In this Section, we describe in details the mild segmentation model proposed by Errunza and Losq (1985). We introduce the following assumptions:
(i) Unequal Access Assumption. The investing population is divided in two subsets: the unrestricted and restricted investors. Restricted investors can trade only eligible as-sets (denoted by e). The ineligible (denoted by i) securities can be held only by the unrestricted investors.
(ii) Perfect Capital Market Assumption. The capital markets are perfect and frictionless.
This assumption includes equal access to information by all market participants, com-pletely rational economic actors, and no transaction costs.
(iii) Mean-Variance Assumption. The expected utility of an investor is function of the expected value of returns and its variance.
(iv) Free Lending and Borrowing Assumption. Investors can borrow or lend any amount of money at the same risk-free rate of return.
Let us define the vector of returns R = [R0i, R0e]0, where Riand Reare the vector of returns on the ineligible and eligible securities, respectively. The returns are supposed to be normally distributed with covariance matrix the partial integration of the market, we introduce the following portfolios (and their corre-sponding notation):
1. the World Market Portfolio: market value M, rate of return RW, representative vector W M P = P = [P0i, P0e]0;
2. the Market Portfolio of Ineligible Securities: market value MI, rate of return RI = R0iPi, representative vector MP IS = [P0i, 00]0.;
3. the Market Portfolio of Eligible Securities: market value ME, rate of return RE= R0ePe, representative vector MP ES = [00, P0e]0.
Since, the portfolio of eligible assets Reis not observable, we estimate the diversified port-folio DP , that is the portport-folio most highly correlated with the market portport-folio of ineligible assets RI. Errunza et al. (1999) estimate DP from the set of industry portfolios. Carrieri et al. (2007) consider also the country funds (CF) and the American Depository Receipts (ADRs) to generate highly correlated return with the market portfolios of their ineligible assets. Given the assumption of normality, DP is the portfolio that minimize Var[RI− α0Re] w.r.t. to α, i.e., the optimal is α∗ = Σ−1ee ΣIe, where ΣIe = Cov[RI, Re]. Since RI = R0iPi,
Thus, the restricted investors can duplicate returns on unavailable assets through home-made diversification.
Errunza and Losq(1985) show that under market segmentation, the expected return on the i-th ineligible security in the I-th market is
E[Ri] = Rf+ AM Cov [Ri, RW] + (Au− A)MICov [Ri, RI|Re]
= Rf+ λWCov [Ri, RW] + λICov [Ri, RI|Re], (8) where asset i in the I-th market is accessible only to nationals. A is the aggregate risk aversion coefficient with A−1 ≡ A−1r + A−1u . Au is the absolute risk aversion coefficient for
unrestricted investors on the I-th market and Ar is the absolute risk aversion coefficient for restricted investors. The prices of risk λW and λI are functions of the relative risk aversions of restricted and unrestricted investors, as showed in Errunza and Losq (1985).
The expected return on the potentially segmented market is proportional to the covariance with a global factor and to the conditional market risk23. The expected excess return on the ineligible security market index can be obtained aggregating over the ineligible set of securities:
E[RI− Rf] = AM Cov [RI, RW] + (Au− A)MIVar [RI|Re].
Under the assumption that returns are jointly normally distributed, we have Var[RI|Re] = Var[RI] − Cov[RI, Re]0Var[Re]−1Cov[RI, Re]
= Var[RI]{1 − ρ2(RI, Re)}, (9)
and
ρ2(RI, Re) = Cov[RI, Re]0Var[Re]−1Cov[RI, Re]
Var[RI] .
ρis the multiple correlation coefficient that can be interpreted as the correlation coefficient between RI and that portfolio of eligible securities which is most correlated with RI, i.e., the DP portfolio. When ρ = 0 the extreme form of market segmentation takes place, i.e. when no correlation exists between RI and the return on any eligible security and the market are completely segmented:
E[Ri] − Rf = AuMICov [Ri, RI].
Let use define r∗,t, ∗ = {I, DP, W } the excess return on the R∗,t return index. From the Errunza and Losq (1985) model, the following system of equations must hold at any point in time,
The first equation in the system is the pricing of the local market index, where two factors are priced: the world market covariance risk and the super risk premium, proportional to the conditional local risk represented by Vart−1[rI,t|rDP,t]. The second equation prices the DP through the covariance risk with the world portfolio return. Finally, the last equation is the pricing equation for the world index portfolio. The theory predicts that the world price of risk should be the same for each country.
The model needs the specification of the law of motion of the conditional covariance matrix. To this purpose we consider alternative specifications in the multivariate GARCH family, see Bauwens et al.(2006). The GARCH model are usually appropriate for modeling conditional variances and covariances for stock market. Assuming a conditional Gaussian distribution of stock returns, the GARCH models allow components of variances and co-variances vary over time depending on the shocks at time t − 1 and on the past values of variances and covariances terms (see Bollerslev, 1986;Engle, 1982). The model's param-eters are estimated by Quasi Maximun Likelihood (QMLE), see Bollerslev and Wooldridge (1992).
C.2 Additional tables
This Section contains additional tables that allow to give an exhaustive description of em-pirical applications.
Table10lists the regressors included in the estimation procedure for the diversified port-folio. Table 12reports the summary statistics for the monthly excess returns of European countries and the World market index. To analyze these data, we consider two subsamples:
from January 1995 to July 2007, the so-called pre-crises subsample, and from August 2007 to August 2016. The European returns on average are positive and large in the pre-crises subsample. The returns display high volatility in the second subsample, as expected. In the full sample, the difference between the two subsamples are mitigate. The data for
23The conditional market risk is defined as the conditional covariance between the return of asset i and the return on the market portfolio of all ineligible securities I, given the returns on all eligible securities. The conditional market risk can be interpreted as a measure of substitutability between a specific ineligible security and the eligible segment of the world market.
Austria and Belgium show a high level of kurtosis and normality test are rejected. The nor-mality tests are not rejected for Italy. We also provide results for the Engle's ARCH test for heteroskedasticity. For most of the countries, this test is rejected. Table 12also provides the descriptive statistic for the world market index and the correlation index between data on European stock market (rI), diversified portfolios (rD) and the world index (rW). On average, the correlation index Corr(rI, rW)is 0.70, thus the data are positive correlated.
x1 MSCI world index,
Table 10: Regressors to estimate the diversified portfolio. List of variables involved in the stepwise regressions to determine the diversified portfolio for each country.
Table 11 shows the model taxonomy that arises from our theoretical framework and the availability of empirical data. We consider two conditional covariance models: the GARCH(1,1) model and GARCH models with cross-sectional market volatility. The GARCH(1,1) model involves the true conditional covariance matrix by mean zero errors in its parameter-ization. This model is the most used in the literature. On the opposite, the GARCH(1,1)-X model involves an estimate of the matrix of quadratic covariations based on the monthly realized variances and covariances.24
GARCH(1,1)
Model Representation Price of risk Frequency Complete Diagonal λ∗ λ∗,t Daily Monthly
1 x x x
Model Representation Price of risk Frequency Complete Diagonal λ∗ λ∗,t Daily Monthly
7 x x x
8 x x x
9 x x x
10 x x x
Table 11: Model taxonomy.
24Technical details on the parametrization of conditional covariance models and estimation of realized covariances are reported inOssola and Rossi(2017a).
ATBEDKFIFRGEIEITNLNOPTESSEUKWorld SubsamplefromJanuary1995toJuly2007 mean3.5623.9187.7667.3214.1202.2643.3762.7271.8543.3114.0877.2735.1990.0072.587 stdev6.1095.6386.15312.0136.1727.5466.0716.8966.3837.1506.6986.9588.0324.3734.614 skewness-0.988-1.132-0.513-0.495-0.586-0.816-0.7630.176-1.119-0.959-0.362-0.841-0.508-0.882-0.786 kurtosis5.2826.1293.5404.4053.7945.4133.6743.7965.5056.0534.3185.8314.0154.3524.267 SubsamplefromAugust2008toAugust2016 mean-12.404-2.4974.631-6.731-3.110-1.238-8.827-9.752-0.816-3.756-12.876-5.755-0.606-0.8000.246 stdev9.6837.9926.7797.7546.0206.8418.8067.6746.3447.7636.7887.6096.2034.9915.968 skewness-1.238-2.228-0.678-0.351-0.354-0.626-0.830-0.156-0.858-1.400-0.648-0.171-0.720-0.479-0.916 kurtosis7.18111.4535.0854.5963.0864.2404.1822.8544.7747.2484.0493.5185.9903.3925.429 FullsamplefromJanuary2007toAugust2016 mean-3.1321.2296.4521.4301.0890.796-1.740-2.5040.7350.348-3.0241.8112.765-0.3311.606 stdev7.8326.7196.41210.4476.1077.2477.3527.2456.3567.4076.7757.2527.3144.6335.215 skewness-1.381-1.996-0.603-0.419-0.486-0.745-0.932-0.016-1.010-1.181-0.478-0.535-0.544-0.682-0.903 kurtosis8.52711.5634.3904.9703.4835.0554.7623.3925.1906.7464.2394.5744.6493.8885.412 JB0.0010.0010.0010.0010.0080.0010.0010.3860.0010.0010.0010.0010.0010.0010.001 LB(2)0.0000.0000.3700.0160.3780.7270.0050.6440.4740.0190.1730.2300.3320.7040.147 LB(4)0.0010.0000.3440.0580.5080.8730.0020.2450.5120.0430.1720.3190.0630.1430.080 LB(8)0.0000.0000.2280.0720.1880.5840.0040.0560.0700.0500.5700.5230.0370.2910.197 ARCH(2)0.0000.0000.0010.0000.0000.0000.0000.1090.0000.0000.0690.0160.0000.0000.000 ARCH(4)0.0000.0000.0030.0000.0010.0000.0000.1910.0000.0000.0360.0790.0010.0000.000 ARCH(8)0.0000.0000.0050.0000.0020.0010.0000.0370.0010.0000.1830.1090.0050.0010.001 Corr(rI,rD)0.7620.7220.6520.7090.8370.8230.6210.7230.7870.7880.6420.7740.7950.845 Corr(rI,rW)0.7100.6890.6520.6270.8220.7900.5980.6890.7760.7510.6050.7380.7290.845 Corr(rD,rW)0.9360.9551.0000.8670.9820.9560.9360.9530.9850.9530.9270.9450.9251.000 Table12:Summarystatisticsofexcessreturns.MonthlydataonreturnsofEuropeanandWorldmarketindexesareinexcessofthe30-dayEurodollardeposit rate.ThefullsamplecoverstheperiodfromJanuary1995toAugust2016.Wealsoreportsomedescriptivestatisticsfortwosubsamples.Meanandstandard deviationsareinannualizedpercentageterms.Thetestforkurtosiscoefficienthasbeennormalizedtozero,JBistheJarque-Beratestfornormalitybasedon skewnessandexcesskurtosis.LB(.)istheLjung-Boxtestforautocorrelationoforder2,4and8.ARCH(.)istheEngle'sARCHtestforresidualheteroscedasticity oforder2,4and8.PairwisecorrelationsfortheexcessportfolioreturnsrI,rDandrWarealsoreported.
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List of figures
Figure 1. Price based indicators. . . 4
Figure 2. Integration index, systematic and idiosyncratic volatilies of Germany and Spain.. . . 9
Figure 3. Integration, systematic and idiosyncratic indexes among EU28. . . 11
Figure 4. II and total risk premium of Austria (AT). . . 18
Figure 5. II and total risk premium of Belgium (BE).. . . 18
Figure 6. II and total risk premium of Denmark (DE). . . 19
Figure 7. II and total risk premium of Finland (FI). . . 19
Figure 8. II and total risk premium of France (FR). . . 20
Figure 9. II and total risk premium of Germany (GE). . . 20
Figure 10.II and total risk premium of Ireland (IE). . . 21
Figure 11.II and total risk premium of Italy (IT). . . 21
Figure 12.II and total risk premium of Netherland (NL). . . 22
Figure 13.II and total risk premium of Norway (NO).. . . 22
Figure 14.II and total risk premium of Portugal (PT).. . . 23
Figure 15.II and total risk premium of Spain (ES). . . 23
Figure 16.II and total risk premium of Sweden (SE). . . 24
Figure 17.II and total risk premium of United Kingdom (UK). . . 24
List of tables
Table 1. Market integration models. . . 1
Table 2. Average correlation index among cluster of EU countries.. . . 8
Table 3. Results from regression over the subsamples (i) and (ii).. . . 10
Table 4. Drivers of integration degree among EU28 countries and EA countries . . . 13
Table 5. Representation of the mild segmentation model . . . 14
Table 6. Financial integration index . . . 15
Table 7. Summary statistics of integration index and risk premia . . . 17
Table 8. Stock prices indexes of the 28 EU countries sample. . . 28
Table 9. Summary statistics of daily returns of EU28. . . 29
Table 10.Regressors involved in the estimation of diversified portfolio . . . 32
Table 11.Model taxonomy . . . 32
Table 12.Summary statistics of excess returns . . . 33
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