WPS6479
Policy Research Working Paper 6479
Partial Consumption Insurance
and Financial Openness Across the World
Constantino Hevia
Luis Servén
The World Bank
Development Research Group
Macroeconomics and Growth Team
May 2013
Policy Research Working Paper 6479
Abstract
This paper examines the extent of international developing countries, and that the gap between both
consumption risk sharing for a group of 50 industrial country groups appears to have widened over the period
and developing countries. The analysis is based on of financial globalization. Moreover, the pattern of
the empirical implementation of a model of partial consumption risk sharing is related to the degree of
consumption insurance whose parameters have the financial openness: countries with larger stocks of foreign
natural interpretation of coefficients of partial risk assets or liabilities exhibit larger degrees of risk sharing.
sharing even when the null hypothesis of perfect risk Furthermore, countries whose foreign asset stocks are
sharing is rejected. Estimation results show that rich more tilted towards foreign direct investment assets also
countries exhibit higher degrees of risk sharing than show higher degrees of consumption risk sharing.
This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors
may be contacted at chevia@worldbank.org and Lserven@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Partial consumption insurance and …nancial openness
across the world
Constantino Heviaa;b , Luis Servéna ,
a
World Bank, Washington DC, USA
b
Department of Economics, Universidad Torcuato Di Tella, Buenos Aires, Argentina
June 2013
Abstract
This paper examines the extent of international consumption risk sharing for a
group of 50 industrial and developing countries. The analysis is based on the empir-
ical implementation of a model of partial consumption insurance whose parameters
have the natural interpretation of coe¢ cients of partial risk sharing even when the
null hypothesis of perfect risk sharing is rejected. Estimation results show that rich
countries exhibit higher degrees of risk sharing than developing countries, and that the
gap between both country groups appears to have widened over the period of …nan-
cial globalization. Moreover, the pattern of consumption risk sharing is related to the
degree of …nancial openness: countries with larger stocks of foreign assets or liabili-
ties exhibit larger degrees of risk sharing. Furthermore, countries whose foreign asset
stocks are more tilted towards foreign direct investment assets also show higher degrees
of consumption risk sharing.
Keywords : Risk sharing, Financial globalization, Permanent Income.
JEL classi…cation : E21; F36; F41
We thank Aart Kraay for useful discussions, and Vivian Norambuena for excellent research assistance.
This work was supported by the World Bank’ s Knowledge Change Program. The views expressed here are
only ours and do not necessarily re‡ect those of the World Bank, its Executive Directors, or the countries
they represent.
1 Introduction
A considerable analytical and empirical literature has been concerned with the extent of
consumption risk sharing across countries. The relatively high sensitivity of aggregate con-
sumption to domestic income shocks – or, equivalently, the low degree of comovement of
aggregate consumption observed across countries –has been singled out as one of the major
puzzles in international macroeconomics (Obstfeld and Rogo¤ 2001). In recent years, it has
attracted renewed interest due to the growing degree of …nancial integration of economies
across the world. The theoretical argument is straightforward: if …nancial markets are com-
plete –in the sense that the available assets su¢ ce to span the set of all idiosyncratic risks
–the ratio of the marginal utilities of consumption of any pair of agents (or countries in our
case) must be constant across dates and states of nature. That is, the economy features per-
fect risk sharing.1 In turn, if risk sharing is imperfect and markets are incomplete, …nancial
innovation that expands the set of tradeable assets (or reduces the costs of trading existing
assets) should lead to enhanced risk sharing. To the extent that the global increase in cross-
border asset holdings over the last quarter century (Lane and Milesi-Ferretti, 2007) re‡ects
such kind of innovation, it should be associated with a rise in consumption risk sharing across
countries.
A number of papers have performed empirical tests of the null hypothesis of perfect
consumption risk sharing. The standard approach is to adopt a parametric assumption for
the utility function –typically, constant relative or absolute risk aversion –and use it to check
if the theoretical prediction is satis…ed in the data. This is often done indirectly, through
least squares regressions testing whether idiosyncratic income shocks have a signi…cant e¤ect
on individual consumption after controlling for the average consumption of all agents (or
countries). Obstfeld (1994), Canova and Ravn (1996), and Lewis (1996) are some leading
1
Strictly speaking, market completeness represents a su¢ cient, but not necessary, condition for perfect
risk-sharing. The same outcome might be achieved by, say, agreement between governments to a suitable
system of transfers across countries.
1
examples of this literature applied to cross-country aggregate data.2
These conventional tests can be informative about whether perfect risk sharing holds
empirically. But once their null hypothesis has been rejected – as is almost invariably the
case – in general they cannot say much about the extent of imperfect risk sharing present
in the data. Put di¤erently, in most cases one cannot draw inferences about the degree of
risk sharing just from the magnitude of consumption correlations or estimated regression
coe¢ cients. To do this in a rigorous and meaningful way, one needs a model describing the
channels through which the risk sharing arrangement is implemented.
One leading example is the model of partial risk sharing developed by Crucini (1999).
It assumes that, prior to any income realization, agents contribute a common fraction of
their incomes to an income pool in exchange for the right to the same common fraction of
the pooled income after shocks are realized. Thus, the fraction of the income that agents
contribute to the pool has the natural interpretation of a coe¢ cient of partial risk sharing.
After income shocks are realized and transfers made according to the risk sharing agreement,
agents act as permanent income consumers borrowing and lending at a …xed interest rate.
This basic framework has been used by Crucini (1999), Crucini and Hess (2000), Asdrubali
and Kim (2008), and Artis and Ho¤man (2008) to obtain estimates of the extent of partial
risk sharing. However, this setting su¤ers from a major limitation, in that all participants
in the risk sharing arrangement are assumed to contribute the same fraction of their income
to the common pool. This makes the framework unsuitable for a situation in which di¤erent
agents or countries may engage in di¤erent extents of risk sharing.
This is precisely the focus of this paper. It analyzes to what extent di¤erent countries are
able to diversify their idiosyncratic risks, and relates their respective degree of risk sharing to
selected country characteristics, in particular regarding their international asset portfolios.
To do this, we develop an expanded version of the model in Crucini (1999) that allows
each agent / country to contribute a di¤erent fraction of their income to the income pool,
2
Cochrane (1991) and Townsend (1994) present early implementations of risk sharing tests based on
household level data.
2
and assume that, after income shocks are realized, the transfer that each agent receives is
determined by her initial relative contribution to the income pool.
While this modi…cation may seem pretty intuitive, it has some important consequences.
On the one hand, it complicates the solution of the model and its empirical implementation.
s (1999) model leads to an estimating equation in which the growth rate
Speci…cally, Crucini’
s consumption depends linearly on the innovation to that agent’
of an agent’ s permanent
income and on the growth rate of average consumption across agents. Moreover, the para-
meters multiplying these variables are linear functions of the common contribution to the
income pool, which makes the model suitable for OLS estimation. In contrast, empirical im-
plementation of the model with heterogeneous contributions to the pool requires estimating
a system of equations in which the path of consumption of each agent depends on the inno-
vation to the permanent income of all agents. Moreover, the coe¢ cients multiplying these
innovations depend in nonlinear fashion on the contributions to the income pool of all agents
participating in the risk sharing agreement.
On the other hand, the expanded model solves a problem that plagues all conventional
risk sharing regressions that include average consumption among the explanatory variables.
As noted by Deaton (1990), pooling the observations on consumption of all agents in a linear
regression that includes the cross-sectional average of the dependent variable among the
regressors leads, mechanically, to a regression coe¢ cient of unity on that variable. Crucini
(1999) dealt with this problem by estimating di¤erent coe¢ cients of partial risk sharing
for each agent (regions or countries in his analysis), even though his model is built on the
assumption of a common coe¢ cient for everyone. To obtain a single estimate of partial risk
sharing, as implied by the model, Crucini then takes the average of the estimated risk sharing
parameters of all agents.3 In contrast, our model does not su¤er from this problem, because
our estimating equation is stated only in terms of innovations to the permanent incomes of
all agents. Unlike conventional risk sharing regressions, it does not include the cross-sectional
3
The paper reports a Monte Carlo experiment to check if this procedure leads to major biases in the
risk-sharing estimate.
3
average of the dependent variable among the regressors.4
We estimate the model using a time-series cross-section dataset comprising 50 industri-
alized and developing countries over the period 1970-2007. We …nd a cross-country average
s theoretical model, this
coe¢ cient of partial risk sharing around 0.42. In terms of the paper’
means that the average country contributes less than half its income to the income pool. The
estimates, however, vary systematically between industrial and developing countries, with
the former exhibiting, on average, signi…cantly higher degrees of risk sharing than the latter
(centered around 0.52 and 0.35, respectively). Re-estimating the model over rolling time sam-
ples, we …nd evidence that the degree of risk sharing has been on the rise among industrial
countries during most of the sample period, while for developing countries average risk shar-
ing remained ‡at or even declined somewhat during the globalization period (although there
seems to be a modest uptick at the end of the sample). This result is consistent with the view
that the bene…ts from …nancial globalization do not accrue evenly to all countries, and may
prove elusive in particular for economies with relatively low levels of …nancial development.5
s framework, these partial risk sharing coe¢ cient estimates can be interpreted
In the paper’
as measuring agents’ chosen contributions to the global income pool. However, the model
is silent on the factors behind those choices. To gain further insight, we assume that they
re‡ect cross-country variation in policies and institutions that help or hinder consumption risk
sharing. Speci…cally, we re-estimate the model expressing the partial risk sharing coe¢ cient
as a function of selected measures of …nancial and trade openness, which are commonly viewed
as re‡ecting the mechanisms through which risk sharing may be actually implemented.
While this adds further complication to an already-challenging nonlinear estimation prob-
lem, it yields fairly robust empirical results. The main conclusion is that international …-
nancial integration is a signi…cant factor behind the observed cross-country patterns of con-
sumption risk sharing. The degree of …nancial integration, as summarized by total foreign
4
Moreover, even if we had written the estimating equation in terms of a weighted consumption average,
the nonlinearity of the model would break the linear relation between right and left-hand side variables
inherent to the linear regression.
5
On the costs and bene…ts of …nancial globalization, see Kose, Prassad, Rogo¤, and Wei (2006).
4
asset and liability positions, is positively correlated with the coe¢ cient of partial risk sharing,
consistent with the view that …nancial integration improves risk sharing across countries. In
addition, the FDI share of the foreign asset position –i.e., the proportion of residents’foreign
assets that takes the form of direct investment abroad –is also positively correlated with con-
sumption risk sharing even after controlling for total foreign assets. The implication is that
FDI assets are more conducive to risk sharing than other kinds of foreign assets. In contrast,
the FDI share of foreign liabilities –i.e., the portion that takes the form of foreigners’direct
investment in the home country – is negatively associated with consumption risk sharing.
This …nding seems to be at odds with the conventional wisdom that capital in‡ows in the
form of FDI are preferable, from the point of view of risk, to other forms of capital in‡ows.6
Importantly, once these measures of international …nancial integration are taken into ac-
count, trade integration, domestic …nancial development, and the level of GDP per capita do
not contribute signi…cantly to explain the cross-country pattern of consumption risk sharing.
Our paper is related to a substantial empirical literature assessing international consump-
tion risk sharing. Sorensen et al. (2007) relate trends in risk sharing among OECD countries
to their foreign assets and liabilities. They …nd that the degree of consumption risk sharing
appears to be positively related to foreign asset holdings, while the relation with foreign liabil-
ities is not robust. Kose, Prasad, and Terrones (2009) argue that risk sharing has risen among
industrial countries (but not among developing countries), and the rise is correlated with the
increase in gross foreign assets and liabilities over the globalization period. Holinski et al.
(2012) also examine how international consumption risk sharing relates to various features of
countries’equity portfolios. These papers base their conclusions on conventional risk sharing
regressions, and thus they are subject to the criticism that, strictly speaking, such regressions
do not provide a solid basis for inferences about the extent of partial risk sharing. Fratzscher
6
We conjecture that the result might re‡ ect the action of third factors that both hinder risk sharing and
tilt the composition of in‡ows towards direct investment. In other words, a large share of FDI liabilities could
be the constrained optimal mix of foreign …nancing for a country with poor institutions and weak governance
and, therefore, with a riskier environment and less opportunities for risk sharing (Albuquerque, 2003, Daude
and Fratzscher, 2008).
5
and Imbs (2009) develop a model with transaction costs and discuss their e¤ects on conven-
tional tests of risk sharing. They …nd that larger holdings of foreign capital (especially in the
form of equity or bonds) are associated with higher consumption risk sharing. In contrast,
larger holdings of FDI or bank loans are not. Flood, Marion, and Matsumoto (2009) use the
s share of world consumption as a measure of consumption risk sharing.
variance of a country’
Perfect consumption risk sharing occurs when this share is constant. Using rolling windows,
they argue that consumption risk sharing rose during the recent era of …nancial globalization,
particularly when the focus is on low-frequency movements in consumption. Finally,Bai and
Zhang (2012) argue that …nancial frictions can explain why risk sharing failed to improve
during the globalization era. They develop a model with incomplete …nancial markets and
enforceability constraints and show that removing capital controls leads to the emergence of
default risk that limits the extent of consumption risk sharing.
The rest of the paper is organized as follows. Section 2 develops the model of partial
risk sharing and section 3 describes the data and the details of the empirical implementation
of the model. Section 4 discusses the main empirical results of the paper and section 5
concludes. The appendices contain some proofs and additional tables.
2 A model of partial risk sharing
In this section we describe a model of partial risk sharing that generalizes the model in
Crucini (1999). Time is discrete and denoted by t = 0; 1; 2; :::; 1. There is a group of N > 1
families. Family i = 1; 2; ::; N is composed of Hi members, each of whom receives a random
‡ow of income Yit in period t. We assume that families enter into a partial risk sharing
agreement before income is realized. The agreement requires each family to contribute a
possibly di¤erent fraction of their income into an income pool in exchange for a claim to
a fraction of the pooled income of all families. In particular, if each member of family i
contributes a fraction i of her income to the pool, the contribution to the fund at time t is
6
PN
equal to i Hi Yit , and the aggregate pooled income equals i=1 Hi i Yit . We assume that the
agreement speci…es that each family i gets back a fraction of the income pool equal to its
share in the total initial contribution, or
i Hi Yi0
i = PN : (1)
j =1 j H j Yj 0
Therefore, total income of family i at time t after the risk sharing agreement is
i Hi Yi0
X
N
Hi Yit = (1 i ) Hi Yit + PN k Hk Ykt :
j =1 j Hj Yj 0 k=1
In per capita terms, each member of family i receives
X
N
! i0
Yit = (1 i ) Yit + i k Ykt ; (2)
k=1
! k0
PN
where ! i0 = Yi0 = j =1 s per capita income in aggregate per-capita
Yj 0 is the share of family i’
income.
We also assume that, on top of the risk sharing agreement, each family member acts as a
permanent income consumer borrowing or lending at a …xed interest rate r. Following Camp-
bell and Mankiw (1990), the permanent income hypothesis is represented by the following
decision rule for the change in consumption,
Cit = i + "it ; (3)
where, for any variable xt , xt = xt xt 1 is the di¤erence operator and "it is the innovation
between time t and t s per-capita permanent disposable income. The latter,
1 in family i’
in turn, is given by
X
1
t
"it = (1 ) Et Yit+k Et 1 Yit+k (4)
t=0
where = 1= (1 + r) is the reciprocal of the gross interest rate and Et is the expectations
7
operator conditional on all the information available to the family at time t. The above
expression measures the innovation to permanent income after the risk sharing agreement.
A related object is the innovation to permanent income before the risk sharing agreement,
denoted by
X
1
t
"it = (1 ) (Et Yit+k Et 1 Yit+k ) : (5)
t=0
Using equations (2), (4), and (5) into equation (3) delivers the following expression for
the change in consumption as a function of the innovation to all families’permanent incomes
before risk sharing
XN
! i0
Cit = i + (1 i ) "it + i j "jt : (6)
j =1
! j 0
s per-capita consumption. The term
The left side of this expression is the change in family i’
on the right side includes the deterministic drift in consumption plus the innovation to the
s permanent disposable income. The latter term, in turn, can be written as
family’
!
XN
! i0
i j "jt "it + "it ;
j =1
! j 0
s has a bad income
which illustrates the risk sharing agreement. In particular, if family i’
realization relative to that of the other families (if the term in parentheses is positive), it
receives a transfer proportional to the di¤erence in the income shocks, where the factor of
proportionality is the contribution of family i to the income pool. If family i has a good income
realization relative to that of the other families (if the term in parentheses is negative), it
transfers resources to the other families in proportion to its contribution i. Furthermore,
note that if i = 1, so that country i shares all of its income with the pool, the idiosyncratic
shock "it disappears as an isolated term from expression (6) and only enters through its
P
impact on the innovation to the pool’s permanent income N ! i0
j =1 ! j 0 j "jt .
Before proceeding, we discuss an important speci…cation issue. As noted in Mankiw and
Campbell (1989,1990) and Crucini (1999), aggregate consumption and income time series
8
seem to follow log-linear rather than linear processes. We thus follow Mankiw and Campbell
(1989,1990), Crucini (1999), and Asdrubali and Kim (2008) and replace all variables by their
logs and consider the resulting expression as a log-linear approximation to the permanent
income hypothesis equations. (The Appendix shows how to log-linearize the permanent
income innovation.) Therefore, (6) is approximated according to
XN
! i0
cit = i + (1 i ) "it + i j "jt ;
j =1
! j 0
where cit = log Cit , yit = log Yit , and
X
1
k
"it = [Et yit+k Et 1 yit+k ] :
k=0
In the econometric analysis below we interpret each family as a country and the number
s population. Our objective is then to estimate the degree
of family members as that country’
of risk sharing i for each country i = 1; 2; :::; N . To that end, we follow the usual procedure
in the rational expectations econometric literature (Hansen and Sargent, 1980) and assume
that, when performing revisions to their permanent incomes, agents have more information
"it denote the innovation to the permanent income estimated
than the econometrician. Let ^
by the econometrician and rewrite (6) as
XN
! i0
cit = i + (1 "it + i
i) ^ "jt + uit ;
j^ (7)
j =1
!j0
where uit is a residual that represents the superior information of agents relative to the
econometrician, and is given by
!
XN
! i0 XN
! i0
uit = (1 i ) " it + i j "jt (1 i ) ^
" it + i j^
"jt ;
j =1
! j 0 j =1
! j 0
s information set at time
It is easy to verify that uit is orthogonal to the econometrician’
9
t 1.7 This information set, however, may omit some relevant information used by the agents
to forecast their future incomes. This omitted information can lead to serially correlated
residuals for a given agent as well as contemporaneously correlated residuals across agents.
The former could be due to a persistent variable that is used by an agent, but unobserved by
the econometrician, to perform revisions to his or her permanent income. The latter could
be due to an aggregate shock observed by all agents, but unobserved by the econometrician,
s permanent income.
that a¤ects everyone’
It is important to stress the key di¤erence between our framework, as summarized by (7),
and that in Crucini (1999). In both cases, the starting point is the assumption that agents
contribute a fraction of their income to an income pool and behave as permanent-income
consumers after transfers are realized. In Crucini (1999), however, all agents are assumed to
contribute the same fraction of their income to the pool, which forces their respective partial
risk sharing coe¢ cients i to be all equal and allows a drastic simpli…cation of the model.8
However, instead of estimating a single for all agents, Crucini estimates i separately for
each agent through OLS regressions, and then computes a single coe¢ cient by averaging
s empirical
the individual coe¢ cients. While this procedure might be justi…ed in Crucini’
application, whose focus is on relatively homogeneous groups of regions or countries –U.S.
states, Canadian provinces, and G-7 countries –it is clearly less defensible in our case given
the likely great degree of heterogeneity across the …fty countries in our sample. Moreover,
one of our objectives is precisely to assess the extent to which the coe¢ cients of partial risk
sharing vary across countries. This requires a more general framework such as (7), in which
di¤erent countries may exhibit di¤erent coe¢ cients of partial risk sharing.
However, the added generality of this framework relative to the conventional model used
in the literature poses additional econometric challenges too. For each country, (7) involves
a nonlinear function of the risk sharing parameters of all countries. Thus, empirical imple-
7
The proof of this result is an application of the law of iterated expectations (Hansen and Sargent, 1980).
In addition, any classical measurement error in consumption can also be included in the error term uit .
8
This simplifying assumption is maintained in subsequent empirical applications of Crucini’ s framework;
see for example Artis and Ho¤man (2008) and Asdrubali and Kim (2008).
10
mentation of the model requires more complex econometric techniques than those found in
earlier literature. This is the topic of the next section.
3 Empirical implementation
We …rst discuss the econometric issues surrounding the estimation of the parameters of the
risk sharing model, as well as those related to the construction of the permanent income
innovations required for such task. Then we brie‡y summarize our data sources.
3.1 Implementing the risk sharing model
The core of our empirical analysis is the estimation of (7) using a cross-country time-series
s consumption path depends
dataset. This poses a challenging problem because each country’
in nonlinear fashion on the risk sharing parameters of all countries. To make this explicit in
what follows, it is useful to rewrite (7) as
X
N
! i0 Hj j !j0
cit = i + i "jt + (1
^ "it
i) ^ + uit (8)
!j0 P
N
j =1 Hk k ! k0
k=1
where uit is potentially heteroskedastic and possibly correlated across countries and over
time.
To express the empirical model in more compact form, it is convenient to introduce some
notation. Let ci = ( ci1 ; :::; ciT )0 and "i = ("i1 ; :::; "iT )0 denote the (column) vectors of
consumption growth and permanent income innovations of country i, and de…ne the T
0
N matrix Z = ("1 ; :::; "N ) : In addition, let = ( 1 ; :::; N) ; the vector of risk sharing
parameters for all countries, and de…ne the N N diagonal matrices H and W0 ; which have
along their main diagonal the population of each country Hj 0 and its per capita GDP share
! j 0 , respectively, with all o¤-diagonal elements equal to zero. After some straightforward
manipulations the observations of (8) corresponding to the i-th country can be compactly
11
written
0 1 0
ci = T i +Z ei e0i ( N )+H ( N HW0 ) e i W0 1 +ui (9)
where denotes a column vector of ones, and ei is an N 1 vector of zeros with an 1 in the
i-th entry:
Finally, stacking the observations on consumption growth for all countries into the N T 1
vector c = ( c01 ; :::; c0N )0 and letting = ( 1 ; :::; 0
N) , the full system of equations can
be written
1
c= T + (IN Z) W0 1 H ( 0
N HW0 ) + vec (IN diag ( )) + u (10)
where denotes the Kronecker product, and diag ( ) is an N N diagonal matrix with its
main diagonal equal to and zeros elsewhere.
Thus, the empirical model amounts to a system of N equations with cross-equation para-
meter restrictions. The restrictions imply that a system estimation procedure is needed, even
though the explanatory variables (the innovations to permanent income in all N countries,
contained in the T N matrix Z) are the same in all equations. In this context, we use
system NLS to estimate the parameters of (10). We …rst partial out by expressing c
and Z as deviations from their country-speci…c means; let ~ denote the transformed
c and Z
~
variables. Then we compute the NLS estimator that solves the problem
0
min c
~ ~ ; H; W0 ; )
f (Z c
~ ~ ; H; W0 ; )
f (Z (11)
~ ; H; W0 ; ) = IN ~ 1
where f (Z Z W0 1 H ( 0
N HW0 ) + vec (IN diag ( )) : Since
the residuals may exhibit heteroskedasticity, serial correlation, cross-sectional dependence, or
any combination of all three, to perform inference on we use the robust covariance matrix
estimator proposed by Driscoll and Kraay (1998) and Vogelsang (2012).9
9
In the formula of the covariance matrix, the usual matrix of regressors is replaced in our case by the
12
This procedure yields a set of unrestricted estimates of the risk sharing coe¢ cients ^ :
However, we are also interested in learning about the covariates of risk sharing. To do this,
we restrict the risk sharing coe¢ cients so that they satisfy = X , where X is an N K
matrix whose i-th row contains the (time-invariant) covariates of risk sharing for country
i (including a constant). Replacing in (10), estimation proceeds along the same lines as
above, with the parameter vector now given by : That is, we solve
0
min c
~ ~ ; H; W0 ; X; )
f (Z c
~ ~ ; H; W0 ; X; )
f (Z (12)
~ ; H; W0 ; X; ) = IN ~ 1
where now f (Z Z W0 1 X HX ( 0
N HW0 X ) + vec (IN diag (X )) .
Like in the unrestricted case, we use the Driscoll and Kraay (1998) robust covariance matrix
estimator to conduct inference on :
3.2 Income prediction
Empirical implementation of the risk sharing model requires suitable forecasts of permanent
income. To construct them, we use a simple time-series setting allowing for common factors
a¤ecting per capita income growth across countries. Speci…cally, we assume that per capita
GDP growth depends on its lagged value as well as the current and lagged values of world
per capita income:
yit = i + ai yit 1 + bi y t + c i y t 1 + vit i = 1; :::; N ; t = 1; :::; T (13)
X
N
where we have de…ned the growth rate of world per capita GDP yt = sj yjt = s0 yt ;
j =1
yt = ( y1t ; :::; yN t )0 ; and s = (s1 ; s2 ; :::; sN )0 is a set of shares, which will be taken as
0
constant in what follows, with Ns =1: In turn, vit is a disturbance assumed independent
across t and i. If bi = ci = 0 for all i, (13) reduces to a set of country-speci…c …rst order
autoregressive models (AR1), similar to that employed by Crucini (1999). In turn, if ci = 0 for
matrix of partial derivatives r f :
13
all i but bi 6= 0; we have an intermediate case in which common factors have contemporaneous
e¤ects but not lagged e¤ects.
Stacking the T observations for country i, we can write
0 0
yi s IT y bi = T i + yi; 1 ai + s IT y 1 ci + vi (14)
0 0 0
where yi ; vi and yi; 1 are T 1 vectors, and y = ( y1 ; :::; yN ) . De…ning the N 1
0 0 0
parameter vectors ; b; and c; the N T 1 vector v = (v1 ; :::; vN ) ;and the matrix diag (a)
with a = (a1 ; :::; aN )0 along the main diagonal and zeros everywhere else, the full system can
be compactly expressed
h 0
i
[(IN bs0 ) IT ] y = ( T) + diag (a) + cs IT y 1 + v: (15)
Like in the case of the risk sharing model described above, estimation of ( 0 ; a0 ; b0 ; c0 )0
can be accomplished by solving the problem
min g( y; y 1 ; s; )0 g( y; y 1 ; s; )
0
where g( y; y 1 ; s; ) = [(IN bs0 ) IT ] y ( T )+ diag (a) + cs IT y 1 : With
vit uncorrelated across i and t, a heteroskedasticity-robust covariance matrix for ^ can be
constructed as
1
Cov (^) = [r g
^0 r g
^] r0 g
^ diag (v ^)0 r g
^) diag (v ^0 r g
^ r g ^ 1;
^) is an N T
where diag (v N T matrix with the estimated residuals along the main diagonal.
Using the parameter estimates, the innovation to permanent income can be constructed
using the recursive formula
^ yt
yt = ~ + P 1 +w
^t (16)
14
1 0 1
^
where P ^ 1
IN + b ^
s0 b s0 ^ +^
A cs , ~ ^ 1
IN + b ^
s0 b s0 ^ , and w
^t
1
^ 1
IN + b ^
s0 b s0 v
^t . Simple algebra shows that the innovation to permanent income
ypt then is
1
ypt Et 1 ypt = IN + IN ^
P ^ w
P ^t
1 1
= IN + IN ^
P ^
P ^ 1
IN + b ^
s0 b s0 v
^t (17)
3.3 Data
The empirical sample is dictated by data availability. We work with the …fty largest countries
in the world (in terms of their respective GDP in U.S. dollars in the year 2000) for which
complete annual data on consumption, income and foreign asset and liability positions (as
described below) could be assembled. Table B1 in the appendix lists the countries included
in the analysis. Taken together, these countries account for over 93 percent of world GDP in
the year 2000.
Data on real GDP growth, aggregate consumption growth and total population are taken
from the Penn World Tables 6.3. Along the time dimension, our sample runs from 1970 to
2007. We use the 1970-71 averages to construct the initial shares W0 and population H.10
The regression sample therefore runs from 1972 to 2007, and comprises 1,800 country-year
observations. For the calculation of permanent income, we set the discount factor = 1=1:05:
For our analysis of the covariates of risk sharing, we use data on international asset
and liability positions drawn from the Lane and Milesi-Ferretti dataset released on August
2009,11 which covers the period 1970-2007. Speci…cally, we consider the ratios of total foreign
assets and liabilities to GDP, as well as the ratios of FDI assets and liabilities to GDP.
These variables relate to countries’international …nancial integration. In addition, we also
10
However, estaimtion results change very little if we use instead the avearges over 1970-74 or the year
1970 only.
11
Speci…cally, the data are drawn from the updated and extended version (as of August 2009) of the
External Wealth of Nations Mark II database developed by Lane and Milesi-Ferretti (2007).
15
experiment with trade openness (as measured by imports plus exports divided by GDP) as a
measure of real integration. We include a variable measuring real integration because, in the
end, any redistribution of endowments due to (implicit or explicit) risk sharing agreements
should be materialized through ‡ows of goods. Lastly, because domestic …nancial depth can
also contribute to risk sharing among domestic agents (perhaps lessening the need for, or the
value of, international risk sharing) we also use data on domestic …nancial depth, as captured
by the ratio of domestic credit to GDP. Like the trade openness indicator, the credit depth
measure is drawn from the World Development Indicators. To avoid simultaneity concerns,
the empirical exercises assessing the covariates of risk sharing employ these variables measured
at the beginning of the sample –that is, in 1972 (or the …rst year in which data is available).
4 Empirical results
We proceed in two stages. First, we estimate the income process given by (15) and construct
the innovations to permanent income (17); then we use the latter to estimate the risk sharing
model (10).
4.1 Estimation of the income process
To estimate (15), we construct the country shares of global GDP s using averages over the
entire sample period.12 We estimate three di¤erent speci…cations of (15). The …rst one
ignores common shocks by setting b = c = 0, and therefore simpli…es to a system of N AR1
equations, similar to that employed by Crucini (1999). The second speci…cation allows for
contemporaneous e¤ects of common factors by leaving b unrestricted while still imposing
c = 0: The third speci…cation allows for both current and lagged common shocks by leaving
both b and c unrestricted.
Table 1 summarizes the estimation results. The …rst column reports the results of the AR1
12
The correlation between the growth rate of global GDP per capita and the sum of individual-country
per capita GDP growth rates weighted by these shares equals 0.95.
16
speci…cation. The average of the country-speci…c autoregressive parameters (denoted ai in
(13)) equals 0.244. The Wald test shows that they are jointly highly signi…cant. Inspection
of the individual-country estimates shows that they are all smaller than one in absolute
value (the largest one equals 0.658), so the estimated growth dynamics are stable. However,
s (2004, 2012) cross-sectional dependence test overwhelmingly rejects the null that
Pesaran’
the estimation residuals are uncorrelated across countries, which suggests the presence of
common factors – although, strictly speaking, the performance of the test in settings like
ours featuring cross-equation parameter restrictions has not been examined in the literature.
Column 2 of table 1 turns to the model with contemporaneous common shocks. Overall,
these are highly signi…cant (as shown by the corresponding Wald test) and relatively large:
the mean of the country-speci…c parameters on global GDP growth (denoted bi in (13)) equals
0.835. In turn, the mean of the country-speci…c autoregressive parameter estimates remains
roughly similar to that in column 1. However, inspection of the characteristic roots of the
^ in (16) now reveals the presence of one negative root outside the
implied transition matrix P
unit circle. Lastly, the cross-section dependence test statistic is not signi…cant anymore at
the 5 percent level, although it still exceeds the 10 percent critical value.
Column 3 reports estimates of the full model including current and lagged global growth.
The lagged e¤ect (ci in (13)) is, on average, negative and small, and not signi…cantly di¤erent
from zero. However, the Wald test statistic reported in the lower half of the table shows that
the country-speci…c parameters on lagged global growth are jointly highly signi…cant, just
like those on current global growth and lagged country-speci…c growth. Inspection of the
individual-country estimates shows that 11 countries exhibit estimates of the ci parameter
signi…cant at the 5 percent level. In turn, the country-speci…c autoregressive parameter
estimates, as well as those on contemporaneous global growth, remain roughly similar to
^ matrix are smaller than one in absolute
those in column 2. Further, all the roots of the P
value (the largest one equals 0.879). Finally, the test statistic of cross-sectional dependence
falls short of 10 percent signi…cance, suggesting that current and lagged global GDP growth
17
su¢ ce to capture common shocks to the countries in the sample.
Using the parameter estimates summarized in Table 1, the innovations to permanent
income can be readily constructed employing (17). We construct two sets of innovations,
corresponding to the …rst and third model speci…cations in the table; we leave aside the second
speci…cation because of the unstable root mentioned earlier. The two sets of innovations
constructed in this way are not very di¤erent from each other: the correlation between them
equals 0.92.
4.2 Estimation of the risk sharing model
Table 2 summarizes the NLS estimates of the risk sharing parameters obtained using the two
sets of permanent income innovations constructed in the previous step. The results are very
consistent across the two speci…cations. They can be summarized in three main facts. First,
46 of the 50 individual-country estimates lie between zero and one, as predicted by theory.
When using the AR1-based innovations, the four remaining point estimates are negative;
in contrast, when using the system-based innovations, three are negative and one exceeds
1. However, in no case can we reject the null hypothesis that these parameters lie between
zero and one, whether individually or jointly. Moreover, the vast majority of the individual
estimates in the admissible region (41 out of 46) are statistically signi…cant at least at the
10 percent level; they are overwhelmingly signi…cant jointly.
Second, the country-speci…c risk sharing parameters are centered around 0.40-0.42. This
is roughly similar to the result obtained by Crucini (1999) using the restricted homogenous
model of partial risk sharing; he reports an average partial risk sharing parameter between
0.37 and 0.60 for the group of G7 countries over the period 1970-1987 (his estimates vary
depending on the assumed income process). In turn, Asdrubali and Kim (2008) obtain higher
coe¢ cients of partial risk sharing, namely 0.77 for a group of 15 European Union countries
and 0.82 for all OECD countries, both over the period 1960-2004.
And third, the data overwhelmingly reject the null hypothesis that the risk sharing coe¢ -
18
cient is the same for all sample countries –as shown by the Wald test reported in the middle
of the table. Furthermore, the estimates di¤er systematically between industrial and develop-
ing economies: on average, the former exhibit higher degrees of risk sharing than the latter.
The same conclusion is reached by Kose, Prasad, and Terrones (2009) using conventional risk
sharing regressions, and by Flood, Marion, and Matsumoto (2012) using a di¤erent measure
of imperfect risk sharing. In our case, the mean for industrial countries exceeds 0.5, while
for developing countries it equals 0.35. As the table also shows, the di¤erence between both
means is highly signi…cant. However, additional tests show that even within these two groups
the individual-country coe¢ cients display signi…cant variation. Indeed, Wald tests of equal-
ity of the risk sharing coe¢ cients of all the countries in each income group overwhelmingly
reject the null, both among developing countries and among industrial countries.
Table B2 in the appendix reports the parameter estimates for the individual countries for
both speci…cations in Table 2. For the majority of countries, the parameters show only minor
changes across speci…cations; indeed, the correlation between the two sets of estimates is 0.91.
One notable exception to this …nding is the U.S., whose risk sharing parameter estimate equals
0.93 when we allow for common factors across countries, but rises to a whooping 1.5 when
we use the independent AR1 processes for income growth. This di¤erence could be due to
the large e¤ect of U.S. growth on global economic activity and, thereby, on growth in all the
other countries –a feature that is built into the system estimates of the income process but
not into the AR1 estimates. Using the latter therefore results in a distorted indication of the
degree of U.S. risk-sharing.
The estimates reported in Table 2 are obtained without imposing any restriction on their
admissible values. As a result, four of the individual point estimates lie outside the [0,1]
interval assumed by theory, although – as already noted – not signi…cantly so. Still, given
the fact that in our empirical model the risk sharing parameters of all countries are linked
through cross-equation restrictions, one may wonder how the estimates would change if they
were explicitly constrained to lie within the unit interval. To investigate this issue, we
19
re-estimated the two speci…cations in Table 2 constraining all parameters to fall within the
theoretically-admissible region. The estimates that result are virtually indistinguishable from
those shown in Table 2. In fact, the correlation between matching pairs of unrestricted and
restricted estimates is around 0.98 in both the AR1 and the common-factor models. Thus,
to save space we do not report the restricted estimates.
4.2.1 Patterns of consumption risk sharing over time
Since the mid-1980s, the world has seen a large and ever-increasing level of …nancial integra-
tion, facilitated by the removal of barriers to international capital movements and re‡ected
in a steady rise in cross-border asset and liability positions (Lane and Milesi-Ferretti, 2007).
This observation leads to a natural question, namely: has the rise in global integration been
associated with a corresponding rise in consumption risk sharing across countries? Put dif-
ferently, are countries doing a better job at sharing their idiosyncratic income shocks? To
address this question, we use our model of partial risk sharing to examine the evolution of
consumption risk sharing over time.
For this purpose, we calculate time-varying estimates of the parameters of the consump-
tion risk sharing model over rolling time samples. Speci…cally, we use moving windows of
21 years each. With 36 years of data in total, this yields 16 sets of estimates. Importantly,
prior to estimation of the risk sharing model we also recalculate the permanent income inno-
vations in a matching way, by estimating the income process over rolling samples of 21 years
of data each; this ensures that both construction of the permanent income innovations and
estimation of the risk sharing parameters use data from the same time period.13
The results from this exercise are summarized in …gures 1 and 2. Figure 1 plots the mean
of the risk sharing parameter estimates obtained in each estimation window, along with their
two standard-error bands. The two graphs correspond to the estimates obtained using the
rolling AR1 GDP growth forecasts and the estimates obtained using the system-based growth
13
Similarly, for each window we recalculate population H and the initial income shares W0 using the
average of the two annual observations preceding the initial year of the window.
20
forecasts over rolling windows.
In both graphs, mean risk sharing displays a cycle: a slight initial decline, which is
reversed in the windows that start in the late 1970s, and then a steady increase beginning
around the windows that start in the mid-1980s. Closer inspection reveals that in the top
graph the mean estimate obtained in the …nal window lies outside the 95-percent con…dence
region of the initial window, suggesting that over the entire sample period there has been a
statistically signi…cant increase in the average degree of consumption risk sharing, at least
when the calculation of the latter is based on income forecasts that allow for common factors.
Examining the individual-country risk sharing parameter estimates we …nd that a majority of
countries (33 when using the common factor-inclusive forecasts, and 27 when using the AR1
forecasts) exhibit higher coe¢ cients in the …nal estimation window than in the initial one.
Further, for 16 countries (15 when using the AR1 forecasts) the increase in the risk sharing
parameter is statistically signi…cant at least at the 10 percent level. At the other end, 10
countries (or eight, when using AR1 income growth forecasts) show statistically signi…cant
declines in risk sharing between the initial and …nal estimation windows.
Figure 2 plots the time path of the average risk sharing coe¢ cients separately for industrial
and developing countries, along with their respective two standard-error bands, for the same
two sets of permanent income innovations as in the previous …gure. Both panels reveal a clear
contrast between the mean estimates of the two country groups. The mean for industrial
countries displays a rising trend over time, particularly marked in the top panel, which is
based on the permanent income forecasts inclusive of common factors. Indeed, this rising
trend of average risk sharing among rich countries is the force behind the trend increase in
overall mean risk sharing found in Figure 1. More formally, in both panels the 95-percent
con…dence region for the industrial country mean corresponding to the …nal window lies fully
outside that obtained in the initial window. In other words, over the period of analysis
there has been a highly signi…cant rise in mean risk sharing among industrial countries. In
contrast, there has been no discernible trend among developing countries. While the …gure
21
suggests, if anything, a slight decline in their mean risk sharing, the 95 percent con…dence
region of the …nal window is contained almost in full in the initial one, which suggests that no
signi…cant change has taken place. The contrasting time path of the two group means re‡ects
the fact that 10 of the 16 countries (11 out of 15, when using the AR1 forecasts) showing
a statistically signi…cant (at the 10 percent level) increase in the coe¢ cient of risk sharing
belong to the industrial country group. In contrast, all of the countries whose coe¢ cient of
risk sharing exhibits a statistically signi…cant decline belong to the developing country group.
4.2.2 Covariates of consumption risk sharing
The empirical results reported earlier unambiguously showed that the coe¢ cients of partial
risk sharing display signi…cant heterogeneity across countries. While the country-level risk
sharing arrangement that underlies our model is admittedly an abstraction, we can interpret
the variation in the estimated coe¢ cients of partial risk sharing as re‡ecting cross-country
variation in policies or institutions that help or hinder consumption risk sharing. The natural
next step is to investigate if the cross-country pattern of coe¢ cients of partial risk sharing
is related to measures of …nancial and trade openness commonly viewed as re‡ecting the
mechanisms through which risk sharing may be actually implemented. This approach has
been used in previous empirical literature, which has examined the relation between summary
measures of the extent and form of international …nancial integration, and conventional
reduced-form estimates of risk sharing coe¢ cients. For example, using this approach, Kose,
Prasad, and Terrones (2009) conjecture that emerging economies have failed to attain the
levels of consumption risk sharing of the developed countries because their international
liabilities have been dominated by foreign debt instead of other, more resilient, liabilities like
FDI or portfolio ‡ows.14
Formally, to explore the covariates of consumption risk sharing we re-estimate the risk
sharing model as speci…ed in (12), that is, with the vector of country-speci…c risk sharing
14
See also Sorensen et al (2007) and Kolinsli et al (2012). In turn, Fratzscher and Imbs (2009) correlate
their coe¢ cients of bilateral risk-sharing to measures of …nancial opennes and institutional quality.
22
coe¢ cients given by = X , where X is a matrix of time-invariant risk sharing determinants,
and now is the vector of parameters to be estimated. To avoid simultaneity concerns, the
variables in X are measured at the beginning of the sample period, and expressed in logs in
15
order to mitigate the in‡uence of extreme observations.
Table 3 reports the estimates of obtained using di¤erent choices of the X variables. All
speci…cations include also a constant, not reported in the table. The results shown correspond
to estimates obtained with permanent income innovations constructed using income forecasts
inclusive of common factors (that is, based on the model in the last column of Table 1); results
using instead AR1-based forecasts are roughly similar but are not reported to save space.
The …rst column of Table 3 relates the degree of risk sharing to total foreign assets and
total foreign liabilities, both as percent of GDP. Both variables carry positive and signi…cant
coe¢ cients, suggesting that higher degrees of international …nancial integration, as measured
by gross foreign asset and liability stocks, come along with higher consumption risk sharing.16
Using the parameter estimates, we can compute the implied risk sharing coe¢ cients of the
di¤erent countries as ^ = X^: As shown in the bottom part of the table, they are centered
around 0.47, with the individual coe¢ cients ranging from .097 to 0.903. Further, on average
they are higher for industrial than for developing countries, in line with the estimation results
reported in Table 2 above. Finally, the last row of Table 3 shows the correlation between
these implied risk sharing coe¢ cients and their unrestricted counterparts (shown in the …rst
column of Table B2). For the speci…cation in column 1, the correlation is just 0.31, which
suggests that foreign asset and liability stocks account for a signi…cant but relatively modest
proportion of the cross-country variation in consumption risk sharing.
Column 2 of Table 3 adds to the total foreign asset and liability stocks a measure of
15
For a few countries in the sample, the initial observation (corresponding to the year 1972) on some of
the foreign assets and liabilities included among the X variables is missing from the Lane-Milesi dataset. In
those cases, we use instead the earliest available value. Additionally, for some countries the ratios of FDI
assets and/or liabilities to total foreign assets and liabilities contain some zeros in the initial sample years.
To allow taking logs, we replace them with very small values (1.e-6).
16
In reduced-form regressions, Kose et al (2009) obtain a similar result for industrial countries, while
Sorensen et al (2007) …nd that gross foreign assets are positively related to risk sharing among OECD
countries, while for gross foreign liabilities the relationship is negative instead.
23
their composition, speci…cally distinguishing between FDI and other types of foreign assets
and liabilities, to test whether direct investment is di¤erent regarding its contribution to risk
sharing.17 In contrast with the preceding column, the estimated coe¢ cients on FDI asset
and liability positions, relative to overall asset and liability stocks, carry opposing signs,
suggesting that residents’direct investment abroad helps risk sharing more than other types
of foreign asset holdings, but non-residents’direct investment at home has the opposite e¤ect.
While this latter result seems to be at odds with theoretical expectations, we note that there
is a body of evidence that documents that the share of FDI in total in‡ows tends to be larger
in countries that are riskier and possess weaker institutions (Hausmann and Fernandez-Arias,
2000; Albuquerque, 2003, and Daude and Fratzscher, 2008). In other words, a large share of
FDI liabilities could be the constrained optimal mix of foreign …nancing for a country with
both a riskier environment and fewer opportunities for risk sharing.
The range of the implied risk sharing coe¢ cients narrows somewhat relative to that in
column 1, but the opposite happens to the gap between the industrial-country and developing-
country averages. Moreover, the correlation with the unrestricted risk sharing coe¢ cient
estimates in Table B2 rises substantially, to 0.47.
Going beyond foreign assets and liabilities, columns 3 to 5 assess the contribution of two
other potentially important variables, namely trade integration, as measured by imports plus
exports over GDP, and domestic …nancial development, as measured by the stock of credit
to the private sector over GDP. However, the estimation results suggest that these variables
add little to the determination of consumption risk sharing patterns, since they are both
insigni…cant, whether individually or jointly. Indeed, a Wald test of the null that in column
5 both variables carry a zero coe¢ cient yields a p-value of 0.80, suggesting that they can
be safely dropped from the speci…cation. On the other hand, the estimated coe¢ cients on
the volume and composition of foreign assets and liabilities show minimal change relative to
those in column 2.
17
The estimation results are virtually identical if instead of FDI we focus on equity plus FDI assets and
liabilities vs the rest of assets and liabilities.
24
Since the unrestricted risk sharing coe¢ cient estimates shown in table 2 above were, on
average, higher for industrial than for developing countries, it is natural to wonder if their
cross-country pattern is somehow related to countries’level of development –as captured by
(log) per capita GDP –even after taking into account their degree of international …nancial
integration. This question is addressed in columns 6 and 7, which add per capita GDP to
the list of regressors of columns 2 and 5, respectively. In both cases the per capita GDP
parameter estimate falls well short of statistical signi…cance. Further, in column 7 a Wald
test fails to reject the null that all the variables other than total foreign assets and liabilities,
and their respective FDI shares, carry zero coe¢ cients (with a p-value of 0.89).
Overall, these empirical exercises suggest that international …nancial integration is a sig-
ni…cant factor behind the cross-country patterns of consumption risk sharing. Speci…cally,
both the degree of overall integration, as re‡ected in total foreign asset and liability posi-
tions, and the FDI asset position, are positively related to countries’performance in terms
of consumption risk sharing, while FDI liabilities show the opposite pattern. Moreover, once
these summary measures of international …nancial integration are taken into account, trade
integration, domestic …nancial development, and the overall level of development have little
to contribute.
5 Concluding remarks
A considerable literature is concerned with assessing the extent to which countries share
their consumption risk. Much of it, however, makes use of empirical models designed to test
the null hypothesis of perfect consumption risk sharing. Once such hypothesis is rejected
– as is almost invariably the case in practice – those models do not o¤er a rigorous basis
for inferences about the degree of imperfect risk sharing present in the data. Drawing such
inferences requires an empirical model of partial risk sharing. Furthermore, unless one is
willing to assume that the extent of consumption risk diversi…cation is the same across all
25
countries in the world, the model needs to allow for cross-country heterogeneity in the degree
of risk sharing.
This paper extends the existing literature by developing an empirical model that …ts
those two requirements. In the model, countries contribute possibly di¤erent fractions of
their income to a common pool, in exchange for a claim on the aggregate income contributed
to the pool by all countries. The fraction of income contributed to the global pool by each
country can be naturally interpreted as its respective degree of risk sharing. Solution of the
s consumption path depends on
model yields a system of equations in which each country’
the innovations to the permanent income of all countries. Moreover, the system features
nonlinear parameter restrictions across equations.
The model is implemented empirically using panel data for industrial and developing
economies. Estimation results show that consumption risk sharing varies signi…cantly across
countries. On the whole, rich countries exhibit higher degrees of risk sharing than developing
countries, and the gap between both country groups appears to have widened over the period
of …nancial globalization. Moreover, the pattern of consumption risk sharing across countries
is signi…cantly related to their degree of …nancial openness. Countries possessing larger stocks
of international assets and/or liabilities exhibit larger degrees of risk sharing. Countries whose
foreign asset stocks are more tilted towards FDI assets show higher consumption risk sharing,
while the opposite happens with countries whose foreign liability stocks involve larger FDI
liabilities. Once these measures of international …nancial integration are taken into account,
trade integration, domestic …nancial development and the level of GDP per capita do not
contribute signi…cantly to explain the cross-country pattern of consumption risk sharing.
26
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Appendix A. Proofs.
Here we show how to log-linearize the innovation to the permanent income. Consider the
present value of a history of future incomes in some country (we omit the subindex i),
X
1
Ytp = (1 ) k
Yt+k
k=0
X
1
k
= Yt 1 (1 ) (1 + yt+k yt 1 ) ;
k=0
where yt log Yt and we use the approximation yt+k yt 1 = (Yt+k Yt 1 ) =Yt 1 . By adding
and subtracting yt+s for s = 1; 2; ::: it follows that Ytp can be written as
" #
X
1 X
k
Ytp = Yt 1 1 + (1 ) k
yt+s
k=0 s=0
" #
X
1
k
= Yt 1 1+ yt+k :
k=0
Taking logs and using the approximation log (1 + x) = x then gives
p
X
1
k
yt = yt 1+ yt+k :
k=0
The approximation to the innovation to the permanent income is de…ned as
p p
X
1
k
Et yt Et 1 yt = [Et yt+k Et 1 yt+k ] :
k=0
so that the innovation in permanent income is approximated by the innovation in future
growth rates.
29
Appendix B. Additional Tables
Table B1
List of countries
Industrial Developing
Australia Algeria
Austria Argentina
Belgium Bangladesh
Canada Brazil
Denmark Chile
Finland China
France Colombia
Germany Dominican Republic
Greece Egypt
Ireland Hong Kong
Italy Hungary
Japan India
Netherlands Indonesia
New Zealand Israel
Norway Korea, Republic of
Portugal Malaysia
Spain Mexico
Sweden Morocco
Switzerland Pakistan
United Kingdom Peru
United States Philippines
Poland
Romania
Singapore
South Africa
Thailand
Turkey
Venezuela
Vietnam
30
Table B2
Individual risk sharing coefficie nts
Income prediction model Income prediction model
Industrial Developing
System AR1 System AR1
Australia 0.457 0.293 Algeria 0.447 0.390
Austria -0.024 0.156 Argentina -0.108 -0.082
Belgium 0.331 0.371 Bangladesh 0.054 0.090
Canada 0.527 0.498 Brazil 0.275 0.265
Denmark 0.250 0.173 Chile 0.277 0.382
Finland 0.696 0.704 China 0.080 0.059
France 0.488 0.524 Colombia 0.221 0.380
Germany 0.681 0.646 Dominican Republic 0.261 0.335
Greece 0.615 0.707 Egypt 0.675 0.658
Ireland 0.908 0.868 Hong Kong 0.518 0.436
Italy 0.158 0.352 Hungary 0.530 0.570
Japan 0.653 0.604 India 0.133 0.069
Netherlands 0.423 0.231 Indonesia 0.454 0.375
New Zealand 0.628 0.455 Israel 0.380 0.394
Norway 0.495 0.330 Korea, Republic of 0.307 0.274
Portugal 0.647 0.712 Malaysia -0.044 -0.220
Spain 0.654 0.680 Mexico 0.419 0.357
Sweden 0.562 0.373 Morocco -0.333 -0.290
Switzerland 0.512 0.354 Pakistan 0.498 0.495
United Kingdom 0.486 0.407 Peru 0.411 0.415
United States 0.931 1.525 Philippines 0.825 0.777
Poland 0.416 0.479
Romania 0.757 0.779
Singapore 0.663 0.659
South Africa 0.650 0.643
Thailand 0.550 0.514
Turkey 0.067 0.008
Venezuela 0.375 0.469
Vietnam 0.411 0.397
31
Table 1
NLS estimate s of the income prediction model
(1) (2) (3)
Means of the country-specific estimates
Lagged GDP growth 0.24 0.23 0.23
(9.39) (9.63) (9.30)
Global GDP growth -- 0.84 0.82
(12.05) (12.22)
Lagged global GDP growth -- -- -0.07
(1.01)
Wald tests of joint significance (p-values)
Lagged GDP growth 0.00 0.00 0.00
Global GDP growth -- 0.00 0.00
Lagged global GDP growth -- -- 0.00
Test of cross-sectional dependence (p-value) 0.00 0.09 0.12
The top half of the table reports the means of the country-specific parameter estimates and their
robust t-statistics in parentheses. The Wald test statistics in the bottom half of the table are
distributed as chi-square with 50 degrees of freedom. The last row in the table reports the test of
cross sectional dependence of Pesaran (2004, 2012).
32
Table 2
NLS estimates of the country-specific risk-sharing parameters
(1) (2)
Income forecasting model System AR1
Statistics of risk sharing e stimates
Number of risk sharing estimates between 0 and 1 46 46
of which significantly positive 41 41
Number of risk sharing estimates greater than 1 0 1
of which significantly greater than 1 0 0
Number of risk sharing estimates smaller than zero 4 3
of which significantly negative 0 0
Median 0.42 0.40
Maximum 0.93 1.53
Minimum -0.33 -0.29
Wald test of equality of all estimates (p-value) 0.00 0.00
Average estimates
All countries 0.42 0.42
(17.1) (11.4)
Industrial countries 0.53 0.52
(20.9) (11.7)
Developing countries 0.35 0.35
(9.9) (8.2)
Wald test of difference in means (p-value) 0.00 0.00
Wald test of joint significance of all risk-sharing parameters 0.00 0.00
Robust t-statistics in brackets, using spatial-correlation consistent standard errors of Driscoll and Kraay (1998). The
Wald test statistics at the bottom of the table are distributed as chi-square with 50 degrees of freedom.
33
Table 3
NLS estimates of the covariates of risk-sharing
(1) (2) (3) (4) (5) (6) (7)
Total foreign assets / GDP 0.121 0.085 0.102 0.085 0.105 0.085 0.105
(6.10) (3.33) (2.67) (3.34) (2.67) (3.33) (2.67)
Total foreign liabilities / GDP 0.105 0.085 0.100 0.085 0.101 0.083 0.100
(9.92) (7.14) (3.48) (6.50) (3.56) (4.76) (2.93)
FDI assets / Total assets 0.016 0.016 0.016 0.016 0.016 0.016
(3.06) (3.00) (2.88) (2.92) (2.39) (2.45)
FDI liabilities / Total liabilities -0.028 -0.031 -0.028 -0.032 -0.028 -0.032
(-4.88) (-4.11) (-4.37) (-3.62) (-4.79) (-3.62)
Trade openness (imports + exports / GDP) -0.033 -0.038 -0.037
(-0.63) (-0.69) (-0.68)
Private sector credit / GDP 0.000 -0.010 -0.011
(-0.01) (-0.23) (-0.24)
GDP per capita 0.003 0.002
(0.12) (0.06)
Wald test of joint significance 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Implied risk-sharing coefficients
Median 0.440 0.477 0.478 0.477 0.475 0.479 0.475
Maximum 0.903 0.825 0.836 0.825 0.832 0.823 0.832
Minimum 0.097 0.102 0.103 0.102 0.104 0.101 0.103
Mean - All countries 0.468 0.483 0.483 0.483 0.483 0.483 0.475
Industrial countries 0.543 0.585 0.589 0.585 0.588 0.586 0.569
Developing countries 0.414 0.410 0.407 0.410 0.406 0.409 0.382
Correlation with unrestricted risk-sharing coefficients 0.306 0.471 0.477 0.471 0.476 0.471 0.476
Robust t-statistics in brackets, using spatial-correlation consistent standard errors of Driscoll and Kraay (1998). All the explanatory variables are
expressed in logs. All specifications include also a constant not reported here. The last row in the table reports the correlation coefficient between the
individual-country risk-sharing parameters predicted by the regression in each column with those reported in the first column of Table B2.
34
Figure 1. Average coefficient of partial risk sharing across countries
A. System income prediction with a common factor
0.55
0.50
0.45
0.40
0.35
0.30
0.25
72-92 74-94 76-96 78-98 80-00 82-02 84-04 86-06
B. Income prediction based on independent AR1 processes
0.55
0.50
0.45
0.40
0.35
0.30
0.25
72-92 74-94 76-96 78-98 80-00 82-02 84-04 86-06
Solid lines represent average point estimate of the coefficients of partial risk sharing. Shaded areas
represent two standard errors around the average estimates.
35
Figure 2. Coefficients of partial risk sharing by income group
A. System income prediction with a common factor
Industrial countries Developing countries
0.80 0.80
0.70 0.70
0.60 0.60
0.50 0.50
0.40 0.40
0.30 0.30
0.20 0.20
0.10 0.10
72- 74- 76- 78- 80- 82- 84- 86- 72- 74- 76- 78- 80- 82- 84- 86-
92 94 96 98 00 02 04 06 92 94 96 98 00 02 04 06
B. Income prediction based on independent AR1 processes
Industrial countries Developing countries
0.80 0.80
0.70 0.70
0.60 0.60
0.50 0.50
0.40 0.40
0.30 0.30
0.20 0.20
0.10 0.10
72- 74- 76- 78- 80- 82- 84- 86- 72- 74- 76- 78- 80- 82- 84- 86-
92 94 96 98 00 02 04 06 92 94 96 98 00 02 04 06
Solid lines represent average point estimate of the coefficients of partial risk sharing. Shaded areas represent two standard errors around the average estimates.
36