Department of Policy and Planning Sciences
Discussion Paper Series

No.1329

Model Uncertainty and International Differences in Risk
Aversionby
Masakatsu OKUBO

May 2015

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANModel Uncertainty and International Differences in Risk Aversion ∗Masakatsu Okubo †University of Tsukuba

Abstract
Existing studies have revealed both remarkably high levels of risk aversion and significant variation in risk aversion across countries. The main purpose of this paper is to investigate whether
Hansen and Sargent’s multiplier preferences can explain both these findings. We show that large
variation in risk aversion can arise under moderate levels of detection error probabilities. We
also present evidence suggesting that differences in cultural traits may partially explain heterogeneity in agent concerns about model misspecification across countries.

Keywords: Risk aversion; Model misspecification; Detection error probability; Ambiguity aversion; Cultural traits
JEL Classification: D81; E21; G12∗First draft, June 2013; Revised, December 2014. An earlier version of this paper was presented atthe 2014 Fall Meeting of the Japanese Economic Association held on October 11–12, 2014. The author is
grateful to the discussant, Masao Ogaki, and other session participants, especially Chiaki Hara, for theirhelpful comments. The author also thanks John Campbell for permitting use of his data. This paper
is based upon work supported by a grant from the Ministry of Education, Culture, Sports, Science andTechnology of the Japanese Government (Grant-in-Aid No. 24730190).
†
Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba,
Ibaraki 305-8573, Japan; Tel.: +81-29-853-5369; Fax: +81-29-853-5070; E-mail: okubo@sk.tsukuba.ac.jp

1

Introduction

The coefficient of relative risk aversion (RRA) is a key parameter in macroeconomic and asset
pricing models. However, as shown by Mehra and Prescott (1985) and a subsequent voluminous
literature, obtaining a plausible value of this parameter (ordinarily thought to be less than 2or 3) is extremely difficult from the available U.S. data. The values sometimes exceed 50 whencalibrated to match asset market data (Tallarini 2000). As Campbell (2003) points out, thisdifficulty—referred to in the literature as the equity premium puzzle—is a robust phenomenon
in at least 11 developed countries, including European countries and Japan. Unfortunately,when international data are used, yet another problem emerges; namely, large variation acrosscountries in the magnitude of the RRA parameter. For instance, Campbell (2003, Table 4)
reports estimated values of RRA that are sometimes negative, and in excess of 100 for somecountries.
There is one possible explanation for the large values of RRA that at first appear implausible
in the U.S. data. It has been recently proposed by Hansen and Sargent (2008a) and Barillas etal. (2009) in an attempt to reinterpret the figure in Tallarini (2000). Both Hansen and Sargent(2008a) and Barillas et al. (2009) assume multiplier preferences as in Hansen and Sargent (2001a,
2001b), and link the risk aversion parameter with a fear of model misspecification. Based on
simulation studies, they show that an agent with an RRA value of around 50 is equivalent toan agent with a plausible level of concern about model misspecification. This positive view
about large values of RRA contrasts sharply with that in the previous literature (e.g., Lucas
2003). The following simple questions arise here. Does this U.S. finding hold for other developed
countries? If so, to what extent does that fact explain the significant variation in risk aversion
across countries?
In this paper, we show that reexamination of Campbell’s (2003) international data yields

1both the following findings that respond to the above questions and another open question. First,
implausibly high levels of risk aversion can substitute for plausible levels of concerns about model
misspecification for some other developed countries, as well as in the United States. Becausea plausible level of agent concern about model misspecification does not necessarily imply thesame value for the RRA coefficient, the significant cross-country variation in RRA can arise.
Second, our results suggest that there may be substantial heterogeneity in concerns about model
misspecification across countries. This finding raises another question. That is, why are theydifferent? We compare both the penalty parameters and the detection error probabilities withmeasures of cultural traits, including Hofstede’s (2001) scores for dimensions of culture, religious
composition, and primary language. We find that whether countries have stronger uncertaintyavoiding cultures following the terminology in Hofstede (2001) or differences in religious beliefs
partly explains the heterogeneity in our sample.
The remainder of the paper is organized as follows. Section 2 briefly reviews Tallarini’s (2000)
risk-sensitive preferences and the reinterpretation of Tallarini’s (2000) preference specification
undertaken by Hansen and Sargent (2008a) and Barillas et al. (2009). As in the previous
literature, we use detection error probabilities to measure the degree of concern about model
misspecification. Our computation procedure, however, relies on a simpler calibration techniquethan what the existing literature has used in that we employ the cumulative distribution function
instead of simulation. Section 3 explains the international data and presents the estimates ofthe parameters used in the computation. Section 4 details the empirical results. In Section 5,
we compare our results with the three types of measures of cultural traits and provide a possible
interpretation of our results. The appendix includes details of the derivation of our formulas forthe detection error probabilities.

2

2

Framework

2.1

Risk-Sensitive Preferences

Consider the recursive preference specification suggested by Epstein and Zin (1989) and Weil
(1990):
Vt =!

(1 − β)Ct1−η + β"

1−γ
Et [Vt+1
]

# 1−η
1−γ

$ 1

1−η

,

(1)

where Ct is consumption in period t, β is the subjective discount factor, γ is the coefficient of
RRA, and η denotes the reciprocal of the intertemporal elasticity of substitution (IES). Assumingthat η → 1, this specification reduces to
Vt = Ct1−β%"1−γ
Et [Vt+1]# 1 &β1−γ.

(2)Let ct denote log consumption. After taking the natural logarithm of both sides, equation (2)implies risk-sensitive recursion of the form'%

Ut = ct − βθ ln Et exp
where Ut ≡ ln Vt /(1 − β) and
θ=−

'

−Ut+1
θ

(&(

,

1
.
(1 − β)(1 − γ)

(3)

(4)

Suppose that a representative agent with recursion (3) values consumption streams generated
from the following random walk with drift model of log consumption in an endowment economy:
ct = µ + ct−1 + σ$ &t , &t ∼ i.i.d.N (0, 1).

(5)Then, solving the utility recursion (3) for Ut yields a value function of the form
!

$

β
σ$2
1
Ut =
µ
−
+
ct .
2
(1 − β)2θ(1 − β)1−β(6)An important relation that we can derive from this value function is the following:
"

#

−Ut+1
'
(
1 2
θ
)
"
#* = exp wRW &t+1 − wRWg(&t+1 ) ≡,2Et exp −Uθt+1

exp

3

(7)

where wRW = −σ$ /θ(1 − β) (see, e.g., Hansen and Sargent (2008a, Ch. 14) for details of thederivation of (6) and (7)).

2.2

Reinterpretation by Barillas, Hansen, and Sargent

Let &t = [&t , &t−1 , . . . , &1 ], and {&t } be a sequence of random shocks with conditional densities
π(&t ) = π(&t |&t−1 , x0 ), where x0 is a given initial state. The state at time t+1, xt+1 , is determinedby the state in the previous period, xt , and the realization of a random shock, &t+1 . The
consumption plan denoted by (5) is assumed to arise from the recursive restriction, xt+1 =µ + xt + σ$ &t+1 and ct = xt . Let Wt ≡ W (xt ) be a value function. Suppose that an agent does
not completely trust π(&t ) because of model uncertainty and chooses some other density π̂(&t ) in
proximity to π(&t ). The agent then imposes a penalty (known as relative entropy) on the choiceof π̂(&t ) and uses the following recursion to value the consumption streams:+t+1 ,
Wt = c t + β W

where+t+1 =
W

subject to,minπ̂($t+1 )≥0%,π̂(&t+1 )Wt+1 d&t+1 + θ∗

-,

(8)

π̂(&t+1 ) ln'(π̂(&t+1 )
d&t+1
π(&t+1 ).&(9)π̂(&t+1 )d&t+1 = 1. Here the parameter θ∗ limits the magnitude of the probability

distortions measured by relative entropy. Hansen and Sargent (2008a) and Barillas et al. (2009)
interpret this parameter as expressing the agent’s distrust of the density π(&t ) or fear of model
misspecification because smaller values of θ∗ allow the agent to choose more distorted densities.
+t+1 = Et [Wt+1 ], as shown by Maccheroni et al.
If the parameter θ∗ goes to infinity, then W(2006). In this case, the agent has no concern about model misspecification and evaluates theconsumption streams according to the utility recursion Wt = ct + βEt [Wt+1 ]. The solution forthe minimization problem (9) is

',

'

(

(

−Wt+1
π(&t+1 )d&t+1 ,
∗
' %
' θ
(&(
−Wt+1
= −θ∗ ln Et exp,
θ∗+t+1 = −θ ∗ ln
W

exp

4

(10)

and the ratio of conditional densities is given by
"

#

−Wt+1
π̂(&t+1 )
θ∗
)
"
#* .
=
π(&t+1 )Et exp −Wθ∗t+1

exp

(11)

As Barillas et al. (2009) argue, three implications arise from these results. First, the agent’s
utility recursion takes the form
'

%

'

−Wt+1
Wt = ct − βθ ln Et exp
θ∗
∗

(&(

.

(12)

It follows from (3) and (12) that an agent with risk-sensitive recursion (henceforth, a type I
agent) is observationally equivalent to an agent who is concerned about model misspecification(henceforth, a type II agent). Second, the parameter θ∗ has two interpretations because thisobservational equivalence implies that θ∗ ≡ θ: first, as a measure of risk aversion for the typeI agent; second, as a fear of model misspecification for the type II agent. Third, because
wRW ≡ −σ$ /θ∗ (1 − β), it follows from (7) and (11) that'(π̂(&t+1 )1 2
g(&t+1 ) == exp wRW &t+1 − wRW.
π(&t+1 )2(13)Hence, if π(&t+1 ) ∼ N (0, 1) as in the random walk model of log consumption, then the distorted
density is specified as
'

(

1 2
π̂(&t+1 ) = π(&t+1 ) exp wRW &t+1 − wRW
,
2 (
'
1
1
= √ exp − (&t+1 − wRW )2 .
2
2π

(14)

That is, the random shock &t+1 follows a normal distribution with mean wRW and unitary
variance.
Hansen and Sargent (2008a) and Barillas et al. (2009) exploited these implications to reinterpret Tallarini’s (2000) finding; i.e., an implausibly high level of γ for the type I agent. This
paper focuses on this observational equivalence between type I and type II agents, and attempts
to reevaluate the international differences in the parameter γ in terms of model misspecification
aversion.
5Implicit in the random walk model is the assumption that there is no long-run risk in log
consumption. Subsequent to the influential work of Bansal and Yaron (2004), long-run risk
models have received increasing attention in the modeling of aggregate consumption growth (see,e.g., Ludvigson (2013, Sections 6.3–6.4 and 7) for a survey). However, as Sargent (2007), Hansen(2007), and Hansen and Sargent (2008b, 2010), among others, have pointed out, it is difficult todistinguish a model with long-run risk from one without. In practice, the estimation of long-run
risk models appears to invoke a variety of empirical concerns when using U.S. aggregate timeseries data (see, e.g., Beeler and Campbell (2012) and Nakamura et al. (2012) for a discussion).
As Okubo (2014) shows, it is possible to incorporate long-run risk into the present frameworkusing a simplistic version of these models; however, there is much room for discussion about theestimation results.1 For the purpose of this paper, we assume that there is no long-run risk inlog consumption and concentrate on the random walk case in the following analysis.

2.3

Detection Error Probabilities

This subsection describes our procedure for calculating detection error probabilities, which weinterpret as a measure of the fear of model misspecification. Unlike the calibration method
using simulation, our procedure enables us to identify the exact value of the detection error
probabilities in an easy and timesaving manner.
Based on the discussion in the previous subsections, suppose that an agent’s baseline approximating model is
ct+1 = ct + µ + σ$ &t+1 , &t+1 ∼ i.i.d.N (0, 1).

(15)

1

Using the international data described in Section 3, we indeed attempted to estimate the simple version of
long-run risk models specified as
ct+1 − ct=µ + zt + σ# "t+1 ,

zt+1

=

ρzt + σz "t+1 , "t+1 ∼ i.i.d.N (0, 1).

The result shows that it is difficult to obtain precise estimates of ρ, σ# , and σz even with this simple model. In
particular, the critical problem is that the point estimate of ρ is too low, which is inconsistent with Bansal andYaron’s (2004) premise concerning the importance of long-run risks. For this reason, we must impose restrictions
to obtain a value of ρ close to one. See Okubo (2014) for details of the estimation results.

6

However, the agent doubts this model and constructs a worst-case model with N (wRW , 1) insteadof N (0, 1):
ct+1 = ct + µ + σ$ wRW + σ$ &t+1 , &t+1 ∼ i.i.d.N (0, 1), wRW = −σ$ /θ∗ (1 − β).

(16)

Our procedure is as follows.
1. Set the values of β, µ, and σ$ . Consider a sample of observations on log consumption ct
with a size of T . Find the value of γ when a pair of the mean and standard deviation of
the stochastic discount factor m, (E(m), σ(m)), succeeds in achieving Hansen and Jagannathan’s (1991) volatility bounds and then compute the value of θ∗−1 associated with thisγ using the relation derived from (4): θ∗−1 = (1 − β)(γ − 1). For the random walk model,
the pair is given by

/

0

σ2
E(m) = β exp −µ + $ (2γ − 1) ,2
1

2 2 23σ(m) = E(m) exp σ$ γ

41/2

−1

(17)

.

2. Suppose that before observing the data, the agent initially assigns a probability of 0.5 to theapproximating model (henceforth, model A) and the worst-case model associated with θ∗−1
(henceforth, model B). Suppose that after observing the data, the agent performs a testfor distinguishing model A from model B. Calculate the log-likelihood ratio, ln(LA /LB ).
The test selects model A if ln(LA /LB ) > 0 and model B if ln(LA /LB ) < 0. Then computethe detection error probability that the agent selects model B when model A generates the
data
'

'

LA
pA = Prob ln
LB

(

(

/ √

0

T
σ$
<0 =Φ −,
2 θ∗ (1 − β)(18)where Φ(·) denotes the standard normal cumulative distribution function.
3. Conversely, compute the detection error probability that the agent selects model A when7

model B generates the data''LA
pB = Prob ln
LB

(

(

>0 =1−Φ

/√

0

T
σ$
.
∗
2 θ (1 − β)(19)4. Compute the overall probability of model detection errors by weighting pA and pB by priorprobabilities of 0.5
1p(θ∗−1 ) = (pA + pB ).
2(20)

Note that pA and pB are equivalent because of the symmetry of the normal distribution.
Hence, the overall detection error probability p(θ∗−1 ) is equal to pA (see Appendix A forthe derivations of (18) and (19)).
The procedure developed by Hansen and Sargent (2008a) and Barillas et al. (2009) conversely
starts from giving a plausible value of p(θ∗−1 ) and then inverts p(θ∗−1 ) to set a value of θ∗−1 .
It is possible to follow the same flow as that of the Barillas–Hansen–Sargent procedure by using√
the inverse function of Φ(·): θ∗−1 = [−2(1 − β)/ T σ$ ]Φ−1 (pA ). In practice, the obtained results
are equivalent. Our procedure, however, is more straightforward in two respects: first, it does
not rely on simulation, and second, it does not require us to repeat the flow for various given
values of p(θ∗−1 ).
Our formula (18) establishes that the overall detection error probability p(θ∗−1 ) is a decreasing function of the penalty parameter θ∗−1 , which is consistent with the finding using thesimulation-based method of Hansen and Sargent (2008a) and Barillas et al. (2009). This meansthat an agent with smaller values of θ∗ can substitute for an agent with smaller model detection
errors. Similarly, we could consider from the formula that an increase in the volatility parameter
σ$ leads to a decrease in p(θ∗−1 ). However, this is not necessarily the case. The intuition is
as follows. An increase in σ$ moves a locus of a pair (E(m), σ(m)) upward. The value of γthat attains the Hansen–Jagannathan bounds will then decrease. This decrease in γ leads to adecrease in θ∗−1 because of the relation θ∗−1 = (1 − β)(γ − 1). That is, the increase in σ$ may
8make the value of p(θ∗−1 ) larger. This example also suggests that if an increase in the sample
size T leads to a decrease in σ$ , the value of p(θ∗−1 ) is not necessarily a decreasing function of
the sample size T .2
The first step of our procedure requires us to determine the value of γ that matches theasset market data using the Hansen–Jagannathan bounds. This step usually relies on a visualassessment of a graph that plots both the Hansen–Jagannathan bounds and pairs (E(m), σ(m))
for various values of γ. In order to obtain more precise values of γ, we use a condition that thestandard deviation σ(m) exceeds the minimum value of the Hansen–Jagannathan bounds:
σ(m) ≥ σ ∗ (m),(21)

where the minimum value is calculated as∗!

$

σ (m) = 1 Var(x)−1(E(x)$ Var(x)−1 1)2
1−
E(x)$ Var(x)−1 E(x)

$1/2

.

(22)

Here, x is a 2×1 vector of gross real returns on stocks and Treasury bills (or their proxies)
and 1 denotes a 2×1 vector of ones in our application. This condition is one way of expressing
the situation in which a pair (E(m), σ(m)) approaches the Hansen–Jagannathan bounds in thesense of Tallarini (2000).3 For a sensitivity analysis, we also employ a stronger condition that
requires that a pair (E(m), σ(m)) be definitely on or inside the Hansen–Jagannathan bounds.

3

Data

In this section, we describe the international data used in this paper and report maximum
likelihood (ML) estimates of the random walk model. The data are from Campbell (2003) and areavailable at quarterly and annual frequencies. We use only the quarterly data because, according
2
Note that our formula suggests that the overall detection error probability p(θ∗−1 ) is a decreasing function of
the sample size T , other things being equal. This is consistent with the Monte Carlo simulation finding in Hansen(2007, Figure 1).
3
In his visual assessment, Tallarini (2000) does not require that pairs (E(m), σ(m)) be definitely inside thebounds. Hansen and Sargent (2008a) and Barillas et al. (2009) also follow the same approach as Tallarini (2000).
Hence, for our base result, we use this condition to determine the value of γ.

9to Campbell (2003, Table 4), there is no significant variation in RRA across countries when using
annual data. The quarterly data are for 11 developed countries (the country code from Campbell(2003)): Australia (AUL), Canada (CAN), France (FR), Germany (GER), Italy (ITA), Japan(JAP), the Netherlands (NTH), Sweden (SWD), Switzerland (SWT), the United Kingdom (UK),and the United States (USA). The main data source is the International Financial Statisticsof the International Monetary Fund and Morgan Stanley Capital International (see Campbell(1998) for a description of the data).
The method and timing convention used for the construction of our dataset follow Yogo
(2004) except for one point (see Appendix B). Log consumption, ct , in the model is defined aslog real consumption per capita. As reported in Table 1, the sample periods differ by country.4
For the United States, we report two sample periods beginning in 1947 and 1970.
There are four points to note concerning the Campbell and Yogo data. First, the consumption
measure is total consumption but only expenditure on nondurables and services for the UnitedStates. Second, we use the GDP deflator when converting nominal to real consumption, andthe corresponding consumption deflator only for the U.S. data. Third, we employ the consumer
price index when calculating real asset returns. Fourth, the timing convention when measuringconsumption is at the beginning of the period (see Campbell (2003, pp. 813–814) for the reason).
It is well known that the treatment of durables can affect estimates of the IES and the RRA
coefficient (see, e.g., Mankiw (1985), Ogaki and Reinhart (1998a, 1998b), Yogo (2006), andPakoš (2011)). In our framework, the lumpiness of durables consumption may affect estimatesof the volatility parameter. Hence, for the United States, we add a case using total consumption
instead of nondurables and services. The data on total consumption and the price index are
from line 2 (labeled “personal consumption expenditures”) in the NIPA (National Income andProduct Accounts) Tables 1.1.4 and 1.1.5 compiled by the Bureau of Economic Analysis in the4
Our samples are just two periods longer than those in Yogo (2004, Table 1), as we do not require twice-lagged
variables in the estimation of the random walk model of log consumption.

10

U.S. Department of Commerce. In Table 1, this case is headed USA (PCE). For USA (PCE),
we report only the truncated sample period that begins in 1970 to avoid the period associatedwith the rapid restocking of durables after World War II.
Table 1 reports the results of the ML estimation of the random walk model. This model canbe regarded as a regression of ∆ct on a constant with normal errors. The ML estimator of µ
therefore is the ordinary least squares estimator, and the ML estimator of σ$2 is the sum of thesquared residuals divided by the sample size. In Table 1, the square root of the ML estimate of
σ$2 is also reported for ease of comparison.
The estimate of µ falls in the range of 0.0023 to 0.0080. The estimate of σ$ is around0.010, except in the United States. As a point of comparison, Barillas et al. (2009) obtain
estimates of µ = 0.00495 and σ$ = 0.0050 using quarterly U.S. data over the period 1948:2–
2006:4. Our estimates of µ and σ$ for the United States are thus very similar to those of
Barillas et al. (2009). As noted above, in the Campbell–Yogo data, the consumption measure
is consumption of nondurables and services only for the United States. This appears to explain
the small estimate of σ$ for the United States because it increases from 0.0046 to 0.0073 whentotal consumption is used. The estimate of σ$ = 0.0073, however, remains the smallest value
among the 11 countries.

4

Results

Table 2 reports the results of γ, θ∗−1 , and p(θ∗−1 ) based on the random walk with drift model
when the values of µ and σ$ are set to those reported in Table 1. Column (1) in Table 2 reports
the value of γ that attains the Hansen–Jagannathan bounds in the sense described in Section
2.3, where E(m) and σ(m) are computed using equation (17). We set β = 0.995. The minimumof γ is 8.04 for Australia (AUL), while the maximum is 51.09 for the United States (USA). Thisfinding reveals that even with risk-sensitive preferences, there is still a large variation in risk

11

aversion among countries when we calibrate it to match the asset market data.
Columns (2) and (3) in Table 2 report the penalty parameter θ∗−1 and the detection error
probability p(θ∗−1 ) associated with the value of γ. For the United States (USA), the value of thepenalty parameter is 0.2505 and the corresponding detection error probability is 0.0264 whenusing the sample period 1947:1–1998:4. This result essentially replicates those in Hansen andSargent (2008a) and Barillas et al. (2009). When we use the sample period 1970:1–1998:4 tofacilitate comparisons with other countries, however, the detection error probability increases
from 0.0264 to 0.1382. The value of p(θ∗−1 ) = 0.1382 is below 0.20; it remains what is called“sensible,” “plausible,” or “moderate” by Barillas et al. (2009, p. 2405), Hansen and Sargent(2008a, p. 322), and Ljungqvist and Sargent (2012, p. 547), respectively. Thus, we verify that
implausibly high levels of risk aversion can substitute for moderate amounts of model uncertainty
as long as we use the U.S. data.
For countries other than the United States, there is no existing work comparable to thatreported here. We observe the following from the comparisons across countries. First, the levelsof the detection error probability for countries other than the United States tend to be larger
than that of the United States. The highest of these is about two and a half times that of theUnited States. Second, even though the levels of the detection error probability are similar,
the magnitude of risk aversion appears to differ across countries. For instance, the value of γ
for Sweden (SWD) is 25.76 with a detection error probability of 0.1076. This detection error
probability is close to the value of 0.1382 for the United States (USA), but its value of γ is 45.31.
So far we have used the condition that σ(m) exceeds the minimum of the Hansen–Jagannathanbounds when determining the value of γ. As already discussed, it is not technically requiredthat the pair be inside the Hansen–Jagannathan bounds. In this sense, this condition is weak.
For a sensitivity analysis, we impose a stronger condition such that the pair (E(m), σ(m)) is on
or inside the Hansen–Jagannathan bounds.
12Table 3 reports the results of γ, θ∗−1 , and p(θ∗−1 ) when this alternative condition is used.
The results exhibit much larger variation in γ across countries. The minimum of γ is 12.38 forFrance (FR), while the maximum is 125.85 for Germany (GER). Comparing the detection error
probability between Tables 2 and 3, the value of p(θ∗−1 ) decreases for all countries because thevalue of the penalty parameter θ∗−1 increases. The use of this strong condition substantiallyalters the results for some countries, in particular Germany (GER) and the Netherlands (NTH).
This suggests that when international data are used, the use of the weak condition along thelines of Tallarini (2000) is not necessarily appropriate for some countries. However, it turns
out that implausibly high levels of risk aversion can substitute for moderate amounts of model
uncertainty for some countries other than the United States.

5

Discussion and Conclusion

One of the main points that Hansen and Sargent (2008a) and Barillas et al. (2009) emphasize using U.S. data is that a large risk-aversion parameter that at first appears implausible
is consistent with agents who have a plausible amount of concern about robustness to model
misspecification. The analysis in the previous section reveals that when applied to internationaldata, we can draw the same conclusion for some developed countries other than the UnitedStates. Consequently, large variation in risk aversion can arise across countries.
Our analysis relies crucially on the assumption that there is no long-run risk in log consumption. As Okubo (2014) shows, as long as we assume the simplistic version of long-run risk
models suggested by Hansen (2007, Example 2) and Hansen and Sargent (2008b, 2010), it ispossible to extend the framework used in this paper and the procedure for computing detection
error probabilities. In that case, lower values of the risk-aversion parameter appear to attain the
Hansen–Jagannathan bounds because we can show that both the mean and standard deviationof the stochastic discount factor increase relative to the case of no long-run risk. This would

13

substantially affect the magnitude of the penalty parameter and detection error probabilities.5
However, we must overcome the issue of imprecise estimates of the long-run risk model, before
drawing any conclusion from the results based on such estimates.6
Another notable feature of our results is that there may be substantial heterogeneity in concerns about model misspecification across countries. In our framework, the penalty parameter
depends on both the volatility of consumption and that of stock and bond markets, from theconventional viewpoint of the type I agent.7 Therefore, one possible explanation may relate to
the reasons that these volatilities differ among countries. Given the observational equivalence
between the type I and type II agents, however, it is not the only way to explain the heterogeneity identified in this paper. From the viewpoint of the type II agent, the issue of interest
can be related to the reasons that the agent’s distrust of the density of random shocks varies
across countries. International differences in the agent’s distrust may partly have caused thosedifferences in volatility for consumption and the stock and bond markets. As readily understood,
it is no simple task to identify the causes of international differences in the agent’s distrust.
Before concluding the paper, we discuss a possible explanation for the international differences in the agent’s distrust. Presumably, the most promising is an explanation based on
the relationship between cultural factors and economic outcomes. The basic logic proposed in
the literature is as follows: a culture trait of a group of people affects their beliefs and prefer-

5Preliminary results are reported in Okubo (2014).
There are two related studies that have attempted to improve the estimates of long-run risk models. One is
Bidder and Smith (2013), who propose estimating a long-run risk model that allows the stochastic volatility of
consumption growth by applying a particle filter. Another is Nakamura et al. (2012), who propose estimating
a long-run risk model that allows for both global and country-specific growth rate shocks, using international
panel data. In view of possible international comparisons, allowing for global shocks, as specified by Nakamura
et al. (2012), would be an important direction for future research. However, this is beyond the scope of thepresent paper. As discussed in the Introduction, our primary interest is in Campbell’s (2003) findings thatsuggest internationally large variation in the risk aversion parameter. Note that in estimating the risk aversion
parameter, Campbell (2003) does not consider shocks common to all countries.
7
Note that we used the relation that θ∗ is a function of γ, and determined the value of γ based on the Hansen–
Jagannathan bounds calculated from the mean and covariance matrix of stock and bond returns. Note also thatthe mean and standard deviation of the stochastic discount factor (E(m), σ(m)) depends on the consumption
volatility parameter σ# .
6

14

ences, and those beliefs and preferences influence their economic decision-making, and thus leadto differences in economic outcomes (see, e.g., Guiso et al. (2003, 2006), Haung (2008), andGorodnichenko and Roland (2010)). This chain of causality has also interested researchers as apossible explanation for cross-country differences in creditor rights, equity holdings, trading volume, volatility, and foreign portfolio investment (see, e.g., Stulz and Williamson (2003), Guiso
et al. (2008, 2009), Chui et al. (2010), Anderson et al. (2011), and Aggarwal et al. (2012)). An
issue of interest is thus whether there is any correlation between a country’s cultural traits andits penalty parameter or overall detection error probability.
Table 4 presents Hofstede’s (2001) scores for four dimensions of culture, which are one of
the most widely used data in the economics and finance literature. Hofstede (2001, p. 161)
defines the uncertainty avoidance index (UAI) as “the extent to which the members of a culture
feel threatened by ambiguous or unknown situations”. The more relevant comparison for our
purpose is thus the correlation between Hofstede’s UAI score and the penalty parameter or
detection error probability. As shown in the last two rows of Table 4, the correlation between
the UAI score and the penalty parameter is negative and, as expected from our formula, thatbetween the UAI score and the detection error probability is positive.8 Figure 1 plots the penalty
parameter and detection error probability against the UAI score to clarify the location of eachcountry. As shown, countries with higher UAI scores tend to have smaller values of θ∗−1 (i.e.,larger values of θ∗ ) and therefore higher detection error probabilities.
As discussed in Section 2, the parameter θ∗ represents the degree of the agent’s concern
about model misspecification. Maccheroni et al. (2006) give some theoretical justification forthis interpretation in a more general framework. More specifically, they show that as the parameter θ∗ becomes larger, the agent focuses more on the approximating model as the true
8

Note that according to Hofstede et al. (2010), the power distance index (PDI) tends to be positively correlatedwith the UAI, which measures “the extent to which the less powerful members of institutions and organizations
within a country expect and accept that power is distributed unequally” (p. 61).

15

model and gives less importance to possible alternative models. Hence, our finding suggeststhat agents in countries with higher UAI scores give less importance to possible worst-casemodels. A relevant finding for countries with uncertainty-avoiding cultures by Hofstede et al.
(2010, pp. 197–198) is “Uncertainty-avoiding cultures shun ambiguous situations. People insuch cultures look for structure in their organizations, institutions, and relationships that makes
events clearly interpretable and predictable”. In other words, countries with higher UAI scores
have a stronger tendency to create institutions and rules to reduce ambiguity. This cultural
trait may partially explain our finding that agents in high-UAI countries give less importanceto worst-case situations.9
Some of the cited literature also divides countries into groups based on religion and language
and investigates the abovementioned causality chain (see Guiso et al. (2003, 2006) and Haung(2008)). As Hofstede (2001) and Hofstede et al. (2010) point out, there is a strong correlation
between the population share of Catholics relative to Protestants and the country’s UAI score.
To see whether this holds for our sample, columns (7) and (8) in Table 4 report the percentage
population share of Catholics and Protestants in each country. The data are from Table 1 in Stulz
and Williamson (2003). A correlation holds roughly for the 11 developed countries: countries
with higher percentage shares of Catholics tend to have higher UAI scores. One exception isJapan, where the percentages of Catholics and Protestants are both extremely low. However,unlike the other countries, the population share of Buddhists (55.4%, not reported in the table)
is very high. Thus, our finding may be partly associated with differences in religious beliefs
9

In a preliminary analysis, we also investigated the uncertainty avoidance index reported in House et al.
(2004). The correlation coefficient was similar to that reported in this paper, but the locations of the countries
on the scatter plot were not identical (see Okubo (2014) for the scatter plot). Hofstede et al. (2010, pp. 198–
199) argue that the uncertainty avoidance index of House et al. (2004) presents no alternative to Hofstede’s
index. This suggests that we may be able to provide a different interpretation for our finding. However, we do
not pursue this here as resolving the issue of inconsistency between the two indexes is beyond the scope of thecurrent analysis. Another widely used dataset in the literature is from the World Values Survey (WVS) led by
a U.S. sociologist, Ronald Inglehart. As Hofstede et al. (2010) discuss, various extensions of Hofstede’s (2001)
cultural dimensions based on the WVS results are attempted, and some of the WVS results are closely relatedwith Hofstede’s scores. However, we concentrated on Hofstede’s scores in this paper, as there seem to be various
arguments and interpretations for those extensions.

16

across countries. On the other hand, as the last column in Table 4 shows, differences in the
primary language are unlikely to explain our finding.
These explanations based on cultural traits require further empirical investigation. Note thatthey do not necessarily hold under the stronger condition for the attainment of the Hansen–
Jagannathan bounds. However, our preliminary investigations in this section suggest that extending a model with Hansen and Sargent’s multiplier preferences to an international environment may connect two different strands of research in economics and finance: robustness studies
and culture–economic studies. Analyzing a richer model of this kind would be an interesting research topic, in addition to straightforward extensions such as increasing the number of countries
used in the analysis, and allowing for long-run risks.

17

Appendix
This appendix describes the derivation of our formulas for the detection error probabilities andthe construction method for the asset returns data.

A. Derivation of Formulas for the Detection Error Probabilities
Consider the following AR(1) process with trend:10
ct = ζ + µt + zt , zt = ρzt−1 + σ$ &t , &t ∼ i.i.d.N (0, 1).

(A1)

Since zt has an AR(1) structure, the (average) log-likelihood function for a sample of t =1, 2, . . . , T takes the formln L =T
1
15
ln f (c1 ) +ln f (ct |ct−1 ).
T
T t=2

(A2)

In the case of the random walk with drift (ρ = 1), (A1) at t = 1 is c1 = ζ + µ + z1 andz1 = z0 + σ$ &1 . Assuming z0 = 0, it follows that z1 = σ$ &1 , so that c1 = ζ + µ + σ$ &1 . Under theapproximating model (model A), therefore, the logarithm of the density f (c1 ) is1
1
1
ln f (c1 ) = − ln 2π − ln σ$2 − 2 (c1 − ζ − µ)2 ,
2
2
2σ$

(A3)

and the logarithm of the conditional density f (ct |ct−1 ) is1
1
1
ln f (ct |ct−1 ) = − ln 2π − ln σ$2 − 2 (ct − ct−1 − µ)2 .
2
2
2σ$

(A4)

Substituting (A3) and (A4) into (A2), we obtain the log-likelihood function under model A:
T
1
1
1 1
15
1
2
2
ln LA = − ln 2π − ln σ$ −(c1 − ζ − µ) −(ct − ct−1 − µ)2 .
22
T 2σ$2
T t=2 2σ$2

10

(A5)

Another way is to begin with the first-differenced form of the model: ∆ct+1 = µ + σ# "t+1 . When ρ = 1, these
two ways lead to the same formulas for the detection error probabilities, as discussed in Okubo (2014). However,when ρ < 1, the first-differenced approach is not valid. Here, we describe a more general one that allows the caseof ρ < 1.

18

Noting that the difference between the approximating and worst-case models is that the meanof &t shifts from 0 to wRW , the log-likelihood function under model B is
T
5
1
1
1 1
1
2 1
ln LB = − ln 2π− ln σ$2 −(c−ζ−µ−σw)−(ct −ct−1 −µ−σ$ wRW )2 . (A6)
1
$ RW
2
2
T 2σ$2
T t=2 2σ$2Thus, we obtain the following log-likelihood ratio:
'

LA
ln
LB

(

1
= −
T
1
+
T

!

T
5
1
1
2
(c
−
ζ
−
µ)
+
(c − ct−1 − µ)2
1
2
2 t
2σ$
2σ
$
t=2

!

$

$

T
5
1
1
2
(c
−
ζ
−
µ
−
σ
w
)
+
(c − ct−1 − µ − σ$ wRW )2 .
1
$ RW
2
2 t
2σ$
2σ
$
t=2

(A7)

To calculate the detection error probability under model A, substituting c1 − ζ − µ = σ$ &1
and ct − ct−1 − µ = σ$ &t for t = 2, . . . , T into (A7) yields
ln

'

LA
LB

(

=
=

%

&

T
15
1
1
− &2t + (&t − wRW )2 ,
T t=1
2
2
T
15
1 2
(−wRW &t ) + wRW
.
T t=1
2

(A8)

Therefore, the detection error probability under model A is
pA

'

'

(

(

LA
= Prob ln
<0 ,
LB
/
0
T
15
1 2
= Prob(−wRW &t ) + wRW < 0 ,T t=1
2= Prob

/
/

'

(

T
15
σ$
1
&t < −
T t=1 θ(1 − β)
2

0

T
15
1
σ$
&t < −,T
2 θ(1 − β)
/ t=1 √0Tσ$
= Prob Z < −,2 θ(1 − β)= Prob'σ$θ(1 − β)

(2 0

,

(A9)

√ 6
where Z ≡ (1/ T ) Tt=1 &t . Because of &t ∼ i.i.d.N (0, 1), Z ∼ N (0, 1). Thus, as in equation(18), the detection error probability pA can be expressed using the cumulative distributionfunction of the standard normal. On the other hand, substituting c1 − ζ − µ = σ$ wRW + σ$ &1
and ct − ct−1 − µ = σ$ wRW + σ$ &t for t = 2, . . . , T into (A7), it reduces to
ln

'

LA
LB

(

=
=

%

&

T
15
1
1
− (&t + wRW )2 + &2t ,
T t=1
2
2
T
15
1 2
(−wRW &t ) − wRW
.
T t=1
2

19

(A10)

Therefore, the detection error probability under model B is
pB

'

'

(

(

LA
= Prob ln
>0 ,
LB
/
0
T
15
1 2
= Prob(−wRW &t ) − wRW > 0 ,
T t=1
2
/

'

(

T
15
σ$
1
= Prob&t >T t=1 θ(1 − β)
2

/

T
15
1
σ$
&t >
T
2 θ(1 − β)
/ t=1√0Tσ$= Prob Z >.
2 θ(1 − β)= Prob0'σ$θ(1 − β)

(2 0

,

(A11)

,

This gives equation (19).

B. Asset Returns Data
The gross real returns on stocks are constructed as
1 + rrt ≡ (1 + Rt )cpit−1
,
cpit

(B1)

where Rt is stock returns in quarter t and cpit is the consumer price index in quarter t. The
gross real returns on relatively riskless assets are constructed as

1 + rrft ≡( 'irt−1 cpit−1


1
+



100
cpit
'
(


irt−1 cpit−1


 1+

400

cpit

for the United States,(B2)
for the other countries,

where irt is the short-term interest rate in period t. Here the timing convention for the interest
rate follows that in Campbell (2003) and Yogo (2004). In Campbell’s (2003) data, the interest
rate is quarterly for the United States, while it is an annual percentage rate for countries otherthan the United States. Hence, note that the short-term interest rate is divided by 400 for
countries other than the United States to convert it to a quarterly series. The 2×1 vector x in
the main text is defined as x = (1 + rrt , 1 + rrft )$ .

20

References
Aggarwal, Raj, Colm Kearney, and Brian Lucey (2012), “Gravity and Culture in Foreign
Portfolio Investment,” Journal of Banking and Finance 36, 525–538.
Anderson, Christopher W., Mark Fedenia, Mark Hirschey, and Hilla Skiba (2011), “Cultural Influences on Home Bias and International Diversification by Institutional Investors,” Journal of Banking and Finance 35, 916–934.
Bansal, Ravi, and Amir Yaron (2004), “Risks for the Long Run: A Potential Resolution ofAsset Pricing Puzzles,” Journal of Finance 59, 1481–1509.
Barillas, Francisco, Lars Peter Hansen, and Thomas J. Sargent (2009), “Doubts or Variability?,” Journal of Economic Theory 144, 2388–2418.
Beeler, Jason, and John Y. Campbell (2012), “The Long-Run Risks Model and Aggregate Asset
Prices: An Empirical Assessment,” Critical Finance Review 1, 141–182.
Bidder, Rhys, and Matthew E. Smith (2013), “Doubts and Variability: A Robust Perspective
on Exotic Consumption Series,” Federal Reserve Bank of San Francisco Working Paper
No. 2013-28.
Campbell, John Y. (1998), “Data Appendix for ‘Asset Prices, Consumption, and the Business
Cycle’,” Unpublished manuscript, Department of Economics, Harvard University.
Campbell, John Y. (2003), “Consumption-Based Asset Pricing,” in George M. Constantinides,
Milton Harris, and René M. Stulz (eds.), Handbook of the Economics of Finance Vol. 1B,
pp. 803–887, Amsterdam: Elsevier.
Chui, Andy C. W., Sheridan Titman, and K. C. John Wei (2010), “Individualism and Momentum around the World,” Journal of Finance 65, 361–392.
21

Epstein, Larry G., and Stanley E. Zin (1989), “Substitution, Risk Aversion, and the Temporal
Behavior of Consumption and Asset Returns: A Theoretical Framework,” Econometrica
57, 937–969.
Gorodnichenko, Yuriy, and Gerard Roland (2010), “Culture, Institutions and the Wealth of
Nations,” NBER Working Paper Series No. 16368.
Guiso, Luigi, Paola Sapienza, and Luigi Zingales (2003), “People’s Opium? Religion andEconomic Attitudes,” Journal of Monetary Economics 50, 225–282.
Guiso, Luigi, Paola Sapienza, and Luigi Zingales (2006), “Does Culture Affect Economic Outcomes?,” Journal of Economic Perspectives 20, 23–48.
Guiso, Luigi, Paola Sapienza, and Luigi Zingales (2008), “Trusting the Stock Market,” Journalof Finance 63, 2557–2600.
Guiso, Luigi, Paola Sapienza, and Luigi Zingales (2009), “Cultural Biases in Economic Exchange?,” Quarterly Journal of Economics 125, 1095–1131.
Hansen, Lars Peter (2007), “Beliefs, Doubts and Learning: Valuing Macroeconomic Risk,”
American Economic Review 97, 1–30.
Hansen, Lars Peter, and Ravi Jagannathan (1991), “Implications of Security Market Data for
Models of Dynamic Economies,” Journal of Political Economy 99, 225–262.
Hansen, Lars Peter, and Thomas J. Sargent (2001a), “Robust Control and Model Uncertainty,”
American Economic Review 91, 60–66.
Hansen, Lars Peter, and Thomas J. Sargent (2001b), “Acknowledging Misspecification in
Macroeconomic Theory,” Review of Economic Dynamics 4, 519–535.

22

Hansen, Lars Peter, and Thomas J. Sargent (2008a), Robustness, Princeton, NJ: Princeton
University Press.
Hansen, Lars Peter, and Thomas J. Sargent (2008b), “Fragile Beliefs and the Price of Model
Uncertainty,” Unpublished manuscript, University of Chicago and New York University.
Hansen, Lars Peter, and Thomas J. Sargent (2010), “Fragile Beliefs and the Price of Uncertainty,” Quantitative Economics 1, 129–162.
Hofstede, Geert (2001), Culture’s Consequences: Comparing Values, Behaviors, Institutions
and Organizations across Nations, 2nd ed., Thousand Oaks, CA: Sage.
Hofstede, Geert, Gert Jan Hofstede, and Michael Minkov (2010), Cultures and Organizations:
Software of the Mind, 3rd ed., New York, NY: McGraw-Hill.
House, Robert J., Paul J. Hanges, Mansour Javidan, Peter W. Dorfman, and Vipin Gupta(2004), Culture, Leadership, and Organizations: The GLOBE Study of 62 Societies, Thousand Oaks, CA: Sage.
Huang, Rocco R. (2008), “Tolerance for Uncertainty and the Growth of Informationally Opaque
Industries,” Journal of Development Economics 87, 333–353.
Ljungqvist, Lars, and Thomas J. Sargent (2012), Recursive Macroeconomic Theory, 3rd ed.,Cambridge, MA: MIT Press.
Lucas, Robert E. Jr. (2003), “Macroeconomic Priorities,” American Economic Review 93,
1–14.
Ludvigson, Sydney C. (2013), “Advances in Consumption-Based Asset Pricing: Empirical
Tests,” in George M. Constantinides, Milton Harris, and René M. Stulz, (eds.), Handbookof the Economics of Finance Vol. 2B, pp. 799–906, Amsterdam: Elsevier.
23

Maccheroni, Fabio, Massimo Marinacci, and Aldo Rustichini (2006), “Ambiguity Aversion,
Robustness, and the Variational Representation of Preferences,” Econometrica 74, 1447–
1498.
Mankiw, Gregory N. (1985), “Consumer Durables and the Real Interest Rate,” Review of
Economics and Statistics 67, 353–362.
Mehra, Rajnish, and Edward C. Prescott (1985), “The Equity Premium: A Puzzle,” Journalof Monetary Economics 15, 145–161.
Nakamura, Emi, Dmitriy Sergeyev, and Jón Steinsson (2012), “Growth-Rate and Uncertainty
Shocks in Consumption: Cross-Country Evidence,” Unpublished manuscript, Columbia
University.
Ogaki, Masao, and Carmen M. Reinhart (1998a), “Measuring Intertemporal Substitution: The
Role of Durable Goods,” Journal of Political Economy 106, 1078–1098.
Ogaki, Masao, and Carmen M. Reinhart (1998b), “Intertemporal Substitution and Durable
Goods: Long-Run Data,” Economics Letters 61, 85–90.
Okubo, Masakatsu (2014), “Appendix for ‘Model Uncertainty and International Differences in
Risk Aversion’,” Unpublished manuscript, University of Tsukuba.
Pakoš, Michal (2011), “Estimating Intertemporal and Intratemporal Substitutions When Both
Income and Substitution Effects Are Present: The Role of Durable Goods,” Journal ofBusiness and Economic Statistics 29, 439–454.
Sargent, Thomas J. (2007), “Commentary,” Federal Reserve Bank of St. Louis Review 89,
301–304.

24

Stulz, René M. and Rohan Williamson (2003), “Culture, Openness, and Finance,” Journal of
Financial Economics 70, 313–349.
Tallarini, Thomas D. (2000), “Risk-Sensitive Real Business Cycles,” Journal of Monetary Economics 45, 507–532.
Weil, Philippe (1990), “Nonexpected Utility in Macroeconomics,” Quarterly Journal of Economics 105, 29–42.
Yogo, Motohiro (2004), “Estimating the Elasticity of Intertemporal Substitution When Instruments Are Weak,” Review of Economics and Statistics 86, 797–810.
Yogo, Motohiro (2006),“A Consumption-Based Explanation of Expected Stock Returns,” Journal of Finance 61, 539–580.

25

Table 1
Estimates of the Random Walk Model

CountryUSA

Sample Period
1947.1–1998.4

AUL

1970.1–1998.4

CAN

1970.1–1999.1

FR

1970.1–1998.3

GER

1978.3–1998.3

ITA

1971.2–1998.1

JAP

1970.1–1998.4

NTH

1977.1–1998.4

SWD

1970.1–1999.2

SWT

1975.4–1998.4

UK

1970.1–1999.1

USA

1970.1–1998.4

USA (PCE)

1970.1–1998.4µ(1)0.0049(0.0004)
0.0052(0.0010)0.0054(0.0009)
0.0039(0.0013)0.0043(0.0013)0.0057(0.0008)
0.0080(0.0012)0.0046(0.0014)0.0025(0.0008)
0.0023(0.0013)0.0056(0.0012)0.0046(0.0004)
0.0058(0.0007)

σ$2(2)0.2874×10−4(0.0282×10−4 )0.1051×10−3(0.0138×10−3 )0.9140×10−4(0.1195×10−4 )0.2019×10−3(0.0266×10−3 )0.1406×10−3(0.0221×10−3 )0.6865×10−4(0.0934×10−4 )0.1617×10−3(0.0212×10−3 )0.1678×10−3(0.0253×10−3 )0.8497×10−4(0.1106×10−4 )0.1511×10−3(0.0222×10−3 )0.1563×10−3(0.0204×10−3 )0.2081×10−4(0.0273×10−4 )0.5273×10−4(0.0692×10−4 )σ$(3)0.0054

#Obs
(4)
208

0.0103116

0.0096117

0.0142115

0.011981

0.0083108

0.0127116

0.013088

0.0092118

0.012393

0.0125117

0.0046116

0.0073116

Note: This table reports maximum likelihood (ML) estimates from a regression of log consumption
growth ∆ct on a constant, where the error term is assumed to be i.i.d.N (0, σ!2 ). Standard errors are inparentheses. The column denoted by σ! reports the square root of the ML estimate of σ!2 . #Obs denotes
the number of observations of ∆ct used in the ML estimation. The country codes follow Campbell(2003): Australia (AUL), Canada (CAN), France (FR), Germany (GER), Italy (ITA), Japan (JAP), theNetherlands (NTH), Sweden (SWD), Switzerland (SWT), the United Kingdom (UK), and the UnitedStates (USA). USA (PCE) denotes that personal consumption expenditures are used as the consumption
measure.

26

Table 2
Risk Aversion and Detection Error Probability under the Random Walk Model

Country
USA
AUL
CAN
FR
GER
ITA
JAP
NTH
SWD
SWT
UK
USA
USA (PCE)

Sample Period
1947.1–1998.4
1970.1–1998.4
1970.1–1999.1
1970.1–1998.3
1978.3–1998.3
1971.2–1998.1
1970.1–1998.4
1977.1–1998.4
1970.1–1999.2
1975.4–1998.4
1970.1–1999.1
1970.1–1998.4
1970.1–1998.4

γ
(1)
51.09
8.04
11.32
11.81
18.68
8.87
9.33
27.25
25.76
24.44
15.91
45.31
28.48

θ∗−1
(2)
0.25050.03520.0516
0.05410.08840.03940.04170.13130.12380.11720.07460.22160.1374p(θ∗−1 )(3)
0.02640.34780.29680.20510.17280.36740.28420.05540.10760.08240.15670.13820.1413

T
(4)
208
116
117
115
81
108
116
88
118
93
117
116
116

Note: This table reports calibration results of the risk aversion parameter γ, the penalty parameter θ∗−1 ,and the detection error probability p(θ∗−1 ) when log consumption follows the random walk with drift
model. The risk aversion parameter γ is chosen to satisfy the condition σ(m) ≥ σ ∗ (m), where σ ∗ (m) is
the minimum of the Hansen–Jagannathan bounds. The discount factor is set to β = 0.995.

27

Table 3
Sensitivity Analysis under the Random Walk Model

Country
USA
AUL
CAN
FR
GER
ITA
JAP
NTH
SWD
SWT
UK
USA
USA (PCE)

Sample Period
1947.1–1998.4
1970.1–1998.4
1970.1–1999.1
1970.1–1998.3
1978.3–1998.3
1971.2–1998.1
1970.1–1998.4
1977.1–1998.4
1970.1–1999.2
1975.4–1998.4
1970.1–1999.1
1970.1–1998.4
1970.1–1998.4

γ(1)
108.30
19.02
17.74
12.38
125.85
18.66
31.05
95.15
25.78
28.07
21.97
107.84
66.96

θ∗−1
(2)
0.53650.09010.0837
0.05690.62430.08830.15030.47080.12390.13540.10490.53420.3298p(θ∗−1 )(3)
1.6773×10−5
0.15990.19340.19301.3581×10−110.22350.01985.3063×10−90.10740.05430.07810.00430.0050

T
(4)
208
116
117
115
81
108
116
88
118
93
117
116
116

Note: This table reports calibration results of the risk aversion parameter γ, the penalty parameter θ∗−1 ,and the detection error probability p(θ∗−1 ) when log consumption follows the random walk with drift
model. The risk aversion parameter γ is chosen to satisfy the condition that the pair (E(m), σ(m)) is on
or inside the Hansen–Jagannathan bounds. The discount factor is set to β = 0.995.

28

Table 4
Comparisons with Cultural Traits

Country
AUL
ITA
JAP
CAN
FR
UK
GER
SWT
SWD
NTH
USA (PCE)θ∗−1(1)0.03520.03940.04170.0516
0.05410.07460.08840.11720.12380.13130.1374p(θ∗−1 )(2)
0.34780.36740.28420.29680.20510.15670.17280.08240.1076
0.05540.1413θ∗−1p(θ∗−1 )1.000-0.903-0.903
1.000

Hofstede’s Scores
Religion (%)
MAS
PDI
UAI Protestant Catholic
(4)
(5)
(6)
(7)
(8)
61
36
51
36.7
28.1
70
50
75
0.8
97.2
95
54
92
0.5
0.4
52
39
48
20.3
41.9
43
68
86
1.6
82.8
66
35
35
53.8
9.6
66
35
65
37.2
34.9
70
34
58
41.6
44.8
5
31
29
94.8
1.9
14
38
53
27.1
35.5
62
40
46
24.3
21.2
Correlation with θ∗−1 and p(θ∗−1 )0.182 -0.052a -0.502 -0.523
0.520-0.331-0.033 0.081a 0.373 0.409-0.514
0.336
IDV
(3)
90
76
46
80
71
89
67
68
71
80
91

Primary
Language(9)
EnglishItalianJapanese
EnglishFrenchEnglishGermanGermanSwedishDutch
Englishʵʵ

Note: This table reports the penalty parameter θ∗−1 , the detection error probability p(θ∗−1 ), Hofstede’s(2001) scores for four dimensions of culture, the percentages of Protestants and Catholics in the population, the primary language, and the correlation coefficients between these variables. Countries are sorted
in ascending order of the values of the penalty parameter. In this table, USA (PCE) is only listed for
the United States for ease of comparison with the other countries (see Section 3 for details). The penalty
parameter and detection error probability are from Table 2. IDV, MAS, PDI, and UAI are the individualism, masculinity, power distance, and uncertainty avoidance indexes, respectively, from the four tables(labeled Exhibit 5.1, Exhibit 6.3, Exhibit 3.1, and Exhibit 4.1) in Hofstede (2001). The percentages forreligion and the primary language are from Table 1 in Stultz and Williamson (2003).
a

SWD and NTH are excluded when computing the correlation coefficient because they are obvious

outliers in the sample.

29

GHMK$$/01()2($,-$'.($*(0"+'3$*")"4('()$

GHMJ$$

6%8$%D(5(0$

GHMI$$

F('.()+"052$$%D7'E()+"05$$

GHMG$$
C()4"03$
GHGL$$

6B$

GHGK$$A)"0&($@"0"5"$
GHGJ$$>"?"0$/'"+3$

8=2')"+7"$

GHGI$$
IG$

NG$

JG$

OG$

KG$

PG$

LG$

QG$

MGG$

R,-2'(5(S2$60&()'"70'3$81,75"0&($/05(9$%&,)($!

!"#$%&"''()$*+,'$,-$'.($/01()2($,-$'.($*(0"+'3$*")"4('()$"05$'.($60&()'"70'3$81,75"0&($/05(9$$GHJG$$/'"+3$$$
GHNO$$

8=2')"+7"$$;('(&'7,0$<)),)$*),:":7+7'3$

GHNG$$@"0"5"$>"?"0$$

GHIO$$A)"0&($$

GHIG$$
C()4"03$$$
6B$

GHMO$$

6%8$$%D(5(0$$GHMG$$%D7'E()+"05$$
F('.()+"052$$

GHGO$$
GHGG$$
IG$

NG$

JG$

OG$

KG$

PG$

LG$

QG$

MGG$

R,-2'(5(S2$60&()'"70'3$81,75"0&($/05(9$%&,)($!:#$%&"''()$*+,'$,-$'.($;('(&'7,0$<)),)$*),:":7+7'3$"05$'.($60&()'"70'3$81,75"0&($/05(9$!

!"#$%&'()'*&+,-./'*,%,0&.&%1'2&.&3."4+'5%%4%'*%46,6"-"./1',+7'8+3&%.,"+./'9:4"7,+3&';+7&<'30!
!

!