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\title{Are Correlations of Stock Returns Justified by Subsequent Changes in
National Outputs?}
\author{Bernard Dumas,\\ \small INSEAD, 77305 Fontainebleau C\'edex, France \and Campbell R.\
Harvey, \\ \small Duke University, Durham, NC\ 27708, USA \and Pierre Ruiz, \\ \small McGill University\
Montreal, Quebec, Canada\thanks{%
Dumas and Ruiz acknowledge gratefully the support of the HEC\ Foundation.\
Dumas is also affiliated with the University of Pennsylvania (as an Adjunct
Professor), the NBER and the CEPR.\ Harvey is also affiliated with the NBER.
The comments of Enrique Arzac, Pierre Batteau, Geert Bekaert, Bruno Biais,
William Brainard, John Campbell, Jacques Cr\'{e}mer, Harris Dellas, Vihang Errunza, Christian Gouri\'{e}roux, Robert Hodrick, Hayne Leland, Jacques Olivier, Jayendu Patel, Franck Potier, Jean-Charles Rochet, Jay Shanken,
Huntley Schaller, Robert Shiller, Alan Stockman, and Philippe Weil are gratefully acknowledged.
We also appreciate the comments of participants at the NBER Asset Pricing workshop at
the University of Pennsylvania, the French Finance Association, and the meeting of Inquire (Europe),
as well as seminar participants the City
University of London, the European Monetary Institute, the European
University Institute in Florence, the University of Rochester, the London School of
Economics, Rice University, Universitat Pompeu Fabra, the
University of Toulouse, Yale University, University of Bern, the Centre de
Recherche en Economie et Statistique (CREST) and the University of Virginia. We thank Jacques Raynauld who
gave us the SCOREM code to estimate the ``dynamic single-index'' model and
Wouter DenHaan for keeping statistical programs available on his website at
the University of California, San Diego. Ben Zhang was very helpful in
developing the econometric test in the final subsection of the paper.}}
\date{\small September 24, 2001}
\maketitle
\vspace{.2in}
\begin{abstract}
In an integrated world capital market, the same pricing kernel is applicable
to all securities. We apply this idea to the stock returns of different
countries. We investigate the underlying determinants of cross-country stock
return correlations. First, we determine, for a given, measured degree of
commonality of country outputs, what should be the degree of correlation of
national stock returns. We propose a framework that contains a
statistical model for output and an intertemporal financial market model for
stock returns. We then attempt to match the correlations generated by the model with
measured correlations.\ Our results show that under the hypothesis of market
segmentation, the model correlations are much smaller than observed correlations.
However, assuming world markets are integrated, our model correlations closely match observed correlations.
\end{abstract}
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Shiller (1981) argues, with a simple asset pricing model, that in the
United States ``stock prices are too volatile to be justified by subsequent
changes in dividends''. This is confirmed by Campbell (1996) for a number of
other countries. The ``excess volatility'' puzzle is commonly ascribed to an
excessive degree of volatility of the pricing kernel.\footnote{%
``Excessive'' is understood relative to the observed degree of volatility of
consumption.}\ In integrated world capital markets, the same pricing
kernel is applicable to all securities. If the kernel is excessively
volatile, this can, under some assumptions, translate into an equally excessive degree of
correlation of world equity returns.
It can be, however, that world capital markets are not integrated.\ We
shall demonstrate in the context of a particular model that, other things
equal, over the relevant parameter range, correlations of stock returns are
larger in an integrated market than they would be in a segmented market.\
Our framework allows us to diagnose the degree of integration of financial
markets. If the market is segmented, the correlation of world equity returns
should be excessively low. When it is based on an asset pricing model which
assumes that markets are completely integrated, one could interpret a
rejection (observed correlations lower than model correlations) as either
evidence against the model or against the hypothesis of market integration.
However, we are able to apply the technique under the hypothesis of market
segmentation and, in this way, draw a meaningful comparison.
The goal of our paper is, therefore, to explore, and draw economic
inferences from, correlations of stock returns, much like Shiller (1981) did
for volatility. We ask the question, for a measured commonality in country
outputs, what should the correlation among equity returns be? Are the equity
correlations higher than can be justified by a dynamic model of the world
financial market equilibrium? And, if they are not found to be higher, we
attempt to answer the alternative question: Are they about what they should
be under financial market integration or are they closer to what they should
be under segmentation? The intuition is that, if most of the variation in
economic activity in two countries is associated with the world business
cycle, then the two countries should have high equity correlations.
Our paper is intermediate between a macroeconomic real business cycle paper
and a finance paper.\footnote{%
See also Canova and de Nicolo (1995).}\ A\ real business cycle paper would
attempt to explain observed facts such as that outputs across countries are
more highly correlated than consumption.\ No equity returns would be
measured.\ A\ finance paper would attempt to explain return correlations
across countries with asset pricing model employing variables like cash flows to
equity that may not be directly or contemporaneously related to output.
To answer the questions posed, we combine a statistical model of the
business cycle with a log-linear asset pricing framework applied to an
exchange economy. After making some choices on the form of the utility
function and some distributional assumptions, we are able to determine the
model's implied level of correlation for two countries' returns.
The modeling of stock returns by means of an exchange economy follows the
tradition of Lucas (1978) and Mehra and Prescott (1985).\ While the
assumption that world output equals world consumption is evidently
simplistic -- and, worse yet, prevents us from drawing useful information
from the physical investment time series\footnote{%
See Cochrane (1991), Restoy and Rockinger (1994).} -- there is no doubt that
it is a useful shortcut and that calibration exercises in this tradition
have produced many insights concerning stock returns. In the international
context, the empirical behavior of each country's consumption is, in fact,
very close to the behavior of the country's production, much more
so than it should be in an internationally integrated world.\footnote{%
See Lewis (1999).}
Our approach offers an opportunity to understand the interplay between the
real economy and stock returns. Three routes have so far been taken to study
this interplay. First, a number of papers (see, for example, Fama, 1990,
Schwert, 1990, Choi \textit{et al.}, 1999) show that there is a relation
between expected output and stock returns.\ This empirical literature is
about the relation between real activity and the conditional first moment
of stock returns, which is not our focus here. Asset pricing tests offer a
second possible route of examination. These tests specify common factors
which each country has sensitivity to (see Ferson and Harvey, 1993, Cheung,
He and Ng, 1997). We can deduce from the estimated sensitivities to the
common factors what the correlation of equity returns should be. Correlation
is determined by a statistical model that determines the relative movement
of each country's return versus some global benchmarks.\ The associated
pricing model is not solved out over time; it only provides expected returns
from covariances but does not endogenize the latter.\ Third, Hamao, Masulis
and Ng (1990) and the many papers that followed this work study the
spillover of information from one economy to another.
These are statistical exercises and it is not possible to use them to
understand correlations of different countries or sectors. While these
studies are important in tracing the type of information that causes common
movement in expected returns and volatility, they do not give us a starting
point. That is, they do not answer the main question in our study: What
should the level of correlation be?
Ammer and Mei (1996) and Campbell and Mei (1993) attack the correlation
issue by decomposing the innovations in stock returns into three components:
news about future dividends, interest rates, and equity premiums. A similar
question had been raised earlier by Shiller (1989) and Beltratti and Shiller
(1993) but their asset pricing model was a ``present-value'' model with
unspecified, potentially stochastic discount rates which only generated wide
bands on the values of correlations.\ In the present paper, the valuation
equation is derived from optimal portfolio choices. Our approach will be
different. We look beyond financial data and tackle the real economy.
Several researchers have examined the correlations of stock returns
internationally.\footnote{%
Among the more recent investigations, see Longin and Solnik (1995), Erb,
Harvey and Viskanta (1994), and Ang and Bekaert (1999). See also Bansal and
Lundblad (2000).} It has been tempting to interpret the measured
correlations as indications of the degree of integration of financial
markets.\ For instance, if one finds that correlations have been rising, one
is tempted to conclude that financial markets are in the process of gradual
integration. However, one needs to control for the degree of correlation of economic fundamentals before
drawing this conclusion.\footnote{%
Bekaert and Harvey (1995) link correlation with the degree of market
integration.\ Freimann (1998) offers an alternative, entirely statistical
procedure based on randomization of industrial sector returns, to compare
country correlations to what they would have been under integration.}
Our work is related to, but unfortunately does not encompass, those studying
time-varying correlations.\ First, we execute
calibrations (and, later, statistical tests) using unconditional moment
conditions. This choice is unrelated to the assumptions of the model%
\footnote{%
Such as the heteroskedasticity assumption.\ See below.} since any model can
be tested on the basis of its unconditional predictions. The choice of
moment conditions is also not a limitation placed on the rational-choice
paradigm.\ In our model, the representative investor dynamically optimizes
his portfolio on the basis of all the information at his disposal.
Second, while we aim to understand why correlations are different across
countries, the correlations are modeled as being constant over time.\
Longin and Solnik (1995) show by means of a statistical model, how
correlations change through time.\footnote{%
Hodrick (1989) derives the multivariate GARCH\ process followed by stock
prices when dividends themselves follow a multivariate GARCH\ process.}\
Both Longin and Solnik (1995) and Erb, Harvey and Viskanta (1994) try
empirically to explain, on the basis of economic variables, how correlations
vary over time.\footnote{%
For instance, non-US stock returns tend to have a higher correlation with US
stock returns while the US is in a recession than while it is in an
expansion. Volatility of returns is also larger while the US is in a
recession.\ See also Perez-Quiros and Timmermann (1996), Ang and Bekaert
(1999) and Carrieri \textit{et al.} (2000).} The analysis in Erb \textit{et al.}
shows that while there is some time-variation in the correlations of the G7
countries' equity returns through time, the ranking of the correlations
rarely changes. That is, while there is variation in both the U.S.-U.K. and
U.S.-Japan correlations through time, the U.S.-U.K. correlation is always
higher than the U.S.-Japan correlation. Similarly, the U.S.-Canada
correlation is always higher than the U.S.-U.K. correlation. While it is
clear that correlations are not constant, our assumption should not
interfere with the main point of our paper, to explain why correlations are
different across different countries.
The paper is organized as follows. Section \ref{firstlook} explores the data
and the phenomena that we are trying to explain. In Section \ref
{common_trend}, we develop the dynamic single-index model of Stock and
Watson (1993) which we will use to define each country's business cycle.\
The log-linear pricing kernel of Restoy and Weil (1993) is explained in
Section \ref{kernel}. Section \ref{combined} applies the log-linear pricing
kernel to the dynamic single-index business cycle model to derive
equilibrium security returns. We then examine, in Section \ref{calibration_}%
, the correlations implied by the model and the actual correlations observed
in the data. Section \ref{test} develops a statistical test of the
hypothesis of financial market integration. Some concluding remarks are
offered in the final section.
\section{\large A first look at the data}
\label{firstlook}In the empirical analysis below, we focus on the behavior
of industrial production and stock returns in twelve OECD\ countries%
\footnote{%
Austria, Belgium, Canada, France, Germany, Italy, Japan, The Netherlands,
Spain, Sweden, United Kingdom and the United States of America.} on a
monthly basis from January 1970 to June 1996.\ The monthly frequency is
dictated by the fact that the time period of the sample cannot be extended
and that more frequent data do improve the precision of estimates of second
moments.\ Figure 1 shows a positive relation between the
correlation of a country's output with OECD output and the correlation of
that country's stock market returns with OECD\ stock returns. We report
a country's correlation with the \textit{other} countries, itself
excluded. This is in order to abstract from the effect of a country's size.\
Had we calculated the correlation of a country with the world, including the
country itself, larger countries would automatically have shown a larger
correlation.\ The index of the other countries' output is calculated with
annually updated GDP\ weights.\footnote{%
To account for the possibility of lags, the correlations of a country's
output with the OECD\ output is, in fact, the square root of the $R^{2}$ in
a multiple regression of the country's industrial production on aggregate
OECD\ industrial production (the country itself excluded), contemporaneous
and with eleven monthly lags. The data period provides us with 315 monthly
returns.\ Taking into the eleven lags, leaves us with 304\ observations.}
The work in our paper is calibrated to industrial production growth. To be
frank, the overriding reason for which we chose industrial production was a
practical one.\ It was the only measure of economic activity that was
available on a monthly basis for all twelve countries in our sample. As will
become clear later, in the context of the model, production proxies for the
cash flows generated by stock securities. Furthermore, current and past
production is the information variable that investors in the model use in
their investment decision, as a way of predicating their decisions on the
stage of the business cycle the economy is in.
There is a severe drawback to using output as a proxy for the cash flows
generated by stock securities.\ In several countries, many of the companies
listed in the stock exchange typically have levels of foreign activities
markedly larger than the share of exports in the corresponding output
series. To assess the extent of the problem, we measured the percent of
foreign sales for the companies in each of the 12 countries in 1997. Using
the Worldscope universe, we constructed country aggregates by
value-weighting these ratios by the total revenues of each firm. Belgium,
Canada, and the Netherlands have the highest proportions (64.7\%, 64.2\% and
65.2\% respectively). The same ratio averages only 40.1\% for the other
countries in our sample.\ The share of exports in GDP is typically a lower
number; in December 1997, it ranges from 7\% for the U.S. to 49\%\ in the
Netherlands.\footnote{%
It is tempting to make a scale adjustment to each country's correlation
based on the level of foreign activity, to bring that level down to equal
the share of exports in GDP. This adjustment would have to change through
time. For example, in 1991 the weighted proportion of foreign sales in
Canada was 47.2\% and it has increased to 64.2\% in 1997. But the proportion
of sales is an imperfect measure because it only measures one part of
earnings -- the revenues. We have no information as to the extranational
costs of the firms.\ We choose not to apply a scaling factor of that type.
First, we felt that imposing the scaling factors based on these measures
would be arbitrary as we do not observe for each country the composition of
foreign trade by destination. Second, we were worried about introducing
another level of estimation error.}
Another possibility that we did not pursue is to use dividends or earnings
for the countries that we study, as in Bansal and Lundblad (2000). While
there are issues with the macroeconomic data, one encounters a different set
of problems using dividends and earnings. Estimates of earnings are only
available quarterly.\ Dividends are paid regularly on stock market indexes
but their time pattern over the year may have nothing to do with fluctuations in
economic activity.\ Instead, they have everything to do with
conventions concerning the dates of payments, made by each individual firm,
of a total yearly payment.\ This yearly payment has been declared at the end
of the previous fiscal year; it contains only stale information and does not
capture the latest news about economic activity.\footnote{%
As an illustration of these problems, both the dividends and earnings
available from Morgan Stanley Capital International are smoothed with a
12-month moving average.}\ In addition, from year to year managers tend to
smooth both dividends and earnings by means of accounting manipulations.
While the macroeconomic data is far from ideal, we elected not to use the
smoothed financial data.\footnote{%
Aside from the smoothing issue, there are two additional differences between
our research and Bansal and Lundblad. First, they allow for
heteroskesdasticity and moving-average processes, which we do not. However, second, they use a static CAPM whereas we use
a more appropriate intertemporal framework.}
We must also evaluate industrial production as a proxy for output and for
contemporaneous information on the business cycle. To address this issue, we
collected real GDP data which was available on a quarterly basis for nine of
twelve countries and on an annual basis in the other three countries. We
then calculated for each country their GDP correlations with the rest of the
world and compared these to the industrial production correlations (these
results are available on request). We found very similar patterns between
the GDP and industrial production correlations. Both GDP and industrial
production correlations are positively related to equity correlations. There
were two countries, Germany and Japan, that had GDP growth correlations with
the rest of the world that were far smaller than the industrial production
correlations. Nevertheless, there seemed to be a reasonable correspondence
between GDP and industrial production.\footnote{%
We also conducted sensitivity analysis on the number of lags and the
frequency of measurement of industrial production. We found that the
multiple correlation measures produced considerably higher correlations than
using two lags or no lags. We also compared quarterly correlations with the
monthly correlations and found broad similarity.}
All in all, industrial production appears to be the only measure of the
pulse of the economy, that is contemporaneous with the business cycle and
available at a high enough frequency.
The data we use for each country are expressed in U.S.\ dollars.\ Randomly
fluctuating exchange rates can cause a disconnection of realized returns
expressed in local currency since, in theory, they ought to be linked by an
equilibrium pricing relationship applicable to returns expressed in a common
currency.\footnote{%
On that count, see the empirical results of Dumas and Solnik (1995). Similar
results were reached by Dumas (1994) who relates the international asset
pricing relationship to business conditions.} In Section \ref{calibration_}
below, we find that the calibrations we conduct are not markedly affected by
the choice of unit. This is a reflection of the fact that stock returns
expressed in dollars exhibit approximately the same measured correlations as
do stock returns expressed in the respective local currencies.
Consistent with the observations in Ammer and Mei (1996), the correlations
of stock returns, in Figure 1, are always higher than the correlations of
industrial productions. Our research has a simple goal.\ We aim to
understand the empirical observations contained in Figure 1.\ Specifically,
for a given degree of commonality in real activity growth, our model will
predict a level of correlation of market returns. We want to know what
mapping connects the correlations of output to the correlations of stock
returns.
\section{\large The ``dynamic single-index'' model}
\label{common_trend}We now describe the model that captures the evolution of
the vector of national outputs.\ This will be the first component of our
overall model.\ It is a purely statistical model of international business
cycles.\ It represents a shortcut for Real Business Cycle (RBC) models,
such as Backus, Kehoe and Kydland (1992), which contain (i) a statistical
model for productivity shocks and (ii) an explicit representation of the
households' consumptions and work decisions and the firms' investment and
production decisions. In our framework, we postulate a pure-exchange economy
in which the dynamics of output is exogenous and people consume the entire
output. It is hoped that not much will be lost by this short cut, since it
is generally agreed that most of the dynamics in RBC\ models comes from the
exogenous dynamics of productivity shocks and very little comes from the
endogenous capital accumulation process.
The statistical model decomposes each country's industrial output growth
into two unobserved components: the ``world'' business cycle which is common
to all and the ``country-specific'' business cycle.\ For reasons of
parsimony, each of the cycles, whether common or specific, is assumed to
follow an autoregressive process of order two.\ We assume that the volatilities of the innovations for
each cycle are constant (homoskedasticity) and that the innovations are
independent of each other across cycle processes.\
Throughout, $\Delta \mathbf{d}_{t}$ denotes a vector of output log-growth rates of
a number of countries.\ We postulate a dynamic single index model.\footnote{%
See Sargent and Sims (1977), Geweke (1977) and Singleton (1980).\ This model
is discussed at length in Stock and Watson (1989, 1991, 1993).} The
comovements at all leads and lags among the output variables are modeled as
arising from a single common source $c_{t}$, a scalar unobserved variable
that portrays the world business cycle.\ The idiosyncratic component, $%
\mathbf{u}_{t}$, which is the part not arising from leads and lags of $c_{t}$%
, is assumed to be stationary and uncorrelated across countries.\ Otherwise,
it follows a general autoregressive process.\ The statistical representation
of the system is:
\begin{align}
\Delta \mathbf{d}_{t}& =\mathbf{\xi \times }c_{t}+\mathbf{u}_{t} \notag \\
\chi (L)c_{t}& =\eta _{t} \label{common} \\
\mathbf{D}(L)\mathbf{u}_{t}& =\mathbf{\varepsilon }_{t} \notag
\end{align}
where $L$ is the lag operator, $(\mathbf{\varepsilon }_{t},\eta _{t})$ are
serially uncorrelated with a diagonal covariance matrix and $\mathbf{D}(L)$
is diagonal.
The model is formulated in terms of demeaned log-growth rates.\ In terms of
levels, the output series of all twelve countries should be integrated
(this is confirmed at the 10\% level by a Dickey-Fuller (1979) test). Had it
been written in terms of levels, the model would say that all twelve series
are cointegrated with one common trend.\ This is a testable proposition.\
The null hypothesis in such a test (see Stock and Watson (1988)), is that
the twelve series are not cointegrated.\ We performed the test, with
ambiguous results; at the 10 percent level, the hypothesis could not be
rejected.\footnote{%
Kasa (1992) presents evidence of a single stochastic trend in GNP in a
sample of five of the markets that we study.} Had it been rejected, the
estimation of the common trend could have been done in level form by
cointegration methods; but such was not the case.\footnote{%
For a similar approach to modeling the world business cycle and a
comparison between estimations in terms of levels or growth rates, see
Gregory \textit{et al.} (1994).} The log-growth rates, on the other hand,
are all stationary at the one percent level, a feature which validates the
estimation method we use. One added advantage of this formulation is that
the log-linear pricing kernel to be used below (Section \ref{kernel})
directly applies to log-growth rates.
The statistical model (\ref{common}) is estimated by means of a linear
Kalman filter.\ We use for the purpose the SCOREM algorithm of Raynauld,
Simonato and Sigouin (1993).\footnote{%
We are very grateful to Jacques Raynauld who generously provided us with the
GAUSS code to run the algorithm, and to Ren\'{e} Garcia who pointed out that
the Raynauld-Simonato-Sigouin code would be useful to us.} The program uses
a few iterations of the ``EM algorithm'' (as described in Watson and Engle
(1983), Shumway and Stoffer (1982) and Shumway (1988)) which is not very
sensitive to initial values, in order to generate a first set of parameter
estimates. That first set is then used as initial values for the ``scoring
algorithm'' (described in Engle and Watson (1981)), which is fast and
accurate.
The results for the countries in our sample are presented in Table 1.
Practically all parameter estimates are significantly
different from zero.\ The autoregressive behavior of the world business
cycle is very different from the autoregressive behavior of the
country-specific cycles.\ The world component is driven by positive
coefficients which sum to 0.705 which implies no deterministic cycle but
random shocks with a persistent behavior, whereas the country cycles mostly
have negative coefficients implying a much more transient (strongly mean
reverting) character. The persistent world cycle will play a driving role in
the determination of stock returns.
Many alternative specifications of the statistical model could have been
considered.\ For instance, a moving-average component might have been useful
in representing the persistence of the world business cycle.\footnote{%
See Bansal and Lundblad (2000).}
The number of
lags in the autoregressive specification could be varied.\footnote{%
A specification with three lags produced parameter estimates for the
additional lags that were not significant.}\ A multi-index model, with a
regional index for Europe, might be considered.\ Or the world business cycle
could have been pre-specified as, for instance, a weighted average of
country output growth rates.\ It is impossible to explore all variations. In addition,
some of them do not lead to convergence of the algorithm so that no
comparison is possible.\ Furthermore, comparison of models, some of which
are not nested, requires the use of somewhat \textit{ad hoc} goodness-of-fit
criteria.\ Ultimately, what we must demonstrate is our ability to capture
almost all the common variation in output by means of the single index,
while the residuals $\eta _{t}$ and $\mathbf{\varepsilon }_{t}$ are almost
uncorrelated.\ As a measure of the descriptive quality of the output model,
we present Figure 2 which compares the correlations of each
country's output with the rest of the world to correlations obtained by a
simulation performed under the assumptions of the model, including the
zero-correlation of residuals assumption. The model does a good job of
matching these correlations. For a small number of countries, the simulated
correlations are slightly lower than the actual ones.\footnote{%
But the reader should keep in mind that, in subsequent analysis, we use the
actual innovations rather than the simulated ones.}
\section{ \large The log-linear pricing kernel}
\label{kernel}Restoy and Weil (1996) follow the lead of Campbell (1993) in
log-linearizing the budget constraint of a household.\ They obtain an
approximate pricing kernel for multiperiod securities which is based solely
on consumption behavior.\ Their economy consists of many identical,
infinitely-lived consumers who are endowed with an intertemporal, recursive
utility of the Epstein-Zin (1989) or Kreps-Porteus (1978) isoelastic form.
This type of utility function allows a distinction, which we find useful
(see below), between two behavioral parameters: (i) the relative risk
aversion on the one hand and (ii) the elasticity of intertemporal
substitution (e.i.s.) on the other.\ The latter is a measure of the person's
willingness to shift her consumption over time.
Let relative risk aversion be denoted $\gamma $, elasticity of intertemporal
substitution be denoted $1/\rho $ and $\beta $ denote the discount factor of
utilities, all of which are assumed to be constant and equal for all
individuals. Let $\Delta x_{t+1}$ denote the increment in the logarithm of
the households' consumption. Epstein and Zin (1989, 1991) and Weil (1990)
have shown that the increment, $\Delta m_{t+1}$, of the logarithm of the
pricing kernel between time $t$ and time $t+1$ is given by:
\begin{equation}
\Delta m_{t+1}=\theta \ln \beta -\rho \theta \Delta x_{t+1}+(\theta
-1)r_{W,t+1}
\end{equation}
\ where: $\theta =\frac{1-\gamma }{1-\rho }$ and $r_{W,t+1}$ denotes the
logarithmic rate of return on aggregate wealth between times $t$ and $t+1$.
This pricing kernel corresponds to an asset pricing model containing two
risk premia: one based on the covariance with consumption, the other based
on the covariance with wealth.\footnote{%
See Epstein and Zin (1989, 1991) and Giovannini and Weil (1989).} But recall
that, in our pure-exchange economy, consumption is equal to output.
Campbell (1993) and Restoy and Weil (1996) point out, however, that in this
expression $\Delta x_{t+1}$ and $r_{W,t+1}$ are not independent quantities
since wealth equals the present value of consumption: $\Delta x$ represents
changes in output and $r_{W}$ captures changes in ``discounted'' future
output.\ It is possible to derive an approximate relationship between these
two quantities.\ Assuming that the households' consumption is one-step-ahead
lognormal and conditionally homoskedastic, Restoy and Weil present an
expression for the financial market pricing kernel which does not involve
the rate of return on wealth and, in fact, allows returns to be endogenous.%
\footnote{%
Whereas Campbell (1993), by the same reasoning, derives an expression for
the pricing kernel which does not involve consumption. Equation (\ref{rw})
below reflects the approximate relationship between changes in output and changes
in wealth, that is being used here.}
Restoy and Weil's (1996) work can be interpreted as meaning that the
increment, $\Delta m_{t+1}$, in the logarithm of the pricing kernel is given
by:
\begin{align}
\Delta m_{t+1}& =\ln \beta -(\rho -\gamma )\frac{1-\gamma }{2}var_{t}\left[
\Delta x_{t+1}+h_{t+1}\right] -\rho E_{t}\left[ \Delta x_{t+1}\right] \\
& -\gamma S_{t+1}\left[ \Delta x_{t+1}\right] +\left( \rho -\gamma \right)
S_{t+1}\left[ h_{t+1}\right] \notag
\end{align}
where, because of homoskedasticity, the conditional variance $%
var_{t}(x_{t+1}+h_{t+1})$ is a constant, to be determined on the basis of
the stochastic process for consumption, and $\delta $ is a linearization
constant (equal to one minus the exponential of the unconditional expected
value of the log-ratio of consumption over wealth) arising in the log-linear
approximation to the budget constraint.\ In addition:
\begin{align}
h_{t+1} & =E_{t+1}\left[ \sum\nolimits_{j=1}^{\infty}\delta^{j}\Delta
x_{t+j+1}\right] \\
S_{t+1}\left[ \Delta x_{t+1}\right] & =\Delta x_{t+1}-E_{t}\left[ \Delta
x_{t+1}\right] \\
S_{t+1}\left[ h_{t+1}\right] & =h_{t+1}-E_{t}\left[ h_{t+1}\right].
\end{align}
$S_{t+1}$ is the ``surprise'' operator.
The above pricing kernel may be used to price any security in an exchange
economy in which production and consumption are equal.\ For instance, the
conditional expected value of the pricing kernel provides the one-period
riskless rate of interest:
\begin{align}
r_{f,t}& =-\ln \beta +\left( \rho -\gamma \right) \frac{1-\gamma }{2}var_{t}%
\left[ \Delta x_{t+1}+h_{t+1}\right] \label{interest} \\
& +\rho E_{t}\left[ \Delta x_{t+1}\right] -\frac{1}{2}var_{t}\left[ -\gamma
\Delta x_{t+1}+\left( \rho -\gamma \right) h_{t+1}\right] . \notag
\end{align}
Because of homoskedasticity, both $var_{t}$ terms are time invariant.
Applying the kernel to an asset that pays aggregate consumption provides a
value for the aggregate stock market return. This last task has been also
undertaken by Restoy and Weil (1996) who show that:
\begin{equation}
r_{W,t+1}=\mu +\rho \Delta x_{t+1}+\left( 1-\rho \right) S_{t+1}\left[
\Delta x_{t+1}+h_{t+1}\right] , \label{rw}
\end{equation}
where:
\begin{equation}
\mu =-\ln \beta -\frac{\left( 1-\gamma \right) \left( 1-\rho \right) }{2}%
var_{t}\left[ \Delta x_{t+1}+h_{t+1}\right].
\end{equation}
Similarly, applying the pricing kernel to an asset which pays a dividend $%
d_{i,t}$ at time $t$, Restoy and Weil get the equilibrium rate of return on
individual assets.
Stock market returns in country $i$ are:
\begin{equation}
r_{i,t+1}=\pi _{i}+\rho \Delta x_{t+1}+S_{t+1}\left[ \Delta
d_{i,t+1}+f_{i,t+1}\right] -\rho S_{t+1}\left[ \Delta x_{t+1}+h_{i,t+1}%
\right] , \label{asset}
\end{equation}
where:\footnote{%
Recall that the $var_{t}$ terms are assumed time invariant.}
\begin{align}
\pi _{i}& =-\ln \beta +\left( \rho -\gamma \right) \frac{1-\gamma }{2}var_{t}%
\left[ \Delta x_{t+1}+h_{t+1}\right] \notag \\
& -\frac{1}{2}var_{t}\left[ \left( \rho -\gamma \right) \left( \Delta
x_{t+1}+h_{t+1}\right) \right. \notag \\
& \left. +\Delta d_{i,t+1}+f_{i,t+1}-\rho \left( \Delta
x_{t+1}+h_{i,t+1}\right) \right] \\
f_{i,t+1}& =E_{t+1}\left[ \sum\nolimits_{j=1}^{\infty }\delta _{i}^{j}\Delta
d_{i,t+j+1}\right] \\
h_{i,t+1}& =E_{t+1}\left[ \sum\nolimits_{j=1}^{\infty }\delta _{i}^{j}\Delta
x_{t+j+1}\right] , \label{h}
\end{align}
and $\delta _{i}$ is a Taylor-expansion coefficient arising from the
log-linearization of the definition of a rate of return.\footnote{%
The Campbell-Shiller linearization of return begins with the observation
that log return equals log dividend growth minus the log of the current
price-dividend ration plus the log of one plus the next-period
price-dividend ratio.\ That identity is then linearized.} This constant is
related to the unconditional expected value of the dividend yield of each
security. In equation (\ref{asset}), observe the respective roles of country
\textit{vs}. world outputs. Out of the three random terms, two (the first
and last one) stand for the current and future behavior of world output
whereas only the center term refers to the future behavior of asset $i$'s
specific output stream. The terms related to world output reflect the
movement in the pricing kernel applicable to all assets worldwide.
A fascinating result falls out of equation (\ref{asset}).\ Whereas the
conditionally expected return on assets ($\pi _{i}+\rho E_{t}\left[ \Delta
x_{t+1}\right] )$ depends on risk aversion $\gamma $, the e.i.s.\ $%
1/\rho $, and on the impatience parameter $\beta ,$ the second
moments (volatilities and correlations) of the asset return \textit{depend
on only one utility parameter: the elasticity of intertemporal substitution}%
. This result would hold exactly in the case of Epstein-Zin utilities with
constant risk aversion and e.i.s., and identically, independently
distributed returns (see Epstein (1988)).\ In the case of our output model,
the result holds under the log-linear approximation made by Restoy and Weil.
This remarkable property is \textit{the} reason why we have chosen to adopt
this type of utility function.\ It will prove most convenient in what
follows.
Here is the intuitive reason for which elasticity of intertemporal
substitution governs the price response to shocks.\ Suppose that output
undergoes a positive shock; if consumers are willing to absorb this shock
into consumption without further ado, there is no need to adjust asset
prices.\ But, if their e.i.s. is low, they will have to be induced to
consume the increased current output by the device of higher market prices
of assets relative to current consumption. This intuition is well known.\ It
is identical to the one Lucas (1978) gives on this issue,\footnote{%
Lucas (1978), page 1439.} except for the fact that Lucas ascribes to risk
aversion the role actually played by the elasticity of intertemporal
substitution (while pointing out in a footnote that, in his case of
time-additive utility, ``the term `risk aversion' is perhaps misleading,
since the curvature of [the utility function] also governs the intertemporal
substitutability of consumption'').
A multivariate rendition of the same intuition goes as follows.\ Suppose
that output in any given country undergoes a positive shock; if world
consumers are willing to absorb this shock into world consumption without
further ado, there is no need to adjust asset prices.\ But, if their e.i.s.
is low, they will have to be induced to consume the increased current output
by the device of higher market prices of \textit{all }assets relative to
current consumption. That is why the e.i.s. (inverse of $\rho $) plays a
crucial role in this model.\ An increase of the parameter $\rho $ (lower
elasticity) implies both more volatility and more covariance across assets.\
The resulting effect on correlations remains ambiguous at this point.\ It
was asserted in the introduction that, were the kernel excessively volatile,
this can translate into an equally excessive degree of correlation of
world equity returns. Figure 2 illustrates that
this is true in the context of our model over the relevant range of values
of the parameter $\rho $.
It may seem surprising that the second moments of rates of return do not
depend on risk aversion. In fact, the ``level'' of rates of returns does
change when people become more risk averse, but, in a homoskedastic world,
it changes by a time invariant amount. As a consequence, the time-series
``volatility'' of each individual return is not affected by changes in risk
aversion. The effect of risk aversion is time invariant in a homoskedastic
world.\footnote{%
We are grateful to Philippe Weil for a helpful discussion on this point.}
One more observation should be made on the basis of equation (\ref{asset}).\
The model induces correlations of stock returns through two channels.\ There
is a common movement in the world interest rate caused by consumption
growth, and there is a forecast of movements of future dividends from the
common component of world growth.\ Despite the homoskedasticity assumption,
it is \textit{not} true that all the common movement in stock returns comes
from the interest rate.\ The second channels still causes \textit{ex post}
equity premia (or excess returns) to fluctuate with a common component (in
Section \ref{test} below, we\ examine the common behavior of excess
returns).\ But it \textit{is }true that conditionally expected excess
returns are constant over time so that there is no equity return variability
caused by variability in equity risk premia.\footnote{%
Heteroskedasticity of output would produce time varying expected returns
which would contribute, to a small extent, to an explanation of the high
volatility of stock returns.}
\section{\large The log-linear pricing kernel combined with the dynamic
single-index model of output}
\label{combined}Rodriguez \textit{et al.} (1996) specialize (\ref{rw}) to
the case in which the growth rate in aggregate consumption is AR(2):
\begin{equation}
\left( 1-\phi _{1}L-\phi _{2}L^{2}\right) \Delta x_{t+1}=\varepsilon _{t+1}
\end{equation}
This particular autoregressive process implies that:\footnote{%
The result can be extended trivially to an autoregressive process of any
order.}
\begin{equation}
S_{t+1}\left( \Delta x_{t+1}+h_{t+1}\right) =\frac{1}{1-\phi _{1}\delta
-\phi _{2}\delta ^{2}}\varepsilon _{t+1}
\end{equation}
and, therefore:
\begin{equation}
var_{t}(\Delta x_{t+1}+h_{t+1})=\left[ \frac{1}{1-\phi _{1}\delta -\phi
_{2}\delta ^{2}}\right] ^{2}var\left( \varepsilon \right)
\end{equation}
Our approach is similarly to apply the pricing kernel to the dynamic
single-index model in equation (\ref{common}) and obtain the behavior of
individual stock returns where stocks are defined as claims on individual
output series.\ Since we have made the assumption of an exchange economy,
aggregate consumption growth is equal to the weighted sum of output growth
rates of individual countries: $\Delta x_{t+1}=\sum_{j}w_{j,t}\Delta
d_{j,t+1}$.
For the dynamic single-index model, the terms of (\ref{asset}) can be
particularized as follows:
\begin{align}
S_{t+1}\left[ \Delta d_{i,t+1}+f_{i,t+1}\right] & =A_{i,i}\eta
_{t+1}+B_{i,i}\varepsilon _{i,t+1} \label{S} \\
S_{t+1}\left[ \Delta x_{t+1}+h_{i,t+1}\right] & =\sum_{j}\left[ A_{j,i}\eta
_{t+1}+B_{j,i}\varepsilon _{j,t+1}\right] \\
var_{t}\left[ \Delta x_{t+1}+h_{t+1}\right] & =var\left\{ \sum_{j}\left[
A_{j,0}\eta +B_{j,0}\varepsilon _{j}\right] \right\}
\end{align}
\begin{align}
A_{j,i}& =\xi _{j}\frac{1}{\sum_{s=0}^{\infty }\chi _{s}\delta _{i}^{s}}%
;\quad A_{j,0}=\xi _{j}\frac{1}{\sum_{s=0}^{\infty }\chi _{s}\delta ^{s}} \\
B_{j,i}& =\frac{1}{\sum_{s=0}^{\infty }D_{j,s}\delta _{i}^{s}};\quad B_{j,0}=%
\frac{1}{\sum_{s=0}^{\infty }D_{j,s}\delta ^{s}}
\end{align}
\begin{equation*}
var_{t}\left[ \left( \rho -\gamma \right) \left( \Delta
x_{t+1}+h_{t+1}\right) +\Delta d_{i,t+1}+f_{i,t+1}-\rho \left( \Delta
x_{t+1}+h_{i,t+1}\right) \right] =
\end{equation*}
\begin{equation}
var\left\{ \left( \rho-\gamma\right) \sum_{j}\left[ A_{j,0}\eta
+B_{j,0}\varepsilon_{j}\right] +A_{i,i}\eta+B_{i,i}\varepsilon_{i}\right.
\left. -\rho\sum_{j}\left[ A_{j,i}\eta+B_{j,i}\varepsilon_{j}\right] \right\}
\label{assetfin}
\end{equation}
Our next goal is to determine whether the second moments of observed stock
returns can be matched with those of the theoretical model above.
\section{ \large Calibration of the model}
\label{calibration_}The system of equations (\ref{common}), coupled with
equations (\ref{asset}-\ref{h}, \ref{S}-\ref{assetfin}), provides a strong
set of restrictions on the output and stock returns series. The unknown
parameters are: those of the dynamic single-index model $\mathbf{\zeta }$%
\textbf{, }$\mathbf{\xi }$\textbf{, }$\mathbf{D}$, $\chi $, the initial
value for the common cycle $c_{0}$, the variances of the residual terms $%
\mathbf{\varepsilon }$ and $\eta $, the utility parameters $\beta $, $\rho $
and $\gamma $ (and the linearization constants $\delta $, $\delta _{i}$,
which, however, are not in principle independent entities).
While it is possible (but extremely difficult) to estimate the full system
of equations, it is not clear how useful such an exercise would be.\ We
already know that a model of an exchange economy, even if coupled with a
generalized utility formulation such as the Epstein-Zin utility function,
has difficulties accounting for observed mean rates of return on equity in
the United States.\ This is the ``equity premium'' puzzle of Mehra and
Prescott (1985).\footnote{%
Goetzmann and Jorion (1997) have pointed out that most countries are not
like the United States and have argued that U.S.\ equity rates of return
presumably represent a repeated sequence of surprises, not to be confused
with high expected returns.} Like Kocherlakota (1996) and Campbell and Koo
(1997), we use Epstein-Zin utility functions.\ Previous research shows that
the freedom separately to choose the risk aversion and the elasticity of
intertemporal substitution of the representative individual allows a
somewhat better fit of first moments (equity returns and interest rate) than
standard time-additive utilities. Nonetheless, it is clear that, at
generally accepted levels of risk aversions, our model would not account for
the U.S.\ equity premium.
In what follows, therefore, we employ the result obtained above (see
equation (\ref{asset})) that, to an approximation, the elasticity of
intertemporal substitution alone determines second moments while, once
second moments have been determined, the two taste parameters jointly
determine the first moments.\ We rely on this argument to separate out the
debate on the equity premium and we focus exclusively on second moments.
Further, we know from Shiller (1981) that stock return volatilities tend to
be larger in reality than can be explained by a simple pricing model based
on dividends. In our model, however, dividends, or even earnings, are not
the basis for the determination of stock returns -- Output is.\ In the real
world, two layers of leverage (operational and financial) should normally
tend to magnify the volatility of dividend growth rates relative to that of
output growth rates. In our dataset, the average taken across countries of
the volatilities of industrial production growth rates is equal to
2.02\% per month whereas the average volatility of dividend growth rates is
equal to 4.7\% per month and the average volatility of earnings is equal to
8.4\% per month.\footnote{%
The average of earnings growth rates given here is based on eight countries
only.\ In the US, these numbers are 0.8\%, 2.4\% and 3.6\% respectively.\
In all cases, earnings and dividends are measured as twelve-month moving
averages.} Without the extensive amount of smoothing that is built into the
dividend and earnings series, the increased volatility created by leverage
would be even more apparent.\ Under the simplifying (probably simplistic)
assumption that dividends and output are exactly linearly related, leverage
magnifies volatilities but leaves correlations unchanged.\footnote{%
We are grateful to Huntley Schaller for helping us articulate this
distinction.}
Below, we report the correlations and the volatilities produced by our
model, knowing, however, that actual volatilities are magnified by leverage,
which is unobserved.\ For this reason, we focus mostly on correlations. The
correlations, viewed as moment conditions, provide us with a calibrated
version of the model. In a later section (Section \ref{test}), we use the
Generalized Method of Moments to test the validity of the moment conditions.
\subsection{\large \it Calibration under integration}
The calibration is carried out in a simple way. We have already observed
that, according to equation (\ref{asset}), the second moments are all dictated by the elasticity of
intertemporal substitution of the market participants.\ Based on the dynamic
single-index statistical model, which we estimate in a first stage, we
select at a second stage the degree of e.i.s.\ that will best match the
levels of a number of correlations between stock returns and output. Once
that is done, we have pinned down all the parameters of the model.\footnote{%
The parameters $\beta $ and $\gamma $ are ignored since they play no role in
determining correlations. The same is true for the linearization constant $%
\delta .\;$For $\delta _{i}$, we use the mean dividend yields calculated
over the entire sample.}\ We then calculate the equilibrium stock returns
for the history of shocks which we have identified statistically and compute
their correlations.\footnote{%
A two-stage procedure raises the problem that first-stage parameter values
are not known for certain. This problem is explicitly taken into account in
the hypothesis test (Section \ref{test}) but not in the calibration.}
The range of reasonable values for the coefficient $\rho $ is dictated by
measurements of e.i.s. ($=1/\rho $) that have been conducted in the past.
Regressing the rate of growth of aggregate consumption on changes in the
rate of interest, Hall (1988) finds an elasticity equal to 0.1 which is
lower than most previous estimates which range as high as e.i.s. = 1.\
Epstein and Zin (1991), in their test of the CAPM implied by their
preferences and applied to stock returns, find values for the elasticity
ranging from 0.2\ to 0.8.\ In short, values for $\rho $ ranging from 1 to 10
seem reasonable.\footnote{%
In order to carry out this type of analysis, it would be best to examine
disaggregated consumption according to social categories (borrowers vs.
lenders, old vs.\ young, employed vs. unemployed, etc.).\ See Deaton (1992).
Attanasio and Weber (1989), placing themselves explicitly in the Epstein-Zin
paradigm, and using a single cohort of household found a value $\rho =0.514$
or an e.i.s. approximately equal to 2.}
Ultimately, three kinds of ``moments'' will have to be matched:
the correlations of a country's stock return with the country's own output,
the correlations of a country's stock returns with the rest of the world
output and the correlations of a country's stock returns with the rest of
the world stock returns.\ In the hypothesis tests of Section \ref{test}, we
evaluate the fit of all three.
But in the calibration, we would like to proceed differently.\ We choose to
calibrate the model (i.e., pick the value of $\rho $) on the basis of
correlations with the rest of the world output.\ Then we show how the other
two, ``overidentifying'' moments are matched.\ Using an objective function
which downweighs the countries with more volatile correlation estimates and
searching for the best fitting value of $\rho $, we find an optimal value
equal to 2.1.\footnote{%
Standard errors of estimates will be provided in the section containing the
hypothesis tests (Section \ref{test}).\ The confidence intervals will prove
to be narrow.}
Figure 3 then illustrates the calibration trade-offs that we
are facing. The figure displays the straight arithmetic average across
countries of the theoretical correlations and the average level of the
corresponding observed correlations that we try to match.\footnote{%
As mentioned earlier, given that we may not observe in the real-world the
exact synchroneity between stock returns and output shocks that is
postulated in the model, the correlations in question are not simple
correlations.\ They are actually square roots of the $R^{2}$ of multiple
regressions of stock returns on contemporaneous, plus eleven lagged, output
growth rates.} The figure confirms that the value of $\rho $ that would best
match the average correlation with world output is about 2.1.\ That value of
the parameter $\rho $ explains remarkably well the average level of
cross-country stock market correlations.\ But the figure also shows that it
does not explain the within-country correlation with output.\ In fact, the
value of $\rho $ that would best match the average within-country
correlation with output is about equal to 6.9.
For each country, Figure 4 also shows the comparison
between actual and model stock market correlations resulting from the value $%
\rho =2.1$.
There are two factors that play a role in the derivation of
model correlations.\ First, the world pricing kernel, which applies to all
securities by construction, has been set in such a way as to match the
observed correlations of stock returns with rest of the world output, as we just explained.\
Secondly, in our dataset, the world business cycle (see Table 1)
has been found to be fairly persistent.\ A component of stock
returns fluctuations comes from the anticipation of discounted future
dividends (see equation (\ref{asset})).\ If a time series is persistent, any
movement occurring today is the harbinger of a lasting movement in future
realizations and produces a large immediate effect on returns.\ In our
statistical model, this large component is common to all countries since it
originates in the world business cycle.
The model correlations turn out to be of a magnitude similar to the observed
ones.\footnote{%
Canova and de Nicolo (1995), in the context of a full-blown, calibrated
model of international business cycles find much larger model correlations.\
The difference is due to the fact that their choice of parameters was
dictated by first and own second moments.}\ Figure 3 shows
that, if the pricing kernel were excessively volatile, it would be the case,
if the same kernel applies worldwide, that international stock returns are
excessively correlated. However, no such excess correlation appears in our
results.
\subsection{\large \it Discussion}
If the above calibration errs, the direction in which it errs is perfectly
clear; the correlations observed in the data can only be viewed as being
lower than (or equal to) those of the model, for two reasons.\ First,
the model correlations that we have calculated are lower bounds.\ By
choosing a relatively low value of $\rho ,$ we can only have \emph{%
understated the model correlations}.\ As has been mentioned, if, instead of
what we did, we had attempted to match a country's stock return correlation
with its own output, we would have picked a value of $\rho $ equal
approximately to 6.9 (see Figure 3). For that value of $%
\rho ,$ the model would have given us an average cross-country correlation
of stock returns equal to: 0.97.\
What if we had tried instead to match volatilities? It is evident from
Figure 5 that the model, at the chosen value $\rho =2.1,$
explains only about 25 percent of the actual volatilities. Based on the
leverage argument that we have put forward, we do not intend to take the
measured volatilities literally.\ Our analysis of the relative
volatilies of output \textit{vs.} dividend or earnings growth rates is
sufficient to explain away the discrepancy in stock return volatilities.\
Had we tried to match the observed volatilites, however, the needed value of
$\rho $ would have been slightly above 9, a value at which the model would
indicate an average cross-country correlation of stock returns equal to
0.98. So, at the level of e.i.s. that matches the excessive volatility,
there would be no evidence whatsoever of excessive correlation in the data.
The second reason why our calibration can only err in the direction we have
stated is that the correlations observed in the data are, if anything
overstated.\ This is because, in several countries, many of the companies
listed in the stock exchange typically have levels of foreign activities
markedly larger than the share of exports in the corresponding output
series.\ When an overstated share of profits originates abroad, one would
surmise, the correlation of the market value of these profits relative to
the rest of the world is also overstated.\ This conjecture is buttressed by
the fact that Netherlands is a clear outlier in Figure 4;
the Amsterdam stock market covers an industrial base which, in fact, is a
world-wide one. Besides the Netherlands, Belgium and Canada are the
countries for which actual correlations fall above their full-integration
levels. We have seen in Section \ref{firstlook} that these countries have
the highest proportions of foreign sales.
Other countries that stand out are Japan, Germany and Austria because their
full integration correlation is much higher than the observed correlation.
But we know from previous research that Japan was segmented from the world
market by regulation until 1981 at least (see Gultekin, Gultekin and Penati
(1989)). We have no explanation for the Austrian and German deviations.
Except for the cases of Japan, Germany and Austria, it seems perfectly
reasonable to conclude that there is no evidence of excess correlation in
the data.\ If anything, the cross-correlations may be lower than what they
should be under full integration,\footnote{%
Within an integrated country such as France, the correlation of stock
returns across industrial sectors is equal to 0.818 whereas the
cross-country correlation, as we saw, is equal to 0.587.\ Freimann (1998),
as mentioned, offers an alternative, entirely statistical procedure based on
randomization of industrial sector returns, to compare country correlations
to what they would have been under integration. He finds that
cross-correlations are lower than they should be under full integration.}
this being especially true if we insist on trying to match the correlation
between a country's stock return and its own output (see Section \ref{test}\
below).
\subsection{\large \it Calibrating local-currency returns, deflated returns and excess
returns}
We need to check whether our results depend on the way we measure the stock
returns. Table 2 compares the results of the calibration,
conducted as above, depending on whether the returns are measured in dollars
(as has been done so far), in local currency, in local currency but deflated
by the local Consumer Price Index or in local currency in excess of the
local rate of interest.
The first set of results in Table 2 show that, whether returns are measured
in dollars, in local currency or in local currency deflated makes very
little difference.\
The results in terms of excess returns are markedly different from the
results in terms of other units.\ The fit of excess stock return correlations is
poorer.\ It is clear that the model does not explain actual one-month
interest rates very well. While interest rates make little difference to the
variance of returns, they have a clear impact on correlations.
When examining returns measured in local currency in excess of the local
rate of interest, we have had to reduce the number of countries and reduce
the length of the sample period because some data on one-month Euro-rates of
interest were not available.\footnote{%
The reduced sample starts in\ October 1978.} Some of the difference in the
results for ``local excess returns'' is accounted for by the change in
sample. An additional discrepancy arises because the output process was not
refitted to the shorter sample.\ But most of the difference is the result of
interest-rate behavior.
The model should probably not have been expected to explain the behavior of
local-currency excess returns since it has not been designed to distinguish
interest rates denominated in different currencies.\ But its failure to
explain the correlations of US dollar excess returns is more disappointing.
\subsection{\large \it Calibration under segmentation}
With the same estimated ``dynamic single-index'' business cycle model as in
Section \ref{common_trend}, we now modify the log-linear pricing kernel,
taking each national stock market as a stand-alone financial market.\ The
required change in the pricing kernel is straightforward: Equation (\ref{rw}%
), where $x$ now stands for each country's output, instead of equation (\ref
{asset}), is used to obtain individual country stock returns.\ In this
formulation, each country lives in autarky.\ The correlation in output
behavior which happens to exist statistically, is the only source of common
behavior in stock returns.\ The pricing kernel is a different one in each
country although the pricing kernels of different countries do exhibit some
degree of cross-correlation since outputs are cross-correlated.
The calibration trade-offs that we face in this case are displayed in Figure
6.\ It is immediately apparent from this picture that no value
of $\rho $ will allow us to match the actual between-country stock return
correlation of 0.595; the model values for these correlations barely reach
the value 4.7 when $\rho $ is as high as 10.\ As far as the correlations
with output are concerned, the correlations of a country's stock returns
with its own output is very large in this model.\ Even with a value of $\rho
=10$, the correlation falls to 0.67 which is still far greater than the
observed correlation of 0.207.
Focusing on the only correlation that can reasonably be matched, we choose
the value of the e.i.s.\ to get the best possible match of correlations of
stock returns with world output, weighted by the reliabilities of
correlation estimates: $\rho =1.42$. The combined result of the
calibration exercise for each country is shown in Figure 7.
With the single exception of the U.S., we find that theoretical stock market
correlations now fall far below actual ones.\footnote{%
Even if the value of $\rho $ selected under integration had been maintained,
the correlations under segmentation would have been similar to what they are
in Figure 7.} This suggest that the observed levels of
international stock returns correlations are inconsistent with the
hypothesis of market segmentation.
In short, we find that correlations are about equal to (or lower than) their
full-integration levels and markedly above their complete-segmentation
levels, given the common behavior of outputs.\ This gives us some reason to
try and see whether we can construct a test of these hypotheses.
\section{\large A\ statistical test of the integration hypothesis}
\label{test}It is evident from the international calibration exercise of
Section \ref{calibration_}\ that reality is very much at variance with the
full segmentation hypothesis.\ Two of the three correlation categories that
we have chosen to look at, present no prospect of coming reasonably close to
their measured counterparts, no matter what value of the crucial parameter $%
\rho $ we choose. Hence, we focus in this section exclusively on the design
and implementation of a test of the full integration hypothesis.\footnote{%
Under the integration hypothesis, it is not easy to allow for a different
value of the parameter $\rho $ for each country's sub-population of
investors.\ This is because the aggregate of a world population of investors
with recursive utility is not a representative investor with recursive
utility (see Dumas, Uppal and Wang (2000)).\ Under the segmentation
hypothesis, it would have been possible, of course, to allow such a
difference from country to country.}
\subsection{\large \it Test design}
\label{Zhou}We construct deviations between model and reality $u_{t}$ in the
following way. Define $y_{t}$ as a variable which has been regressed on a
set of explanatory variables and call $\varepsilon _{t}$ the residuals of
that regression.\ Let,
\begin{equation}
\xi _{t}=\frac{\left( \varepsilon _{t}\right) ^{2}}{\text{var}(y)}.
\end{equation}
Notice that $\sum_{t}\xi _{t}$ is equal to one minus the $R^{2}$ of the
regression. In our application, $y_{t}$ is each country's monthly rate of
stock return and the regressors are the contemporaneous and eleven lagged
values of either the country's own output growth rate, or the rest of the
world output growth rate or the rest of the world stock return, as the case
may be. We calculate this variable in two versions; one denoted $\xi _{t}$
is based on realized observations; the other $\widehat{\xi _{t}}$ is
calculated from model outputs and is, therefore, a function of the unknown
parameter $\rho $.
The deviations between model and reality are calculated as:
\begin{equation}
u_{t}=\xi _{t}-\widehat{\xi _{t}}.
\end{equation}
Since we have twelve countries and three categories of residuals that we try
to match, we have thirty six such deviations at each point in time.\ We
stack them in a 36-element vector which we then use to construct a weighted
objective function in the manner of the Generalized Method of Moments. The
moments form a vector $g=\sum_{t}u_{t}$ and the weighting matrix is the
inverse of the variance-covariance matrix of $u$.\ This objective function
can be minimized to obtain an estimate of the single unknown parameter $\rho
$.
Asymptotically, the minimized objective function is $\chi ^{2}$ distributed
with 35 degrees of \ freedom.\ We do not know the distribution of the
minimized objective function in a finite sample but that is not a
problem in our case. The total number of numerical items to be
estimated, equal to the number of parameters plus the elements of the weight
matrix, is equal to 631 (i.e., 1\ +\ 36$\times $35/2)\ while, with 304 monthly
observations, the total number of stock returns,
domestic output and rest-of-the-world output observations is equal to 10944\ (i.e., 304$\times $%
12$\times $3).\ The adjustment for finite-sample size suggested by Ferson
and Foerster (1994, 1995), albeit in a somewhat different context, implies that estimated variances of estimates
should be multiplied by 1.06\ (i.e., 10944/(10944 - 631)), or that the standard error of the
estimate of $\rho $ should be increased by 3\% only.
We make two amendments to the procedure.\ The first one aims to take account
of the possible serial dependence of the vector $u$.\ The adjustment
involves an optimal number of lags of the vector $u$. It follows the method
proposed by DenHaan (1996). We allow a maximum lag of fifteen months.\ We
only perform a univariate correction: a series' own past values only are
considered in this correction for serial dependence.\ The lag length is
chosen on the basis of the Schwarz Bayesian Information Criterion. Call $w$
the inverse variance-covariance matrix of $u$ after adjustment for serial
correlation.
Our second adjustment takes first-stage parameter uncertainty into account.\
Recall that, in a first stage of our procedure, the dynamic single-index
model has been estimated to model output behavior while, in a second stage
of the estimation, we now estimate the preference parameter $\rho $.\ This
is acceptable because the output model is independent of the financial model.%
\footnote{%
Some improvement in efficiency could still be achieved if the two model
components were estimated jointly but that is not feasible.\ }\ While the
structure of the procedure is sound, the parameter uncertainty of the first
stage must be taken into account at the second stage.\ This is easily
achieved by first computing the Jacobian matrix $\partial g/\partial \theta $
where $\theta $ stands for all the first-stage parameters, and then
adjusting the weighting matrix $w$ as follows:
\begin{equation}
w_{1}^{-1}=w^{-1}+\left[ \partial g/\partial \theta \right] \Omega \left[
\partial g/\partial \theta \right] ^{\prime }+\left[ \partial
g/\partial \theta \right] \Gamma + \Gamma^\prime\left [\partial
g/\partial \theta \right]^\prime
\end{equation}
where $\Omega $ is the 50$\times $50 variance-covariance matrix of the
first-stage parameter estimates and $\Gamma $ is the 50$\times $36 matrix
of covariances between the first-stage parameter estimates and the moments.
The GMM\ iterates over the choice of the parameter $\rho $ and over the
choice of the weighting matrix $w$.\ Once that is done, the matrix $w$ is
replaced by the matrix $w_{1}$ and one more iteration series is performed
over the choice of $\rho $.
\subsection{\large \it Test results}
Table 3 presents the results of the tests conducted over the thirty six moment conditions of the twelve
countries of our sample. In the first row, we reject the hypothesis of financial market integration for all
twelve countries against an unspecified alternative. It is notable that our
study of correlations has produced a test powerful enough to reject the
integration hypothesis whereas extant tests based on the first moment and a
partial equilibrium model such as an international Capital Asset Pricing
Model, for the most part, have had too little power to reject.\footnote{%
For an exception, see Jorion and Schwartz (1986).\ Note that the
equity-premium puzzle would not cause partial-equilibrium models to be
rejected, except if they are based on consumption behavior.}
This result is subject to an important caveat.\ It is possible that our rejection is
a result of imposing the assumption that $\rho $ is the same for all
countries.\ An extension of the test to accommodate the possibility of
country-specific $\rho _{i}$s will await future research.
\subsection{\large \it Robustness: breaking the sample and other variations}
The model rejection is very much
dependent on the fact that we tried to match three types of correlations:
correlations of stock returns with each country's output growth, with the
rest of the world output and with the rest of the world stock returns. The
rejection of the integration hypothesis is a rejection of the adequate match
of these three moments. If the
integration model is tested with only the two moment categories involving the
rest of the world,\footnote{%
Namely, the correlation of stock returns with the rest-of-the-world stock
returns and the correlation of output with the rest-of-the-world output.}
we no longer reject the
hypothesis at the usual 1\% level (even though we still reject it at the 5\%
level) in the second row of Table 3.\ A test of the pair of moment conditions involving cross-country
stock returns and stock return correlation with own output would lead to a
rejection (p-value = 0.003) but the moment pair involving each country's
stock return correlation with world output and with own output would not lead
to a rejection (p-value = 0.99). It is the cross-country correlation of
stock returns which is not compatible with the within-country stock/output
correlation.
We are also able to use our test to determine whether the world financial market
has evolved over time towards more integration.\ We split the
sample into two halves.\ Taking output behavior as given and independent
of the workings of the financial market, we estimate the
second-stage financial component of the model over two subsamples.\
The third row of Table 3 shows that we reject the integration hypothesis for all twelve countries over
both subsamples.
It is also interesting to see which country, if any, causes the
full-integration hypothesis to fail.\ While commenting the calibration
results, we had second thoughts about including the Netherlands in the
sample and we recognized that Japan may have been segmented
from the world financial market.\ However, excluding one country at a time,
we still reject the integration hypothesis at the 1\% level (the complete
results are available on request). One thus finds no evidence in favor of
the idea that one country, being perhaps segmented financially from the rest
of the world, would have caused the overall integration test to fail. The
calibration exercise that we performed provides us with a reason to try and
exclude one pair of countries that may both have been at some point
segmented away from the world financial market, namely Japan and Austria.\
But excluding these two countries together also does not allow us to accept
the hypothesis at the 1\% level.
Finally, we checked in the calibration section that our results are not very
sensitive to the way we measure stock returns. In order to confirm this
intuition, we conducted a test for each convention (local currency, real local currency, excess of the risk
free rate) we considered.
The results in Table 3 indicate that the conclusions remains the same: we reject the integration hypothesis at the
1\% level.
\section{\large Conclusion}
In this paper, we have linked the correlations of stock returns to their
fundamental determinants. These determinants were taken to be the behavior
of output in the various countries.\ We have represented the behavior of
output by means of a ``dynamic single-index'' statistical model, designed to
capture the ``covariation'' of outputs in a dynamic framework, over the
business cycle.\ The coefficients of the statistical model seem reasonable,
and produce a common world cycle which is fairly persistent.
The theory of integrated stock markets which we have applied to the
estimated behavior of output, has yielded levels of theoretical correlations
of rates of return about equal to the measured correlations and, above all,
the alternative hypothesis of financial-market segmentation hypothesis has
produced correlations markedly lower than the actual ones.\ The likely
interpretation is that the stock markets of the world are reasonably
integrated.
One type of correlation, however, has not been explained satisfactorily by
our model.\ It is the correlation of each country's stock return with the
own-country industrial production.\ The theoretical value is quite a bit
higher than the observed one, at the value of the unknown parameter
(elasticity of intertemporal substitution) that matches the other
correlation moments.\ This is the single reason for which the
full-integration model was rejected by the data.
One often hears the assertion that increased global integration implies
higher global stock market correlations.\ This assertion is problematic
because it does not control for the economic fundamentals of each country.\
This is exactly the motivation of our paper.\ Our framework allows us to
give international stock market correlations an interpretation in terms of
degree of \ integration \textit{vs. }segmentation, albeit not in a
time-varying fashion.\pagebreak
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\section{\large Data appendix}
The data used in this article are monthly time series covering the
industrial production and the stock returns of a subset of OECD countries.
The data are from two different sources, both available on DATASTREAM: OECD
for industrial production series and Morgan Stanley Capital International
(MSCI) for stock return data.
We selected the following twelve countries based on data availability and on
the joint sizes of their economy and their stock market during the last
twenty years: Austria, Belgium, Canada, France, Germany, Italy, Japan,
Netherlands, Spain, Sweden, the United Kingdom and the United States.\ There
exist no monthly output series of any kind for Switzerland and Australia.
Some summary statistics are presented in Appendix Table A1.\ Panel A shows
the equity capitalization to GDP\ ratios for the twelve countries in 1971,
1980, 1990\ and 1995.\ The proportional size of the equity market increases
in every country in our sample except for Canada and Spain.\ Panel B shows
the equity capitalization as a ratio of the MSCI\ world.\ The twelve
countries in our sample encompass 81\%\ of world market capitalization in
1995.\ Finally, we examine the GDP as a proportion of OECD\ GDP.\ The twelve
countries we choose have 91\%\ of OECD\ GDP (reported in Panel C) in 1995.
\subsubsection{\large \it Industrial Production}
We have used the monthly time series of real industrial production with a
1990 basis year, deseasonalized, as published for each of the twelve
countries by the OECD.
The series codes of the series in DATASTREAM\ are: OEOCIPRDG, BGOCIPRDG,
CNOCIPRDG, FROCIPRDG, BDOCIPRDG, ITOCIPRDG, JPOCIPRDG, NLOCIPRDG, ESOCIPRDG,
SDOCIPRDG, UKOCIPRDG and USOCIPRDG.
For the weighting of each country in the world aggregate economy, we have
used the yearly values of Gross Domestic Product (GDP), with a 1990\ basis
year for the prices and for the exchange rates, as published by the OECD.
The series codes of the series in DATASTREAM\ are: OEGDP90, BGGDP90,
CNGDP90, FRGDP90, BDGDP90, ITGDP90, JPGDP90, NLGDP90, ESGDP90, SDGDP90,
UKGDP90 and USGDP90.
\subsubsection{\large \it Stock returns}
We have used the monthly time series of MSCI\ indices, measured in U.S.\
dollars, with dividends re-invested, with a 1970\ basis year, for the twelve
countries.
For the weighting of each country in the world stock market, we have used
the yearly values of Gross Domestic Product (GDP).
The average dividend yields of each country come also from MSCI.\ We should
caution that in some countries, the dividend yield of the index is not
available in the early years of the sample period.\ We have assumed the
yield to be constant over the period with missing data.
The Consumer Price Indices (CPI) used to deflate the stock returns and the
risk free rates for the excess returns also come from DATASTREAM. The
interest rates are the one month Eurodollar deposit rate.
\end{document}