Global stocks and contemporaneous market risk.
This paper provides a comprehensive set of contemporaneous estimates of the market risk for globally listed stocks on 59 world equity markets. The results reveal that the estimates of the systematic risk of a stock are distinct from market to market, and from the stock's corresponding estimate of its global beta. The limitations of these estimates and, in particular, the constraints under which they are computed are pointed out. The results also indicate the need to first refine upon the single-market country estimate of the beta in the original capital asset pricing model (CAPM) before extending and or analyzing such estimates in determining the corresponding global beta.
JEL Classification: G15
Keywords: Global beta; Market risk; International equity markets
There is a consensus that in estimating the relative systematic risk of a firm's stock the use of the standard market model is inappropriate, if the stock is listed in multiple markets (see, among others, Stulz (1981), Harvey (1991), and Bernard and Bruno (1995)). This consensus arises from the specification of the standard market model wherein the market index is bounded to represent the stocks in a single country only. Though it is possible that the indices of the country equity exchanges where the stock is traded on may be highly correlated with one another, yet whether this correlation will be identically equal to one is highly improbable. Hence, using the market index from only one country may not include the available information set that is contained (or conveyed) by the indices of other countries.
In a seminal paper, Roll (1977) contends that, irrespective of how an index is structured econometrically, it is impossible to construct a market index that will include all the assets in the universe. Thus, it is not possible to have a "global" market index. In fact, Roll argues, even if one is able to construct the global market index, it may not be possible to correctly measure the global rate of return of an individual stock if it is traded in multiple markets.
Roll's position is revisited by Prakash, Reside and Smyser (1993) who suggest a procedure to obtain the BLUE estimator of a global beta under the usual wide-sense stationarity assumptions of the linear regression models. Later on, Ghai, de Boyrie, Hamid and Prakash (2001) provide a detailed procedure to obtain such estimators when the wide-sense stationarity assumptions are violated.
Under both of the above extensions of Roll's work, an attempt is made to employ all the relevant available information in the global market, thereby resolving the measurement problems to some extent. Notwithstanding these extensions, Prakash et al. and Ghai et al., do not provide any empirical evidence as to whether there is any statistically robust significant difference between the estimates of beta obtained using the standard procedures (as suggested by Markowitz (1959) and Sharpe (1963)) vis-a-vis the procedures suggested by them.
The purpose of this paper is to estimate and statistically compare the betas of multiple-listed firms using the Markowitz and Sharpe procedures as well as the one forwarded by Prakash, Reside and Smyser (1993). The paper is organized as follows. In section II we note briefly the Markowitz-Sharpe as well as the Prakash et al., procedures. In section III, we present the data selection procedure. The empirical findings and the concluding remarks are in sections IV and V, respectively.
A. Markowitz's Procedure (1)
The testable ex-post version of the market model is expressed as:
[R.sub.it] = [[alpha].sub.i] + [[beta].sub.i][R.sub.mt] + [[epsilon].sub.it]; t = 1, ..., T (1)
where [[epsilon].sub.it] is the random error term for the ith security, or the residual portion of [R.sub.it] which is unexplained by the regression of the ith stock during the tth time period. The random error term [[epsilon].sub.it] is assumed to follow the wide-sense stationarity assumptions (2).
B. Prakash et al.'s Procedure (3)
For simplicity of exposition, we will consider only two markets. The extension to more than two markets is considered next. Assume a stock is being listed in markets K and J. Let: [R.sub.kt] = rate of return of the underlying security in market K during time t (t = 1,2, , m); [R.sub.Kt] = rate of return of the Kth market index during time t (t = 1,2, ..., m); [R.sub.jt] = rate of return of the underlying security in market J during time t (t = 1,2, ..., n); [R.sub.Jt] = rate of return of the Jth market index during time t (t = 1, 2, ..., n).
Let [beta] be the global measure of the systematic risk. Since this measure of beta will be the same in the two markets, the underlying return generating process for the security in each of two markets is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
The number of bivariate observations ([R.sub.k], [R.sub.K]) and ([R.sub.j], [R.sub.J]) available in markets K and J, that is, m and n, respectively, may or may not be the same as long as, by assumption, [beta] remains the same in the two markets. Econometrically, if there is reason to believe that, intertemporally, beta might change if m [not equal to] n, then m should be taken equal to n and observations must be chosen contemporaneously in each market. The properties of the estimators obtained below, however, are unaffected by whether or not m = n.
Econometrically, there is no loss of generality if the returns are measured from their respective means. Thus, the return generating process reduces to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4b)
where [r.sub.kt] = [R.sub.kt] - [[bar.R].sub.k], etc.
Our purpose is to obtain, in the Gauss-Markov sense, the best estimator for [beta]. Prakash et al. provide the BLUE estimator for [beta], i.e. [beta], as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
A variant of relationship (5) could be derived based on the covariance and variance of each security with the market index. For example, for a security traded in the kth market, the covariance and variance are defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
Relationship (5) could then be cast into:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (6)
The estimator in relationship (6) can be easily extended to multi (more than two) markets case. Specifically, if there are p markets with [n.sub.1], [n.sub.2], ..., [n.sub.p] observations on the stock, then the multi-market BLUE estimator of beta will be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (7)
where [r.sub.i] and [r.sub.mi] are, respectively, the rates of return measured from the means on the stock, and the market index in market i, i = 1, 2, ... , p; and [n.sub.i] is the number of observations in market i.
C. Our Step-by-Step Procedures
To sum up the methodology and our step-by-step procedures, for each group of the securities we first estimate the betas for each security in each market using the market model described in equation (1). Then, using relationship (7), the global beta for each security is estimated. Next, for each security, the estimates of its betas that are separately obtained in each market are compared and tested for equality using the W-test statistic (Welch, 1953) (4). To confirm the appropriateness of our estimates, we also test one of the underlying wide-sense stationarity assumptions that are the subject of the Prakash et al., extensions, i.e., that the error variances of the market model are the same (homoskedasticity). We use the Bartlett's M-test for this purpose. This test as well as the Welch's W-test, as they are cast within the framework of our analysis, are described below.
a. Welch's W-test
Suppose a stock is traded in k exchanges (markets). Also assume that [[beta].sub.i] is the computed estimate of beta in the ith market with standard error of the estimate [s.sub.i]. Under the null hypothesis that all [[beta].sub.i]'s (i = 1, 2, ...., k) are same, Welch (1951) test requires the computation of the W-statistic given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
where [n.sub.i] is the number of observations in the ith market (which may be non-overlapping with the other markets), [s.sub.i] is the standard error of [beta], and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
Under the null hypothesis: [H.sub.0]: [[beta].sub.1] = [[beta].sub.2] = ... = [[beta].sub.k], the W-statistic will follow Snedecor's F distribution with (M-1) and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
degrees of freedom, where M is the number of regression coefficients in the market model. Therefore, M-1 will equal to one, in our case.
b. Bartlett Test for Homogeneity of Variances
An assumption underlying our methodology, as well as those of Prakash et al.'s, is the homogeneity of the variances of each security across the k-markets wherein it is cross-listed. To examine this assumption, Bartlett's test statistic (Snedecor and Cochran, 1983) is used to check if the k-market samples have equal variances. For each security, letting [[sigma].sup.2.sub.i] be the variance of the error terms of the market model in the equity exchange i, the Bartlett's statistic is defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
where [s.sup.2.sub.i] is the variance of the ith group, n is the total sample size (n = [k.summation over (i)] [n.sub.i]), [N.sub.i] is the sample size of the ith group, and [s.sup.2.sub.q] is the pooled variance defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (12)
We want to test the null hypothesis:
[H.sub.0]: [[sigma].sup.2.sub.1] = [[sigma].sup.2.sub.2] = [[sigma].sup.2.sub.3] = ... = [[sigma].sup.2.sub.k], againe the alternative
[H.sub.a]: [[sigma].sup.2.sub.i] [not equal to] [[sigma].sup.2.sub.j] for at least one pair (i, j). (13)
If the null hypothesis is true, then the B-statistics will be distributed as a chi-square distribution with (k-1) degrees of freedom.
Scanning for cross listed stocks in 59 world wide stock exchanges, seven hundred and four company stocks are identified to have multiple listing on two or more equity exchanges. Weekly price series for these stocks (2) are compiled from DataStream for the period of June 1998 to June 2003. Table 1 summarizes the data set. Among the multiple-listed stocks, 636 companies have their stocks listed in two equity exchanges, 55 companies are listed in three exchanges, 13 companies are listed in four exchanges, 2 companies are listed in seven different international markets, and one company, Bayer AG, is listed in eight different exchanges.
Since a "market portfolio" is required to compute the beta for each security, we also obtain the corresponding weekly index series for the 59 equity exchanges from DataStream. Table 2 contains the names of the 59 international stock exchanges and their respective market indices. If more than one index is available for an equity exchange in a country, an effort is being made to select the most comprehensive index in that country. In the rare cases wherein data is deficient for the market index of a particular country, we select the Morgan Stanley Composite index for that country.
IV. EMPIRICAL RESULTS
Table 3 provides a summary of the various estimates of the betas. The firm's stock is cited in column 1 and the estimate of its global beta is included in column 5. The second column of this Table includes the number of equity markets wherein each stock is traded on. To manage space and to provide a concise summary of the values of the various betas in the seven group cross-listed markets (see Table 1), we have included in this Table a limited number (the first ten) of stocks, and have provided only the range of their beta estimates (columns 3 and 4) (3). A cursory examination of the estimates of the global betas and the single country betas reveals substantial differences between them. This statement, as is stated, is ad hoc at this point and needs to be statistically scrutinized via the application of the W-test that is reported below.
Table 4 summarizes the results of the W-test statistics for the multiple-listed stocks in the seven group cross-listed markets. For Group 1, i.e. wherein each security is listed on two stock exchanges, the null hypotheses for 289 out of 636 stocks are rejected, i.e., these stocks show different beta values in different stock exchanges at the10 percent or below significance (33+60+196=289). The null hypothesis for the remaining 347 stocks in this group cannot be rejected, thus leaving us to conclude that each of these stocks possesses similar betas in the two different exchanges it is cross-listed. In Groups 2 through 7, wherein each security is listed in three through eight stock exchanges, the W-test statistics for each group indicate that most of the betas in each of the exchanges are statistically significantly different. There are only two securities in Group 3 (cross-listing in 4 markets) and one security in Group 4 (cross-listing in 5 markets) that exhibit similar betas.
The above results indicate that for multiple-listed stocks, the estimates of beta computed from the market index in one exchange is significantly different from the beta computed using the market index from another exchange. This provides ample evidence not to rely on the betas computed from local market indices in decisions involving international investments. A global beta for a cross-listed stock, as specified in relationship (7), is more suitable under a global investment setting.
Table 5 provides a summary of the Bartlett test statistics for the homogeneity of the error variances for all the securities that are cross-listed in the various group markets. Note that K varies across the groups, i.e., K = 2, 3, ....., 8. The total number of company betas that show homoskedasticity of variance in all markets is 401. Thus, the results reported in Table 4 are subject to the caveat that they are based on estimates that may not be BLUE. Hence, we conclude that employment of Ghai et al. (2001) approach that adjusts for some of these caveats is more appropriate.
To further elaborate on the above Bartlett test results, it should be pointed out that a measure of the information content of a market is often provided by the inverse of its variance. Hence, our use of the Bartlett tests above provides us a venue, in addition to our prime purpose to check on the homogeneity of the error variances, to examine the similarity of the information contents of the markets in each cross-listed group. In other words, the results of the Bartlett tests are indicative as to whether the markets in each group reveal the same information. Thus, an interpretation of the null hypothesis in relationship (13) is that the markets in each group reveal the same information, against the alternative hypothesis that at least one market in the group has different information from the rest of the markets in the group. For example, in the two markets case, if a security is traded in markets i and j, and another security is traded in markets i and k, we have tested, respectively, the null hypotheses that [[sigma].sup.2.sub.i] = [[sigma].sup.2.sub.j] and [[sigma].sup.2.sub.i] = [[sigma].sup.2.sub.k]. The results in Table 5 could thus be viewed in the context of the diversity in the information contents of the markets.
Since the majority of the multiple-listed stocks show strong statistical evidence of heteroskedasticity, we opted to check, as an aside, the equality of the variances of the equity markets that appear in each "group" of the markets. More specifically, we tested the equality of the variances on the market indices that appear in each of the groups. These groupings are exactly the same as the trading locations (exchanges) of the cross-listed stocks. For example, if stock j is traded in New York (NYSE) and London (FTALLSH), then the bi-variate observation (NYSE, FTALLSH) will constitute a member of the group 1 markets.
Table 6 presents the results of the heteroskedasticity of the various groups. In the case of the two-market groupings, the Bartlett test statistics for 70 out of 86 groups are statistically significant at the 10 percent level or below, i.e., rejecting the null of the equality of the market index variances. The majority of these test statistics, i.e., 60 of them, are statistically significant at the one percent level or below. Similarly, in the remaining groups of three to eight market groupings, the test statistics for only four out of 51 groups are found to be insignificant, i.e., their respective equity exchanges exhibit similar variances.
As was mentioned above, in uni-variate analysis the inverse of the variance is a measure of the information content of the data population that underlies the variance. Thus, the above strong evidence of heteroskedasticity in the various market groupings suggests that the information provided by the various markets is asymmetric. That is, one market disseminates more (or less) information in comparison to another market.
V. CONCLUDING REMARKS
In this paper we provided a comprehensive set of contemporaneous estimates of the market risk for globally listed stocks across various world equity markets. We nearly exhausted a sample of 704 globally listed stocks in 59 international exchanges that were available in Datastream. Our results reveal that the estimates of the systematic risk of a stock are distinct from market to market, and from the stock's corresponding estimate of the global beta that are computed using Prakash et al.'s procedure.
Using Welch's W-test and Bartlett's B-test statistics, we pointed out the limitations of our estimates and, in particular, we examined the constraints under which such estimates were computed. The refined procedure provided by Ghai et al., that addresses some of these limitations, i.e., the assumption of the homogeneity of the error variances across various markets, is expected to provide better estimates of global beta.
Irrespective of the value of the estimates rendered by either Prakash et al.'s or Ghai et al.'s procedures, we would like to conclude that it is imperative to first refine upon the single-market country estimate of the beta in the original capital asset pricing model (CAPM) before extending and or analyzing such estimates in determining the corresponding global beta. The extent of such refinements is purely empirical and is dictated by the level of the accuracy desired. At minimum, the standard application of a few simple econometric techniques, e.g., adjustments for multicollinearity and heteroskedasticity, will substantially improve the resultant estimates of the global beta.
(1.) For a general discussion of the historical development of the market model see Prakash et al. (1999).
(2.) The wide-sense stationarity assumptions are (see Reinmuth and Wittink, 1974): E([[epsilon].sub.it])=0 (zero mean), var([[epsilon].sub.it])= [[sigma].sup.2.sub.[epsilon]] (homoskedasticity), cov([[epsilon].sub.it], [[epsilon].sub.it+k])=0 for all k [not equal to] 0, and cov([[epsilon].sub.it], [R.sub.mt])=0.
3. This section draws upon Prakash, Reside and Smyser (1993).
4. The list of these stocks is available upon request from the first author.
5. The estimate of each single stock beta is available upon request from the first author.
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Harvey, C., 1991, "The World Price of Covariance Risk," Journal of Finance 46 (1), 111-158.
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Prakash, A.J., M.A. Reside, and M.W. Smyser, 1993, "A Suggested Simple Procedure to Obtain Blue Estimator of Global Beta," Journal of Business Finance and Accounting 20 (5), 755-760.
Prakash, A.J., R. Bear, K. Dandapani, G. Ghai, T.E. Pactuwa, and A.M. Parhizgari, 1999, The Return Generating Models in Global Finance, Oxford, UK: Pergamon, Elsevier Science Ltd.
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Reinmuth, J.E., and D.R. Wittink, 1974, "Recursive Models for Forecasting Seasonal Processes," Journal of Financial and Quantitative Analysis 9 (4), 659-684.
Roll, R., 1977, "A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of Theory," Journal of Financial Economics 4 (2), 129-176.
Scheffe, H., 1959, The Analysis of Variance, New York, John Wiley & Sons.
Sharpe, W.F., 1963, "A Simplified Model for Portfolio Analysis," Management Science 9 (Jan.), 277-293.
Snedecor, G.W., and C. William, 1989, Statistical Methods, Eighth Edition, Iowa State University Press.
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Leyuan You (a), Chun-Hao Chang (b), Ali M. Parhizgari (c) and Arun J. Prakash (d)
(a) Department of Business, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK, 99508, firstname.lastname@example.org
(b) Department of Finance, Florida International University,11200 SW 8th Street, Miami, FL 33199, email@example.com
(c) Department of Finance, Florida International University, 1200 SW 8th Street, Miami, FL 33199, firstname.lastname@example.org
(d) Department of Finance, Florida International University, 1200 SW 8th Street, Miami, FL 33199, email@example.com
Table 1 Number of securities cross-listed in two or more equity markets Number of Securities Listed In Number of Securities Group1: 2 markets 636 Group 2: 3 Markets 47 Group 3: 4 Markets 13 Group 4: 5 Markets 5 Group 5: 6 Markets 0 Group 6: 7 Markets 2 Group 7: 8 Markets 1 Total 704 Note: The name of the companies and the equity exchanges wherein their securities are cross-listed are available upon request from the first author. Table 2 Major international equity exchanges that host multiple listing of stocks and their respective indices (Total number of exchanges = 59) Exchange Index AMERICAN AMXIXAX AMSTERDAM (AEX) NLALSHR AMSTERDAM (AEX) NLALSHR BERLIN DAXINDX BOMBAY IBOMBSE BOMBAY IBOMBSE BRUSSELS BRUSIDX COLOMBO SRALLSH COPENHAGEN CHAGENZ DUBLIN ISEQUIT FRANKFURT DAXINDX HAMBURG DAXINDX HELSINKI HEXINDX HONG KONG HNGKNGI ISTANBUL TRKISTB JASDAQ JASDAQI JOHANNESBURG JSEOVER KARACHI PKSE100 KOREA KORCOMP KUALA LUMPUR KLPCOMP LILLE FSBF120 LIMA PEGENRL LISBON POPSI20 LONDON FTALLSH LUXEMBOURG LXLUXXI LYON FSBF120 MADRID-SIBE MADRIDI MANILA MANCOMP MILAN MILANBC MILAN MILANBC MUNICH DAXINDX NASDAQ EUROPE BGBEL20 NASDAQ NM NASCOMP NASDAQ SMALLCAP NASCOMP NATIONAL INDIA IBOMBSE NATIONAL INDIA IBOMBSE NEW YORK NYSE NEW ZEALAND NZ40CAP OSLO OSLOASH OTC BULL.BD.NASD NASCOMP OTHER OTC NASDAQ NASCOMP PARIS-SBF FSBF250 PRAGUE CZPX50I SANTIAGO IGPAGEN SHANGHAI CHSCOMP SHENZEN DJSHENZ SINGAPORE SNGPORI STOCKHOLM AFFGENL STUTTGART BDSTUTT TAIWAN TACOMPT TEL AVIV ISTGNRL THAILAND TOTMKTH TOKYO TOKYOSE TORONTO TTOCOMP VIENNA WBKINDX VIRT-X SWISSMI XETRA DAXINDX ZIMBABWE ZIMINDS ZURICH SWISSMI Table 3 Estimates of each stock's "exchange-" and "global-" betas (1) # of Range of Betas Name Exchanges Low 1-800 CONTACTS 2 0.3273 24/7 MEDIA (FRA) 2 0.6959 3COM 2 0.8166 8X8 2 0.8483 A B WATLEY GP. 2 -0.1305 A D A M 2 -0.0646 AAON 2 -0.6872 AASTROM BIOSCIENCES 2 0.7139 AB SOFT 2 -0.7664 ABAXIS 2 -0.0846 ABBOTT LABS. 3 0.0068 ABER DIAMOND 3 0.0108 ABN AMRO HOLDING 3 1.0313 AFLAC 3 0.0007 AGNICO-EAGLE MNS. 3 -0.1135 AGRIUM 3 0.0028 AJINOMOTO 3 -0.0051 ALCAN 3 0.0009 ALCOA 3 0.0009 ALLEGHENY EN. 3 0.0090 AEGON (FRA) 4 0.2697 ALLIANZ 4 0.3710 AT & T (FL) (AMS) 4 0.2590 BELLSOUTH 4 -0.1336 BHP BILLITON 4 0.5250 BOEING 4 -0.1199 CATERPILLAR 4 -0.0003 CLARIANT 4 0.5589 COMMERZBANK 4 0.3084 COREL 4 0.2623 AKZO NOBEL 5 -0.0130 ALTRIA GP. 5 -0.0022 AMER.INTL.GP. 5 -0.3094 BARRICK GOLD 5 -0.0524 SANTANDER CTL.HISP.(FRA) 5 0.3070 DAIMLERCHRYSLER 7 0.0019 DEUTSCHE BANK 7 0.6603 BAYER 8 0.3150 Range of Betas Name High Global Beta 1-800 CONTACTS 0.5034 0.4342 24/7 MEDIA (FRA) 1.9390 1.4505 3COM 0.9412 0.8656 8X8 1.4540 1.2160 A B WATLEY GP. 0.5645 0.2914 A D A M 0.8773 0.5072 AAON 0.2502 -0.1181 AASTROM BIOSCIENCES 0.9335 0.8472 AB SOFT 0.1303 -0.2586 ABAXIS 0.5930 0.3267 ABBOTT LABS. 0.5873 0.0085 ABER DIAMOND 0.4589 0.1890 ABN AMRO HOLDING 1.3716 1.1630 AFLAC 0.5219 0.0027 AGNICO-EAGLE MNS. -0.0779 -0.1014 AGRIUM 0.1944 0.0034 AJINOMOTO 0.6418 0.2418 ALCAN 0.9812 0.0035 ALCOA 0.9006 0.0049 ALLEGHENY EN. 0.9545 0.0120 AEGON (FRA) 1.7952 1.1629 ALLIANZ 1.5738 1.1509 AT & T (FL) (AMS) 1.0693 0.7773 BELLSOUTH 0.5770 0.0019 BHP BILLITON 1.4259 1.0210 BOEING 0.7103 0.0052 CATERPILLAR 0.6743 0.0035 CLARIANT 1.3182 0.7878 COMMERZBANK 1.2665 0.9533 COREL 1.2534 0.7753 AKZO NOBEL 0.6754 0.5230 ALTRIA GP. 0.6572 0.0008 AMER.INTL.GP. 0.9535 0.0009 BARRICK GOLD -0.0003 -0.0001 SANTANDER CTL.HISP.(FRA) 1.0354 0.8546 DAIMLERCHRYSLER 0.9406 0.0133 DEUTSCHE BANK 1.6223 1.1727 BAYER 1.3128 0.8399 To manage space and to provide a concise summary of the values of the various betas in the seven group cross-listed markets (se Table 1), we have provided in this Table only the range of such estimates (columns 3 and 4), and only a limited (the first ten) number of stocks. Table 4 Summary of W-statistics for multiple-listed stocks Number of Multiple-Listed Stocks with Different Betas at Significance Level 0.05 <[alpha] 0.01 <[alpha] [alpha][less [less than or [less than than or equal to] 0.10 or equal equal to] 0.05 to] 0.01 Group 1: 2 Markets 33 60 196 Group 2: 3 Markets 1 5 27 Group 3: 4 Markets 3 8 Group 4: 5 Markets 1 3 Group 6: 7 Markets 1 1 Group 7: 8 Markets 1 Total 34 70 236 Not Significant (= Same Betas) Total Group 1: 2 Markets 347 636 Group 2: 3 Markets 14 47 Group 3: 4 Markets 2 13 Group 4: 5 Markets 1 5 Group 6: 7 Markets 0 2 Group 7: 8 Markets 0 1 Total 364 704 Note: No stocks were cross-listed in Group 5, i.e., in 6 markets. Table 5 Summary of Bartlett test statistics on the equality of the error variances of the cross-listed stocks Number of Stocks Significant at the Levels of 0.05<[alpha][less than 0.01<[alpha][less than or equal to]0.10 or equal to]0.05 Group 1: 2 Markets 30 43 Group 2: 3 Markets 1 3 Group 3: 4 Markets 1 1 Group 4: 5 Markets Group 6: 7 Markets Group 7: 8 Markets Total 32 47 Number of Stocks Significant at the Levels of [alpha][less than Not Significant Total or equal to]0.01 Group 1: 2 Markets 193 370 636 Group 2: 3 Markets 18 25 47 Group 3: 4 Markets 7 4 13 Group 4: 5 Markets 3 2 5 Group 6: 7 Markets 2 0 2 Group 7: 8 Markets 1 0 1 Total 224 401 704 Table 6 Summary of Bartlett test statistics on the equality of the variances of the market indices that appear in various groups Number of Stocks Significant at the Levels of 0.05<[alpha][less than 0.01<[alpha][less than or equal to]0.10 or equal to]0.05 Group 1: 2 Markets 2 8 Group 2: 3 Markets 1 Group 3: 4 Markets 1 Group 4: 5 Markets Group 6: 7 Markets Group 7: 8 Markets Total 2 10 Number of Stocks Significant at the Levels of [alpha][less than Not Significant Total or equal to]0.01 Group 1: 2 Markets 60 16 86 Group 2: 3 Markets 27 3 31 Group 3: 4 Markets 11 0 12 Group 4: 5 Markets 5 0 5 Group 6: 7 Markets 2 0 2 Group 7: 8 Markets 0 1 1 Total 105 20 137 Note: No stocks were cross-listed in Group 5, i.e., in 6 markets.
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|Author:||You, Leyuan; Chang, Chun-Hao; Parhizgari, Ali M.; Prakash, Arun J.|
|Publication:||International Journal of Business|
|Date:||Jun 22, 2007|
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