Department of Policy and Planning Sciences
Discussion Paper Series

No.1375

Model Uncertainty, Economic Development,and Welfare
Costs of Business Cycles

byMasakatsu OKUBO

May 2021

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANModel Uncertainty, Economic Development, and Welfare Costs of Business
Cycles ∗Masakatsu Okubo †University of Tsukuba

Abstract

Some researchers argue that the welfare gains from eliminating consumption ﬂuctuations for theUnited States are not small once model uncertainty is taken into account. This paper presents new
evidence on the welfare gains from eliminating model uncertainty using a data set from a broad
range of countries. It quantiﬁes exactly the eﬀect of model uncertainty on the welfare gains using
an analytical formula. The results indicate that most countries derive much larger gains from thereduction of model uncertainty compared with the United States. Countries at higher stages of
economic development tend to have lower welfare gains because their gains from eliminating model
uncertainty become smaller. This relationship does not depend on country size or trade openness.

Keywords: Consumption risk; Detection error probability; Model uncertainty; Welfare costs
JEL Classiﬁcation: D81; E21; E32∗

First draft, June 2020; Revised, March 2021. This paper is based on a revised and extended version of
the main part of the paper formerly circulated as “International Evidence on Welfare Gains from Eliminating
Model Uncertainty.” The author is grateful to two anonymous readers for constructive and critical commentson the previous version of the paper. This work was supported by a grant-in-aid from the Japan Society for
the Promotion of Science (JSPS KAKENHI Grant No. 18K01538).
†
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

Since the seminal work of Lucas (1987), researchers have reconsidered the striking ﬁnding that thewelfare gains from eliminating consumption ﬂuctuations calculated from US post-World War II data
are trivially small. As Lucas (2003) reviews, in the ﬁrst 15 years after his ﬁnding, many researchers
have found that modiﬁcations of the Lucas calculation can lead to larger, but still small, welfare
gains. One exception among these early studies is Tallarini (2000), whose welfare-gain estimates
are two orders of magnitude larger than those found by Lucas.
Barillas et al. (2009) focus on the observational equivalence between Tallarini’s risk-sensitiveand multiplier preferences, and argue that Tallarini’s large gains arise from eliminating model
uncertainty, not from reducing consumption risk drawn from a known probability distribution.
Ellison and Sargent (2015) also argue that the large gains are associated with agents’ concerns
about model misspeciﬁcation, and that incorporating idiosyncratic risk into the framework leads
to larger gains. These conclusions are based on evidence from US data.
In this paper, we present additional evidence on the welfare gains from eliminating model
uncertainty. We examine the following two questions:(i) Are there large welfare gains from eliminating model uncertainty in a broad range of countries
other than the United States?
(ii) If so, how does the gain diﬀer at various stages of economic development, e.g., between rich,emerging, and poor countries?
There is existing evidence on the ﬁrst question. Engel et al. (2018) show that there aresubstantial gains in emerging countries. An earlier version of this paper (Okubo (2019)) showsthat relatively large gains are observable in other developed and emerging countries compared with
the United States.1 However, as the sample size of countries is small in these studies (15 and1

It includes a numerical example based on Barillas et al.’s (2009) parameter settings, a replication of Ellison and1

22, including the United States, in Engel et al. (2018) and Okubo (2019), respectively), further
investigation is required.
The second question is motivated by the literature on business cycles in emerging and poor
countries. Pallage and Robe (2003) follow the tradition of Lucas (1987) and show that there arevery large welfare costs of consumption ﬂuctuations in developing countries relative to the UnitedStates. Uribe and Schmitt-Grohé (2017) document excess output volatility and higher relative
consumption volatility in emerging and poor countries as key business-cycle facts.2 We aim toreevaluate these diﬀerences between poor, emerging, and developed countries through the lens ofmodel uncertainty.
We use annual data on 64 countries for the period 1970–2018. Our welfare-gain calculations are
based on an analytical formula, which combines Okubo’s (2018a) closed-form solutions of detection
error probabilities with the formula proposed by Barillas et al. (2009). The advantage of using
this analytical formula is that the eﬀect of agents’ fear of model misspeciﬁcation on welfare gains isquantiﬁed exactly. We examine questions (i) and (ii) in two ways. First, we classify the 64 countries
into three groups—rich, emerging, and poor—based on a measure of economic development andcompare their sample statistics within and between the groups, as well as with the US welfare-gain
estimates. Then, we carry out regression analysis with our welfare-gain measures as dependent
variables to characterize the relationship between welfare gains and economic development. In thisanalysis, we control for the eﬀects of two factors—country size and trade openness—that may aﬀectbusiness cycles, as pointed out in the literature (e.g., Kose et al. (2003), Furceri and Karras (2007),
di Giovanni and Levchenko (2009, 2012), Haddad et al. (2013), and Uribe and Schmitt-Grohé(2017)).
Sargent (2015) using an analytical solution, and comparisons of results between quarterly and annual data.
2
More speciﬁcally, Uribe and Schmitt-Grohé (2017, p. 9) summarize these facts as “Business cycles in rich countries
are about half as volatile as business cycles in emerging or poor countries” and “The relative consumption volatility
is higher in poor and emerging countries than in rich countries.” They argue that explaining these facts is one of
the most important unﬁnished subjects in business-cycle theory. Indeed, it has led to a large body of literature, as
discussed later in Section 4.3.

2

A summary of our ﬁndings is as follows. The welfare cost of model uncertainty is substantial
relative to that of consumption risk for a broad set of countries. Most countries have welfare
gains from eliminating model uncertainty that are considerably higher than those of the UnitedStates. As for the second question, the comparison of the sample statistics between the groups
shows that the welfare gains from eliminating model uncertainty are, on average, at least two times
higher in emerging or poor countries than in rich countries. This suggests that higher levels of
economic development are associated with lower welfare costs of model uncertainty. The regression
analysis shows that this negative relationship is robust to controlling for country size and trade
openness, and that it is stronger than the negative relationship between the level of economic
development and the welfare cost of consumption risk. In addition, it shows that as agents’ fearof model misspeciﬁcation becomes milder, the negative relationship between the level of economic
development and the welfare cost of model uncertainty becomes weaker.
There is a limitation to the formula of Barillas et al. (2009) and it deserves special mention at
the outset. According to their formula (shown in formula (4) or (5) below), the welfare gain from
eliminating model uncertainty is positively linked to the volatility of consumption. This meansthat cross-country comparisons of the gains from reducing model uncertainty are in fact equivalent
to comparing the magnitudes of consumption volatility across countries. Thus, the ﬁrst part ofour ﬁndings simply says that countries with higher consumption volatility, which are largely the
emerging and poor countries, have higher welfare gains from eliminating model uncertainty, otherthings being equal. However, the other part of our ﬁndings cannot be drawn from such cross-country
comparisons. The use of our formula (formula (7) below), which quantiﬁes exactly the eﬀect of
agents’ fear of model misspeciﬁcation on welfare gains, allows us to draw these other ﬁndings.
The remainder of the paper is organized as follows. Section 2 describes our formula after
reviewing the welfare-gain measures proposed by Barillas et al. (2009). Section 3 describes the dataand classiﬁcation of the sample countries. Section 4 presents our ﬁndings. We also provide potential3explanations for the ﬁndings by discussing how they can be related to the existing literature. Section
5 explores the eﬀect of idiosyncratic risk. Section 6 provides a brief conclusion. The separate
appendix contains details on parameter settings and estimates for individual countries as well as
the regression results that were omitted from the main paper.

2

Framework

2.1

The Barillas–Hansen–Sargent Formula

The notation is the same as that of Barillas et al. (2009). Let {ϵt } denote a sequence of random
shocks with conditional densities π(ϵt ) ≡ π(ϵt | ϵt−1 , x0 ), where ϵt−1 ≡ [ϵt−1 , ϵt−2 , . . . , ϵ1 ], and x0
is a given initial state. Two models are assumed for log consumption streams {ct }. One is therandom walk with drift model:ϵt ∼ i.i.d.N (0, 1).

ct = µ + ct−1 + σϵ ϵt ,(1)Another is the trend stationary model:
ct = ζ + µt + zt ,zt = ρzt−1 + σϵ ϵt ,|ρ| < 1,ϵt ∼ i.i.d.N (0, 1).

(2)

The key assumption of Barillas et al.’s framework is that a representative agent with log period
utility does not completely trust π(ϵt ) because of model uncertainty. Consequently, the agent
imposes a penalty based on relative entropy and chooses some other density π̂(ϵt ) in proximity toπ(ϵt ). Then, the agent’s preferences over consumption streams can be expressed by the following
value function recursion:
[

(

Ut+1
Ut = ct − βθ ln Et exp −
θ

)]

,

(3)

where Ut is the value function, β is the discount factor, and θ is the penalty parameter attached to
the relative entropy. In their framework, a smaller value of the parameter θ implies a higher degreeof agent concern about model misspeciﬁcation (see Maccheroni et al. (2006) for details).

4

The Barillas–Hansen–Sargent formula for measuring welfare gains from the removal of consumption ﬂuctuations is based on this value function recursion. The idea is to consider the adjustment toinitial consumption that makes the agent indiﬀerent between a deterministic consumption path andrisky consumption paths (1) or (2). Additionally, let {cdt } denote the deterministic consumption
path that has the same mean as the risky consumption path. Then, the Barillas–Hansen–Sargent
formula is given by
βσϵ2
β σϵ2
+
1−β 2
2θ(1 − β)2

(4)

β
σϵ2
βσϵ2
+
1 − βρ2 2
2θ(1 − βρ)2(5)

λ ≡ c0 − cd0 =for the random walk model, and byλ ≡ c0 − cd0 =for the trend stationary model (see Table 3 of Barillas et al. (2009)). The ﬁrst term on the righthand side of the formulas corresponds to Lucas’s (1987, 2003) welfare-gain measure (multiplied byβ/(1 − β) or β/(1 − βρ2 )) under log utility, i.e., the welfare gain from eliminating consumption risk.
The second term captures the welfare gain from eliminating model uncertainty.

2.2

Welfare-Gain Calculations Based on Closed-Form Solutions

To implement welfare-gain calculations using the formula above, it is necessary to set the parameterθ, the magnitude of which is associated with the degree of the agent’s fear of model misspeciﬁcation.
The strategy for quantifying that degree developed in the literature (see, e.g., Anderson et al. (2003)
and Hansen and Sargent (2008)) is to utilize detection error probabilities. The procedure relies
entirely on simulation. Barillas et al. (2009) use this simulation-based method in calculating thewelfare gain λ. Alternatively, we use an analytical method, which combines Okubo’s (2018a) closedform solutions for the detection error probabilities with the Barillas–Hansen–Sargent formula.
The main point of the method is that the overall detection error probability developed in Hansen
and Sargent (2008) (denoted by p(θ−1 )), which measures the degree to which the agent fears model5

misspeciﬁcation, can be represented in a closed form as
( √
)


σ
T
ϵ

−1

Φ −
θ
for the random walk model,



2
1−β

p(θ−1 ) =
( √
)



T −1 σϵθfor the trend-stationary model, Φ −2
1 − βρ

(6)

where Φ(·) is the standard normal cumulative distribution function and T is the length of time
period (more precisely, the number of observations on ∆ct in the current setting) (see footnote
3 of Okubo (2018a) for the intuition of why this formula holds, and see Okubo (2018b) for thederivation). Thus, for a given value of p(θ−1 ) (denoted by p(θ−1 )), we can write the Barillas–
Hansen–Sargent formula as

λ=









β σϵ −1 ( −1 )
β σϵ2
√ Φ
−
p(θ )
1−β 2
1−β T








β
σϵ −1 ( −1 )
β
σϵ2
√
−
Φ
p(θ )
1 − βρ2 2
1 − βρ T

for the random walk model,(7)
for the trend-stationary model,

where Φ−1 (·) is the inverse function of Φ(·).
The advantages of using formula (7) are twofold.3 First, it facilitates calculations because
there is no need to implement simulation.4 Second, it makes the results easier to interpret in
two aspects. The ﬁrst is that the formula clariﬁes that the welfare gain from eliminating model
uncertainty is a function of the volatility parameter σϵ and the length of time period T . For example,
formula (7) tells us that the length of the sample period used for analysis must be common to all
countries in making cross-country comparisons. In contrast, formulas (4) and (5) do not provide
any such guidance. The second aspect is that the formula clariﬁes how the degree of the agent’s
fear of model misspeciﬁcation aﬀects the welfare gain from eliminating model uncertainty through()

the term Φ−1 p(θ−1 ) . To see this, let p(θ−1 ) = 0.5. This means that there is no fear of model
misspeciﬁcation. Then, the second term on the right-hand side of formula (7) is clearly zero because3
Okubo (2019) shows that there is another advantage of formula (7), as it allows us to compute standard errorsof λ, λf ear , and λnof ear by applying the delta method.
4
In the simulation-based method, the overall detection error probability function (i.e., the values of p(θ−1 ) for
various values of θ−1 ) must be computed by simulation. See Chapter 9 of Hansen and Sargent (2008) or Section 2 ofOkubo (2018a) for an exposition of the simulation procedure. The use of formula (6) enables us to skip this step.

6Φ−1 (0.5) = 0. Thus, the welfare-gain measure λ coincides with Lucas’s (1987, 2003) calculation(multiplied by β/(1 − β) or β/(1 − βρ2 )). In the case of 0 < p(θ−1 ) < 0.5 (which means that(the agent fears model misspeciﬁcation), the term Φ−1 p(θ−1 ))takes a negative value. Hence,the second term is positive as long as the agent fears model misspeciﬁcation. It determines anincrement in the welfare gain associated with a change in an amount of model uncertainty. Forease of notation, we refer to the ﬁrst and second terms on the right-hand side of formula (7) as
λnof ear and λf ear , respectively.
As is clear from a comparison between the two measures in formula (7), if ρ is close to one, thenthe measure for the trend stationary model is approximately equal to that for the random walkmodel. Otherwise, because β/(1 − βρ2 ) < β/(1 − β) and β/(1 − βρ) < β/(1 − β), λnof ear and λf ear(the ﬁrst and second terms, respectively, on the right-hand side) for the trend stationary model aresmaller than those for the random walk model, other things being equal.
In our welfare-gain calculations, we must give the overall detection error probability, p(θ−1 ).
Following the literature,5 we choose p(θ−1 ) = 0.05, 0.10, and 0.20 in our empirical analysis.

3

The Data

The data are from the World Bank’s World Development Indicators (WDI) database. The WDI
data are annual and the sample period that we use is 1970–2018.6 One advantage of using thisdata is that they cover many more countries relative to those from other international databases.
For every country, we require data on private consumption, total population, and purchasing power
parity (PPP)-converted GDP. After removing the countries with missing data for the sample period,
our sample consists of 64 countries. A list of the countries included is provided in Appendix A and5

For example, Barillas et al. (2009, p. 2405) state: “. . . we think it is sensible for a decision maker to want to guard
against possible misspecications whose detection error probabilities are 0.2 or even less.” Meanwhile, Ljungqvist andSargent (2018, p. 584) state: “From our own experience ﬁtting models to data, a person whose specication doubts
include perturbed models with a detection error of .25 or .1 or even .05 could be said to have a plausible amount ofmodel uncertainty.”
6
The WDI data are available from 1960. However, for many countries including developed ones, data on private
consumption start from 1970. Thus, we set 1970 as the beginning of the sample period.

7

a detailed description of the data is given in Appendix B.
For each country, we use real private consumption per capita as consumption in the model.
In the WDI data, consumption is measured as total consumption expenditure for all countries.
Unfortunately, separate data on nondurable and durable consumption expenditure are not available.
Hence, an important caveat to our analysis is that estimates of the volatility parameter σϵ may
become somewhat higher than those based only on expenditure on nondurables and services becauseof the lumpiness of durable consumption.
We classify the countries in the sample into three groups (i.e., poor, emerging, and rich) bystages of economic development. Following Uribe and Schmitt-Grohé (2017), we use the geometricaverage of PPP-converted GDP per capita in 2011 US dollars over the period 1990–2018 as themeasure of development. We deﬁne poor, emerging, and rich countries with the geometric averageof PPP-converted GDP per capita up to 3,000 dollars, between 3,001 and 25,000 dollars, and morethan 25,000 dollars, respectively. Our sample contains 10 poor countries, 28 emerging countries,
and 26 rich countries.
As pointed out by Uribe and Schmitt-Grohé (2017), among others, the choice of thresholds forthe classiﬁcation is somewhat arbitrary. Another classiﬁcation adopted by the World Bank is also
applicable. It divides countries into four groups (i.e., high, upper-middle, lower-middle, and low
income) based on gross national income per capita.7 Table 1 shows the relationship between thetwo classiﬁcation schemes using a cross tabulation. Combining the upper-middle and lower-middle
income groups into one middle income group, we can see that our three groups (rich, emerging,and poor) almost coincide with the high, middle, and low income groups based on the World Bank
classiﬁcation. Thus, we concentrate only on our PPP-converted GDP-based classiﬁcation below.

7
The thresholds for the four groupings are updated every year. For more information on that method, see theWorld Bank data help desk (https://datahelpdesk.worldbank.org/).

8

Table 1: Relationship between Two Classiﬁcations

Rich
Emerging
Poor
Total

The World Bank classiﬁcation by income
High Upper-middle Lower-middle Low
26
0
0
0
3
16
9
0
0
0
4
6
29
16
13
6

Total
26
28
10
64

Note: The number of countries is shown in each cell. The classiﬁcation (i.e., rich, emerging, and poor) is based on thegeometric average of PPP-converted GDP per capita in 2011 US dollars over the period 1990–2018. Following Uribeand Schmitt-Grohé (2017, Chapter 1), we deﬁne poor, emerging, and rich countries as those for which the geometricaverage of PPP-converted GDP per capita is up to 3,000 dollars, between 3,001 and 25,000 dollars, and more than
25,000 dollars, respectively. See Appendix A for details of the sample countries.

4

Results

4.1

Welfare Gains in Rich, Emerging, and Poor Countries

We begin by looking at a simple scatterplot between welfare gains and a measure of economic
development. In what follows, we set the discount factor and the sample size to β = 0.950 and T =48, respectively. The choice of the discount factor follows Ellison and Sargent (2015). The valuesof the volatility and autoregressive parameters, σϵ and ρ, are set to maximum likelihood estimatespresented in Appendix Table B. To save space, we provide results only for the case of a detection
error probability of 0.10.
Figures 1 and 2 plot the welfare gains against the geometric average of PPP-converted GDP
per capita over the period 1990–2018. Panels (a), (b), and (c) of each ﬁgure correspond to thecases where the vertical axis indicates λ, λf ear , and λnof ear , respectively. Figure 1a, which isbased on the random walk model, shows that there is a negative relationship between the welfare
gains and the measure of economic development: rich countries, i.e., countries at higher stages of
economic development, have lower welfare gains from eliminating consumption ﬂuctuations. The
comparison between Figures 1b and 1c also indicates that this negative relationship arises mainly
from the negative relationship between the welfare gain from eliminating model uncertainty andthe measure of economic development. Figure 2 shows that these results are also observable for9the welfare gains based on the trend stationary model.
As is clear from Figure 1, there are three extreme values in the random walk case. In thefollowing analysis, we present results both including and excluding the three countries (Congo,
Puerto Rico, and Togo) that produce those extreme cases to illustrate their inﬂuence. Furthermore,
there are two inappropriate cases in the trend stationary model, Bolivia and the Philippines, as
their autoregressive parameters are above one, as conﬁrmed in Appendix Table B. Figure 2c showsthat, of the two cases, one leads to a negative value of λnof ear . Therefore, for the trend stationarymodel, we present results both including and excluding these two countries.
Table 2 reports the mean and standard deviation of the welfare gains for each group as well as
estimates for the United States. Panels A and B of Table 2 show the results from the random walkand trend stationary models, respectively. The second column for each group (i.e., columns (3),(5), (7), and (9)) reports the results for the sample excluding the extreme or inappropriate cases.
The standard deviations are shown in square brackets.
Table 2 reveals three interesting facts. First, comparisons within groups show that the welfare
gain from eliminating model uncertainty, λf ear , is considerably higher than that from eliminating
consumption risk, λnof ear . For example, the average λf ear -to-λnof ear ratio at p(θ−1 ) = 0.10 for all
countries is approximately 5 (= 0.160/0.031) and 7 (= 0.141/0.020) in the random walk model forthe samples including and excluding outliers, respectively, whereas it is almost 10 (= 0.053/0.005
and 0.049/0.005 for the two samples) in the trend stationary model.
Second, comparisons between groups show that the welfare gains, λ and λf ear , are at least
two times higher on average in emerging or poor countries than in rich countries. For the case ofp(θ−1 ) = 0.10 in Panel A, for example, the average total welfare gain for rich countries is λ = 0.108
and 0.086 for the samples including and excluding outliers, respectively, whereas it is λ = 0.229
and 0.198 for emerging countries and λ = 0.302 and 0.256 for poor countries. This fact is true forthe other values of detection error probability, irrespective of whether the random walk or trend
10(a) Welfare gains from eliminating consumption ﬂuctuations, λ(b) Welfare gains from eliminating model uncertainty, λf ear(c) Welfare gains from eliminating consumption risk, λnof ear

Figure 1: Relationship between welfare gains based on the random walk model and PPP-converted
GDP per capita under the 10% detection error probability
11

(a) Welfare gains from eliminating consumption ﬂuctuations, λ(b) Welfare gains from eliminating model uncertainty, λf ear(c) Welfare gains from eliminating consumption risk, λnof ear

Figure 2: Relationship between welfare gains based on the trend stationary model and PPPconverted GDP per capita under the 10% detection error probability
12

Table 2: Welfare Gains in Rich, Emerging, and Poor Countries
Detection
error prob

US
(1)

All
(2)

(3)

Mean and standard deviation
Rich
Emerging
(4)
(5)
(6)
(7)

Poor
(8)

(9)

A. Results based on the random walk model
λ
5%0.077 0.237
0.201[0.212] [0.130]10%0.060 0.192
0.161[0.178] [0.106]
20%
0.041 0.136
0.113[0.136] [0.078]
λf ear5%
0.074 0.206
0.180[0.157] [0.106]
10%
0.058 0.160
0.141[0.122] [0.082]
20%
0.038 0.105
0.092[0.080] [0.054]
λnof ear0.003 0.031
0.020[0.058] [0.025]# of countries
64
61

0.136
[0.140]0.108[0.116]0.075[0.088]0.123[0.106]0.096[0.083]0.063[0.054]0.012[0.034]26

0.109[0.038]0.086[0.030]0.059[0.021]0.104[0.034]0.081[0.027]0.053[0.018]0.006[0.004]25

0.283[0.239]0.229[0.201]0.164[0.156]0.243[0.170]0.189[0.132]0.124[0.087]0.040[0.071]28

0.247[0.144]
0.198[0.119]0.140[0.088]0.218[0.114]0.170[0.089]0.112[0.058]0.028[0.030]27

0.372[0.186]0.302[0.156]0.217[0.121]0.317[0.133]0.247[0.103]0.162[0.068]0.054[0.054]10

0.318[0.078]0.256[0.064]0.181[0.047]0.280[0.062]0.218[0.048]0.143[0.032]0.038[0.016]
9

B. Results based on the trend stationary model
λ
5%0.028 0.073
0.067[0.062] [0.050]
10%
0.022 0.058
0.053[0.049] [0.040]
20%
0.015 0.040
0.037[0.034] [0.028]
λf ear5%
0.028 0.069
0.063[0.059] [0.045]
10%
0.022 0.053
0.049[0.046] [0.035]
20%
0.014 0.035
0.032[0.030] [0.023]
λnof ear0.001 0.005
0.005[0.006] [0.006]# of countries
64
62

0.042
[0.023]0.033[0.018]0.022[0.012]0.040[0.022]0.031[0.017]0.021[0.011]
0.002[0.001]
26

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–

0.098
[0.076]0.077[0.060]0.053[0.041]
0.091[0.073]0.071[0.057]0.047[0.037]0.007[0.007]28

0.085[0.057]0.068[0.046]0.047[0.032]0.079[0.051]
0.061[0.039]0.040[0.026]0.007[0.007]26

0.086[0.056]0.069[0.045]0.048[0.032]0.079[0.051]
0.062[0.040]0.041[0.026]0.007[0.006]
10

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–

Note: Column (1) shows the welfare gains for the United States. In column (1), there are cases in which the sumof the estimates λnof ear and λf ear is not completely equal to the estimate of λ due to rounding errors. Columns(2)–(9) show the mean and standard deviation of the welfare gains for each group. The standard deviation is insquare brackets. The second column for each group (columns (3), (5), (7), and (9)) shows the results for the sample
excluding outliers. The symbol “–” denotes that the mean and standard deviation are the same as those in the ﬁrst
column for each group.

13

stationary model is used.
Third, comparisons between column (1) and the others indicate that many other countries havewelfare gains that are considerably higher than those of the United States. As shown in Appendix
Tables C.1 and C.2, there are only four countries with smaller welfare gains than the United Statesin the random walk model and nine countries in the trend stationary model.
Our estimated value of λnof ear for the United States in column (1) is comparable with Lucas(2003), who reports a welfare gain from eliminating consumption risk of 0.0005 using annual US
data for the period 1947–2001. If we use another parameter estimate (σϵ = 0.022) mentioned infootnote 5 of Lucas (2003), then his welfare-gain estimate becomes 0.0002 (≈ (0.022)2 /2). For
direct comparison with these estimates by Lucas (2003), for example, in the random walk case,
we must divide λnof ear by 19 (= 0.950/(1−0.950)), as indicated by formula (7). This adjustment
produces a welfare-gain estimate of approximately 0.0002 (≈ 0.003/19).
One fact that emerges clearly from the ﬁgures and table is the negative relationship between thewelfare gains and the extent of economic development, although it does not indicate any causation.
In the next subsection, we further examine this negative relationship.

4.2

Evidence from Cross-Sectional Regressions

At least two factors must be controlled to establish the negative relationship between welfare gains
and the extent of economic development noted in the previous subsection. The ﬁrst is that alarger country size (i.e., a larger population) may be associated with lower output volatility (see,e.g., Furceri and Karras (2007) for empirical evidence and di Giovanni and Levchenko (2012) for a
detailed discussion of this mechanism). This implies that, in our context, a larger country size may
reduce the welfare cost of business cycles. The second factor is that diﬀerences in international
trade openness may aﬀect business cycles because even a shock common to all countries can havediﬀerent impacts across countries through trade (see, e.g., Kose et al. (2003), di Giovanni and14

Levchenko (2009, 2012) and Haddad et al. (2013)). Uribe and Schmitt-Grohé (2017) examine theeﬀects of these two factors on business cycles using regressions based on the standard deviation of
output. They conclude that there are both country-size and trade-openness eﬀects, but that they
are less statistically signiﬁcant than economic development levels. In this subsection, we estimate
the same type of regression for our sample.
The key to our speciﬁcation is that the standard deviation of output, which is a typical measureof business cycles in the literature, is replaced with our welfare-gain measures. The regression that
we run across countries is
λi = β0 + β1 · ln(PPP GDPi ) + β2 · ln(popi ) + β3 · opennessi + ui ,i = 1, . . . , 64,(8)where λi is the welfare gain for country i, PPP GDPi is the geometric average of PPP-converted
GDP per capita over the period 1990–2018 for country i, popi is country i’s total population in2018 (the end of the sample period), opennessi is country i’s trade openness in 2018, and ui isthe error term.8 As the measure of trade openness, we use the ratio of exports plus imports to
GDP (see Appendix B for more details). In the literature, countries with higher values of thismeasure are interpreted as being more open to trade. We also estimate regressions in which λf ear,iand λnof ear,i are used as the dependent variable. There are three cases for each of λi and λf ear,ibecause we consider three diﬀerent values for the detection error probability (i.e., p(θ−1 ) = 0.05,
0.10, and 0.20). As before, we report results only for the case of p(θ−1 ) = 0.10 below.
An important caveat to this speciﬁcation is that there may be a negative and linear relationshipbetween the country-size and trade-openness variables (see, e.g., Alesina and Wacziarg (1998) andRam (2009) for empirical evidence and discussions).9 This point has not been considered in Uribeand Schmitt-Grohé’s regression. To see this eﬀect, we consider three types of regressions with log
8Here, we use variations in 2018 for the country-size and trade-openness variables. We also tried using the averages
over the sample period. However, as the results were similar to those reported below, we have omitted them fromthe paper.
9
When we implement a regression of the trade-openness variable on a constant and the country-size variable as in
Alesina and Wacziarg (1998), the coeﬃcient of the country-size variable is negative and signiﬁcant at the 1% level.

15

population, openness, and both of these control variables (as in (8)) for each of the three dependent
variables, λi , λf ear,i , and λnof ear,i , as well as a base regression with no such control variables. Hence,
12 regressions are run for each of the random walk and trend stationary models.10
The regression results for the random walk model are presented in Table 3. Panel A of the table
reports the results from all 64 countries, whereas Panel B gives the results for 61 countries, after
excluding the three outliers mentioned in the previous subsection. The ﬁrst regression for the total
welfare gain λ, reported in column (1) in Panel A, shows that the slope for the log of average PPPconverted GDP per capita is negative and signiﬁcant at the 1% level. Columns (2)–(4) conﬁrm that
including additional variables that control for country size and trade openness mitigates an upwardbias in the slope estimate of interest. They also show that the coeﬃcient on the trade-openness
variable is not statistically signiﬁcant if it is added together with the country-size variable.
The results for λf ear (columns (5)–(8)) and λnof ear (columns (9)–(12)) indicate that the slope
for the log of average PPP-converted GDP per capita is steeper in the λf ear regressions than in
the λnof ear regressions. The slope estimates range from −0.059 to −0.051 in the λf ear regressions,whereas they fall in a range from −0.020 to −0.017 in the λnof ear regressions. In other words,
higher levels of economic development (higher income per capita) tend to be associated with lower
welfare costs of model uncertainty, and this negative relationship is stronger for welfare costs ofmodel uncertainty than for welfare costs of consumption risk.

10

We do not include dummy variables for regions in the regressions. Note that diﬀerences in stages of economic
development are closely associated with the regions to which countries belong. This holds true for our sample.
According to the classiﬁcation of regions by the World Bank, of the 26 rich countries in our sample, 24 countries
are located in three regions (“Europe and Central Asia,” “East Asia and Paciﬁc,” and “North America”). Of the 28
emerging countries, 17 countries are in the two regions of “Latin America and the Caribbean” and the “Middle Eastand North Africa,” while the 10 poor countries are in the two regions of “South Asia” and “Sub-Saharan Africa.”
There is no overlap of regions between the rich and poor countries and the overlaps between the emerging and poor
countries and between the rich and emerging countries are very restricted. Thus, the inclusion of such region-dummy
variables controls for diﬀerences in economic development. If we include the region-dummy variables in equation (8),the coeﬃcient of the log of average PPP-converted GDP per capita becomes insigniﬁcant.

16

17

(1)

(2)

(3)

-0.076(0.020)-0.045(0.013)0.008(0.028)
0.149
0.331

(4)

-0.046(0.006)-0.020(0.006)

0.061
0.463

0.067
0.350

0.098
0.378

0.108
0.238

-0.043(0.007)-0.056(0.011)-0.033(0.009)-0.051(0.011)(5)

0.017(0.007)
0.067
0.368

-0.047(0.007)0.035(0.019)
0.106
0.274

-0.059(0.014)

Dependent variable
λf ear(6)(7)-0.046(0.007)-0.021(0.007)-0.003(0.011)
0.062
0.464

-0.057(0.013)-0.032(0.009)0.004(0.018)
0.099
0.378

(8)

0.021
0.249

-0.011(0.002)

0.055
0.113

-0.017(0.006)

(9)

0.020
0.369

-0.012(0.002)-0.006(0.002)

0.052
0.223

-0.019(0.006)-0.014(0.005)0.004(0.002)
0.021
0.263

-0.012(0.003)0.017(0.012)
0.054
0.150

-0.020(0.008)

λnof ear(10)(11)-0.011(0.002)-0.007(0.002)-0.002(0.003)
0.020
0.371

-0.019(0.007)-0.013(0.004)0.004(0.010)
0.052
0.224

(12)

coeﬃcient on the constant term in the regression is not reported. SER is the standard error of the regression.

2018, and openness is the trade openness measured by the ratio of exports plus imports to GDP in 2018. See Appendix B for the details of each variable. The

is the geometric average of PPP-converted GDP per capita over the period 1990–2018, a measure of economic development, pop is the total population in

Note: Heteroskedasticity-robust standard errors are in parentheses. The welfare gains, λ and λf ear , are those for a detection error probability of 0.10. PPP GDP

B. Results based on 61 countries excluding outliersln(PPP GDP) -0.053-0.058-0.058-0.057(0.009) (0.008) (0.010) (0.009)
ln(pop)-0.026-0.027(0.008)(0.009)
openness
0.021-0.005(0.010) (0.014)
SER
0.088
0.081
0.088
0.081
R2
0.329
0.445
0.346
0.446

A. Results based on all 64 countriesln(PPP GDP) -0.068-0.075-0.079(0.017) (0.017) (0.021)
ln(pop)-0.047(0.014)
openness
0.052(0.030)
SER
0.161
0.148
0.158
R2
0.198
0.331
0.235

Regressor

λTable 3: Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, Country Size, and Openness(Random Walk Model)

We ﬁnd the same results for the other cases of p(θ−1 ) = 0.05 and 0.20, although we do notreport the results here to save space (see Appendix Tables D.1 and D.2 of the separate appendix).
In addition, we obtain an additional ﬁnding that the slope for the log of average PPP-converted
GDP per capita becomes much steeper for p(θ−1 ) = 0.05; conversely, it is ﬂatter for p(θ−1 ) = 0.20.
This ﬁnding suggests that as agent fear of model misspeciﬁcation becomes milder, the negative
relationship between levels of economic development and the welfare gains of eliminating model
uncertainty becomes weaker.
A point to note concerning the regression results presented so far is that the sample size is
only 64 in our regressions; thus, the three outliers lying far above the other values may make theregression line steeper. Panel B of Table 3 conﬁrms that excluding the three countries with extreme
values of λ (i.e., Congo, Puerto Rico, and Togo) reduces the slope estimate for the log of average
PPP-converted GDP per capita in terms of the absolute value. Except for this reduction in themagnitude of the slope, however, the regression results in Panel B are in line with those in Panel
A. Thus, the previous ﬁndings hold in the regressions without the outliers.
Do they also hold in the trend stationary model? Table 4 presents regression results based on thewelfare-gain calculations in the trend stationary model. Panel A gives results for all 64 countries,
whereas Panel B gives results for 62 countries, excluding the two countries (i.e., Bolivia and thePhilippines) for which the autoregressive parameters are above one. Table 4 and the results shownin Appendix Tables E.1 and E.2 conﬁrm the same three ﬁndings. First, the λ, λf ear , and λnof ear
regressions all produce negative slopes for the log of average PPP-converted GDP per capita.
Second, the negative slopes are steeper in the λf ear regressions than in the λnof ear regressions.
Third, as the agent’s fear of model misspeciﬁcation becomes milder, the negative slopes become
ﬂatter. Thus, the regression results of Table 4 lend additional support to the negative relationshipbetween levels of economic development and the welfare gains of eliminating model uncertainty.

18

19

(1)

(2)

(3)

-0.017(0.005)0.000(0.005)0.009(0.005)
0.047
0.143

(4)

Note: See the note to Table 3.

B. Results based on 62 countries excluding outliersln(PPP GDP) -0.012-0.013-0.014-0.014(0.004) (0.004) (0.005) (0.004)
ln(pop)-0.004-0.003(0.003)(0.003)
openness
0.009
0.005
(0.004) (0.003)
SER
0.038
0.038
0.038
0.038
R2
0.127
0.152
0.148
0.158

A. Results based on all 64 countriesln(PPP GDP) -0.015-0.015-0.017(0.005) (0.005) (0.005)
ln(pop)-0.002(0.004)
openness
0.009(0.004)
SER
0.046
0.047
0.046
R2
0.129
0.132
0.143

Regressor

λ-0.011(0.004)-0.003(0.003)

0.033
0.137

0.033
0.118

0.044
0.114

0.043
0.114

-0.010(0.004)-0.013(0.004)-0.001(0.004)-0.013(0.004)(5)

0.007(0.004)
0.033
0.138

-0.012(0.004)0.008(0.004)
0.043
0.126

-0.015(0.005)-0.012(0.004)-0.002(0.003)0.005(0.003)
0.033
0.144

-0.015(0.005)0.001(0.005)0.009(0.005)
0.044
0.127

Dependent variable
λf ear
(6)
(7)
(8)

0.005
0.160

-0.002(0.0006)

0.005
0.148

-0.002(0.0005)

(9)

0.005
0.227

-0.002(0.0006)-0.001(0.0004)

0.005
0.229

-0.002(0.0005)-0.001(0.0004)

0.001(0.0007)
0.005
0.183

-0.002(0.0006)

0.001(0.0007)
0.005
0.168

-0.002(0.0006)

λnof ear(10)(11)-0.002(0.0006)-0.001(0.0004)
0.0003(0.0006)
0.005
0.228

-0.002(0.0006)-0.001(0.0004)
0.0001(0.0006)
0.005
0.229

(12)

Table 4: Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, Country Size, and Openness(Trend Stationary Model)

Finally, a few caveats are in order. The coeﬃcients themselves (especially their magnitudes)in the regressions cannot be compared directly between the random walk and trend stationarymodels. As shown in (7), the formulas for λ, λf ear , and λnof ear diﬀer between these models. The
main message here is that the negative relationship suggested in the previous subsection holds evenafter adding control variables, irrespective of whether the random walk or trend stationary modelis used. However, we do not intend to push the existence of a causal eﬀect from levels of economic
development to welfare gains too hard.

4.3

Interpreting the Diﬀerence between Rich, Emerging, and Poor Countries

The ﬁndings presented so far include: (i) the gains from eliminating consumption ﬂuctuations inemerging and poor countries are larger than in rich countries, and (ii) emerging and poor countries
have larger gains from eliminating model uncertainty, which is the main source for (i).
Why does this diﬀerence arise? As formula (7) indicates, the gain from reducing model uncertainty is proportional to the volatility of consumption. In fact, the diﬀerence is attributable to thecross-country diﬀerence in the magnitude of the standard deviation of consumption shocks, σϵ (seethe last three rows of Appendix Table B). Therefore, the question can be rephrased as a familiar
one: why is consumption in emerging and poor countries more volatile than in rich countries?
A large literature exploring business cycles in emerging countries provides at least two viewson this question. One is that emerging countries are subject to permanent and/or more volatile
exogenous shocks, including permanent and transitory productivity shocks, as well as shocks tointerest-rates, preferences, country premiums, and terms-of-trade (see, e.g., Aguiar and Gopinath(2007), Neumeyer and Perri (2005), Uribe and Yue (2006), Garcı́a-Cicco et al. (2010), FernándezVillaverde et al. (2011), Chang and Fernández (2013), Fernández and Gulan (2015), Ben-Zeev etal. (2017), and Schmitt-Grohé and Uribe (2018)). Another view, which is not mutually exclusive,
concerns the existence of ﬁnancial frictions in emerging economies (see, e.g., Garcı́a-Cicco et al.

20

(2010), Chang and Fernández (2013), Álvarez-Parra et al. (2013), Hevia (2014), and Fernández andGulan (2015)). The main message from this line of research is that ﬁnancial frictions in emerging
economies amplify the impact of the exogenous shocks that they encounter.
In light of these views, our ﬁndings, summarized in (i) and (ii) above, may be interpreted asfollows: (i) there are larger welfare gains in countries with more ﬁnancial frictions (i.e., emerging
and poor countries) than in countries with less ﬁnancial frictions (i.e., rich countries); (ii) the
larger gains arise because the inﬂuence of model uncertainty is ampliﬁed by the large consumption
volatility in emerging and poor countries that is a result of ﬁnancial frictions.11 Thus, the existing
literature suggests that we may be able to focus on ﬁnancial frictions rather than on economic
development. However, as the issues are complicated, a reinterpretation that focuses on ﬁnancial
frictions is left to future research.12

5

The Eﬀect of Idiosyncratic Risk Revisited

Our ﬁndings reinforce those of Barillas et al. (2009). As mentioned in the Introduction, anotherposition put forward by Ellison and Sargent (2015) is that a model incorporating both idiosyncraticrisk and agent fear of model misspeciﬁcation yields large welfare-gain estimates even under log
preferences. Therefore, a question that arises is to what extent the inclusion of idiosyncratic risk
aﬀects the contribution of model uncertainty to welfare gains. This section compares the welfare
gains computed with the closed-form solution of this paper and the version obtained in Ellison andSargent (2015) to see the eﬀect of idiosyncratic risk.

()

Note the second term of formula (7) can be regarded as the volatility σϵ multiplied by the term Φ−1 p(θ−1 ) .
12
According to this line of thinking, the following questions arise: why do emerging and poor countries lack ﬁnancial
markets to diversify shocks and relevant policies to counter such shocks? Are the weak shock absorbers in emerging
and poor countries the result of their fragile ﬁnancial systems and their economic and political institutions? Answering
this question is one of the most important subjects in open-economy macroeconomics, but it is beyond the scope of
this paper. We hesitate to draw further implications about such questions from our welfare-gain estimates.
11

21

Following Ellison and Sargent (2015), we assume the following consumption process:cit = ct + δti ,
√
∆ct =
ϵw ,
√ 1t
∆δti =
ϵw2t ,
[

w1t
w2t

]

([

∼N

g − τ12 /2−τ22 /2

] [

,

(9)

τ12 0
0 τ22

])

,

where cit is log consumption for an individual i and δti is an individual-speciﬁc (i.e., idiosyncratic)
shock. The value function recursion that Ellison and Sargent (2015) used isŨt = (1 − β)cit −

[
(
)]
1
ln Et exp −σβ Ũt+1 ,σ(10)where Ũt and σ in (10) are related to Ut and θ in (3) as Ut = Ũt /(1 − β) and σ = 1/βθ(1 − β),respectively. In their paper, σ is called the robustness parameter.
The crucial point here is that speciﬁcation (9) can be rewritten as a random walk model,
cit+1 = cit +√ϵ(w1t+1 + w2t+1 ). This means that we can derive a closed-form solution for welfare

gains in the same way as for (7). Note that the case of no idiosyncratic risk that we have discussed
so far corresponds to cit = ct and ct+1 = ct +√ϵw1t+1 . Therefore, using the notation of Ellison

and Sargent (2015), our welfare-gain measure can be expressed by replacing σϵ2 with ϵτ12 (i.e., thevariance of aggregate shocks√
ϵw1t+1 ) as
√

(
)
ϵτ12
β ϵτ12β√ Φ−1 p(σ) .
λ=
−
1−β 2
1−β T

(11)

In computing welfare gains under the model with idiosyncratic risk, the deﬁnition of an economy
without aggregate risk adopted by Ellison and Sargent (2015) is the same as that of De Santis(2007). That is, aggregate consumption growth in that economy equals the expected growth ofaggregate consumption in the economy with aggregate shocks. This is equivalent to considering adeterministic process only for aggregate consumption ct in (9). Then, the welfare-gain measure isλ=()β ϵτ12β1ϵτ12√ √Φ−1 p(σ) .
−1−β 2
1 − β T ϵ(τ 2 + τ 2 )
1

22

2

(12)

Table 5: The Eﬀect of Idiosyncratic Risk
50%

Detection error probability
45%
40%
20%
10%

A. Model with aggregate shocks only
λ0.80% 1.63% 2.48% 6.38%
λnof ear
0.80
0.80
0.80
0.80
λf ear
0
0.83
1.68
5.58

9.30%
0.80
8.50

5%

11.71%
0.80
10.91

B. Model with both aggregate and idiosyncratic shocks
λ0.80% 1.03% 1.27% 2.35% 3.17% 3.84%
λnof ear
0.80
0.80
0.80
0.80
0.80
0.80
λf ear
0
0.23
0.47
1.55
2.37
3.04
Note: The results in panels A and B are based on equations (11) and (12), respectively. The welfare-gain estimates
are multiplied by 100 to express them in percentage terms.

The derivation of (12) is given in Appendix C. We can establish the validity of this formula in two
ways. First, the closed-form solution of the detection error probability p(σ) used in this formula
replicates the calibration results of the robustness parameter σ reported in Table 2 of Ellison andSargent (2015) (see Appendix D for details). Second, as shown below, this formula replicates theresults for the case of log preferences reported in Table 3 of Ellison and Sargent (2015).
Table 5 reports the values of welfare gains based on these two formulas. Here, we use thefollowing parameter settings based on Ellison and Sargent (2015):√√ϵτ1 = 0.029; ϵτ2 = 0.10; β =0.95; and T = 69. Panels A and B of the table show the results from (11) and (12), respectively.
In addition, for ease of comparison with Table 3 in Ellison and Sargent (2015), we present resultsfor detection error probabilities of 0.50, 0.45, and 0.40, as well as for the previous three cases. We
report the results in percentage terms.13 The welfare-gain measures in Panel B, λnof ear and λf ear ,correspond to “Baseline cost” and “Contribution of fear of model misspeciﬁcation,” respectively,
in Table 3 of Ellison and Sargent (2015).
The major ﬁndings are as follows. First, the result for the no idiosyncratic risk case in Panel
13
The welfare-gain measure is multiplied by 100 to express it as a percentage because it is deﬁned as the diﬀerenceof log consumption.

23

A of Table 5 is comparable with that for the United States in column (1) of Table 2 based on the
WDI data. Two factors account for the diﬀerence between them: one is the sample size used in
calculating detection error probabilities; the other is the magnitude of the variance of aggregate
shocks. In our case, where the sample size T increases from 48 to 69 and the estimate of σϵ
increases from 0.016 to 0.029, the latter eﬀect dominates the former one. Hence, the estimates of
λf ear become higher than those based on the WDI data.
Second, the results for detection error probabilities of 0.50, 0.45, and 0.40 in Panel B of Table 5
are entirely consistent with those for log preferences in Table 3 of Ellison and Sargent (2015). For
detection error probabilities of 0.20, 0.10, and 0.05 (which are not reported in Ellison and Sargent(2015)), as expected from our formula (12), smaller detection error probabilities (i.e., larger fears
of model misspeciﬁcation) lead to larger estimates of λf ear . For example, λf ear rises from 0.23%
at p(σ) = 0.45 to 2.37% at p(σ) = 0.10.
Third, the comparison between Panels A and B of Table 5 indicates that the exclusion of
idiosyncratic risk by assumption results in overestimating the contribution of model uncertainty to
welfare gains. For example, for a detection error probability of 0.10, the value of λf ear rises from
2.37% in Panel B (the idiosyncratic risk case) to 8.50% in Panel A (the no idiosyncratic risk case).
However, this ﬁnding should not be taken as evidence against the claim of Ellison and Sargent(2015) because, as is clear from Panel B, λf ear still raises the overall value of the welfare gain
substantially under the idiosyncratic risk model.
The result that the value of λ is smaller for the idiosyncratic risk case (Panel B) than for theno idiosyncratic risk case (Panel A) depends on the assumption of log preferences. We can see
from Table 3 of Ellison and Sargent (2015) that the value of λ increases by relaxing the assumptionof log preferences. For example, the value of λ at p(σ) = 0.45 is 1.03% in Panel B of our table(or equivalently, the ﬁrst panel of Table 3 in Ellison and Sargent (2015)). By contrast, it rises to
1.428% under constant relative risk aversion preferences with a relative risk aversion coeﬃcient of24

1.5, owing to the contribution of idiosyncratic risk and the combined eﬀect of idiosyncratic risk andfear of model misspeciﬁcation (see the third panel of Table 3 in Ellison and Sargent (2015)). Thus,by relaxing the assumption of log preferences, the value of λ at p(σ) = 0.45 becomes closer to thevalue of 1.63% shown in Panel A of our table.
To summarize, if we restrict our speciﬁcation of consumption processes to the random walk,
trend stationary, and idiosyncratic risk models used in the literature, the estimates of the welfare
gain from the random walk model represent the upper bounds. Although the result here is limited
to the United States, we can state with certainty that the welfare gain from eliminating model
uncertainty is not trivial, irrespective of the presence or absence of idiosyncratic risk.

6

Conclusion

A broad range of countries tends to have large welfare gains from eliminating model uncertainty
relative to the United States. In addition, there is a negative relationship between the welfare
gains and the level of economic development. These ﬁndings indicate that the welfare gains from
eliminating aggregate consumption ﬂuctuations are much larger than indicated by the previous
Lucas-style calculations, and that countries at higher stages of economic development have lower
welfare gains because the gains from eliminating model uncertainty are smaller in those countries.
For developing countries, we agree with Pallage and Robe’s (2003, p. 695) conclusion thatthe welfare cost in those countries is “a large multiple of that in the United States.” However,
the evidence presented in this paper shows that the source for it is model uncertainty, not only
consumption risk drawn from a known probability distribution. Our ﬁnding concerning the crosscountry diﬀerence in the welfare gains from eliminating model uncertainty suggests that policies thatchange the amount of model uncertainty may have signiﬁcantly diﬀerent eﬀects among countries.

25Appendix
A. List of Sample Countries and Classiﬁcation
The sample countries and their PPP-converted GDP-based classiﬁcations are as follows. The
classiﬁcation by the World Bank is presented in Appendix Table A.
Poor Countries: Benin, Burkina Faso, Lesotho, Mali, Rwanda, Togo, Cameroon, Kenya, Madagascar, Bangladesh
Emerging Countries: Bolivia, Chile, (Republic of) Congo, Costa Rica, Dominican Republic,
Ecuador, El Salvador, Gabon, Guatemala, Mauritania, Nicaragua, Paraguay, Uruguay, Algeria,
Colombia, South Korea, Malaysia, Peru, South Africa, Sri Lanka, Thailand, Venezuela, Brazil,
Egypt, India, Indonesia, Mexico, the Philippines
Rich Countries: Austria, Belgium, Denmark, Finland, Greece, Hong Kong, Ireland, Israel,
Luxembourg, the Netherlands, New Zealand, Norway, Portugal, Puerto Rico, Singapore, Sweden,
Switzerland, Australia, Canada, France, Italy, Spain, the United Kingdom, Germany, Japan, theUnited States

B. Data Description
The World Development Indicators (WDI) database is available from the World Bank’s website(https://databank.worldbank.org/) as of March 2020. The deﬁnition of each variable used in thepaper is as follows.
Real private consumption per capita is based on “Household ﬁnal consumption expenditure(constant LCU)” (code: NE.CON.PRVT.KN) and “Population, total” (code: SP.POP.TOTL),
where LCU means that the data are in local currency. Final consumption expenditure includes
expenditure on durables as well as that on nondurables and services.
PPP-converted GDP per capita is “GDP per capita, PPP (constant 2011 international dollars)”(code: NY.GDP.PCAP.PP.KD). This series was available from 1990 to 2018 at the time of writing

26

(March 2020). Hence, its geometric average was calculated for the period 1990–2018, as describedin the text.
Trade openness is deﬁned as the ratio of exports plus imports to GDP. This variable wascalculated as the sum of “Exports of goods and services (% of GDP)” (code: NE.EXP.GNFS.ZS)
and “Imports of goods and services (% of GDP)” (code: NE.IMP.GNFS.ZS) divided by 100. These
data on exports and imports were not available for Venezuela in 2018. Therefore, 2017 values wereused for Venezuela instead.
C. Derivation of Equation (12)Consider the consumption process (9) reproduced here as
cit = ct + δti ,
√
∆ct =
ϵw ,
√ 1t
ϵw2t ,∆δti =
where

[

w1t
w2t

]

([

∼N

To simplify the notation below, let ϵ1t ≡g − τ12 /2−τ22 /2

(C1)

] [

,

τ12 0
0 τ22

])

.

√
√
ϵw1t and ϵ2t ≡ ϵw2t .

We ﬁrst derive a deterministic process for aggregate consumption, for which consumption growthis equal to the expected growth of the random process {Ct }, where Ct = exp(ct ). Because Ct+1 =exp(ct + ϵ1t+1 ) = Ct · exp(ϵ1t+1 ), the expected growth of Ct is E[Ct+1 /Ct ] = E[exp(ϵ1t+1 )] =)(exp E(ϵ1t+1 ) + 21 Var(ϵ1t+1 ) . Therefore, letting Ctd be the level of aggregate consumption in theeconomy without aggregate shocks, it follows that
(

(

)

(

d
√
Ct+1
1
τ12
=
exp
E(ϵ
)
+
Var(ϵ
)
=
exp
ϵ
g
−
1t+1
1t+1
2
2
Ctd

)

)

1
+ ϵτ12 .
2(C2)Thus, taking logarithms of both sides of (C2), we have√
cdt+1 = cdt + ϵ

(

τ2
g− 1
2

where cdt = ln Ctd .

27

)

1
+ ϵτ12 ,2(C3)

Next, we replace ct in (C1) with cdt and derive the analytical solution to the corresponding valuefunction recursion. More speciﬁcally, we consider the following consumption process:c̄it = cdt +(δti ,
)
2
√
τ
1
1
∆cdt =ϵ g−+ ϵτ12 ,2
2∆δti = ϵ2t .

(C4)Substituting the second and third equations of (C4) into the ﬁrst one (after taking the ﬁrstdiﬀerence), the consumption process can be rewritten as(√τ2c̄it+1 = c̄it + ϵ g − 12)1+ ϵτ12 + ϵ2t+1 .
2

(C5)

[
(
)]
1
ln Et exp −σβ Ūt+1 .
σ(C6)

The value function recursion deﬁned by this c̄it isŪt = (1 − β)c̄it −To solve for Ūt , guess Ūt = k0 + k1 c̄it . Then,Ūt+1 = k0 + k1 c̄[it+1 ,= k0 + k1(√
τ2c̄it + ϵ g − 1
2

)

]

1
+ ϵτ12 + ϵ2t+1 .
2(C7)

Substituting (C7) into (C6) yields
Ūt

[

(

[

{

√
c̄it + ϵ

(

)

]})]

τ2
1
=
exp −σβ k0 + k1,
g − 1 + ϵτ12 + ϵ2t+1
σ{
2
2
[
(
)
]}
√
1
1
τ2
= (1 − β)c̄it + β k0 + k1 c̄it + ϵ g − 1 + ϵτ12− ln Et [exp (−σβk1 ϵ2t+1 )] .
2
2
σ
(C8)
1
(1 − β)c̄it − ln Et

Noting that exp (−σβk1 ϵ2t+1 ) is a log-normally distributed random variable with mean −σβk1 E(ϵ2t+1 )and variance σ 2 β 2 k12 Var(ϵ2t+1 ) and rewriting the last term of the right-hand side, (C8) can be written as
{

Ūt

[

(

)

]}

(

)

√
√ τ2 1τ2
1
1= (1 − β)c̄it + β k0 + k1 c̄it + ϵ g − 1 + ϵτ12−
σβk1 ϵ 2 + σ 2 β 2 k12 ϵτ22 ,
2
2 )
σ
2 ] 2
[
(
2 + τ2√
τ1
1
2= [(1 − β) + βk1 ] c̄it + βk0 + βk1
ϵ g− 1+ ϵτ12 − σβk1 ϵτ22 .
2
2
2
(C9)

28

Matching the coeﬃcients in (C9) and Ūt = k0 + k1 c̄it , we have
k1 = 1,
k0 =

[

(

√
β
τ 2 + τ22
ϵ g− 11−β2Thus, we obtain

[

(

√
β
τ 2 + τ22Ūt =ϵ g− 1
1−β
2

)

)

]

1
1
+ ϵτ12 − σβϵτ22 .
2
2

(C10)

]

1
1
+ ϵτ12 − σβϵτ22 + c̄it .
2
2

(C11)

Following the same steps, we can derive the following solution to the value function recursion (10)(i.e., equation (11) of Ellison and Sargent (2015, p. 47)):
[

(

√
β
τ 2 + τ22Ũt =ϵ g− 1
1−β
2

)

]

1
− σβϵ(τ12 + τ22 ) + cit .
2

(C12)

Finally, we solve Ũ0 = Ū0 to obtain the welfare gain λ, deﬁned as the diﬀerence of the initialconsumption, ci0 − c̄i0 . This gives
λ ≡ ci0 − c̄i0 =

β
1
β 1 2
ϵτ1 +
σβ ϵτ12 .
1−β2
1−β 2

(C13)

One more step is required to obtain equation (12). As Okubo (2018a) showed, the overall
detection probability under the consumption process (C1) is given by√ √
2 + τ 2)
ϵ(τ
T
1
2
.
p(θ−1 ) = Φ −
θ−1

1−β

2

(C14)

A proof of (C14) is provided in Okubo (2018b) (in Appendix D, we present additional evidence that
demonstrates the validity of (C14), rather than outlining the proof). Because θ−1 = σβ(1 − β),(C14) can be rewritten as

( √

)

T √ 2
p(σ) = Φ −σβ ϵ(τ1 + τ22 ) .
2

(C15)

Therefore, given any value of the overall detection error probability (denoted by p(σ) in the text),
we have()
21√
σ = −√
Φ−1 p(σ) .
T β ϵ(τ 2 + τ 2 )1

Substituting (C16) into (C13) gives equation (12).
29

2

(C16)

D. Replication of the Robustness Parameter σThis appendix shows that our formula (C16) can replicate the calibration results of σ for the caseof log preferences reported in Table 2 of Ellison and Sargent (2015). Here, we use the following
parameter settings of Ellison and Sargent (2015):√ϵτ1 = 0.029;√
ϵτ2 = 0.10; β = 0.95; and T =69.
The calculation results of the robustness parameter σ based on (C16) are reported in Appendix
Table F. The ﬁrst row of the table presents the results from our formula, and the second row
presents the results from Table 2 of Ellison and Sargent (2015). As Φ−1 (0.5) = 0 by the deﬁnitionof the standard normal cumulative distribution function, it is obvious from (C16) that σ = 0 forp(σ) = 0.5. The other two cases are calculated as follows:12√
σ = −√ ·· (−0.1256613) ≈ 0.31
69 0.95 (0.029)2 + (0.10)2

for p(σ) = 0.45,

(D1)

1
2
√
σ = −√ ·· (−0.2533471) ≈ 0.62
69 0.95 (0.029)2 + (0.10)2

for p(σ) = 0.40.

(D2)

It turns out that the diﬀerences between Ellison and Sargent’s results and ours are due to rounding
errors.

30

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34

For online publication:

Appendix for “Model Uncertainty, Economic Development, and Welfare
Costs of Business Cycles”
Masakatsu Okubo †University of Tsukuba

This appendix has several tables.
• Table A lists the 64 countries in our sample and the country classification used in the paper,
along with the World Bank classification.
• Table B reports parameter estimates of the random walk and trend stationary models forindividual countries.
• Tables C.1 and C.2 report welfare-gain estimates for individual countries based on the random
walk and trend stationary models, respectively.
• Tables D.1 and D.2 report regression results using the welfare-gain estimates from the random
walk model under detection error probabilities of 0.05 and 0.20, respectively.
• Tables E.1 and E.2 report regression results using the welfare-gain estimates from the trend
stationary model under detection error probabilities of 0.05 and 0.20, respectively.
• Table F reports the calculation results of the robustness parameter σ based on (C16) inAppendix D of the paper.

†
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Appendix Table A
List of Sample Countries and Classification
No. Country
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

Algeria [DZA]
Australia [AUS]
Austria [AUT]Bangladesh [BGD]
Belgium [BEL]
Benin [BEN]
Bolivia [BOL]
Brazil [BRA]
Burkina Faso [BFA]
Cameroon [CMR]
Canada [CAN]
Chile [CHL]
Colombia [COL]
Congo [COG]
Costa Rica [CRI]
Denmark [DNK]
Dominican Republic [DOM]
Ecuador [ECU]
Egypt [EGY]El Salvador [SLV]
Finland [FIN]France [FRA]
Gabon [GAB]
Germany [DEU]
Greece [GRC]
Guatemala [GTM]
Hong Kong [HKG]
India [IND]Indonesia [IDN]
Ireland [IRL]
Israel [ISR]
Italy [ITA]
Japan [JPN]
Kenya [KEN]
Korea [KOR]
Lesotho [LSO]
Luxembourg [LUX]
Madagascar [MDG]Malaysia [MYS]
Mali [MLI]
Mauritania [MRT]Mexico [MEX]Netherlands [NLD]
New Zealand [NZL]
Nicaragua [NIC]
Norway [NOR]
Paraguay [PRY]
Peru [PER]
Philippines [PHL]
Portugal [PRT]This paper
Emerging
Rich
Rich
Poor
Rich
Poor
Emerging
Emerging
Poor
Poor
Rich
Emerging
Emerging
Emerging
Emerging
Rich
Emerging
Emerging
Emerging
Emerging
Rich
Rich
Emerging
Rich
Rich
Emerging
Rich
Emerging
Emerging
Rich
Rich
Rich
Rich
Poor
Emerging
Poor
Rich
Poor
Emerging
Poor
Emerging
Emerging
Rich
Rich
Emerging
Rich
Emerging
Emerging
Emerging
Rich
2Classification
World Bank
Upper middle income
High income
High income
Lower middle income
High income
Low income
Lower middle income
Upper middle income
Low income
Lower middle income
High income
High income
Upper middle income
Lower middle income
Upper middle income
High income
Upper middle income
Upper middle income
Lower middle income
Lower middle income
High income
High income
Upper middle income
High income
High income
Upper middle income
High income
Lower middle income
Lower middle income
High income
High income
High income
High income
Lower middle income
High income
Lower middle income
High income
Low income
Upper middle income
Low income
Lower middle income
Upper middle income
High income
High income
Lower middle income
High income
Upper middle income
Upper middle income
Lower middle income
High income

51
52
53
54
55
56
57
58
59
60
61
62
63
64

Appendix Table A (continued)
Puerto Rico [PRI]
Rich
Poor
Rwanda [RWA]Rich
Singapore [SGP]
Emerging
South Africa [ZAF]Rich
Spain [ESP]
Emerging
Sri Lanka [LKA]Rich
Sweden [SWE]Rich
Switzerland [CHE]
Emerging
Thailand [THA]
Poor
Togo [TGO]Rich
United Kingdom [GBR]Rich
United States [USA]
Emerging
Uruguay [URY]
Emerging
Venezuela [VEN]

High income
Low income
High income
Upper middle income
High income
Upper middle income
High income
High income
Upper middle income
Low income
High income
High income
High income
Upper middle income

Note : The country code in the WDI database is in square brackets.

3Appendix Table B
Parameter Estimates of the Random Walk and Trend Stationary Models

No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49

Country
Algeria [DZA]
Australia [AUS]
Austria [AUT]Bangladesh [BGD]
Belgium [BEL]
Benin [BEN]
Bolivia [BOL]
Brazil [BRA]
Burkina Faso [BFA]
Cameroon [CMR]
Canada [CAN]
Chile [CHL]
Colombia [COL]
Congo [COG]
Costa Rica [CRI]
Denmark [DNK]
Dominican Republic [DOM]
Ecuador [ECU]
Egypt [EGY]El Salvador [SLV]
Finland [FIN]France [FRA]
Gabon [GAB]
Germany [DEU]
Greece [GRC]
Guatemala [GTM]
Hong Kong [HKG]
India [IND]Indonesia [IDN]
Ireland [IRL]
Israel [ISR]
Italy [ITA]
Japan [JPN]
Kenya [KEN]
Korea [KOR]
Lesotho [LSO]
Luxembourg [LUX]
Madagascar [MDG]Malaysia [MYS]
Mali [MLI]
Mauritania [MRT]Mexico [MEX]Netherlands [NLD]
New Zealand [NZL]
Nicaragua [NIC]
Norway [NOR]
Paraguay [PRY]
Peru [PER]
Philippines [PHL]

Random walk
σε
0.053
(0.006)0.015(0.002)0.019(0.002)0.054(0.006)0.017(0.002)0.046(0.005)0.025(0.003)0.046(0.005)0.065(0.007)0.075(0.008)0.017(0.002)0.068(0.007)0.023(0.002)
0.198(0.020)0.039(0.004)0.024(0.002)0.043(0.004)0.042(0.004)0.029(0.003)0.057(0.006)0.024(0.003)0.014(0.001)0.104(0.011)
0.016(0.002)0.035(0.004)0.019(0.002)0.040(0.004)0.027(0.003)0.034(0.003)0.032(0.003)0.035(0.004)0.022(0.002)0.020(0.002)0.070(0.007)0.039(0.004)0.077(0.008)0.020(0.002)0.042(0.004)0.047(0.005)0.053(0.005)0.103(0.011)
0.035(0.004)0.019(0.002)0.025(0.003)0.103(0.011)
0.021(0.002)0.035(0.004)0.057(0.006)0.018(0.002)

4

Model
Trend stationary
σε
ρ
0.919
(0.041)
0.051
0.914(0.062)0.014
0.921(0.033)0.014
0.919(0.030)0.039
0.873(0.036)0.012
0.712(0.102)0.042
1.015(0.035)0.023
0.803(0.062)0.040
0.766(0.093)0.061
0.811(0.083)0.071
0.851(0.064)0.016
0.873(0.054)0.063
0.935(0.051)0.023
0.443(0.144)0.172
0.944(0.053)0.037
0.821(0.086)0.022
0.817(0.092)0.041
0.833(0.073)0.040
0.837(0.063)0.026
0.925(0.052)0.056
0.885(0.066)0.023
0.899(0.035)0.010
0.738(0.090)0.095
0.918(0.033)0.013
0.974(0.042)0.031
0.967(0.046)0.019
0.935(0.038)0.037
0.997(0.032)0.021
0.888(0.063)0.032
0.954(0.049)0.032
0.808(0.088)0.033
0.989(0.027)0.018
0.963(0.028)0.016
0.662(0.103)0.064
0.999(0.042)0.036
0.987(0.031)
0.059
0.981(0.034)0.017
0.934(0.049)0.039
0.871(0.080)0.046
0.921(0.072)0.052
0.612(0.116)0.093
0.816(0.081)
0.033
0.965(0.044)0.018
0.931(0.051)0.024
0.929(0.052)0.099
0.849(0.076)0.020
0.902(0.051)0.034
0.979(0.043)0.054
1.043(0.028)0.015(0.005)(0.001)(0.002)(0.004)(0.001)(0.004)(0.002)(0.004)(0.006)(0.007)(0.002)(0.006)(0.002)(0.018)(0.004)(0.002)(0.004)(0.004)(0.003)(0.006)(0.002)(0.001)(0.010)(0.001)(0.003)(0.002)(0.004)(0.002)(0.003)(0.003)(0.003)(0.002)(0.002)(0.007)(0.004)(0.006)(0.002)(0.004)(0.005)(0.005)(0.010)(0.003)(0.002)(0.002)(0.010)(0.002)(0.004)(0.006)(0.002)

50 Portugal [PRT]
51 Puerto Rico [PRI]
52 Rwanda [RWA]
53 Singapore [SGP]
54 South Africa [ZAF]
55 Spain [ESP]56 Sri Lanka [LKA]
57 Sweden [SWE]
58 Switzerland [CHE]59 Thailand [THA]
60 Togo [TGO]
61 United Kingdom [GBR]
62 United States [USA]63 Uruguay [URY]64 Venezuela [VEN]
Mean :
Rich countries
Emerging countries
Poor countries

Appendix Table B (continued)
0.035(0.004)0.883
0.137(0.014)0.240
0.077(0.008)0.866
0.031(0.003)0.915
0.026(0.003)0.938
0.025(0.003)0.938
0.056(0.006)0.719
0.019(0.002)0.894
0.013(0.001)0.797
0.036(0.004)0.920
0.146(0.015)0.561
0.022(0.002)0.972
0.016(0.002)0.918
0.056(0.006)0.880
0.088(0.009)
0.960
0.027
0.054
0.070

(0.058)(0.140)(0.065)(0.053)(0.050)(0.046)(0.095)(0.065)(0.068)(0.064)(0.112)(0.050)(0.060)(0.063)(0.078)

0.033
0.108
0.071
0.028
0.026
0.024
0.051
0.019
0.011
0.035
0.125
0.022
0.016
0.053
0.085

(0.003)(0.011)(0.007)(0.003)(0.003)(0.002)(0.005)(0.002)(0.001)(0.004)(0.013)(0.002)(0.002)(0.005)(0.009)

0.024
0.050
0.062

Note : Standard errors are in parentheses. The sample period is from 1970 to 2018. The sample size is 48,
which corresponds to the number of observations on Δc t .

5Appendix Table C.1
Welfare-Gain Estimates for Individual Countries: Random Walk Model
5%
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49

Country
Algeria [DZA]
Australia [AUS]
Austria [AUT]Bangladesh [BGD]
Belgium [BEL]
Benin [BEN]
Bolivia [BOL]
Brazil [BRA]
Burkina Faso [BFA]
Cameroon [CMR]
Canada [CAN]
Chile [CHL]
Colombia [COL]
Congo [COG]
Costa Rica [CRI]
Denmark [DNK]
Dominican Republic [DOM]
Ecuador [ECU]
Egypt [EGY]El Salvador [SLV]
Finland [FIN]France [FRA]
Gabon [GAB]
Germany [DEU]
Greece [GRC]
Guatemala [GTM]
Hong Kong [HKG]
India [IND]Indonesia [IDN]
Ireland [IRL]
Israel [ISR]
Italy [ITA]
Japan [JPN]
Kenya [KEN]
Korea [KOR]
Lesotho [LSO]
Luxembourg [LUX]
Madagascar [MDG]Malaysia [MYS]
Mali [MLI]
Mauritania [MRT]Mexico [MEX]Netherlands [NLD]
New Zealand [NZL]
Nicaragua [NIC]
Norway [NOR]
Paraguay [PRY]
Peru [PER]
Philippines [PHL]

λ
0.268
0.067
0.090
0.270
0.077
0.225
0.119
0.225
0.335
0.390
0.078
0.349
0.110
1.265
0.188
0.112
0.211
0.208
0.137
0.290
0.114
0.063
0.568
0.074
0.168
0.091
0.198
0.128
0.162
0.153
0.170
0.102
0.095
0.364
0.190
0.404
0.094
0.207
0.235
0.264
0.567
0.167
0.087
0.117
0.566
0.100
0.171
0.286
0.084

Detection error probability
10%20%
λ fearλ fearλ fear
λ
λ
0.241
0.215
0.188
0.151
0.123
0.065
0.053
0.051
0.035
0.033
0.087
0.071
0.068
0.048
0.045
0.243
0.216
0.189
0.152
0.124
0.075
0.061
0.058
0.041
0.038
0.206
0.180
0.160
0.125
0.105
0.113
0.094
0.088
0.064
0.058
0.205
0.179
0.160
0.125
0.105
0.294
0.270
0.229
0.191
0.151
0.337
0.315
0.262
0.225
0.172
0.075
0.061
0.059
0.041
0.039
0.306
0.282
0.238
0.200
0.157
0.105
0.087
0.082
0.059
0.054
0.893
1.067
0.696
0.829
0.457
0.174
0.149
0.135
0.103
0.089
0.107
0.088
0.083
0.060
0.055
0.194
0.169
0.151
0.117
0.099
0.191
0.166
0.149
0.115
0.098
0.130
0.109
0.101
0.074
0.066
0.259
0.233
0.202
0.164
0.133
0.109
0.090
0.085
0.061
0.056
0.062
0.050
0.048
0.033
0.031
0.467
0.465
0.364
0.341
0.239
0.071
0.058
0.056
0.039
0.037
0.156
0.133
0.122
0.091
0.080
0.088
0.072
0.068
0.048
0.045
0.182
0.157
0.142
0.109
0.093
0.121
0.101
0.094
0.069
0.062
0.151
0.128
0.118
0.088
0.077
0.144
0.122
0.112
0.083
0.074
0.159
0.135
0.124
0.093
0.081
0.098
0.081
0.076
0.055
0.050
0.091
0.075
0.071
0.050
0.047
0.317
0.294
0.247
0.209
0.162
0.176
0.151
0.137
0.104
0.090
0.347
0.327
0.271
0.234
0.178
0.090
0.074
0.070
0.050
0.046
0.190
0.165
0.148
0.114
0.097
0.214
0.188
0.167
0.131
0.109
0.237
0.211
0.185
0.148
0.122
0.466
0.464
0.363
0.339
0.238
0.155
0.132
0.121
0.091
0.080
0.084
0.069
0.065
0.046
0.043
0.111
0.092
0.087
0.063
0.057
0.465
0.463
0.362
0.339
0.238
0.096
0.079
0.075
0.053
0.049
0.159
0.136
0.124
0.093
0.081
0.256
0.230
0.199
0.162
0.131
0.081
0.066
0.063
0.045
0.042

6

λ nofear
0.027
0.002
0.004
0.027
0.003
0.020
0.006
0.020
0.040
0.053
0.003
0.044
0.005
0.372
0.014
0.005
0.018
0.017
0.008
0.031
0.006
0.002
0.102
0.002
0.011
0.004
0.016
0.007
0.011
0.010
0.012
0.005
0.004
0.047
0.014
0.056
0.004
0.017
0.021
0.026
0.101
0.011
0.003
0.006
0.101
0.004
0.012
0.031
0.003

50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

Portugal [PRT]
Puerto Rico [PRI]
Rwanda [RWA]
Singapore [SGP]South Africa [ZAF]
Spain [ESP]
Sri Lanka [LKA]
Sweden [SWE]
Switzerland [CHE]Thailand [THA]
Togo [TGO]
United Kingdom [GBR]
United States [USA]Uruguay [URY]
Venezuela [VEN]

Appendix Table C.1 (continued)
0.170
0.159
0.135
0.124
0.797
0.618
0.660
0.482
0.403
0.346
0.326
0.270
0.147
0.138
0.116
0.107
0.125
0.118
0.099
0.092
0.120
0.114
0.095
0.089
0.283
0.253
0.227
0.197
0.091
0.087
0.072
0.068
0.059
0.058
0.047
0.045
0.174
0.162
0.138
0.126
0.858
0.656
0.713
0.511
0.106
0.101
0.084
0.079
0.077
0.074
0.060
0.058
0.282
0.253
0.227
0.197
0.473
0.399
0.385
0.311

0.093
0.495
0.233
0.079
0.067
0.064
0.160
0.048
0.031
0.095
0.537
0.057
0.041
0.159
0.278

0.081
0.316
0.177
0.070
0.061
0.058
0.130
0.045
0.030
0.083
0.336
0.052
0.038
0.129
0.204

0.012
0.179
0.056
0.009
0.007
0.006
0.030
0.004
0.002
0.012
0.201
0.005
0.003
0.030
0.074

Note : There are cases in which the sum of the estimates λ nofear and λ fear is not completely equal to the estimate of
λ , owing to rounding errors.

7

Appendix Table C.2
Welfare-Gain Estimates for Individual Countries: Trend Stationary Model
Detection error probability
10%20%
λ fearλ fearλ fearλ nofearλλ0.0901 0.0764 0.0702 0.0523 0.0461 0.0062
0.0244 0.0195 0.0190 0.0129 0.0125 0.0005
0.0260 0.0208 0.0203 0.0138 0.0133 0.0005
0.0691 0.0574 0.0538 0.0390 0.0353 0.0036
0.0165 0.0131 0.0128 0.0087 0.0084 0.0003
0.0293 0.0245 0.0228 0.0166 0.0150 0.0016
0.1454 0.1251 0.1133 0.0862 0.0744 0.0119
0.0380 0.0316 0.0296 0.0214 0.0194 0.0020
0.0506 0.0435 0.0394 0.0299 0.0259 0.0040
0.0696 0.0606 0.0542 0.0420 0.0356 0.0064
0.0184 0.0147 0.0143 0.0098 0.0094 0.0004
0.0830 0.0714 0.0646 0.0492 0.0425 0.0068
0.0462 0.0374 0.0360 0.0251 0.0236 0.0015
0.0671 0.0697 0.0523 0.0517 0.0344 0.0174
0.0809 0.0673 0.0630 0.0456 0.0414 0.0042
0.0229 0.0185 0.0179 0.0124 0.0117 0.0007
0.0412 0.0343 0.0321 0.0233 0.0211 0.0022
0.0432 0.0359 0.0337 0.0243 0.0221 0.0022
0.0288 0.0234 0.0225 0.0157 0.0148 0.0010
0.1037 0.0887 0.0808 0.0609 0.0531 0.0079
0.0322 0.0260 0.0251 0.0174 0.0165 0.0010
0.0160 0.0127 0.0125 0.0084 0.0082 0.0002
0.0718 0.0649 0.0560 0.0457 0.0368 0.0089
0.0224 0.0179 0.0175 0.0119 0.0115 0.0004
0.0948 0.0786 0.0739 0.0532 0.0485 0.0047
0.0533 0.0431 0.0415 0.0288 0.0273 0.0016
0.0750 0.0623 0.0585 0.0423 0.0384 0.0039
0.0878 0.0720 0.0684 0.0485 0.0449 0.0036
0.0462 0.0379 0.0360 0.0256 0.0236 0.0019
0.0754 0.0622 0.0588 0.0420 0.0386 0.0035
0.0321 0.0264 0.0250 0.0178 0.0164 0.0014
0.0668 0.0542 0.0521 0.0363 0.0342 0.0022
0.0427 0.0343 0.0333 0.0229 0.0219 0.0010
0.0386 0.0333 0.0300 0.0230 0.0197 0.0033
0.1619 0.1384 0.1262 0.0951 0.0829 0.0122
0.2118 0.1869 0.1650 0.1302 0.1084 0.0219
0.0556 0.0449 0.0433 0.0300 0.0285 0.0016
0.0786 0.0655 0.0613 0.0445 0.0402 0.0043
0.0600 0.0503 0.0468 0.0343 0.0307 0.0036
0.0931 0.0791 0.0725 0.0542 0.0476 0.0065
0.0500 0.0454 0.0390 0.0320 0.0256 0.0064
0.0327 0.0268 0.0254 0.0181 0.0167 0.0014
0.0483 0.0389 0.0376 0.0260 0.0247 0.0013
0.0457 0.0371 0.0356 0.0249 0.0234 0.0015
0.1888 0.1727 0.1471 0.1222 0.0966 0.0256
0.0235 0.0189 0.0183 0.0126 0.0120 0.0006
0.0531 0.0438 0.0414 0.0296 0.0272 0.0024
0.1736 0.1506 0.1352 0.1042 0.0888 0.0154
0.3619 0.2789 0.2819 0.1821 0.1851 -0.0031

5%
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49

Country
Algeria [DZA]
Australia [AUS]
Austria [AUT]Bangladesh [BGD]
Belgium [BEL]
Benin [BEN]
Bolivia [BOL]
Brazil [BRA]
Burkina Faso [BFA]
Cameroon [CMR]
Canada [CAN]
Chile [CHL]
Colombia [COL]
Congo [COG]
Costa Rica [CRI]
Denmark [DNK]
Dominican Republic [DOM]
Ecuador [ECU]
Egypt [EGY]El Salvador [SLV]
Finland [FIN]France [FRA]
Gabon [GAB]
Germany [DEU]
Greece [GRC]
Guatemala [GTM]
Hong Kong [HKG]
India [IND]Indonesia [IDN]
Ireland [IRL]
Israel [ISR]
Italy [ITA]
Japan [JPN]
Kenya [KEN]
Korea [KOR]
Lesotho [LSO]
Luxembourg [LUX]
Madagascar [MDG]Malaysia [MYS]
Mali [MLI]
Mauritania [MRT]Mexico [MEX]Netherlands [NLD]
New Zealand [NZL]
Nicaragua [NIC]
Norway [NOR]
Paraguay [PRY]
Peru [PER]
Philippines [PHL]λ0.09630.02490.02650.07270.01670.03090.15720.03990.0546
0.07600.01880.08970.04760.08450.08510.0236
0.04340.04540.02980.1116
0.03310.01620.08080.02280.09950.05480.07890.09140.04810.07890.03350.06900.04380.04180.17420.23370.05720.08290.0636
0.0996
0.05640.03400.04960.04720.21440.02410.05550.18900.3588

8

50
51
52
53
54
55
56
57
58
59
60
61
62
63
64

Portugal [PRT]
Puerto Rico [PRI]
Rwanda [RWA]
Singapore [SGP]South Africa [ZAF]
Spain [ESP]
Sri Lanka [LKA]
Sweden [SWE]
Switzerland [CHE]Thailand [THA]
Togo [TGO]
United Kingdom [GBR]
United States [USA]Uruguay [URY]
Venezuela [VEN]

Appendix Table C.2 (continued)
0.0477 0.0457 0.0376 0.0356
0.0374 0.0315 0.0304 0.0246
0.0990 0.0906 0.0790 0.0706
0.0500 0.0482 0.0393 0.0375
0.0554 0.0535 0.0436 0.0417
0.0500 0.0484 0.0393 0.0377
0.0390 0.0366 0.0310 0.0285
0.0289 0.0282 0.0226 0.0219
0.0105 0.0104 0.0082 0.0081
0.0658 0.0628 0.0519 0.0489
0.0712 0.0605 0.0578 0.0472
0.0663 0.0641 0.0521 0.0499
0.0284 0.0278 0.0222 0.0216
0.0778 0.0728 0.0617 0.0567
0.2424 0.2153 0.1948 0.1678

0.02530.02200.05470.02650.02930.02640.02120.01510.00550.0351
0.04160.03500.01480.04230.13720.02340.0161
0.04640.02460.02740.02480.01870.01440.00530.03210.03100.03280.01420.03720.11020.00200.00590.00840.00180.00190.00160.00250.00070.00010.00300.01060.00220.00060.00500.0270Note : There are cases in which the sum of the estimates λ nofear and λ fear is not completely equal to the estimate of
λ , owing to rounding errors.

9

10

(1)

(2)

0.191
0.204

-0.082(0.020)

0.176
0.339

-0.090(0.020)-0.056(0.017)0.062(0.036)
0.188
0.241

-0.096(0.025)

(3)

0.107
0.333

-0.065(0.011)

0.098
0.448

-0.071(0.010)-0.032(0.009)0.026(0.012)
0.106
0.350

-0.072(0.012)-0.070(0.011)-0.033(0.011)-0.006(0.017)
0.099
0.449

-0.092(0.024)-0.054(0.016)0.009(0.033)
0.177
0.339

(4)

0.086
0.350

-0.055(0.008)

0.138
0.238

-0.066(0.015)

(5)

0.079
0.463

-0.059(0.008)-0.026(0.007)

0.126
0.378

-0.072(0.014)-0.042(0.012)0.021(0.010)
0.086
0.368

-0.060(0.010)0.045(0.024)
0.136
0.274

-0.076(0.017)

Dependent variable
λ fear(6)(7)-0.059(0.009)-0.026(0.009)-0.004(0.014)
0.080
0.464

-0.073(0.016)-0.041(0.012)0.005(0.023)
0.127
0.378

(8)

0.021
0.249

-0.011(0.002)

0.055
0.113

-0.017(0.006)

(9)

0.020
0.369

-0.012(0.002)-0.006(0.002)

0.052
0.223

-0.019(0.006)-0.014(0.005)

(10)

(11)

0.004
(0.002)
0.021
0.263

-0.012(0.003)0.017(0.012)
0.054
0.150

-0.020(0.008)

λ nofear-0.011(0.002)-0.007(0.002)-0.002(0.003)
0.020
0.371

-0.019(0.007)-0.013(0.004)0.004(0.010)
0.052
0.224

(12)

Note : Heteroskedasticity-robust standard errors are in parentheses. The welfare gains, λ and λ fear, are those for a detection error probability of 0.05. PPP_GDP is
the geometric average of PPP-converted GDP per capita over the period 1990--2018, a measure of economic development. pop is the total population in 2018.
openness is the trade openness measured by the ratio of exports plus imports to GDP in 2018. See Appendix B for details of each variable. The coefficient on theconstant term in the regression is not reported. SER is the standard error of the regression. The results for λnofear (columns (9)--(12)) are the same as those in Table
3.

SER
R2

opennessln popln PPP_GDP

B. Results based on 61 countries excluding outliers

SER
R2

opennessln popln PPP_GDP

A. Results based on all 64 countries

Regressor

λAppendix Table D.1
Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, Country Size, and Openness:
Random Walk Model (the 5% Detection Error Probability)

11

(1)

(2)

0.124
0.185

-0.050(0.013)

0.115
0.316

-0.055(0.013)-0.035(0.011)0.040(0.024)
0.122
0.222

-0.059(0.016)

(3)

0.065
0.321

-0.039(0.006)

0.060
0.438

-0.042(0.006)-0.019(0.006)0.015(0.007)
0.065
0.338

-0.042(0.007)-0.041(0.007)-0.020(0.007)-0.004(0.010)
0.060
0.439

-0.057(0.016)-0.034(0.010)0.007(0.022)
0.115
0.316

(4)

0.044
0.350

-0.028(0.004)

0.071
0.238

-0.034(0.007)

(5)

0.040
0.463

-0.030(0.004)-0.013(0.004)

0.064
0.378

-0.037(0.007)-0.022(0.006)0.011(0.005)
0.044
0.368

-0.031(0.005)0.023(0.012)
0.070
0.274

-0.039(0.009)Dependent variable
λ fear(6)(7)-0.030(0.004)-0.014(0.004)-0.002(0.007)
0.041
0.464

-0.037(0.008)-0.021(0.006)0.002(0.012)
0.065
0.378

(8)

0.021
0.249

-0.011(0.002)

0.055
0.113

-0.017(0.006)

(9)

0.020
0.369

-0.012(0.002)-0.006(0.002)

0.052
0.223

-0.019(0.006)-0.014(0.005)

(10)

(11)

0.004
(0.002)
0.021
0.263

-0.012(0.003)0.017(0.012)
0.054
0.150

-0.020(0.008)

λ nofear-0.011(0.002)-0.007(0.002)-0.002(0.003)
0.020
0.371

-0.019(0.007)-0.013(0.004)0.004(0.010)
0.052
0.224

(12)

Note : Heteroskedasticity-robust standard errors are in parentheses. The welfare gains, λ and λ fear, are those for a detection error probability of 0.20. PPP_GDP is
the geometric average of PPP-converted GDP per capita over the period 1990--2018, a measure of economic development. pop is the total population in 2018.
openness is the trade openness measured by the ratio of exports plus imports to GDP in 2018. See Appendix B for details of each variable. The coefficient on theconstant term in the regression is not reported. SER is the standard error of the regression. The results for λnofear (columns (9)--(12)) are the same as those in Table
3.

SER
R2

opennessln popln PPP_GDP

B. Results based on 61 countries excluding outliers

SER
R2

opennessln popln PPP_GDP

A. Results based on all 64 countries

Regressor

λAppendix Table D.2
Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, Country Size, and Openness:
Random Walk Model (the 20% Detection Error Probability)

12

(1)

(2)

0.059
0.126

-0.019(0.006)

0.059
0.128

-0.019(0.006)-0.002(0.005)0.011(0.005)
0.059
0.140

-0.021(0.007)

(3)

0.047
0.126

-0.015(0.005)

0.047
0.149

-0.016(0.005)-0.005(0.004)0.011(0.006)
0.047
0.146

-0.017(0.006)-0.017(0.006)-0.004(0.004)0.007(0.004)
0.047
0.155

-0.021(0.007)0.0004(0.006)0.011(0.007)
0.059
0.140

(4)

0.042
0.118

-0.013(0.004)

0.056
0.114

-0.017(0.006)

(5)

0.042
0.137

-0.014(0.005)-0.004(0.003)

0.056
0.114

-0.017(0.006)-0.001(0.005)0.009(0.005)
0.042
0.138

-0.015(0.005)0.010(0.005)
0.056
0.126

-0.019(0.006)

Dependent variable
λ fear(6)(7)-0.015(0.005)-0.003(0.003)0.006(0.004)
0.042
0.144

-0.019(0.006)0.002(0.006)0.011(0.007)
0.056
0.127

(8)

(10)

(11)

(12)

-0.002-0.002-0.002-0.002(0.0006) (0.0006) (0.0006) (0.0006)-0.001-0.001(0.0004)(0.0004)
0.001
0.0003(0.0007) (0.0006)
0.005
0.005
0.005
0.005
0.160
0.227
0.183
0.228

-0.002-0.002-0.002-0.002(0.0005) (0.0005) (0.0006) (0.0006)-0.001-0.001(0.0004)(0.0004)
0.001
0.0001(0.0007) (0.0006)
0.005
0.005
0.005
0.005
0.148
0.229
0.168
0.229

(9)

λ nofear

Note : Heteroskedasticity-robust standard errors are in parentheses. The welfare gains, λ and λ fear, are those for a detection error probability of 0.05. PPP_GDP is
the geometric average of PPP-converted GDP per capita over the period 1990--2018, a measure of economic development. pop is the total population in 2018.
openness is the trade openness measured by the ratio of exports plus imports to GDP in 2018. See Appendix B for details of each variable. The coefficient on theconstant term in the regression is not reported. SER is the standard error of the regression. The results for λnofear (columns (9)--(12)) are the same as those in Table
4.

SER
R2

opennessln popln PPP_GDP

B. Results based on 62 countries excluding outliers

SER
R2

opennessln popln PPP_GDP

A. Results based on all 64 countries

Regressor

λAppendix Table E.1
Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, County Size, and Openness:
Trend Stationary Model (the 5% Detection Error Probability)

13

(1)

(2)

0.032
0.135

-0.011(0.003)

0.032
0.140

-0.011(0.003)-0.002(0.003)0.006(0.003)
0.032
0.150

-0.012(0.004)

(3)

0.026
0.131

-0.009(0.003)

0.026
0.158

-0.009(0.003)-0.003(0.002)0.006(0.003)
0.026
0.152

-0.010(0.003)-0.010(0.003)-0.002(0.002)0.004(0.002)
0.026
0.164

-0.012(0.004)-0.0004(0.003)0.006(0.004)
0.032
0.150

(4)

0.022
0.118

-0.007(0.002)

0.028
0.114

-0.009(0.003)

(5)

0.022
0.137

-0.007(0.002)-0.002(0.002)

0.029
0.114

-0.009(0.003)-0.0005(0.003)0.005(0.003)
0.022
0.138

-0.008(0.003)0.005(0.003)
0.028
0.126

-0.010(0.003)

Dependent variable
λ fear(6)(7)-0.008(0.003)-0.001(0.002)0.003(0.002)
0.022
0.144

-0.010(0.003)0.001(0.003)0.006(0.003)
0.029
0.127

(8)

(10)

(11)

(12)

-0.002-0.002-0.002-0.002(0.0006) (0.0006) (0.0006) (0.0006)-0.001-0.001(0.0004)(0.0004)
0.001
0.0003(0.0007) (0.0006)
0.005
0.005
0.005
0.005
0.160
0.227
0.183
0.228

-0.002-0.002-0.002-0.002(0.0005) (0.0005) (0.0006) (0.0006)-0.001-0.001(0.0004)(0.0004)
0.001
0.0001(0.0007) (0.0006)
0.005
0.005
0.005
0.005
0.148
0.229
0.168
0.229

(9)

λ nofear

Note : Heteroskedasticity-robust standard errors are in parentheses. The welfare gains, λ and λ fear, are those for a detection error probability of 0.20. PPP_GDP is
the geometric average of PPP-converted GDP per capita over the period 1990--2018, a measure of economic development. pop is the total population in 2018.
openness is the trade openness measured by the ratio of exports plus imports to GDP in 2018. See Appendix B for details of each variable. The coefficient on theconstant term in the regression is not reported. SER is the standard error of the regression. The results for λnofear (columns (9)--(12)) are the same as those in Table
4.

SER
R2

opennessln popln PPP_GDP

B. Results based on 62 countries excluding outliers

SER
R2

opennessln popln PPP_GDP

A. Results based on all 64 countries

Regressor

λAppendix Table E.2
Cross-Sectional Regressions of Welfare Gains on Variables that Measure Economic Development, County Size, and Openness:
Trend Stationary Model (the 20% Detection Error Probability)

Appendix Table F
Replication of the Robustness Parameter
Detection error probability
50%
45%
40%
Formula (C16)

0

0.31

0.62

Ellison and Sargent (2015)

0

0.3

0.6

Note : The results in the second row are from Ellison and Sargent (2015, Table 2)14