the vector of
response errors ε is ignorable (redundant)
in the underlying structural model.
Substituting h for θ: z is correlated
with the unobserved part of RRT
• y = hδθ + zδz + η
• η = (θ-h)δθ + v = uδθ + v
• E(η|h) = 0
• But E(η|z) = δθ E[(θ-h)|z] ≠ 0
Assumptions Behind the
Measurement Error Correction
1. z = θβ + ζ
• z is linear in its relationship to true risk
tolerance
• more generally, z could be a linear
combination of a small set of specific
functions of risk tolerance (< # categories)
2. E(ζ|θ,ε)=0
• ζ is uncorrelated to the response error
• E(ζ|h)= E(ζ|f(θ,ε))= 0
Adjusting Covariances for
Measurement Error
• Since E(ζ|θ)=0 and E(ζ|h)=0, β in the
equation z = θβ + ζ is both the population
OLS estimate and the population IV
estimate using h as an instrument:
β = Cov(θ,z)/Var(θ) = Cov(h,z)/Cov(h,θ)
• Cov(θ,z)=[Var(θ)/Cov(h,θ)] Cov(h,z)
• Var(θ) = λ Var(h)
• Cov(h,θ) = Cov(h,h+u) = Var(h)
• Cov(θ,z)= λ Cov(h,z)
(λ≈5)
Method of Moments
• E(hη)=0
• E(z’ω)=0
• η = y - hδθ –zδz
• ω = y - λhδθ –zδz
Implied R2 if θ were observed
Persistent response error:
xw=x+κ+εw; τ=σx2/[σx2+σκ2]=.5
Looking for the Effect of z on θ
• Suppose if we had a perfect measure of θ,
we would want to know the OLS estimate
of b in the equation θ = z b + ξ.
• If h is the closest we can get to θ, one
might be tempted to estimate h = z d + υ
using OLS.
• Under the assumptions above,
b = Cov(z,θ)/Var(z)
= λ Cov(z,h)/Var(z) =λd
Is there really that big a difference
between men and women in RRT?
Men
μ
σx
σε
-1.971
(.039)
1.007
(.119)
1.517
(.080)
Women
-1.948
(.032)
1.047
(.079)
1.292
(.063)
Results of the Unrestricted MLE
•
•
•
Men and women look alike in the mean
and variance of their true risk tolerance.
However, the difference in σε is
significant (t=2.2). Women have less
response error. (They may answer the
questions more carefully than men.)
The higher response error generates
more apparently risk-tolerant responses
for men.
What went wrong in Table 13?
•
•
•
We assumed z unrelated to ε.
(E(ζ|θ,ε)=0, where z = θβ + ζ)
This assumption is violated: If men have
larger response errors, ε2 is correlated
with being male.
This variation can be handled by
1. Defining h=E(θ|c,s), where s= sex.
2. Giving a special role to s in the second-step
corrections.
What if a separate transitory
error ψ affects behavior?
• Suppose y = (ex+ψ)δθ + zδz + ν
• ψ┴ε since ψ is a different transitory error
• Then the appropriate cardinal proxy is
h*=E(ex+ψ|c) =Eeψ E(ex|c)=ΨE(θ|c)=Ψh,
a constant Ψ= Eeψ times h.
• Also, use λ* in place of λ, where
λ*= λ[exp(σx2+σψ2)-1]/ [exp(σx2)-1].
Otherwise, the procedure is the same.
What if a permanent error (distinct
from κ) affects behavior?
• Note that we implicitly assumed that κ did not
affect behavior.
• If there is a piece of the permanent error that
does affect behavior, it is observationally
indistinguishable from any other component of θ.
• In other words, if it acts like a component of true
risk aversion in all respects, then it might as well
be a part of risk aversion.
• This logic makes the effective permanent error
variance smaller since it is limited to the
permanent errors that affect only survey
responses and not real-life behavior.
Controlling for Risk Preferences
• One of the big motivations for measuring
RRT is to “control for risk tolerance” in
estimation.
• Clearly, one could control for RRT if one
had a perfect measure of θ. Not a likely
event!
• We show how to control for risk tolerance
and obtain consistent estimates even
with an imperfect measure.
Global Theme
• Theoretical concepts are often measured
imperfectly.
• When using an imperfect measure, it is
essential to correct for measurement error.
Researchers usually fail to do this and
make improper inferences. But it is often
possible to correct for these problems by
taking enough care.