DP2020-32

Bilateral Integration Measures
and Risk Attitudes in Large Stock
Markets
Shoka HAYAKI

December 15, 2020

Bilateral Integration Measures and Risk Attitudes
in Large Stock Markets1
Shoka Hayaki2

Abstract
This paper examines whether developed markets are more internationally integrated thanemerging markets. A new bivariate regime switching model is constructed in order to take into accountboth international integration regime and segmentation regime, capture the endogenous and interactive
effects between large markets, and pay attention to the economic structure of the price of variance risk.
We estimated such regime switching model for 24 large stock markets and the US market as a reference
market. That is, the regime reflects whether each of the 24 markets is integrated with the US. As aresult, the structures of representative investor’s risk attitude, or that of the price of variance risk, ineach of the following 15 markets are almost the same; Canada, France, Italy, Australia, Hong Kong,
Netherlands, Spain, Sweden, Switzerland, Brazil, South Korea, Taiwan, Indonesia, Mexico and Saudi
Arabia. In such markets, these “international integration measures” defined as the (smoothed)probability of international integration regime are on average high, declining before the 2008 global
financial crisis, but rising again after the crisis. This means that non-home-biased strategies such as aninternational diversification have advantages over home-biased strategies such as a domestic
concentration except just before the crisis. In addition, the difference between the international
integration measures of developed and emerging markets included in these markets is extremely small.
In other words, being an emerging market does not mean that the market is segmented. Summing up
the above results, it can be concluded that the international diversification is strongly recommended
in these markets regardless of country or period.

1

I would like to thank Kohei Aono, Masaru Chiba, Naoya Mori, Takashi Hatakeda, Takashi
Yamasaki, and participants at the 28th Annual Meeting of Nippon Finance Association, and the 2020
West Japan Meeting of Japan Finance Association for helpful comments. Remaining errors are myown.
2Graduate School of Business Administration, Kobe University, Japan; and Junior Research Fellow,
Research Institute for Economics and Business Administration, Kobe University, Japan.

1. Introduction
The issue of “market scope” is a crucial issue in financial economics. Investors who
misunderstand the true market scope may endure suboptimal strategies for their stockholdings, even
if they have the ability to execute optimal strategies. We need to know the true scope of stock markets
in order to avoid such failures and construct a new model considering the following three points;
researchers’ arbitrary assumptions about market scopes, interdependence between stock markets, andthe time-variability of the price of variance risk.
First, we consider the arbitrariness of market scopes. Many of the models for analyzing stock
prices, such as the standard CAPM (capital asset pricing model) or standard multi-factor models, target
the domestic stock market and do not consider the effects of foreign stock markets. If we use thesemodels, we are implicitly assuming that the stock market is independent of the foreign stock market.
In other words, investors recognize an internationally segmented market of only one country. On theother hand, with the rapid globalization in recent years, more and more evidence of interaction between
stock markets has been observed. Some researchers think that all stock markets should be considereda single, integrated market (e.g. international CAPM). That is, investors recognize an internationallyintegrated market including multiple countries. In any case, it is a problem to “arbitrarily” decide thatthe market is internationally segmented or integrated.
Studies on international comparison between stock markets often assume that developed markets
are integrated with each other and emerging markets are at least partially segmented. De Jong and deRoon (2005) classified stocks in emerging stock markets based on the investable asset index and thenon-investable asset index provided by IFC (International Finance Corporation). They defined theratio of non-investable assets in the total of both assets as the degree of international segmentation.
All stocks in developed stock markets are treated as investable assets in their setting so that thesemarkets are perfectly integrated. Karolyi and Wu (2018) defined “globally accessible securities” assecurities traded on the market with few barriers to foreign investors. These securities are chosen in
the spirit of the “global, long-term view” in Sarkissian and Schill (2016). In their definition, major
stock markets in the US and Europe are perfectly integrated with each other and other markets are atleast partially segmented. These studies clearly show the interaction between stock markets and theseare different from conventional studies on the only domestic stock market. However, these have
similar problems to such conventional studies in that these “arbitrarily” assume market scopes. It isnecessary to construct a model that estimates market scopes instead of assuming them in order to findthe optimal strategy.
Bekaert and Harvey (1995) estimated the degree of market integration between global market
and emerging markets using a two-regime switching model. In their model, the international CAPM
is applied when the global market and the emerging market are integrated and the domestic CAPM isapplied when these are segmented. One of the only minor drawbacks of their model is that they can2only analyze emerging markets and not developed markets, that is, most of the world market
capitalization.
Second, we discuss the interdependence between stock markets. Pukthuanthong and Roll (2009)
and Akbari, Ng and Solnik (2020) quantified market integration in a simple and elegant way. They
expressed market integration by using the coefficient of determination when each market return orpart of it was regressed on one of the global factors. Such global factors are constructed from thevalue-weighted returns of major developed markets or all markets analyzed. These are strongly
affected by large stock markets such as the G20 markets and cannot be treated as pure exogenous
variables when analyzing such markets. This problem of simultaneity is also the reason why the
Bekaert and Harvey (1995) model cannot be applied to the developed markets.
It is ideal to construct a multivariate time series model with the returns of all markets as the
explained variables in order to consider the endogeneity of the global factor. Then, the correlations of
the error terms will be set to non-zero values. However, it is very difficult to estimate a large
multivariate time series model, and even if it can be estimated, there is concern that the estimation
accuracy will deteriorate. Therefore, this paper builds a bivariate model with the US market as areference market and analyzes whether the US integrates with any market. The basic structure of themodel is a bivariate GARCH-in-Mean (generalized ARCH-in-Mean) model with two regimes
modified from the Bekaert and Harvey (1995) model. The smoothed probability of the integration
regime was interpreted as an “international integration measure”. A high international integration
measure means that non-home-biased strategies such as international diversification have advantages
over home-biased strategies such as domestic concentration.
Third, we regard the time-variability of the price of variance risk. This paper stands in the position
that market integration should be estimated without arbitrary assumptions. The specification for thetwo models that make up the regime switching model, the international CAPM and the domestic
CAPM, is refined in order to make a more reasonable estimate. We focus on the “price of variancerisk”, which is an important parameter in CAPM and is defined as the ratio of the market’s expectedexcess return to its variance. The price of variance risk is treated as a constant in the one-period CAPM,
but not in the multi-period or conditional CAPM. The price of (conditional) variance risk is definedas the ratio of the market’s conditional expected excess return to its conditional variance and is timevarying. Harvey (1991), Bekaert and Harvey (1995) and De Santis and Gerard (1997) show empiricalresults that the price of variance risk is time-varying. However, they do not show “how” and “why”
the price of variance risk is time-varying. Their analysis did not provide a sufficient economic
interpretation of its time-variation. Therefore, this paper uses the Chan and Kogan (2002) model(hereafter, CK model), which is one of the continuous-time habit-formation consumption-based asset
pricing models, to specify the time-variation of the price of variance risk. They defined the “state of
the economy” as a weighted average of aggregate consumption in the past and showed that the relative3

risk aversion of representative investors in that economy is counter-cyclical in their model. Convertingthe consumption-based asset pricing model to the conventional CAPM format by replacing the growthof the aggregate endowment with the market return, the relative risk aversion of a representative
investor equals the price of variance risk. The state of the economy is defined as a weighted averageof past market returns and can be rewritten as a weighted average of market returns from each past
date to the current date. These weights of the latter can be interpreted as expressing the distribution of
rebalancing cycles of investors participating in the market.
The relative risk aversion of the representative investor, or the price of variance risk, is expressed
as a function of the state of the economy and is expected to be a monotonically decreasing function
according to the CK model. This means the price of variance risk is countercyclical. It would be
sufficient to express the price of variance risk by a linear function of the state of the economy if it isalways a decreasing function. It may, however, not always hold. The model assumes that individuals
may have different utility functions, but these are all risk averse. On the other hand, this assumption
is extremely restrictive from the perspective of prospect theory by Kahneman and Tversky (1979) andTversky and Kahneman (1992). One of their most basic arguments is that people tend to be risk averse
for bets that are prospected to exceed a certain reference point, but loss averse for bets that areprospected to be below that. To put it more simply, the value function associated with the utility
function is concave for gains and convex for losses.
We extend the CK model from the viewpoint of prospect theory. First, we extend the model tothe case of the loss averse utility function. As a result, we prove that the risk attitude of therepresentative investor, or the price of variance risk, is a monotonically increasing function of the stateof the economy. This means the price of variance risk is procyclical. It is suggested that the relationshipbetween the price of variance risk and the state of the economy is unclear in a market where riskaverse individuals and loss-averse individuals coexist. Second, we provide a rational interpretation of
the state of the economy in line with prospect theory. As mentioned earlier, the state of the economy
is expressed as a weighted average of market returns from each past date to the present date, and its
weight can be interpreted as representing the distribution of investors' rebalancing cycles. The negative
state of the economy means that more investors will recognize the loss than those who recognize thegain, and vice versa. Investors who recognize losses are loss averse, according to prospect theory. It is
expected that the influence of the extended CK model with a loss-averse utility function will exceedthat of the original model with a risk-averse utility function when the state of the economy is negative.
The price of variance risk is expected to be procyclical or increasing. On the contrary, the price ofvariance risk is expected to be countercyclical or decreasing when the state of the economy is positive.
When the state of the economy is near zero, it is expected that the influences of risk-averse individuals
and loss-averse individuals will antagonize. The procyclicality or countercyclicality of the price ofvariance risk will diminish and its slope will be gradual. Therefore, a graph with the state of the4economy on the horizontal axis and the price of variance risk on the vertical axis is expected to be a
curve with the maximum value at the point where the state of the economy is zero. We use polynomial
approximation to verify the structure of the price of variance risk. Specifically, we construct a modelwith a linear function, a quadratic function, a cubic function, and a quartic function, and selected themost appropriate model using the Akaike information criterion.
As a result, the structures of representative investor’s risk attitude, or that of the price of variancerisk, in each of the following 15 markets are almost the same; Canada, France, Italy, Australia, Hong
Kong, Netherlands, Spain, Sweden, Switzerland, Brazil, South Korea, Taiwan, Indonesia, Mexico andSaudi Arabia. It is considered that the effect of the time-variation of the price of variance risk in eachmarket on the time-variation of the international integration measure estimated in such a market isconsidered to be almost eliminated.
International integration measures in such markets are on average high (except Australia),
declining before the 2008 global financial crisis (except Australia and Mexico), but rising again after
the crisis (except Australia). This means that non-home-biased strategies have advantages over homebiased strategies except just before the crisis. In addition, the difference between the international
integration measures of developed and emerging markets was extremely small. In other words, being
an emerging market does not mean that the market is segmented. Summing up the above results, weconclude that the international diversification is strongly recommended in these 15 markets (except
Australia) regardless of country or period. The contribution of this paper is to draw this conclusion on
the globally influential stock markets, paying attention to the distribution of investors’ risk attitudes
and behavioral bias suggested by prospect theory, without making arbitrary assumptions about market
integration.
In Section 2, we examine a model that simultaneously expresses market integration andsegmentation based on the standard discrete-time CAPM, while relating with the continuous-time
habit-formation consumption-based asset pricing model of Chan and Kogan (2002). In Section 3, we
discuss the characteristics of market return data. We provide some evidence of market integration insection 4 and conclude in section 5.
2. Theoretical background and empirical modelThis section discusses many theoretical arguments and constructs an empirical model, focusing
on three important points in preventing misidentification of market scopes; researchers’ arbitrary
assumptions about market scopes, interdependence between stock markets, and the time-variability ofthe price of variance risk.
2.1. Arbitrariness of market scopes
Our aim is to identify the market scope. If the true market scope spans the stock markets of5

multiple countries, it is advisable to adopt the returns of such integrated markets as the pricing factor.
The set of countries included in the integrated market 𝑊 is represented by 𝒲. Let 𝑟𝑊𝑡 be theintegrated market returns, 𝑟𝑖𝑡 be the market returns for each country 𝑖 ∈ 𝒲, 𝑟𝑓𝑡 be the risk-free rate,and Ω𝑡 be the information set at time 𝑡. A multi-period discrete CAPM suggests the following
equation.
E[𝑟𝑖,𝑡+1 |Ω𝑡 ] = 𝑟𝑓𝑡 + 𝜆𝑊 ,𝑡+1|𝑡 Cov[𝑟𝑖,𝑡+1 , 𝑟𝑊 ,𝑡+1 |Ω𝑡 ] ,∀𝑖 ∈ 𝒲.

(1)

The coefficient of the conditional covariance, 𝜆𝑊 ,𝑡+1|𝑡 , is represented by the conditional moments ofthe conditional pricing factor.
𝜆𝑊 ,𝑡+1|𝑡 =E[𝑟𝑊 ,𝑡+1 |Ω𝑡 ] − 𝑟𝑓𝑡.
Var[𝑟𝑊 ,𝑡+1 |Ω𝑡 ](2)

𝜆𝑊 ,𝑡+1|𝑡 is referred to as the price of variance risk for the integrated market 𝑊 .
On the other hand, if the true market scope is limited to the domestic market, it is the market
return of each country 𝑖 that becomes the conditional pricing factor. Eq. (1) can be rewritten as
follows.
E[𝑟𝑖,𝑡+1 |Ω𝑡 ] = 𝑟𝑓𝑡 + 𝜆𝑖,𝑡+1|𝑡 Var[𝑟𝑖,𝑡+1 |Ω𝑡 ] .

(3)

The coefficient of the conditional variance is
𝜆𝑖,𝑡+1|𝑡 =E[𝑟𝑖,𝑡+1 |Ω𝑡 ] − 𝑟𝑓𝑡.
Var[𝑟𝑖,𝑡+1 |Ω𝑡 ](4)

It is clear that 𝜆𝑖,𝑡+1|𝑡 is the price of variance risk for the market 𝑖. Of course, eq. (3) and eq. (4) are
the same equation.
It can be said that such a formulation arbitrarily assumes the market scope. Such arbitrariness canbe eliminated by using the regime switching model. Two opposite regimes at the time 𝑡 are
represented by a binary variable 𝜙𝑡 . The conditional expectation of 𝑟𝑖𝑡 can be decomposed as
E[𝑟𝑖,𝑡+1 |Ω𝑡 ] = Pr[𝜙𝑡+1 = 1|Ω𝑡 ] E[𝑟𝑖,𝑡+1 |Ω𝑡 , 𝜙𝑡+1 = 1] + Pr[𝜙𝑡+1 = 0|Ω𝑡 ] E[𝑟𝑖,𝑡+1 |Ω𝑡 , 𝜙𝑡+1 = 0].
The integration regime is represented by 𝜙 = 1, and it is assumed that eqs. (1) and (2) hold in thisregime. Similarly, the segmentation regime is represented by 𝜙 = 0, and it is assumed that eqs. (3)and (4) hold in this regime. Then, the following equation holds.
(1)

(0)

E[𝑟𝑖,𝑡+1 |Ω𝑡 ] = 𝑟𝑓𝑡 + Pr[𝜙𝑡+1 = 1|Ω𝑡 ] 𝜆𝑊 ,𝑡+1|𝑡 𝜎𝑖𝑊 ,𝑡+1|𝑡 + Pr[𝜙𝑡+1 = 0|Ω𝑡 ] 𝜆𝑖,𝑡+1|𝑡 𝜎𝑖𝑖,𝑡+1|𝑡(𝑘)where 𝜎𝑙𝑚,𝑡+1|𝑡= Cov[𝑟𝑙,𝑡+1 , 𝑟𝑚,𝑡+1 |Ω𝑡 , 𝜙𝑡+1 = 𝑘] .

(5)

2.2. Interdependence between stock markets
Set 𝒲 should include all markets around the world. However, the verification of eq. (1) requires
an “about 200-variate model” because the eq. (1) must hold for all 𝑖 ∈ 𝒲 in the integrated regime.
One way to solve this problem is to build a bivariate model of 𝑟𝑖 and 𝑟𝑊 . Suppose that each return𝑟𝑖 affects all foreign returns 𝑗 ∈ 𝒲 ∖ {𝑖} in the integration regime, but the effects will be negligible

6when aggregated into 𝑟𝑊 . Under this supposition, the “200-variate model” can be reduced to abivariate model. Thus, the integrated market return 𝑟𝑊 is typically set to be exogenous to each market
return 𝑟𝑖 (e.g. Bekaert and Harvey, 1995). Pukthuanthong and Roll (2009) and Akbari, Ng and Solnik(2020) also set global factors as exogenous variables, and such settings are very common.
The integrated market return or global factor is, however, strongly affected by extremely large
stock market returns such as the US and China. In addition, shocks in countries with relatively large
economies, even though not as large as these two countries, can lead to global shocks; for example,
Thailand, Indonesia and South Korea in the Asian financial crisis, and Russia in the subsequent
Russian financial crisis. Shocks in major countries such as the G20 are likely to cause global shocks.
The integrated market return 𝑟𝑊 can no longer be said to be exogenous to such market returns.
Another problem is that the setting of the above integration regime is extremely strict. The above
setting requires that the markets 𝑖 ∈ 𝒲 are all integrated. In other words, a regime in which 𝑖 ∈ 𝒲
and 𝑗 ∈ 𝒲 ∖ {𝑖} are integrated but 𝑖 and 𝑘 ∈ 𝒲 ∖ {𝑖, 𝑗} are segmented is not an integration
regime. The probability that such integration regime will be applied, Pr[𝜙𝑡+1 = 1|Ω𝑡 ], is not expected
to be large. The bivariate model of 𝑟𝑖 and 𝑟𝑊 may underestimate the integration probability when
there is a market 𝑘 that disturbs the integration of the world market.
We analyze bilateral integration to clarify the interdependence between stock markets. In otherwords, the set of bilateral integrated markets of 𝑖 and 𝑗 is 𝒲𝑖𝑗 = {𝑖, 𝑗}, and the regime with its
integrated market return 𝑟𝑊𝑖𝑗 as the pricing factor is interpreted as the integration regime. The
important assumption here is that the effects from 𝑖 to 𝑗 (or 𝑗 to 𝑖) are only direct effects. That is,
it is assumed that the indirect effect from 𝑖 to 𝑗, which is the sum of the effect from 𝑖 to the third
country 𝑘 and the effect from 𝑘 to 𝑗, is negligible. Effects from third countries to 𝑖 and 𝑗 are also
typically ignored. However, the shock from the third countries may affect the correlation structure
between 𝑖 and 𝑗. Allowing time variability in the correlation between the two countries may capture
shocks from third countries. Note that all investors evaluate the returns using the US dollar as thenuméraire, and do not consider the exchange rate in this paper.
The bivariate regime switching model is constructed based on the above discussion. In the caseof integration regime,
𝑟𝑓𝑡 + 𝜆𝑊𝑖𝑗 ,𝑡+1|𝑡 𝜎𝑖𝑊 ,𝑡+1|𝑡𝜖(1)𝑟𝑖,𝑡+1𝑖,𝑡+1(𝑟)=() + ( (1) ) ,𝑗,𝑡+1
𝑟𝑓𝑡 + 𝜆𝑊𝑖𝑗 ,𝑡+1|𝑡 𝜎𝑗𝑊 ,𝑡+1|𝑡𝜖𝑗,𝑡+1
where the error term vector (𝜖(1)
𝑖,𝑡+1

⊤

(1)
)
𝜖𝑗,𝑡+1(6)

follows a bivariate normal distribution;

𝜖(1)
𝑖,𝑡+1

𝜎𝑖𝑖,𝑡+1|𝑡0( (1) ) ∼ 𝒩 (( ) , (𝜎𝑗𝑖,𝑡+1|𝑡0
𝜖𝑗,𝑡+17

𝜎𝑖𝑗,𝑡+1|𝑡
𝜎𝑗𝑗,𝑡+1|𝑡 )) .

(7)

In the case of segmentation regime,(0)
𝜖𝑖,𝑡+1
𝑟𝑓𝑡 + 𝜆𝑖,𝑡+1|𝑡 𝜎𝑖𝑖,𝑡+1|𝑡𝑟𝑖,𝑡+1(𝑟)=() + ( (0) ) ,𝑗,𝑡+1
𝑟𝑓𝑡 + 𝜆𝑗,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡𝜖𝑗,𝑡+1

where the error term vector (𝜖(0)
𝑖,𝑡+1

⊤

(0)
)
𝜖𝑗,𝑡+1(8)

follows a bivariate normal distribution;

𝜖(0)
𝑖,𝑡+1

𝜎𝑖𝑖,𝑡+1|𝑡0( (0) ) ∼ 𝒩 (( ) , (00
𝜖𝑗,𝑡+10
𝜎𝑗𝑗,𝑡+1|𝑡

)) .

(9)

The error term vector at time 𝑡 + 1 is expressed as follows.
(1)

(0)

𝜖𝑖,𝑡+1𝜖𝑖,𝑡+1𝜖𝑖,𝑡+1(𝜖) = Pr[𝜙𝑖𝑗,𝑡+1 = 1|𝜓𝑖𝑗𝑡 ] ( (1) ) + Pr[𝜙𝑖𝑗,𝑡+1 = 0|𝜓𝑖𝑗𝑡 ] ( (0) )𝑗,𝑡+1
𝜖𝑗,𝑡+1𝜖𝑗,𝑡+1(10)

where 𝜙𝑖𝑗,𝑡+1 equals 1 if market 𝑖 and 𝑗 are integrated and 0 otherwise, 𝜓𝑖𝑗𝑡 is the history of 𝑟𝑖
and 𝑟𝑗 . Eqs. (6) - (10) imply that conditional variances do not depend on the regime and conditional
covariances are zero in the segmentation regime because expected returns in such regime should be
estimated independently, as if they were seemingly uncorrelated. The dynamics of these conditional
covariance matrices are specified by GARCH (1, 1).3 The elements of these matrices are specified by
the constant conditional correlation model (CCC model) proposed by Bollerslev (1990) as follows.
𝜎𝑖𝑖,𝑡+1|𝑡 = 𝛼𝑖0 + 𝛼𝑖1 𝑒2𝑖𝑡 + 𝛽𝑖 𝜎𝑖𝑖,𝑡|𝑡−1 ,(11)

𝜎𝑗𝑗,𝑡+1|𝑡 = 𝛼𝑗0 + 𝛼𝑗1 𝑒2𝑗𝑡 + 𝛽𝑗 𝜎𝑗𝑗,𝑡|𝑡−1 ,(12)

𝜎𝑖𝑗,𝑡+1|𝑡 = 𝜎𝑗𝑖,𝑡+1|𝑡 = 𝜌𝑖𝑗 √𝜎𝑖𝑖,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡(13)

where 𝑒𝑖𝑡 and 𝑒𝑗𝑡 are the residuals correspond to the error terms 𝜖𝑖𝑡 and 𝜖𝑗𝑡 . The parameters of
𝛼𝑖0 and 𝛼𝑗0 are constant terms for the dynamics of the volatility, the parameters of 𝛼𝑖1 and 𝛼𝑗1
capture the effects of past residuals, and the parameters of 𝛽𝑖 and 𝛽𝑗 capture the dependence of thepast volatility. Assuming that market integration does not affect the intrinsic volatility of stocks, theseparameters are independent on regimes. Consequently, these are considered as common parameters
between regimes.4 The parameter 𝜌𝑖𝑗 is the constant correlation between 𝜖𝑖𝑡 and 𝜖𝑗𝑡 . In this model,
we assume that the correlation coefficient is not time-varying, since the change over time of thebilateral interaction effect should be captured in the dynamics of the switching variable 𝜙𝑖𝑗𝑡 . At this3

The GARCH model with regime switching requires careful formulation when conditional volatility depends on an
infinite past regime (Haas, Mittnik and Paolella, 2004). This problem does not occur in this paper because theconditional volatility does not depend on the regime.
4
If we set these parameters are not common between regimes, the transition of the switching variable cannot beinterpreted well. That is because we cannot tell whether the regime shift is due to a shift between the integration andsegmentation regimes, or a shift between the large and small volatility regimes like the SWARCH (Switching ARCH)
model by Hamilton and Susmel (1994).

8

time, the variance-covariance matrix of the error term vector is
𝜎𝑖𝑖,𝑡+1|𝑡⎛⎜⎜Pr[𝜙𝑖𝑗,𝑡+1 = 1|𝜓𝑖𝑗𝑡 ] 𝜌𝑖𝑗 √𝜎𝑖𝑖,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡⎝Pr[𝜙𝑖𝑗,𝑡+1 = 1|𝜓𝑖𝑗𝑡 ] 𝜌𝑖𝑗 √𝜎𝑖𝑖,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡
𝜎𝑗𝑗,𝑡+1|𝑡

⎞
⎟
⎟.
⎠

Therefore, the time variability of Pr[𝜙𝑖𝑗,𝑡+1 = 1|𝜓𝑖𝑗𝑡 ] makes the bilateral correlation substantiallytime-varying. The covariance with the integrated market 𝑊𝑖𝑗 is specified as follows.
𝜎𝑖𝑊𝑖𝑗 ,𝑡+1|𝑡 = 𝑤𝑡 𝜎𝑖𝑖,𝑡+1|𝑡 + (1 − 𝑤𝑡 )𝜌𝑖𝑗 √𝜎𝑖𝑖,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡 ,(14)

𝜎𝑗𝑊𝑖𝑗 ,𝑡+1|𝑡 = 𝑤𝑡 𝜌𝑖𝑗 √𝜎𝑖𝑖,𝑡+1|𝑡 𝜎𝑗𝑗,𝑡+1|𝑡 + (1 − 𝑤𝑡 )𝜎𝑗𝑗,𝑡+1|𝑡 ,where 𝑤𝑡 is the value-weight of market 𝑖 at the time 𝑡, that is, the ratio of market capitalization ofmarket 𝑖 to the market capitalization of market 𝑊𝑖𝑗 .
(𝑖,𝑗)The dynamics of 𝜙𝑖𝑗𝑡 are characterized by the probability of staying integration regime, 𝑝1|1 ≡(𝑖,𝑗)Pr[𝜙𝑖𝑗,𝑡+1 = 1|𝜙𝑖𝑗,𝑡 = 1] ∀𝑡 , and that of staying segmentation regime, 𝑝0|0 ≡ Pr[𝜙𝑖𝑗,𝑡+1 =0|𝜙𝑖𝑗,𝑡 = 0] for all 𝑡. The transition probability matrix 𝑃𝑖𝑗 is composed of them as follows.
(𝑖,𝑗)

𝑝1|1

⎛
𝑃𝑖𝑗 = ⎜
(𝑖,𝑗)
⎝1 − 𝑝1|1(𝑖,𝑗)

1 − 𝑝0|0
(𝑖,𝑗)

𝑝0|0

⎞
⎟
⎠

(15)

See Hamilton (1988), Hamilton (1989), Hamilton (1990) and Kim and Nelson (1999) for detail.
2.3. Time-varying price of variance risk
The price of variance risk is defined as the ratio of the conditional expected excess return on themarket portfolio to its conditional variance. The price of variance risk is obviously time-varying, butmany empirical models for CAPM and multi-factor models ignore that time variability. Harvey (1991),
Bekaert and Harvey (1995) and de Santis and Gerard (1997) provided a lot of evidences that the priceof variance risk is time-varying. They have shown the fact that the price of variance risk is timevarying but have not analyzed its economic background.
For this purpose, this paper uses the Chan and Kogan (2002) model (CK model), which is one of
the continuous-time habit-formation consumption-based asset pricing models. Furthermore, we
generalize the CK model so that we can take into account the behavioral bias suggested by prospect
theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992). As a result, it can betheoretically shown that the price of variance risk is countercyclical in a boom, and procyclical in arecession.
2.3.1. Chan and Kogan model
Chan and Kogan (2002) construct a habit-formation consumption-based asset pricing model inwhich the assumptions of a representative individual is relaxed. In their model, individuals have9constant but different relative risk aversions. This model implies that the aggregate risk aversion in theeconomy, corresponding to the risk aversion of a representative individual, is time-varying and its
time-variation is countercyclical.
They solve the following representative individual’s utility function 𝑈 (𝑌𝑡 , 𝑋𝑡 ) (see also Chanand Kogan (2002) or Munk (2013) for details).
∞

𝑈 (𝑌𝑡 , 𝑋𝑡 ) =sup[∫ 𝑓(𝛾{𝑐𝑡 (𝑌𝑡 ,𝑋𝑡 ;𝛾 𝑖𝑛𝑑 )}

1

𝑖𝑛𝑑

1
𝑐𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾 𝑖𝑛𝑑 ))()
1 − 𝛾 𝑖𝑛𝑑𝑋𝑡

1−𝛾 𝑖𝑛𝑑𝑑𝛾 𝑖𝑛𝑑 ,
(16)

∞

s. t.

∫ 𝑐𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾

𝑖𝑛𝑑

)𝑑𝛾

𝑖𝑛𝑑

≤ 𝑌𝑡 ]

1𝛾 𝑖𝑛𝑑 ∈ (1, ∞) is a constant relative risk aversion of individual where the subscript “𝑖𝑛𝑑” means∞“individual”, and 𝑓(𝛾 𝑖𝑛𝑑 ) normalized as ∫1 𝑓(𝛾 𝑖𝑛𝑑 )𝑑𝛾 𝑖𝑛𝑑 = 1 denote the weight of individuals
with the relative risk aversion 𝛾 𝑖𝑛𝑑 . 𝑌𝑡 and 𝑋𝑡 are an aggregate endowment and a benchmark forhabit-formation at time 𝑡, respectively5. The individual consumption 𝑐𝑡 is the function of them. The
log-benchmark 𝑥𝑡 = ln 𝑋𝑡 is specified as follows.
𝑡𝑥𝑡 = exp(−𝜅𝑡) 𝑥0 + 𝜅 ∫ exp(−𝜅(𝑡 − 𝑠)) 𝑦𝑠 𝑑𝑠,0where 𝑦𝑡 = ln 𝑌𝑡 ,

𝜅 is positive and small.

The parameter 𝜅 can be interpreted as the degree of history dependence in 𝑋𝑡 according to Chanand Kogan (2002).
Given the discrete time period 𝑡 − 1 is correspond to [𝑡 − 1, 𝑡] , and considered 1 − 𝜅 ≈exp(−𝜅) for small 𝜅, this equation can be approximated in discrete time as follows.
𝑡−1

𝑥𝑡 ≈ (1 − 𝜅)𝑡 𝑥0 + 𝜅 ∑(1 − 𝜅)𝑡−1−𝑠 𝑦𝑠 .
𝑠=0Consequently,
𝑡−2

𝑥𝑡 ≈ (1 − 𝜅) ((1 − 𝜅)𝑡−1 𝑥0 + 𝜅 ∑(1 − 𝜅)𝑡−2−𝑠 𝑦𝑠 ) + 𝜅𝑦𝑡−1
𝑠=0= (1 − 𝜅)𝑥𝑡−1 + 𝜅𝑦𝑡−1 .

(17)This approximation provides the useful intuition that the log-benchmark 𝑥𝑡 is a weighted average of
its lagged value 𝑥𝑡−1 and the lagged log-endowment 𝑦𝑡−1 . In addition, they introduce the state of theeconomy as the difference between the log-endowment and the log-benchmark, 𝑧𝑡 ≡ 𝑦𝑡 − 𝑥𝑡 . Thiscan be rewritten by using eq. (17) as follows.
𝑧𝑡 ≈ 𝑦𝑡 − ((1 − 𝜅)𝑥𝑡−1 + 𝜅𝑦𝑡−1 )= 𝑦𝑡 − 𝑦𝑡−1 + (1 − 𝜅)(𝑦𝑡−1 − 𝑥𝑡−1 )= ln 𝑌𝑡 /𝑌𝑡−1 + (1 − 𝜅)𝑧𝑡−1
5

Chan and Kogan (2002) assume that the dynamics of the aggregate endowment follow a geometric Brownian
motion and the log-benchmark is expressed as a weighted geometric average of past realizations of logconsumptions.

10≡ 𝑟𝑌𝑡 + (1 − 𝜅)𝑧𝑡−1(18)where 𝑟𝑌𝑡 is the growth rate of the aggregate endowment. This equation means the state of theeconomy can be specified as like AR (1) model with exogeneous variable. The initial value is 𝑧0 =𝜇𝑌 /𝜅 where 𝜇𝑌 is unconditional mean of 𝑟𝑌𝑡 . Substituting eq. (18) successively,
𝑧𝑡 ≈ 𝑟𝑌𝑡 + (1 − 𝜅)𝑟𝑌 ,𝑡−1 + (1 − 𝜅)2 𝑟𝑌 ,𝑡−2 + ⋯ + (1 − 𝜅)𝑡−1 𝑟𝑌 ,1 + (1 − 𝜅)𝑡 𝜇𝑌 /𝜅

(19)Therefore, the state of the economy can be written by the function of the set 𝜓𝑌𝑡 ={𝑅𝑌𝑡 , 𝑅𝑌 ,𝑡−1 , 𝑅𝑌 ,𝑡−2 , … , 𝑅𝑌0 } which is the history of the change rate of the aggregate endowment.
To explicitly show this property, it can be written as 𝑧𝑡 ≡ 𝑧𝑌𝑡 = 𝑧(𝜓𝑌𝑡 ; 𝜇𝑌 , 𝜅). Eq. (19) can be
rewritten as:
𝑧𝑌𝑡 ≈ 𝜅(𝑟𝑌 ,[𝑡−1,𝑡] + (1 − 𝜅)𝑟𝑌 ,[𝑡−2,𝑡] + (1 − 𝜅)2 𝑟𝑌 ,[𝑡−3,𝑡] + ⋯ )(20)

where 𝑟𝑌 ,[𝑡−𝑠,𝑡] is the growth of aggregate endowment from the time 𝑡 − 𝑠 to 𝑡. In the case of 𝜅 =0.02, the state of the economy can be rewritten as 𝑧𝑌𝑡 ≈ 0.02 × (𝑟𝑌 ,[𝑡−1,𝑡] + 0.980 × 𝑟𝑌 ,[𝑡−2,𝑡] +0.960 × 𝑟𝑌 ,[𝑡−3,𝑡] + 0.941 × 𝑟𝑌 ,[𝑡−4,𝑡] + ⋯ ). The influence of investors with a rebalancing cycle of 4periods is 94.1% of the investors with a rebalancing cycle of 1-period, even though they only appear
in the market once every four periods. That is, the density of 4-period investors is 0.941 × 4 ≈ 3.8
times that of 1-period investors. The distribution of investor rebalancing cycles is given below.
𝑔(𝑠; 𝜅) = 𝑠𝜅2 (1 − 𝜅)𝑠−1 ,wherelim ∑𝑡𝑠=1 𝑔(𝑠; 𝜅) = 1𝑡→∞𝑠 ∈ {1,2, … }(21)holds.

Figure 1 shows the distribution of investor rebalancing cycles under various 𝜅. The density ofinvestors with a rebalancing cycle of 𝑠-periods is 𝑠(1 − 𝜅)𝑠−1 times that of 1-period investors. If 𝑠
that maximizes this is 𝑠∗ , then 𝑠∗ = −1/ ln(1 − 𝜅) ≈ 1/𝜅. For example, 𝑠∗ ≈ 50 when 𝜅 = 0.02,𝑠∗ ≈ 4 when 𝜅 = 0.25, and so on. Therefore, this assumption implies that the distribution has thehighest number of investors with a 50-periods rebalancing cycle. In the discrete-time model, 𝜅 plays
the role of parameters that specify the distribution of investor rebalancing cycles. At this time, 1/𝜅represents the mode of the distribution.
[Figure 1]
Chan and Kogan indicate that the utility function of an aggregate (representative) individual isspecified as follows, under the optimal consumption allocation.
∞

𝑈 (𝑌𝑡 , 𝑋𝑡 ) = ∫1

1
1 − 𝛾 𝑖𝑛𝑑𝑖𝑛𝑑 1/𝛾 𝑖𝑛𝑑
𝑓(𝛾
)
exp
ℎ(𝑧𝑡 )) 𝑑𝛾 𝑖𝑛𝑑 ,(−1 − 𝛾 𝑖𝑛𝑑𝛾 𝑖𝑛𝑑∞

where the function ℎ is implicitly defined as ∫ 𝑓(𝛾 𝑖𝑛𝑑 )1/𝛾
1

𝑖𝑛𝑑

(22)

exp(− 𝛾1 ℎ(𝑧𝑌𝑡 ) − 𝑧𝑌𝑡 ) 𝑑𝛾 𝑖𝑛𝑑 = 1.

They have proved that the function ℎ is convex and decreasing in 𝑧𝑡 and the first derivative issatisfied as −ℎ′ (𝑧𝑌𝑡 ) = −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )/𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 ) > 1 where 𝑈𝑌 and 𝑈𝑌𝑌 are the first andsecond derivative of 𝑈 with respect to 𝑌 , respectively. Therefore, −ℎ′ (𝑧𝑌𝑡 ) is interpreted as the11relative risk aversion of the aggregate individual. In order to emphasize this characteristic, it isexpressed as −ℎ′ (𝑧𝑌𝑡 ) ≡ 𝛾(𝑧𝑌𝑡 ) where the function 𝛾(. ) means the aggregate relative risk aversion.
It is decreasing in the state of the economy (because of the convexity of ℎ) so that we can consider its
fluctuations to be time-varying and counter-cyclical. Intuitively, individuals with low risk aversions
hold a large fraction of their portfolio in risky assets such as stocks and a small fraction in non-risky
assets such as treasury bills than individuals with high risk aversions. When the stock prices go up, thewealth of individuals with low risk aversions will grow more than that of individuals with high risk
aversions. Consequently, the aggregate relative risk aversion will be low. Conversely, when the stock
prices fall, the wealth of individuals with low risk aversions will shrink more than others so that theaggregate relative risk aversion will be high. Therefore, the relative risk aversion of the aggregate(representative) individual is time-varying and counter-cyclical.
In their model, the Sharpe ratio of a risky asset 𝑖 is𝑟𝑖𝑡 − 𝑟𝑓𝑡
= −ℎ′ (𝑧𝑌𝑡 )𝜎𝑌𝑡 = 𝛾(𝑧(𝜓𝑌𝑡 ; 𝜇𝑌 , 𝜅))𝜎𝑌𝑡 ,𝜎𝑖𝑡where 𝑟𝑓𝑡 , 𝑟𝑖𝑡 , 𝜎𝑖𝑡 and 𝜎𝑌𝑡 denote the instantaneous risk-free rate, the (expected) return of asset 𝑖,6
its volatility (or its sensitivity to external shock) and the volatility of the aggregate endowment 𝑌 ,respectively. 𝛾(𝑧(𝜓𝑌𝑡 ; 𝜇𝑌 , 𝜅)) is the relative risk aversion of an aggregate individual.
Their general model includes the continuous-time CAPM like Merton (1973) as a special case by
applying the market capitalization 𝑀 as a proxy of 𝑌 . At this time, the growth of aggregate
endowment, 𝑟𝑌𝑡 , is replaced by the stock market return 𝑟𝑀𝑡 . Thus, the Sharpe ratio of the stock
market return is (𝑟𝑀𝑡 − 𝑟𝑓𝑡 )/𝜎𝑀𝑡 = 𝛾(𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅))𝜎𝑀𝑡 . Therefore,
𝛾(𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅)) =𝑟𝑀𝑡 − 𝑟𝑡𝑓
2
𝜎𝑀𝑡

This equation implies that the ratio of expected excess market return to its volatility equals to therelative risk aversion of the aggregate (representative) individual. Approximated from a continuoustime

model

to

a

discrete-time

model

as

2
(𝑟𝑀𝑡 − 𝑟𝑓𝑡 )/𝜎𝑀𝑡≈ (E[𝑟𝑀𝑡 |𝜓𝑀,𝑡−1 ] − 𝑟𝑓𝑡 )/Var[𝑟𝑀𝑡 |𝜓𝑀,𝑡−1 ], it corresponds to eq. (2) and (4). Therefore,
𝛾(𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅)) ≈ 𝜆𝑀,𝑡+1|𝑡(23)

Thus, the price of variance risk, 𝜆𝑀,𝑡+1|𝑡 , can be roughly interpreted as the relative risk aversion of arepresentative individual in the market 𝑀, and considered to be time-varying and counter-cyclical.
2.3.2. Prospect theory, and Chan/Kogan model
Chan and Kogan (2002) assume that all individuals are risk averse. This assumption is reflectedin the condition that 𝛾 𝑖𝑛𝑑 > 1 . However, this does not apply if individual preferences change
6

Strictly speaking, we should consider the (expected) return and dividend of asset 𝑖. In this paper, capital gains andincome gains are not distinguished and are collectively referred to as 𝑟𝑖𝑡 .

12depending on profit or loss, as indicated by Kahneman and Tversky (1979) and Tversky and
Kahneman (1992). They propose the following value function.
𝑣(𝑥) = {𝑥𝛼−𝜃(−𝑥)𝛽

if
if

𝑥≥0
,
𝑥<0

𝛼, 𝛽 > 0.

This implies that an individual becomes risk averse when the outcome is above the reference point (inthe gain domain) and risk loving or “loss averse” when the outcome is below the reference point (inthe loss domain). Their study suggests that 𝜃 > 1, that is, the slope is steeper in the loss region thanin the gain region.
Interestingly enough, the idea of reference points in prospect theory matches that of habit levels
in habit formation models. Habit formation models focus on the difference between consumption andhabit level just as prospect theory focuses on the difference between outcomes and reference points.
However, in the habit formation model, individuals do not change their risk attitudes even if
consumption falls below habit levels. We take this into account and extend the CK model.
In the CK model, the risk attitude of a representative individual (to be exact, an aggregate
individual) depends on the state of the economy 𝑧𝑌𝑡 = 𝑧(𝜓𝑌𝑡 ; 𝜇𝑌 , 𝜅). Eq. (20) can be rewritten as
follows.
𝑧𝑌𝑡 ≈ ∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] + ∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡]
𝑠∈𝒮+

𝑠∈𝒮−

𝑟𝑌 ,[𝑡−𝑠,𝑡] ≥ 0 if
where {
𝑟𝑌 ,[𝑡−𝑠,𝑡] < 0 if𝑠 ∈ 𝒮+.
𝑠 ∈ 𝒮−

Consequently, the following relations approximately hold.
𝑧𝑌𝑡 ≥ 0⇔∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] ≥ ∣ ∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] ∣𝑠∈𝒮+𝑧𝑌𝑡 < 0⇔𝑠∈𝒮−∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] < ∣ ∑ 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] ∣
𝑠∈𝒮+

𝑠∈𝒮−

The reference point of investor with 𝑠-periods rebalancing cycle corresponds to 𝑌𝑡−𝑠 and theiroutcome corresponds to 𝑌𝑡 . The difference between them corresponds to 𝑟𝑌 ,[𝑡−𝑠,𝑡] . Investors with𝑠 ∈ 𝒮+ -periods rebalancing cycle will recognize positive returns or “gains”. That is, ∑𝑠∈𝒮+ 𝜅(1 −𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] means the weighted average influence of investors with gains. 7 Similarly,∣∑𝑠∈𝒮− 𝜅(1 − 𝜅)𝑠−1 𝑟𝑌 ,[𝑡−𝑠,𝑡] ∣ means the weighted average influence of investors with losses.
Therefore, the positive (or negative) state of the economy implies the influence of investors with gains
is greater (or smaller) than that of investors with losses. Intuitively, the representative investor would
be expected to be risk averse as before when the state of the economy is positive. On the other hand,when the state of the economy is negative, the representative investor may become loss averse.
To confirm this expectation, we focus on individual risk attitudes 𝛾 𝑖𝑛𝑑 , rather than representative
7

The weight is the (positive) returns recognized by such investors.

13

investor’s risk attitude 𝛾(𝑧𝑌𝑡 ). Chan and Kogan (2002) assume 𝛾 𝑖𝑛𝑑 ∈ (1, ∞) since they only
assume risk-averse individuals. However, we also consider loss-averse individuals with 𝛾 𝑖𝑛𝑑 ∈(−∞, 0) and the rest of (risk-averse) individuals with 𝛾 𝑖𝑛𝑑 ∈ (0,1). The case where each of theseinvestor groups becomes dominant in the market is expressed using the regime switching variable
𝐺𝑡 ∈ {1,2,3}. Considering the correspondence with prospect theory, it would be natural to assume
that the dynamics of such regime variables depend on the state of the economy; 𝐺𝑡 = 𝐺(𝑧𝑌𝑡 ). The
utility function of the representative individual in our extended CK model from eq. (22) are as follows.
3

𝑈 (𝑌𝑡 , 𝑋𝑡 ) = ∑ Pr[𝐺𝑡 = 𝑖] 𝑈 (𝑖) ,𝐺𝑡 = 𝐺(𝑧𝑌𝑡 ) ∈ {1,2,3}

(24)

𝑖=1

where
∞

𝑈 (1) (𝑌𝑡 , 𝑋𝑡 ) =𝑈 (2) (𝑌𝑡 , 𝑋𝑡 ) ={𝑐𝑡 ;𝛾 𝑖𝑛𝑑 ∈(1,∞)}1
1

1−𝛾 𝑖𝑛𝑑{∫ 𝑓(𝛾 𝑖𝑛𝑑 )sup{𝑐𝑡 ;𝛾 𝑖𝑛𝑑 ∈(0,1)}

0

1
𝑐
( 𝑡)
𝑖𝑛𝑑
1−𝛾
𝑋𝑡

0

𝑈 (3) (𝑌𝑡 , 𝑋𝑡 ) ={𝑐𝑡 ;𝛾 𝑖𝑛𝑑 ∈(−∞,0)}𝑐𝑡 = 𝑐𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾𝑈(1)(𝑌𝑡 , 𝑋𝑡 ) and 𝑈

(2)

𝑖𝑛𝑑

−∞

∞

d𝛾 𝑖𝑛𝑑s. t. ∫ 𝑐𝑡 d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 }1
1d𝛾 𝑖𝑛𝑑s. t. ∫ 𝑐𝑡 d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 }01−𝛾 𝑖𝑛𝑑𝑖𝑛𝑑{∫ 𝑓(𝛾 𝑖𝑛𝑑 )

sup

𝑖𝑛𝑑

1
𝑐𝑡 1−𝛾()
1 − 𝛾 𝑖𝑛𝑑 𝑋𝑡{∫ 𝑓(𝛾 𝑖𝑛𝑑 )sup𝜃(𝛾 ) 𝑐𝑡
( )1 − 𝛾 𝑖𝑛𝑑 𝑋𝑡

0d𝛾 𝑖𝑛𝑑s. t. ∫ 𝑐𝑡 d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 }−∞)(𝑌𝑡 , 𝑋𝑡 ) are the utility functions of the representative individuals composed ofrisk averse individuals. 𝑈 (3) (𝑌𝑡 , 𝑋𝑡 ) is the utility function of the representative individual composedof loss averse (or risk loving) individuals. We assume that the slope adjustment parameter 𝜃𝑖𝑛𝑑 > 0
in the loss averse individual’s utility function is a function of 𝛾 𝑖𝑛𝑑 , that is, 𝜃𝑖𝑛𝑑 = 𝜃(𝛾 𝑖𝑛𝑑 ).8
Then we can prove the following properties (see Appendix for detail):(1)𝛾 (1) (𝑧𝑌𝑡 ) ≡ −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )(1)

𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )∈ (1, ∞),d𝛾 (1) (𝑧𝑌𝑡 )
<0
d𝑧𝑌𝑡

(25)

∈ (0,1),d𝛾 (2) (𝑧𝑌𝑡 )
<0
d𝑧𝑌𝑡

(26)

(2)

𝛾 (2) (𝑧𝑌𝑡 ) ≡ −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )(2)

𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )(3)

𝛾 (3) (𝑧𝑌𝑡 ) ≡ −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )(3)

𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )d𝛾 (3) (𝑧𝑌𝑡 )>0d𝑧𝑌𝑡∈ (−∞, 0),(27)Consequently, 𝛾(𝑧𝑌𝑡 ) will be represented by eqs. (25), (26), and (27) in each regime 1, 2 and 3.
Pr[𝐺(𝑧𝑌𝑡 ) = 1] and Pr[𝐺(𝑧𝑌𝑡 ) = 2] are expected to be large when 𝑧𝑌𝑡 ≥ 0. At this time, 𝛾(𝑧𝑌𝑡 )8This assumption allows us to adopt 𝑓 ⋆ (𝛾 𝑖𝑛𝑑 ) defined as follows as the distribution of individuals.
0𝑓 ⋆ (𝛾 𝑖𝑛𝑑 ) ≡ 𝑓(𝛾 𝑖𝑛𝑑 )𝜃(𝛾 𝑖𝑛𝑑 )/ ∫ 𝑓(𝛾 𝑖𝑛𝑑 )𝜃(𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑
0

−∞

0

0

where ∫ 𝑓 ⋆ (𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = ∫ 𝑓(𝛾 𝑖𝑛𝑑 )𝜃(𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 / ∫ 𝑓(𝛾 𝑖𝑛𝑑 )𝜃(𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 = 1.
−∞

Therefore,

−∞

0

−∞
0

𝑈 (3) (𝑌𝑡 , 𝑋𝑡 ) = ∫ 𝑓(𝛾 𝑖𝑛𝑑 )𝜃(𝛾 𝑖𝑛𝑑 )d𝛾 𝑖𝑛𝑑 sup {∫ 𝑓 ⋆ (𝛾 𝑖𝑛𝑑 )
−∞

{𝑐𝑡 }

−∞

14

1
𝑐 1−𝛾
( 𝑡)
𝑖𝑛𝑑
1−𝛾
𝑋𝑡

𝑖𝑛𝑑

0

d𝛾 𝑖𝑛𝑑s. t. ∫ 𝑐𝑡 d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 }.
−∞will be countercyclical from eqs. (25) and (26). Conversely, when 𝑧𝑌𝑡 < 0 ,𝛾(𝑧𝑌𝑡 ) will beprocyclical from eq. (27). It is expected that 𝛾(𝑧𝑌𝑡 ) will be maximum when 𝑧𝑌𝑡 is near zero.
From the above discussion, the structure of relative risk aversion of representative investor,𝛾(𝑧𝑌𝑡 ) , cannot be approximated by linear function. The structure of the price of variance risk,
𝜆𝑀,𝑡+1|𝑡 ≈ 𝛾(𝑧𝑀𝑡 ) from eq. (23), is similar. We consider the following four models that approximate
the structure of the price of variance risk.
Linear model: 𝜆𝑀,𝑡+1|𝑡 = 𝛿0 + 𝛿1 𝑧𝑀𝑡̂
2Quadratic model: 𝜆𝑀,𝑡+1|𝑡 = 𝛿0 + 𝛿1 𝑧𝑀𝑡
̂ + 𝛿2 𝑧𝑀𝑡̂
32̂ + 𝛿2 𝑧𝑀𝑡
̂ + 𝛿3 𝑧𝑀𝑡
̂
Cubic model: 𝜆𝑀,𝑡+1|𝑡 = 𝛿0 + 𝛿1 𝑧𝑀𝑡

(28)

3
2
4
Quartic model: 𝜆𝑀,𝑡+1|𝑡 = 𝛿0 + 𝛿1 𝑧𝑀𝑡
̂ + 𝛿2 𝑧𝑀𝑡
̂ + 𝛿3 𝑧𝑀𝑡
̂ + 𝛿4 𝑧𝑀𝑡
̂where 𝑧𝑀𝑡
̂ ≡ 1 + 𝑧𝑀𝑡 in order to align the units of 𝛿s. We select the most appropriate model fromthese using criteria such as AIC (Akaike’s information criterion).
2.3.3. Extension to international analysis
The argument so far holds for international analysis as it is by replacing 𝑀 with the market
capitalization of each country. However, some interstate assumptions need to be imposed on theparameters such as 𝜅 in eq. (23) and 𝛿s in eq. (28). It is natural that investors’ rebalancing cycles canvary significantly from country to country. We assume the following:𝑧𝑀𝑡 = 𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅) ≡ 𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅𝑀 )(29)

where 𝜅𝑀 is a parameter that defines the distribution of investor rebalancing cycles in market 𝑀.
On the other hand, we assume that 𝛿s, parameters that define such behavioral tendencies, do notdepend on the country (or market) 𝑀. This means that the function of risk attitude regime switchingvariable 𝐺(. ), the individual consumption function 𝑐𝑡 (. ), and the distribution function of individual
risk attitudes 𝑓(. ) do not depend on the country. Human behavior regarding loss avoidance andconsumption may be considered universal, but the distribution of investor risk attitudes may vary fromcountry to country. Note that this is a rather restrictive assumption.
3. Data
In Section 2, we have constructed an empirical model based on many theoretical arguments. Ourmodel requires stock market returns and market capitalization for each country, 𝑟𝑀𝑡 and 𝑤𝑡 ,respectively. We can use the weekly stock market indices of each country (market) as a proxy variable
for the former, but the market capitalization is not known from these indices. Thus, we use the Market
Capitalization Index (weekly) provided by Bloomberg as a proxy variable for stock market
capitalization. This index cannot be used as a proxy for market returns as the effects of ex-dividends
are not adjusted. Furthermore, the moving average value for the past one year is used in order not tohave an effect of ex-dividend on 𝑤𝑡 . Treasury bill rates as risk free rates are also from Bloomberg.

15

The sample period is from October 17, 2003 to June 26, 2020. In fact, we set a burn-in period of abouta year and a half when constructing 𝑧𝑀𝑡 . The main analysis period is from July 1, 2005 to June 26,
2020 (783 periods, 15 years).
We analyze 25 markets, including G20 countries and comparable large markets. S&P 500 (theUS) is set to reference market 𝑖 in eqs. (6) - (15) to prevent complication due to considering a huge
combination of various markets. The remaining 24 markets are set to market 𝑗 and divided into fourgroups: G7 ex the US (“G7”), other large developed markets (“LDM”), large emerging markets“LEM”, and the rest of the G20 emerging markets (“G20”).9 “G7” includes TSX (Canada), CAC 40(France), DAX (Germany), FTSE MIB (Italy), TOPIX (Japan), and FTSE 100 (UK). “LDM” includes
ASX (Australia), HSI (Hong Kong), AEX (Netherlands), IBEX (Spain), OMXS (Sweden), and SMI(Switzerland). “LEM” includes Bovespa (Brazil), SSE (China), SENSEX (India), RTS (Russia),
KOSPI (South Korea), and TWSE (Taiwan). “G20” includes Merval (Argentina), JCI (Indonesia),
Mexican Bolsa (Mexico), Tadawul (Saudi Arabia), JSE (South Africa), and XU 100 (Turkey).
Table 1 shows the summary statistics of weekly returns [%] on these large stock markets.10 The
mean of the US market return is 0.156%, almost all developed market returns are smaller than the US
and almost all emerging market returns are larger than the US. Markets with large average returns tend
to have large standard deviations. Most skewness values indicate that the distribution is negatively
skewed, and all kurtosis values indicate that the distribution is fat-tail. Correlation with the US tends
to be high in developed markets and low in emerging markets. Japan and Hong Kong are classified as
developed markets but have low correlation, and Brazil and Mexico are classified as emerging markets
but have high correlation. Intuitively, developed markets seem to be integrated with the US, andemerging markets are segmented from the US. Alternatively, it seems that American and European
markets are integrated, and Asian markets are segmented. This intuition may certainly be correct as
one aspect of describing stock market integration. However, this simple analysis does not distinguishbetween spurious correlations and intrinsic correlations based on market integration.
[Table 1]
4. Results
4.1. Setting and model selection
To get started, we will describe how to construct 𝑧𝑀𝑡 = 𝑧(𝜓𝑀𝑡 ; 𝜇𝑀 , 𝜅𝑀 ). The initial value isdefined as 𝜇𝑀 /𝜅𝑀 but we cannot calculate the exact value of 𝜇𝑀 . The average return for all sample
periods can be considered as a proxy variable for 𝜇𝑀 , but it is not desirable that the initial value be a
9

The classification of developed market and emerging market is based on the MSCI (Morgan Stanley CapitalInternational) classification.
10
This paper sets the US dollar as the numéraire. In other words, the return of each market includes the growth rateof each country’s currency.

16

function of future returns. Alternatively, it is possible to use the average of the period from the initialpoint to several years ago as a proxy variable, but this is not desirable because it wastes the sample
size. After all, the effect of the initial value almost disappears by setting a certain burn-in period whenconstructing 𝑧𝑀𝑡 . The mode of investor rebalancing cycle is expressed as 1/𝜅𝑀 . Thus, it is advisable
to set a burn-in period that greatly exceeds 1/𝜅𝑀 . We consider the following 9 patterns as candidates
for 𝜅𝑀 (the mode of rebalancing cycle); 𝜅𝑀 = 0.02 (about 1 year), 𝜅𝑀 = 0.025 (about 9 months),𝜅𝑀 = 0.0325 (about 7 months), 𝜅𝑀 = 0.0375 (about 6 months), 𝜅𝑀 = 0.045 (about 5
months), 𝜅𝑀 = 0.055 (about 4 months), 𝜅𝑀 = 0.075 (about 3 months), 𝜅𝑀 = 0.115 (about 2
months), and 𝜅𝑀 = 0.25 (about 1 months). Figure 1 shows these distributions. The data we haveavailable are the 872 weekly stock market indices and the market capitalization index from October
17, 2003 to June 26, 2020. By setting a burn-in period of 89 weeks, a 𝑧𝑀𝑡 of 783 weeks (15 years)
from July 1, 2005 to June 26, 2020 will be constructed. This greatly exceeds the rebalancing cycle
mode of 50 weeks in the candidate of 𝜅𝑀 = 0.02, and the effect of the initial value is almost negligible
in all candidates. Therefore, we set zero as the initial value 𝑧𝑀0 or 𝜇𝑀 .
Next, we analyze one-market model of the US market in order to get appropriate 𝛿s and 𝜅𝑖 .
Table 2 shows AICs of each polynomial model expressed as eq. (28). Each row shows the value of agiven 𝜅𝑖 . The AIC is the smallest in the cubic function model given 𝜅𝑖 = 0.0375. As expected, thelinear function is extremely poorly fitted. On the other hand, the quadratic function is also very poorly
fit, and it can be seen that the approximation of 𝛾(𝑧𝑀𝑡 ) ≈ 𝜆𝑀,𝑡+1|𝑡 (𝑧𝑀𝑡 ) requires an order of threeor more. Focusing on the column of the cubic function, it can be seen that AIC decreases as 𝜅𝑖approaches 0.0375. This implies that the most typical investors in the US have rebalancing cycles ofaround 6 months.
[Table 2]
Based on the above results, we will analyze the two-market ( 𝑖 = US and 𝑗 ), two-regime(integration and segmentation regime) model. Table 3 shows AICs of each country and each 𝜅𝑗 . It is
assumed that 𝜆𝑗,𝑡+1|𝑡 (𝑧𝑗𝑡 ) is represented by a cubic function of 𝑧𝑗𝑡 and parameters that determine
the distribution of rebalancing cycles for US investors, 𝜅𝑖 , is 0.0375 in all markets 𝑗 ∈{G7, LDM, LEM, G20}. The model with the smallest AIC is underlined. For example, Canada has
the lowest AIC of 6,295.499 when 𝜅𝑗 = 0.0375. This means that the most typical investors in the
Canadian market have rebalancing cycles of around 6 months, similar to the US. Unfortunately, giving
any 𝜅𝑗 does not give any results in Japan and the UK. It can be seen that the rebalancing cycle ofabout half of all (12 markets) is as long as or longer than that in the US. Markets with relatively slow
growth tend to have longer rebalancing cycles, but there are many exceptions.

17[Table 3]
4.2. Estimated parameters
Tables 4 shows the estimated parameters of the two-regime switching model in “G7”, “LDM”,“EDM” and “G20”. All parameters related to the GARCH structure (𝛼𝑖0 , 𝛼𝑗0 , 𝛼𝑖1 , 𝛼𝑗1 , 𝛽𝑖 and 𝛽𝑗 ) are
statistically significant in any markets. The estimates of 𝛼𝑖0 , 𝛼𝑖1 , and 𝛽𝑖 should be equal in all
markets since the market 𝑖 is fixed in the US market in any analysis of the market 𝑗. In almost all
countries, 𝛼𝑖0 ≈ 0.45, 𝛼𝑖1 ≈ 0.2 and 𝛽𝑖 ≈ 0.7. Therefore, it can be said that the GARCH structurein the US is well estimated. On the other hand, the estimates of 𝛼𝑗0 , 𝛼𝑗1 , and 𝛽𝑗 may be different
for each market.
The estimates of the constant conditional correlation with the US, 𝜌𝑖𝑗 , are generally high indeveloped markets and low in emerging markets. However, Australia and Hong Kong have low 𝜌𝑖𝑗
even in developed markets (0.508 and 0.589, respectively)., and Brazil, Mexico, Saudi Arabia andSouth Africa have high 𝜌𝑖𝑗 even in emerging markets (0.668, 0.752, 0.709 and 0.695, respectively).
(𝑖.𝑗)

The probabilities of staying the integration regime, 𝑝1|1 , are above 0.9 in all markets except Australia
and South Africa. It is indicated that once the market integrates with the US, its regime is extremely(𝑖.𝑗)sustainable. On the other hand, the probabilities of staying the segmentation regime, 𝑝0|0 , are lower(𝑖.𝑗)than 𝑝1|1 except Australia but these heights are polarized. France, Italy, Netherlands, Spain, South(𝑖.𝑗)

Korea, Taiwan, Argentina and Indonesia have extremely low 𝑝0|0 . This implies that the segmentation
(𝑖.𝑗)

(𝑖.𝑗)

regime of these markets hardly lasts. In markets with such high 𝑝1|1 and low 𝑝0|0 , the integrated(𝑖.𝑗)

regime will almost always apply. On the other hand, the 𝑝0|0 of other countries is relatively high,(𝑖.𝑗)and once it becomes a segmentation regime, it continues for a while. In markets with high 𝑝1|1 and(𝑖.𝑗)high 𝑝0|0 , the regime transition will be smooth. Using smoothed probabilities makes time-series ofregimes easier to understand. Details will be described later in Subsection 4.5.
The estimated values of 𝛿0 , 𝛿1 , 𝛿2 and 𝛿3 (and 𝛿4 ) all show significant values in almost all
markets except China, Russia and South Africa. It is likely that the structures of risk-pricing, 𝛾(𝑧) ≈𝜆(𝑧), have been successfully captured in these markets. It is common that 𝛿0 and 𝛿2 are positive and𝛿1 and 𝛿3 are negative while the coefficient values vary greatly from market to market. We will
analyze these parameters in more detail in the next subsection.
[Table 4]
4.3. Structure of risk-pricing.
In this subsection, we analyze whether the structures of risk-pricing, 𝛾(𝑧) ≈ 𝜆(𝑧), are equal ineach market. For that purpose, it is not enough to compare the values of the coefficients shown in theprevious subsection. What is important is not the parameters themselves estimated in market 𝑗,

18

(𝑗)
(𝑗)
(̂ 𝑗)
(𝑗)
̂ , 𝛿(𝑗)
̂
̂
̂
̂
̂ (𝑗)
(𝛿0(𝑗)
1 , 𝛿2 , 𝛿3 ) ≡ 𝜹 , but what value 𝜆(𝑧; 𝜹 ) ≡ 𝜆 (𝑧) takes when 𝑧 is given.

Figure 2 illustrates the shape of 𝜆̂(𝑗) (𝑧) in the domain [−0.33, 0.14]. This is because 99% of thestate of the economy in the US, 𝑧𝑖𝑡 , is included in this interval. It can be seen that the 𝜆̂(𝑗) (𝑧)sestimated in the G7 markets have very similar shape except Germany (and missing Japan and the UK).
In these markets, 𝜆̂(𝑗) (𝑧) is a decreasing function of 𝑧 less than 0.23, an increasing function of 𝑧
in the interval [−0.23, 0], and a decreasing function of 𝑧 above 0. These characteristics also apply
to almost all other markets except China and South Africa. From the results in the previous subsection,
̂ s are not statistically significant in China, Russia, and South Africa. This suggests
some or all of 𝛿(𝑗)that the distribution of investor rebalancing cycles in the stock markets of China and South Africa has
not been well approximated, or that the distribution of investor risk attitudes 𝑓(. ) differs significantly
from that of the US.
[Figure 2]
Next, we interpret the shape of 𝜆(𝑧) ≈ 𝛾(𝑧) based on the Chan and Kogan (2002) model (CK
model). Their model suggests that 𝛾(𝑧) is a decreasing function of 𝑧, and 𝛾(𝑧) > 1. These
conditions are satisfied from the local maximum point of 𝑧 ≈ 0 to the point of 𝑧 ≈ 0.1. The
positive state of the economy suggests that most recent market returns are positive since the state of
the economy is a cumulative weighted average of market returns. Therefore, when 𝑧 is in thisinterval [0, 0.1), the economy is likely to be upward but not overheated, that is a Goldilocks
economy. Since the CK model holds in this interval, 𝛾 is counter-cyclical and can be interpreted as
the relative risk aversion. The third column in Table 5, “Goldilocks”, shows the interval of 𝑧 suchthat 𝛾(𝑧) has such characteristics and the ratio of 𝑧𝑗𝑡 of each market included in the interval. Theinterval of Goldilocks, for example, in Canadian market is [−0.018, 0.085). The ratio of 𝑧𝑗𝑡
included in this interval is 43.0%. The average interval in developed markets (G7 & LDM) is[−0.001, 0.096) and the average ratio of 𝑧𝑗𝑡 in the Goldilocks interval is 32.8%. Similarly, theaverage interval in emerging markets (LEM & G20) is [0.007, 0.111) and the average ratio of 𝑧𝑗𝑡
in the Goldilocks interval is 23.5%. Therefore, the period for which the CK model holds is less thanhalf of the total even in developed markets.
We consider the case where 𝑧 is less than about 0. The price of variance risk 𝜆(𝑧) ≈ 𝛾(𝑧) isan increasing function of 𝑧 in the interval from the minimum point of 𝑧 ≈ −0.23 to the point of𝑧 ≈ 0. The CK model partially fails in this interval because the condition that 𝛾(𝑧) is a decreasing
function of 𝑧 is not satisfied. This result is as expected from the perspective of prospect theory. The
negative state of economy means that investors who perceive that the current stock price is lower
than the reference stock price are more influential. Consequently, the probability of a loss-averse
regime, Pr[𝐺𝑡 = 3], increases, and 𝛾(𝑧) becomes procyclical (see also 2.3.2 in this paper, for19detail). The second column in Table 5, “Recession”, shows the interval of 𝑧 such that 𝛾(𝑧) has
such characteristics. The ratio of 𝑧𝑗𝑡 included in this interval is, for example, 16.6% in Canada. The
average of developed markets is about 24.2%, and that of emerging markets is about 29.2%.
Interpreting 𝛾 as the mean-variance ratio 𝜆, the reason for this is thought to be that investors are
pessimistic and the excess expected returns will not be required to the level originally required, eventhough volatility increases with the economic downturn.
The above results are as expected, but in reality, unexpected results appeared at both ends.
When 𝑧 exceeds about 0.1, 𝛾(𝑧) has a negative value while maintaining the countercyclical
property. On the other hand, when 𝑧 is below about 0.23, 𝛾(𝑧) has countercyclical property again.
The former is associated with the overheated economy and the latter is associated with the crisis.
Figure 3 shows the state of the economy and these economic stages. Especially in the crisis interval,
most markets respond to the global financial crisis of 2008 and the Coronavirus crisis of 2020. The
first and forth columns in Table 5, “Crisis” and “Overheated”, show these intervals. The ratios of 𝑧𝑗𝑡
included in the crisis and overheated intervals are, for example, 4.0% and 36.4% in Canada,
respectively. The averages of developed markets are about 3.3% and 39.8%, and these of emerging
markets is about 6.4% and 40.9%. If 𝛾(𝑧) is interpreted as the mean-variance ratio or pseudoSharpe ratio, 𝛾(𝑧) below 0 means that the short-term expected excess return becomes negative due
to overheating of the economy.11 It is a so-called “overbought” market, and it is expected that shortterm selling will be expected. On the other hand, the interpretation of the crisis interval is very
difficult. Loss-averse investors will be screened out when a shock like a financial crisis occurs. As aresult, 𝛾 ≈ 𝜆 may become countercyclical.
[Table 5][Figure 3]
4.4. The accuracy of estimation resultŝ ) ≈ 𝜆̂(𝑗) (𝑧 ) ∀𝑡 must be the same
Due to the design of the model in this paper, 𝛾(𝑧US𝑡 ; 𝜹(𝑗)US𝑡in all markets. Note that 𝑖 is the United States. Using this property, it is analyzed whether or not thestructure of 𝛾 is the same in each market by comparing 𝜆̂(𝑗) (𝑧US𝑡 ) ∀𝑡 with respect to the market 𝑗.
In particular, the characteristics of investors in Canada are similar to those of the US, and the economic
ties between the two countries are deep. Table 5 compares the correlation and mean absolute distance11

In the CAPM, the negative expected excess return cannot exist under equilibrium. This is because investors are
implicitly constrained to have the same rebalancing cycle. When a negative excess return is expected, the price level
will be unbuyable. As a result, the price falls to the level where the expected excess return is positive. On the other
hand, this does not apply if the rebalancing cycle of investors is different. Even if the expected excess return forinvestors with short-term rebalancing cycle is negative, if the expected excess rate of return for investors with longerterm rebalancing cycle is positive, then the transaction is completed at that price. Therefore, the “short-term”
expected excess return can be negative.

20

between 𝜆̂(Canada) (𝑧US𝑡 ) and 𝜆̂(𝑗) (𝑧US𝑡 ) for 𝑗 ∈ {France, Germany, … , Turkey} and all 𝑡 .
“Group I” includes markets with correlations above 0.7, and the shape of 𝜆̂(𝑗) (𝑧US𝑡 ) in such markets
may be considered relatively close to that in Canadian market. “Group II” includes markets where thecorrelation is greater than 0.7, the mean absolute distance is less than 2.0, and all 𝛿s are significant at
the 5% level. All of the markets classified as Group II have a correlation of more than 0.95 with theCanadian market. It can be considered that the 𝜆̂(𝑗) (𝑧US𝑡 ) in such a market has a shape very close to
the 𝜆̂(Canada) (𝑧 ).
US𝑡

The most accurate group, Group II, includes 15 markets; Canada, France, Italy, Australia, Hong
Kong, Netherlands, Spain, Sweden, Switzerland, Brazil, South Korea, Taiwan, Indonesia, Mexico andSaudi Arabia. Results for the remaining 9 markets, Germany, Japan, the UK, China, India, Russia,
Argentina, South Africa and Turkey, need to be carefully interpreted. The distribution of investor risk
attitudes in these markets may differ significantly from that in the US.
[Table 6]
4.5. International integration measure – the smoothed probability of the integration regime
We analyze the existence of international integration, which is the subject of this paper. In theregime switching model, the smoothed probability of regime 𝑆 ∈ {0,1,2, … } defined as
Pr[𝜙𝑡 = 𝑆|𝜓𝑇 ] for 𝑡 ∈ {0,1, … , 𝑇 }, where 𝜙𝑡 denotes the switching variable and 𝜓𝑇 indicates theinformation available at the terminal time 𝑇 , represents the probability of realizing the regime at time
𝑡 from the posterior viewpoint at time 𝑇 . See also Hamilton (1988), Hamilton (1989), Hamilton(1990), and Kim and Nelson (1999) for the definition and detail. The smoothed probability ofinternational integration regime, Pr[𝜙𝑖𝑗𝑡 = 1|𝜓𝑊𝑖𝑗 𝑇 ] , is defined to be “international integration
measure” in this paper.
If the markets 𝑖 and 𝑗 are internationally integrated, a model using the return on the integrated
market 𝑊𝑖𝑗 as a pricing factor is likely to hold. In this case, the rational investors hold the factor
mimicking portfolio of the integrated market 𝑊𝑖𝑗 so that they can be interpreted as non-home-biased
investors. Therefore, a large international integration measure means that non-home-biased investors
have a greater influence on the market, or that the non-home-biased strategies such as international
diversification have an advantage over home-biased strategies such as domestic concentration at thattime.
Figure 4 shows the international integration measure in each country from 2003 to 2019 and thecolor coding based on each interval defined in Table 5. International integration measures in mostmarkets are very high. It is suggested that these markets are highly integrated with the US. On theother hand, international integration measures in Australia, China and South Africa tend to be low.
Restricting to markets classified as Group II suggests segmentation only in Australia and integration21

in the remaining 14 markets. In addition, the international integration measure tends to decrease justbefore the crisis in 2008 and increase during the crisis. This trend is noticeable in the following 12
markets; Canada, France, Germany, Italy, Hong Kong, Netherlands, Spain, Sweden, Brazil, South
Korea, Saudi Arabia and Turkey. Some similar trends can be seen in Switzerland, India, Russia, Taiwan
and Indonesia. This implies that the influence of home-biased investors was increasing just before thecrisis. That is, international diversification during this period was not an effective strategy. However,when a crisis occurred immediately after that, the positions of both were exchanged. Considering thistendency, it can be said that international diversification is a more effective strategy than domestic
concentration over the entire period. This is also the case when considering only Group II.
[Figure 4]
4.6. Are developed markets more integrated than emerging markets?
The first column of Table 7 shows the time series average of international integration measure ineach market. Similar to 4.5, it turns out to be a very high level except for Australia, China and South
Africa. Remaining 5 columns, “Diff” (difference), “Welch” (Welch test), “BM” (Brunner-Munzel test),“Ex Prob” (excess probability) and “Corr” (correlation) are calculated with Canadian market as thereference market. Six (two) developed markets showed significantly higher (lower) international
integration measure than Canada, and Hong Kong’s international integration measure is not
significantly different from Canada. Four (six) emerging markets showed significantly higher (lower)international integration measure than Canada, and international integration measure in Brazil andIndia is not significantly different from Canada. The results suggest that developed markets tend to beslightly more integrated with the US than emerging markets.
The Brunner-Munzel test (Brunner and Munzel, 2000) has a null hypothesis that the probability
that the randomly sampled international integration measure of the market will be greater than the
randomly sampled international integration measure of Canadian market is 0.5. “Ex Prob” shows suchexcess probability. From this perspective, the markets that are more integrated with the US than
Canada are Sweden, Argentina, Indonesia and Mexico. Randomly sampled international integration
measures in Germany, Brazil, South Korea and Saudi Arabia do not differ significantly from randomly
sampled international integration measures in Canada. The average excess probability for developed
markets is 0.366, and that for emerging markets is 0.415. Therefore, it is suggested that emerging
markets are more integrated with the US than developed markets. Little correlation with Canadianinternational integration measure is observed (highest correlation is 0.558 in Saudi Arabia).
Consequently, the dynamics of international integration measures in each market are largely
independent of each other, except for the financial crisis of 2008.
Table 8 compares international integration measures across developed markets and these across22

emerging markets. When compared in all markets, the Welch test shows that the developed market ismore integrated with the US than the emerging market, and the Brunner-Munzel test shows the
opposite result. This is almost the same as the result of Table 7. The results, however, will be different,
limiting to markets where the results are reliable as discussed in 4.4. In both Group I and Group II, theWelch and Brunner-Munzel tests suggested that emerging markets are more integrated with the US
than developed markets. It is expected that the distribution of investor risk attitudes in these markets
will be similar to the US. Therefore, we conclude that the traditional notion that developed markets
are integrated and emerging markets are segmented is false, at least in markets where the distributionof investor risk attitudes is similar to the US.
The US investors are encouraged to make international diversified portfolio in Group II markets
with high international integration measure if the results of this paper continue to hold. Such markets
are following 14 markets; Canada, France, Italy, Hong Kong, Netherlands, Spain, Sweden,
Switzerland, Brazil, South Korea, Taiwan, Indonesia, Mexico and Saudi Arabia. The average of mostrecent international integration measures in these 14 markets is 0.948, which makes international
diversification extremely advantageous. Furthermore, the average staying probability of theintegration regime is 0.968, and it can be said that this integration tendency is likely to continue in the
future. On the other hand, the average staying probability of the segmentation regime is 0.389, andeven if domestic concentration becomes advantageous, that situation will not last long.
[Table 8]
5. Conclusions
In this paper, we analyzed the degree of integration between large stock markets by using abivariate GARCH-in-Mean model with two regimes in which the integration regime and thesegmentation regime of the market are switched. The smoothed probability of the integration regime
was interpreted as an “international integration measure”. A high international integration measuremeans that non-home-biased strategies such as international diversification have advantages overhome-biased strategies such as domestic concentration. In addition, the model was estimated by paying
attention to the characteristic of the price of variance risk (or, relative risk aversion) suggested by the
Chan and Kogan (2002) model and prospect theory (Kahneman and Tversky, 1979; Tversky and
Kahneman, 1992).
The main results of the paper are reported below. First, we found that the structures ofrepresentative investor’s risk attitude, or that of the price of variance risk, in each of the following 15
markets are almost the same; Canada, France, Italy, Australia, Hong Kong, Netherlands, Spain,
Sweden, Switzerland, Brazil, South Korea, Taiwan, Indonesia, Mexico and Saudi Arabia. Second, theinternational integration measure in these markets was on average high (except Australia), declining

23

before the 2008 global financial crisis (except Australia and Mexico), but rising again after the crisis(except Australia). This means that non-home-biased strategies have advantages over home-biased
strategies except just before the crisis. Third, the difference between the international integration
measures of developed and emerging markets was extremely small, both intuitively and statistically.
In other words, being an emerging market does not mean that the market is segmented. Summing up
the results, international diversification is strongly recommended in these 15 markets (except
Australia) regardless of market or period.
Note that the effect of exchange rates is not fully considered in this paper. This is because thispaper uses the US dollar as the numéraire and the US treasury-bill as the risk-free rate. Consideringthat the home bias is affected by the volatility of the real exchange rate as the Fidora, Flatzscher andThimann (2007) pointed out, the analysis in this paper can be improved. One way to deal with thisproblem is to use a conditional two-factor model with market and exchange factors as proposed byBekaert and Harvey (1995). Alternatively, a new three regime switching model may solve this problem.
In this model, three regimes are switched: a segmentation regime, a first-country dominant integration
regime, and a second-country dominant integration regime. In the segmentation regime, the excess
return of each country’s market is denominated in that country’s currency, calculated at that country’s
risk-free rate, and formulated based on its volatility. In the dominant integration regime, the excess
return of the country’s market is denominated in that country’s currency, calculated at that country’s
risk-free rate, and formulated based on the covariance between the market return of the country andthe market return of the other country. The ratio of the probabilities of the first-country and secondcountry dominant integration regime represents the ratio of stock purchasing power of both countries.
Therefore, this is expected to be close to the ratio of the market capitalization of them. If these aredifferent, we can measure the under and over foreign investment between the two countries.

24Appendix
We extend the Chan and Kogan (2002) model (CK model) and consider the case where theindividual’s risk attitude 𝛾 𝑖𝑛𝑑 is included in the real-valued set excluding 0 and 1. That is, we proveeqs. (26) and (27). Eq. (25) does not need to be proved because it is the same setting as the original
CK model.
To get started, we consider the following utility function of representative investor for any
𝑎, 𝑏 ∈ ℝ such that 𝛾 𝑖𝑛𝑑 ∈ (𝑎, 𝑏) satisfies 𝛾 𝑖𝑛𝑑 ∈ ℝ ∖ {0,1}.
𝑏

𝑈(𝑌𝑡 , 𝑋𝑡 ; 𝑎, 𝑏) = sup {∫ 𝑓(𝛾 𝑖𝑛𝑑 )
{𝑐𝑡 }

𝑎

1
𝑐 1−𝛾
( 𝑡)
𝑖𝑛𝑑
1−𝛾
𝑋𝑡

𝑖𝑛𝑑

where 𝑐𝑡 = 𝑐𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾𝑏d𝛾 𝑖𝑛𝑑s. t.

∫ 𝑐𝑡 d𝛾 𝑖𝑛𝑑 ≤ 𝑌𝑡 }
𝑎

𝑖𝑛𝑑

).

Chan and Kogan show that the optimal consumption ratio, 𝛼𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾) = 𝑐𝑡 (𝑌𝑡 , 𝑋𝑡 ; 𝛾)/𝑌𝑡 , isrepresented by:𝑏11
𝛼𝑡 (𝑧𝑡 ; 𝛾) = 𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } where ∫ 𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 = 1𝛾𝛾𝑎

The following equation is obtained by differentiating both sides of above equation by 𝑧𝑡 andrearranging.
−1

𝑏

1
1
𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 )𝛾𝛾𝑎ℎ′ (𝑧𝑡 ; 𝑎, 𝑏) = − (∫≡ −𝐴(𝑧𝑡 ; 𝑎, 𝑏)−1(A1)They also prove the following equation.
−ℎ′ (𝑧𝑡 ; 𝑎, 𝑏) = −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 ; 𝑎, 𝑏)≡ 𝛾(𝑧𝑡 ; 𝑎, 𝑏)
𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 ; 𝑎, 𝑏)

(A2)

𝑏

Let 𝐵(𝑧𝑡 ; 𝑎, 𝑏) ≡ ∫𝑎 𝛾12 𝑓(𝛾)1/𝛾 exp{− 𝛾1 ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 . The following inequality holds from the
Cauchy-Schwartz inequality.
𝐵(𝑧𝑡 ; 𝑎, 𝑏) ≥ 𝐴(𝑧𝑡 ; 𝑎, 𝑏)2(A3)

From eq. (A1),ℎ′ (𝑧𝑡 ; 𝑎, 𝑏) = −𝐴(𝑧𝑡 ; 𝑎, 𝑏)−1 ⇔ 𝐴(𝑧𝑡 ; 𝑎, 𝑏)ℎ′ (𝑧𝑡 ; 𝑎, 𝑏) = −1
The following equation is obtained by differentiating both sides of above equation by 𝑧𝑡 andrearranging.
𝐴(𝑧𝑡 ; 𝑎, 𝑏)ℎ′′ (𝑧𝑡 ; 𝑎, 𝑏) = 𝐴(𝑧𝑡 ; 𝑎, 𝑏)ℎ′ (𝑧𝑡 ; 𝑎, 𝑏) + 𝐵(𝑧𝑡 ; 𝑎, 𝑏){ℎ′ (𝑧𝑡 ; 𝑎, 𝑏)}2
Equations and inequation (A1) - (A4) is not affected by the integral interval of 𝛾

𝑖𝑛𝑑

(A4)

, that is, (𝑎, 𝑏).

Next, we consider the case of 𝛾 𝑖𝑛𝑑 ∈ (0,1). In this case, from eq. (A1),1

𝐴(𝑧𝑡 ; 0,1) = ∫
0

1
1
1
1
𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 > ∫ 𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 = 1𝛾𝛾𝛾0⇔ 0 < 𝐴(𝑧𝑡 ; 0,1)−1 < 1⇔ 0 < −(−𝐴(𝑧𝑡 ; 0,1)−1 ) < 1∴ 0 < −ℎ′ (𝑧𝑡 ; 0,1) < 1
From eq. (A2),250 < 𝛾(𝑧𝑡 ; 0,1) < 1(A5)

From inequation (A3),𝐵(𝑧𝑡 ; 0,1) ≥ 𝐴(𝑧𝑡 ; 0,1)2⇔ −𝐵(𝑧𝑡 ; 0,1) ≤ −𝐴(𝑧𝑡 ; 0,1)2⇔ −𝐵(𝑧𝑡 ; 0,1){−𝐴(𝑧𝑡 ; 0,1)}−1 ≥ 𝐴(𝑧𝑡 ; 0,1)

∵ −𝐴(𝑧𝑡 ; 0,1) < 0′⇔ −𝐵(𝑧𝑡 ; 0,1)ℎ (𝑧𝑡 ; 0,1) ≥ 𝐴(𝑧𝑡 ; 0,1)
⇔ 𝐴(𝑧𝑡 ; 0,1) + 𝐵(𝑧𝑡 ; 0,1)ℎ′ (𝑧𝑡 ; 0,1) ≤ 0⇔ 𝐴(𝑧𝑡 ; 0,1)ℎ′ (𝑧𝑡 ; 0,1) + 𝐵(𝑧𝑡 ; 0,1){ℎ′ (𝑧𝑡 ; 0,1)}2 ≥ 0 ∵⇔ 𝐴(𝑧𝑡 ; 0,1)ℎ′′ (𝑧𝑡 ; 0,1) ≥ 0 ∵′′⇔ ℎ (𝑧𝑡 ; 0,1) ≥ 0 ∵ℎ′ (𝑧𝑡 ; 0,1) < 0

eq. (A4)

𝐴(𝑧𝑡 ; 0,1) > 1∴ ℎ′′ (𝑧𝑡 ; 0,1) ≥ 0 (ℎ is convex. )Finally, we consider the case of 𝛾

𝑖𝑛𝑑

(A6)

∈ (−∞, 0). In this case, from eq. (A1),011
𝑓(𝛾)1/𝛾 exp {− ℎ(𝑧𝑡 ) − 𝑧𝑡 } d𝛾 < 0𝛾𝛾−∞𝐴(𝑧𝑡 ; −∞, 0) = ∫⇔ 𝐴(𝑧𝑡 ; −∞, 0)−1 < 0⇔ −(−𝐴(𝑧𝑡 ; −∞, 0)−1 ) < 0−ℎ′ (𝑧𝑡 ; −∞, 0) < 0∴
From eq. (A2),∴𝛾(𝑧𝑡 ; −∞, 0) < 0(A7)

From inequation (A3),𝐵(𝑧𝑡 ; −∞, 0) ≥ 𝐴(𝑧𝑡 ; −∞, 0)2⇔ −𝐵(𝑧𝑡 ; −∞, 0) ≤ −𝐴(𝑧𝑡 ; −∞, 0)2⇔ −𝐵(𝑧𝑡 ; −∞, 0){−𝐴(𝑧𝑡 ; −∞, 0)}−1 ≤ 𝐴(𝑧𝑡 ; −∞, 0)∵ −𝐴(𝑧𝑡 ; −∞, 0) > 0⇔ −𝐵(𝑧𝑡 ; −∞, 0)ℎ′ (𝑧𝑡 ; −∞, 0) ≤ 𝐴(𝑧𝑡 ; −∞, 0)⇔ 𝐴(𝑧𝑡 ; −∞, 0) + 𝐵(𝑧𝑡 ; −∞, 0)ℎ′ (𝑧𝑡 ; −∞, 0) ≥ 0⇔ 𝐴(𝑧𝑡 ; −∞, 0)ℎ′ (𝑧𝑡 ; −∞, 0) + 𝐵(𝑧𝑡 ; −∞, 0){ℎ′ (𝑧𝑡 ; −∞, 0)}2 ≥ 0 ∵⇔ 𝐴(𝑧𝑡 ; −∞, 0)ℎ′′ (𝑧𝑡 ; −∞, 0) ≥ 0 ∵⇔ ℎ′′ (𝑧𝑡 ; −∞, 0) ≤ 0 ∵ℎ′ (𝑧𝑡 ; −∞, 0) > 0

eq. (A4)

𝐴(𝑧𝑡 ; −∞, 0) < 0∴ ℎ′′ (𝑧𝑡 ; −∞, 0) ≤ 0 (ℎ is concave. )(A8)

From inequations (A5) and (A6),(2)

𝛾 (2) (𝑧𝑌𝑡 ) ≡ −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )(2)
𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )∈ (0,1),d𝛾 (2) (𝑧𝑌𝑡 )
<0
d𝑧𝑌𝑡

(26)

From inequations (A7) and (A8),(3)

𝛾 (3) (𝑧𝑌𝑡 ) ≡ −𝑌𝑡 𝑈𝑌𝑌 (𝑌𝑡 , 𝑋𝑡 )(3)

𝑈𝑌 (𝑌𝑡 , 𝑋𝑡 )∈ (−∞, 0),d𝛾 (3) (𝑧𝑌𝑡 )
>0
d𝑧𝑌𝑡

(27)
Q.E.D.

26

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28

Table 1

Summary statistics
Mean

SD

2
2.418

Skew

3
-0.754Kurt

4
11.529Min

5-18.195

Med

7
0.238

Max

9
12.102

Corr
10
1.000

Ref

US

1
0.156

G7

Canada

0.109

2.307

-1.15813.359-16.089

0.327

13.675

0.816

France

0.091

2.892

-1.047

10.823-22.159

0.282

13.2380.809

Germany

0.194

3.003

-0.812

10.537-21.6100.423

16.116

0.809

Italy

0.023

3.231

-1.1429.490-23.2960.304

11.040

0.727

Japan

0.090

2.810

-0.7557.895-19.7630.347

13.737

0.608

UK

0.074

2.400

-1.15315.506-21.0470.210

13.411

0.823

Australia

0.096

2.210

-1.091

9.182-15.649

0.300

9.542

0.682

Hong Kong

0.131

2.885

-0.1345.744-16.3190.313

12.433

0.588

Netherlands

0.107

2.813

-1.30713.715-24.9900.263

13.287

0.798

Spain

0.056

3.117

-0.9108.736-21.201

0.362

11.740

0.723

Sweden

0.159

2.761

-0.821

8.980-20.1710.324

13.0600.766

Switzerland

0.106

2.394

-1.39517.413-22.2770.314

14.071

0.734

Brazil

0.269

3.600

-0.2277.103-20.0100.465

18.345

0.678

China

0.145

3.344

-0.0575.389-13.8410.118

14.964

0.174

India

0.282

2.956

-0.2736.440-15.9540.424

14.0780.540

Russia

0.206

4.762

0.454

14.787-21.1670.367

46.0400.529

South Korea

0.167

2.779

-0.67010.417-20.4900.356

18.568

0.613

Taiwan

0.115

2.514

-0.672

5.359-10.650

0.262

9.867

0.540

Argentina

0.569

4.855

-0.611

7.495-31.4470.702

19.904

0.525

Indonesia

0.283

2.891

-0.9369.091-20.7820.418

12.2850.437

Mexico

0.217

2.721

0.098

10.080-16.4130.273

20.416

0.723

Saudi Arabia

0.126

3.550

-1.0718.996-22.0410.454

16.8080.356South Africa

0.242

2.750

0.037

8.697

-15.577

0.315

19.6270.648

Turkey

0.310

3.659

-0.261

5.014-17.530

0.535

17.067

0.482

LDM

LEM

G20

Note: Results are based on market index returns [%, weekly] from 17 Oct 2003 to 26 Jun 2020 (872 weeks). The reference
market (Ref) is the US stock market (S&P 500). Analyzed markets are the G7 markets, large developed markets (LDM),
large emerging markets (LEM), and the rest of the G20 markets. The statistics of “Corr” means the correlation with the US
stock returns.

Table 2

AIC of each polynomial model (US / domestic)κ = 0.0200

LinearQuadratic

Cubic

Quartic

1

1
3,326.589

23,328.525

33,323.229

43,325.165

23,326.528

3,328.409

3,322.157

3,324.531

33,326.481

3,328.230(about 1 year)κ = 0.0250(about 9 months)

κ = 0.0325

3,323.733(about 7 months)

κ = 0.0375

4

3,328.111

3,321.535

53,327.950

3,321.932

3,323.981

3,322.537

3,324.502(about 6 months)

κ = 0.0450(about 5 months)

κ = 0.0550

63,327.703

3,327.806

73,326.692

3,327.792

8

3,326.957

3,328.313

93,327.166

3,329.159(about 4 months)

κ = 0.0750(about 3 months)

κ = 0.1150

3,326.366(about 2 months)

κ = 0.2500(about 1 month)

3,328.546

3,330.176

Australia
Hong Kong
Netherlands
Spain
SwedenSwitzerland

BrazilChina
India
RussiaSouth Korea
TaiwanArgentina
Indonesia
MexicoSaudi Arabia
South Africa
Turkey

LDM

LEM

G20

6,927.712
7,146.632
7,121.067
7,801.027

7,486.310
7,044.303

7,869.453
7,204.589
6,927.886
7,145.421
7,120.667
7,801.286

7,614.482
7,303.616
7,098.556
7,487.413
7,044.143
6,825.465

6,745.517
6,346.408

6,745.011
6,348.646

7,613.441
7,303.463

6,643.390
6,860.686

6,644.013
6,860.696

6,815.635
6,966.166

κ = 0.0250
26,296.211

Note: The model with the smallest AIC is underlined.

CanadaFranceGermany
Italy
Japan
UK

G7

κ = 0.0200
1
6,296.305
6,684.174
6,816.012
6,960.292

Table 3

7,844.306
7,215.750
6,927.854
7,143.814
7,120.277
7,801.276

7,612.065
7,299.233
7,107.781
7,488.641
7,047.053
6,826.714

6,745.927
6,346.628

6,643.047
6,862.529

κ = 0.0325
36,295.755
6,683.271
6,815.113
6,965.369

6,927.722
7,143.938
7,118.987
7,801.252

7,870.200

7,144.551
7,118.167

7,856.265
7,241.561

7,613.419
7,296.479
7,107.320
7,489.368

6,745.815
6,347.465

6,873.462
6,745.932
6,346.998
7,611.750
7,302.304
7,107.391
7,490.148

6,642.198
6,862.882

6,814.054

κ = 0.0450
56,295.688

6,642.663
6,862.749

6,814.815

κ = 0.0375
46,295.499

Model selection (AIC)7,859.302
7,242.162
6,927.629
7,144.711
7,116.192
7,798.961

7,614.543
7,299.396
7,107.496
7,488.832
7,051.933

6,745.599
6,348.277

6,711.177

6,642.001

κ = 0.0550
66,295.755
6,691.188
6,813.807

6,927.464
7,145.311
7,114.409
7,790.890

7,865.195

7,487.868

7,617.384
7,297.703

6,745.387
6,349.208

6,641.615

6,812.545

κ = 0.0750
76,296.293

7,146.060
7,113.539
7,788.905

7,146.724
7,112.838
7,789.149

7,236.398

6,828.500

6,827.905

7,242.632

7,621.737
7,293.411
7,106.833
7,483.609

6,744.272
6,350.405

6,642.111
6,860.331

6,960.374

κ = 0.2500
96,296.851

7,619.706
7,297.520
7,108.042
7,488.820

6,744.936
6,350.028

6,642.148

6,809.704

κ = 0.1150
86,296.792

Table 4

Estimated parameters (G7)Canada1
0.490 ***
(0.110)France20.418 ***
(0.093)Germany30.440 ***
(0.095)Italy
40.396 ***
(0.091)𝛼𝑗00.406 ***
(0.098)0.422 **
(0.138)0.592 **
(0.190)0.369 *
(0.146)𝛼𝑖1

0.182 ***
(0.028)0.186 ***
(0.030)0.195 ***
(0.031)0.189 ***
(0.030)𝛼𝑗10.163 ***
(0.024)0.100 ***
(0.019)0.088 ***
(0.019)0.096 ***
(0.019)𝛽𝑖

0.714 ***
(0.041)0.726 ***
(0.039)0.715 ***
(0.039)0.729 ***
(0.039)𝛽𝑗

0.789 ***
(0.028)0.855 ***
(0.029)0.853 ***
(0.032)0.872 ***
(0.026)

𝜌𝑖𝑗

0.810 ***
(0.016)0.712 ***
(0.020)0.706 ***
(0.021)0.645 ***
(0.024)𝑝1|10.983 ***
(0.013)0.972 ***
(0.022)0.990 ***
(0.007)0.974 ***
(0.022)𝑝0|00.840 ***
(0.131)0.001 (0.019)

0.871 ***
(0.080)0.002 (0.087)

𝛿0961 **(322)

1,550 ***
(341)

1,093 **
(352)

1,095 ***
(305)

𝛿1

-3,430 **(1,123)-5,374 ***(1,173)-3,686 **(1,176)-3,863 ***(1,063)

𝛿2

4,025 **
(1,283)

6,124 ***
(1,326)

4,078 **
(1,289)4,475 ***
(1,215)

𝛿3-1,547 **(481)-2,291 ***(495)-1,479 **(466)-1,699 ***
(457)

𝛼𝑖0

Japan
5

UK
6

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 0.1, 1 and 5% levels,
respectively.

Table 4

Estimated parameters (LDM)
Australia
70.369 ***
(0.092)

Hong Kong
8
0.481 ***
(0.107)

Netherlands
90.420 ***
(0.094)Spain
10
0.393 ***
(0.088)

Sweden11
0.487 ***
(0.103)

Switzerland
120.459 ***
(0.100)𝛼𝑗00.191 *
(0.084)0.325 **
(0.126)

0.539 **
(0.181)

0.422 *
(0.167)0.584 **
(0.188)0.549 ***
(0.167)𝛼𝑖1

0.244 ***
(0.037)0.219 ***
(0.034)0.195 ***
(0.032)0.196 ***
(0.032)0.171 ***
(0.027)0.207 ***
(0.031)𝛼𝑗10.059 ***
(0.015)0.074 ***
(0.020)0.129 ***
(0.028)0.106 ***
(0.023)0.100 ***
(0.020)0.141 ***
(0.031)𝛽𝑖

0.695 ***
(0.042)0.684 ***
(0.042)0.718 ***
(0.040)0.725 ***
(0.038)0.720 ***
(0.038)0.716 ***
(0.038)𝛽𝑗

0.897 ***
(0.028)0.884 ***
(0.031)0.810 ***
(0.043)0.854 ***
(0.034)0.847 ***
(0.033)0.780 ***
(0.048)𝜌𝑖𝑗

0.508 ***
(0.117)0.589 ***
(0.046)

0.688 ***
(0.025)0.679 ***
(0.029)0.760 ***
(0.016)0.800 ***
(0.020)𝑝1|10.716 ***
(0.150)0.953 ***
(0.055)0.950 ***
(0.037)0.923 ***
(0.050)0.992 ***
(0.006)0.930 ***
(0.026)𝑝0|00.735 ***
(0.108)0.481 (0.343)

0.002 (0.026)

0.000 (0.000)

0.823 ***
(0.136)

0.569 ***
(0.129)𝛿01,051 ***
(261)

1,850 ***
(344)

1,076 **
(391)

1,194 *
(526)

1,297 *
(550)

1,037 ***
(306)

𝛿1

-3,649 ***(901)-6,462 ***(1,182)-3,782 **(1,363)-4,142 *(1,819)-4,591 *(1,905)-3,636 ***(1,053)

𝛿2

4,162 ***
(1,022)7,431 ***
(1,340)4,368 **
(1,559)4,719 *
(2,062)

5,339 *
(2,162)

4,184 ***
(1,190)𝛿3-1,557 ***(383)-2,809 ***(503)-1,653 **(587)-1,764 *(768)-2,036 *(806)-1,578 ***
(442)

𝛼𝑖0

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 0.1, 1 and 5% levels,
respectively.

Table 4

Estimated parameters (LEM)
Brazil130.493 ***
(0.112)China
140.426 ***
(0.100)India
15
0.425 ***
(0.096)

Russia16
0.414 ***
(0.098)South Korea
17
0.427 ***
(0.100)

Taiwan180.448 ***
(0.105)𝛼𝑗01.735 **
(0.559)0.292 *
(0.116)

0.417 **
(0.137)0.930 ***
(0.258)1.046 ***
(0.273)0.227 **
(0.079)𝛼𝑖10.188 ***
(0.030)0.264 ***
(0.040)0.227 ***
(0.035)0.219 ***
(0.034)0.186 ***
(0.030)0.213 ***
(0.034)

𝛼𝑗1

0.099 ***
(0.018)0.105 ***
(0.025)0.116 ***
(0.022)0.134 ***
(0.022)0.149 ***
(0.027)0.078 ***
(0.017)𝛽𝑖

0.709 ***
(0.042)0.669 ***
(0.043)0.694 ***
(0.040)0.705 ***
(0.041)0.723 ***
(0.040)0.697 ***
(0.043)𝛽𝑗

0.824 ***
(0.035)0.868 ***
(0.029)0.841 ***
(0.030)0.816 ***
(0.028)0.744 ***
(0.045)0.891 ***
(0.023)

𝜌𝑖𝑗

0.668 ***
(0.025)0.390 ***
(0.048)0.562 ***
(0.036)

0.621 ***
(0.036)

0.569 ***
(0.025)0.516 ***
(0.032)𝑝1|10.981 ***
(0.008)0.995 ***
(0.004)0.965 ***
(0.018)0.940 ***
(0.031)0.979 ***
(0.010)0.952 ***
(0.038)𝑝0|00.841 ***
(0.075)0.994 ***
(0.005)0.613 **
(0.194)0.702 ***
(0.142)0.011 (0.077)

0.000 (0.009)

𝛿0813 **
(264)

539 *
(273)

721 ***(198)

643 (333)

1,245 ***
(254)

1,270 ***
(284)

𝛿1

-2,933 **(931)-1,773 (931)-2,536 ***(668)-2,280 (1,193)-4,435 ***(878)-4,497 ***
(982)

𝛿2

3,477 **
(1,070)1,904 (1,036)

2,926 ***
(737)

2,658 (1,398)

5,184 ***
(993)

5,222 ***
(1,115)

𝛿3-1,348 ***(403)-666 (379)-1,103 ***(268)-1,014 (537)-1,984 ***(369)-1,985 ***
(416)

𝛼𝑖0

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 0.1, 1 and 5% levels,
respectively.

Table 4

Estimated parameters (G20)
Argentina
24
0.427 ***
(0.102)Indonesia
19
0.461 ***
(0.108)Mexico25
0.472 ***
(0.102)

Saudi Arabia
26
0.479 ***
(0.106)South Africa
27
0.365 ***
(0.088)Turkey28
0.441 ***
(0.100)𝛼𝑗00.703 **
(0.225)0.779 **
(0.267)1.237 ***
(0.323)

1.073 **
(0.341)0.311 ***
(0.079)3.346 *
(1.698)𝛼𝑖10.179 ***
(0.029)0.230 ***
(0.036)

0.166 ***
(0.026)

0.198 ***
(0.030)0.226 ***
(0.036)0.231 ***
(0.037)

𝛼𝑗1

0.124 ***
(0.021)0.186 ***
(0.036)0.144 ***
(0.024)0.110 ***
(0.019)0.203 ***
(0.026)0.100 ***
(0.029)𝛽𝑖

0.731 ***
(0.039)0.682 ***
(0.044)0.729 ***
(0.037)0.700 ***
(0.040)0.712 ***
(0.040)0.691 ***
(0.041)𝛽𝑗

0.859 ***
(0.022)0.746 ***
(0.051)0.758 ***
(0.040)0.814 ***
(0.036)

0.785 ***
(0.025)0.753 ***
(0.095)𝜌𝑖𝑗

0.549 ***
(0.028)0.412 ***
(0.032)0.752 ***
(0.017)0.709 ***
(0.026)

0.695 ***
(0.046)

0.565 ***
(0.038)𝑝1|10.988 ***
(0.005)0.982 ***
(0.008)0.995 ***
(0.004)0.987 ***
(0.013)0.345 *
(0.137)0.983 ***
(0.013)𝑝0|00.007 (0.050)

0.000 (0.000)

0.968 ***
(0.030)0.912 ***
(0.082)0.391 **
(0.145)0.919 ***
(0.059)𝛿0

1,818 ***
(332)

945 ***(209)

1,110 **
(350)

1,008 **
(326)

693 (488)

971 *
(425)

𝛿1

-6,513 ***(1,150)-3,368 ***(702)-3,857 **(1,212)-3,541 **(1,133)-2,328 (1,698)-3,378 *(1,478)𝛿2

7,656 ***
(1,302)

3,937 ***
(771)

4,405 **
(1,377)

4,092 **
(1,290)2,564 (1,936)

3,848 *
(1,684)

𝛿3-2,950 ***(485)-1,505 ***(278)-1,650 **(515)-1,550 **(482)-925 (725)-1,434 *
(629)

𝛼𝑖0

Note: Values in parentheses denote standard errors. ***,** and * denote statistical significance at 0.1, 1 and 5% levels,
respectively.

Table 5

Inflection points and four economic stages (G7 & LDM)

G7

0

CrisisRecession

Goldilocks

Overheated(countercyclical)(procyclical)

12[-0.248, -0.018)
16.6%(counter & γ > 1)2
3[-0.018, 0.085)
43.0%(counter & γ ≤ 1)[-0.219, 0.000)
20.7%[0.000, 0.090)
34.6%[0.090, 0.394]
40.5%[-0.200, 0.040)
23.5%[0.040, 0.140)
34.1%[0.140, 0.475]
40.6%[-0.237, -0.008)
31.3%[-0.008, 0.093)
29.9%[0.093, 0.396]
31.8%[-0.252, -0.221)
0.3%[-0.221, 0.003)
33.7%[0.003, 0.095)
24.5%[0.095, 0.277]
41.5%[-0.466, -0.221)
2.8%[-0.221, -0.013)
17.8%[-0.013, 0.080)
31.8%[0.080, 0.479]
47.6%[-0.463, -0.235)
4.3%[-0.235, -0.005)
15.5%[-0.005, 0.098)
41.3%[0.098, 0.424]
39.0%[-0.327, -0.221)
2.7%[-0.221, 0.003)
36.9%[0.003, 0.095)
28.6%[0.095, 0.355]
31.8%[-0.677, -0.237)
3.4%[-0.237, -0.015)
24.4%[-0.015, 0.085)
25.3%[0.085, 0.337]
46.9%[-0.422, -0.232)
2.2%[-0.232, 0.000)
21.3%[0.000, 0.098)
35.0%[0.098, 0.236]
41.5%[-0.441, -0.227)
3.3%[-0.227, -0.001)
24.2%[-0.001, 0.096)
32.8%[0.096, 0.369)
39.8%

4
1
[-0.569, -0.248)
4.0%

Canada4 (κ = 0.0375)
0.0375
France[-0.342, -0.219)
4.2%
3 (κ = 0.0325)
0.0325Germany[-0.292, -0.200)
1.8%
8 (κ = 0.1150)
0.115
Italy
[-0.596, -0.237)
7.0%
1 (κ = 0.0200)
0.02
Japan

3
5
[0.085, 0.317]
36.4%

UK

LDM

Australia
7 (κ = 0.0750)
0.075
Hong Kong
9 (κ = 0.2500)
0.25
Netherlands
6 (κ = 0.0550)
0.055
Spain
4 (κ = 0.0375)
0.0375
Sweden9 (κ = 0.2500)
0.25
Switzerland
2 (κ = 0.0250)
0.025

Average G7 & LDM(κ = 0.0898)

Table 5

Inflection points and four economic stages (LEM & G20)

LEM

Brazil4 (κ = 0.0375)
0.0375China
9 (κ = 0.2500)
0.25
India
2 (κ = 0.0250)
0.025
Russia9 (κ = 0.2500)
0.25
South Korea
2 (κ = 0.0250)
0.025
Taiwan2 (κ = 0.0250)

CrisisRecession

Goldilocks

Overheated(countercyclical)(procyclical)(counter & γ > 1)

(counter & γ ≤ 1)[-0.672, -0.258)
10.0%[-0.258, -0.023)
24.4%[-0.023, 0.090)
10.1%[0.090, 0.604]
55.6%[-0.413, -0.190)
3.2%[-0.190, 0.095)
64.4%[0.095, 0.209)
12.4%[0.209, 0.906]
20.1%[-0.601, -0.242)
3.6%[-0.242, 0.011)
23.2%[0.011, 0.130)
30.3%[0.130, 0.585]
42.9%[-1.158, -0.245)
10.2%[-0.245, -0.008)
26.9%[-0.008, 0.103)
22.2%[0.103, 0.672]
40.6%[-0.810, -0.245)
3.4%[-0.245, -0.013)
23.1%[-0.013, 0.087)
27.6%[0.087, 0.445]
45.8%[-0.561, -0.240)
3.4%[-0.240, -0.008)
16.7%[-0.008, 0.093)
40.0%[0.093, 0.291]
39.8%[-0.734, -0.245)
10.0%[-0.245, -0.023)
16.5%[-0.023, 0.066)
9.6%[0.066, 0.670]
64.0%[-0.750, -0.248)
5.0%[-0.248, -0.008)
15.7%[-0.008, 0.101)
30.5%[0.101, 0.508]
48.8%[-0.614, -0.224)
4.9%[-0.224, 0.003)
32.7%[0.003, 0.101)
20.8%[0.101, 0.463]
41.6%[-0.737, -0.232)
5.2%[-0.232, -0.008)
25.2%[-0.008, 0.095)
30.1%[0.095, 0.432]
39.5%[-0.597, -0.200)
7.8%[-0.200, 0.048)
46.1%[0.048, 0.148)
30.7%[0.148, 0.675]
15.5%[-0.751, -0.227)
10.7%[-0.227, 0.016)
35.0%[0.016, 0.109)
17.6%[0.109, 0.605]
36.7%[-0.700, -0.233)
6.4%[-0.233, 0.007)
29.2%[0.007, 0.111)
23.5%[0.111, 0.571)
40.9%

0.025
G20

Argentina
3 (κ = 0.0325)
0.0325Indonesia
2 (κ = 0.0250)
0.025
Mexico7 (κ = 0.0750)
0.075
Saudi Arabia
3 (κ = 0.0325)
0.0325South Africa
9 (κ = 0.2500)
0.25
Turkey8 (κ = 0.1150)
0.115

Average LEM & G20(κ = 0.0952)

Table 6

Accuracy of estimated price of risk

G7

Correlation
Canada1
1.000

Abs. Distance20.000

Significance3100%Group I
5x

Group II
6
x

France

0.961

1.111

100%

x

x

Germany

0.489

4.035

100%

Italy

0.991

1.002

100%

x

x

Australia

0.951

1.311

100%

x

x

Hong Kong

0.985

1.944

100%

x

x

Netherlands

0.986

1.362

100%

x

x

Spain

0.950

1.235

100%

x

x

Sweden

0.998

0.754

100%

x

x

Switzerland

0.973

1.244

100%

x

x

Brazil

0.998

0.850

100%

x

x

China

-0.277

4.021

25%

India

0.890

3.222

100%

x

Russia

0.990

1.519

0%

x

South Korea

0.998

1.161

100%

x

x

Taiwan

0.990

1.180

100%

x

x

Argentina

0.998

3.839

100%

x

Indonesia

0.987

1.892

100%

x

x

Mexico

0.959

1.457

100%

x

x

Saudi Arabia

0.990

0.943

100%

x

x

South Africa

0.336

3.229

0%

Turkey

0.847

2.079

100%

Japan
UK
LDM

LEM

G20

x

Note: Correlation and absolute distance are calculated with Canada as the reference country. “Significance” indicates theratio of significant δs at the 5% level. Group I includes markets with correlations above 0.7. Group II includes markets
where the correlation is greater than 0.7, the absolute distance is less than 2 and the coefficients δs are all significant at 5%
level.

Table 7

International integration measureWelch
50.000 19.203 ***
2.4E-192.750 **
0.006029.609 ***
7.6E-21BM
90.000 1
-4.524 ***6.8E-06
1.048 0.29481-3.760 ***0.00018

Ex Prob
11
0.500

Corr
12
1.000

0.428

-0.0370.515-0.0880.440-0.013-48.165 ***2E-3061.359 0.1744
6.224 ***
7.3E-102.799 **
0.005226.157 ***
9.7E-10-4.784 ***1.9E-06-54.259 ***0-17.405 ***0-11.447 ***0-16.073 ***08.602 ***
0-20.743 ***
0

0.070

-0.145

0.257

0.275

0.326

-0.0460.269-0.010

0.621

0.204

0.230

0.174

-1.482 0.13854-23.047 ***1.7E-95
1.118 0.26365-8.015 ***2.2E-15
9.998 ***
2.1E-226.604 ***
6.8E-11-1.092 0.2749-24.556 ***0-9.783 ***0-24.060 ***00.883 0.37724-11.449 ***
0

0.484

0.076

0.218

0.278

0.357

0.093

0.201

0.347

0.514

0.055

0.325

0.064

10.676 ***2.8E-25
9.825 ***
7.9E-22-2.473 *0.01351-2.752 **0.006-43.310 ***6E-270-6.726 ***2.5E-1116.428 ***05.193 ***
2.4E-0714.309 ***00.040 0.96815-53.190 ***0-14.072 ***
0

0.711

0.068

0.578

0.077

0.702

0.037

0.501

0.558

0.075

0.095

0.310

-0.038-0.018

0.366

0.131

0.843

-0.064

0.415

0.142

0.864

-0.043

0.392

0.137

Mean

Diff

Canada2
0.907

France

0.973

0.066

Germany

0.932

0.025

Italy

0.975

0.068

Australia

0.482-0.425Hong Kong

0.917

0.010

Netherlands

0.952

0.045

Spain

0.928

0.021

Sweden

0.958

0.051

Switzerland

0.861-0.046

Brazil0.892-0.015

China

0.500

-0.407

India

0.916

0.009

Russia0.834-0.073

South Korea

0.979

0.072

Taiwan

0.954

0.047

Argentina

0.986

0.079

Indonesia

0.980

0.073

Mexico0.877-0.030

Saudi Arabia

0.878-0.029South Africa

0.483-0.424

Turkey0.835-0.072Average G7 & LDM

0.889

Average LEM & G20
Total average

G7

3
0.000

Japan
UK
LDM

LEM

G20

Note: Diff (difference), Welch (Welch test), BM (Brunner-Munzel test), Ex Prob (excess probability) and Corr (correlation)
are calculated with Canada as the reference country.

Table 8

International integration measure between DM and EM
G7 & LDM
10.889

LEM & G2020.843Group I

0.884

0.913

-0.029Group II

0.884

0.927

-0.043

All

Diff

3
0.046

Welch
5
13.131 ***3.4E-39-9.641 ***6.3E-22-12.341 ***BM
9-5.911 ***3.5E-09-24.174 ***0-34.061 ***Ex Prob
11
0.474
0.389
0.325

Note: Diff (difference), Welch (Welch test), BM (Brunner-Munzel test), Ex Prob (excess probability) are calculated between
developed markets (G7 & LDM) and emerging markets (LEM & G20)

Figure 1

Distribution of investors’ rebalancing cycles
Distribution of investors' rebalancing cycle (2)

0.06

0.08

kappa = 0.0550
kappa = 0.0750
kappa = 0.1150
kappa = 0.2500

0.000

0.00

0.02

0.04

density

0.010

0.015

kappa = 0.0200
kappa = 0.0250
kappa = 0.0325
kappa = 0.0375
kappa = 0.0450

0.005

density

0.10

Distribution of investors' rebalancing cycle (1)

0

20

40

60

80

100

0

20

40

rebalancing cycle (week)

60

80

100

rebalancing cycle (week)

Figure 2

Structure of risk-pricing
G7

Large Developed Market
10

10

Recession

5

Goldilocks−5

CrisisGermany
Italy−0.2

Japan (missing)
UK (missing)

−0.1

0.0

−10

CanadaFrance

0.1

−0.3

−0.2

SwedenSwitzerland

−0.1

0.0

G20

0.1

5
−5

0

price of risk

510Large Emerging Market

0Brazil
China
−0.3

India
Russia−0.2

S. Korea
Taiwan−0.1
state of the economy−10

price of risk

Netherlands
Spainstate of the economy−5−10

Australia
Hong Kongstate of the economy

10

−0.3

0

price of risk

0
−10

−5

price of risk

5Overheated

0.0

0.1

Argentina
Indonesia−0.3

MexicoSaudi Arabia−0.2South Africa
Turkey−0.1
state of the economy

0.0

0.1

Figure 3

State of the economy and economic stages (1/3)
FranceGoldilocks

2010

2015

Overheated

0.2
0.0

Crisis2005

2020

Recession

Goldilocks

2010

2015

date (weekly)

date (weekly)

Germany

Italy

Overheated2020

Goldilocks

2010

2015

Crisis

2020

2005

Goldilocks

2010

2015
date (weekly)

Australia

Hong Kong

Overheated
2020

2010

2015

0.2
0.0
−0.2

state of the economy

Goldilocks−0.6

Recession

−0.4

0.2
0.0

Overheated2020

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Netherlands

Spain

Overheated
2020

−0.6

−0.4

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated
2020

0.2
0.0
−0.2

−0.2

0.0

state of the economy

0.2

0.4

0.4

date (weekly)−0.4

state of the economy

Recessiondate (weekly)

−0.2
−0.4

Crisis2005

state of the economy

0.2

Overheated

0.4

2005

Recession

0.0

state of the economy

Crisis−0.6 −0.4 −0.2

0.2
0.0
−0.2
−0.4

state of the economy

0.4

0.4

2005

Recession−0.4

Crisis−0.2

state of the economy

0.0
−0.2
−0.4
−0.6

state of the economy

0.2

0.4

0.4

CanadaCrisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020

Figure 3

State of the economy and economic stages (2/3)
Switzerland

2010

2015

Overheated

0.0
−0.2

state of the economy

Goldilocks

Crisis2020

2005

Recession

Goldilocks

2010

2015

date (weekly)

date (weekly)

BrazilChina

Overheated2020

Goldilocks

2010

2015

Overheated

0.5

2020

Crisis2005

Recession

Goldilocks

2010

2015

date (weekly)

date (weekly)

India

RussiaOverheated
2020

0.0
−1.0

−0.5

state of the economy

0.2
0.0
−0.4

Crisis−0.8

state of the economy

0.4

0.5

0.6

2005

Recession

0.0

state of the economy

Crisis

−0.5

0.0
−0.5

state of the economy

0.5

1.0

2005

Recession−0.6

Crisis−0.4

0.0−0.8 −0.6 −0.4 −0.2

state of the economy

0.2

0.2

0.4

SwedenGoldilocks

2010

2015

OverheatedCrisis2020

2005

Recession

Goldilocks

2010

2015

date (weekly)

date (weekly)

South Korea

TaiwanOverheated2020

Goldilocks

2010

2015
date (weekly)

Overheated
2020

0.0
−0.2
−0.4

state of the economy

2005

Recession

−0.6

0.0
−0.5

Crisis−1.0

state of the economy

0.2

0.5

2005

Recession

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020

Figure 3

State of the economy and economic stages (3/3)
Indonesia

Goldilocks

2010

2015

OverheatedCrisis

2020

2005

2010

Overheated2015
date (weekly)

MexicoSaudi Arabia

2020

Goldilocks

2010

2015

0.2
0.0
−0.8

state of the economy

Recession

−0.4

0.4
0.2
0.0

Overheated2020

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

South Africa

Turkey2020−0.5

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

0.0

0.0

state of the economy

0.5

0.5

date (weekly)

Overheated−0.5

state of the economy

Goldilocks

date (weekly)

−0.4
−0.8

Crisis2005

state of the economy

Recession

0.4

2005

Recession

0.0

state of the economy

Crisis

−0.5

0.0
−0.5

state of the economy

0.5

0.5

Argentina

Overheated2020

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020

Figure 4

International integration measure (1/3)
France

2010

2015

1.0
0.8
0.6
0.4
0.2

smoothed probability of integration regime

0.0

2015

1.0
0.8
0.6
0.4

2005

Recession

Goldilocks

2010

2015

2015

0.8
0.6
0.4
0.2

2020

Crisis2005

Recession

Goldilocks

2010

2015

Spain

2010

2015
date (weekly)

Overheated
2020

2020

0.8
0.6
0.4
0.2
0.0

0.8
0.6
0.4
0.2

Goldilocks

Overheated

1.0

Netherlandssmoothed probability of integration regime

date (weekly)

1.0

date (weekly)

Recession

2020

0.0
−0.2

0.8
0.6
0.4
0.2

2010

Overheated

1.0

Hong Kongsmoothed probability of integration regime

Australia
1.0

date (weekly)

Overheated

2020

0.2

2020

Crisisdate (weekly)

Goldilocks

Overheated0.0

smoothed probability of integration regime

Overheated

−0.2

1.0
0.8
0.6
0.4

Goldilocks

0.0
−0.2

2005

2010

Italy

Recession

CrisisGoldilocks

Germany

2010

2005

2005

2020

Recessiondate (weekly)

Recession

CrisisCrisisdate (weekly)

0.0
−0.2

smoothed probability of integration regime

2015

Overheated

0.2

Crisis2005

smoothed probability of integration regime

Goldilocks

0.0
−0.2

smoothed probability of integration regime

2005

Recession−0.2

Crisis

−0.2

1.0
0.8
0.6
0.4
0.2
0.0
−0.2

smoothed probability of integration regime

CanadaCrisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020

Figure 4

International integration measure (2/3)

2015

2015

1.0
0.8
0.6
0.4
0.2

2015

1.0
0.8
0.6
0.4

Goldilocks

2010

2015

2015

1.0
0.8
0.6
0.4

2005

Recession

Goldilocks

2010

2015

2015
date (weekly)

2020

2020

0.8
0.6
0.4
0.2
0.0
−0.2

0.8
0.6
0.4
0.2

2010

Overheated

Overheated

1.0

Taiwansmoothed probability of integration regime

South Korea
1.0

date (weekly)

Goldilocks

2020

0.2

2020

Crisisdate (weekly)

Recession

Overheated

0.0
−0.2

0.8
0.6
0.4
0.2

2010

smoothed probability of integration regime

Russia1.0

IndiaOverheated

2020

0.2

2005

Recessiondate (weekly)

Goldilocks

Overheated

0.0

2020

Crisisdate (weekly)

0.0
−0.2

−0.2

Overheatedsmoothed probability of integration regime

1.0
0.8
0.6
0.4

Goldilocks

0.0
−0.2

2005

2010

China

Recession

CrisisGoldilocks

Brazil

2010

2005

2005

Recessiondate (weekly)

Recession

Crisis0.0

2020

Crisisdate (weekly)

0.2

Crisis−0.2Overheated

0.0
−0.2

smoothed probability of integration regime

Goldilocks

2010

2005

smoothed probability of integration regime

Recessionsmoothed probability of integration regime

1.0
0.8
0.6
0.4
0.2

Crisis2005

smoothed probability of integration regime

Switzerland

0.0
−0.2

smoothed probability of integration regime

SwedenCrisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020

Figure 4

International integration measure (3/3)

2010

2015

Overheated

1.0
0.8
0.6
0.4
0.2

2010

2015

2010

2015

Overheated

1.0
0.8
0.6
0.4

2005

2020

Recession

Goldilocks

2010

2015

Turkey2015
date (weekly)

2020

2020

0.8
0.6
0.4
0.2
0.0
−0.2

0.8
0.6
0.4
0.2

2010

Overheated

Overheated

1.0

South Africasmoothed probability of integration regime

date (weekly)

Goldilocks

2020

0.2

Crisisdate (weekly)

Recession

Overheated0.0

smoothed probability of integration regime

Goldilocks

−0.2

0.4

0.6

0.8

1.0

Saudi Arabia

0.0
−0.2

2005

Goldilocks

MexicoRecession

Crisis2005

Recessiondate (weekly)

1.0

2005

0.0

2020

Crisisdate (weekly)

0.2

Crisissmoothed probability of integration regime

Goldilocks

0.0
−0.2

smoothed probability of integration regime

Recession

−0.2

1.0
0.8
0.6
0.4
0.2

Crisis2005

smoothed probability of integration regime

Indonesia

0.0
−0.2

smoothed probability of integration regime

Argentina

Crisis2005

Recession

Goldilocks

2010

2015
date (weekly)

Overheated2020