6. the International Transmission of Conditional Volatility *

Chapter 6

6. The International Transmission of Conditional Volatility*

6.0 Introduction

Shiller's1 research on volatility stimulated increased interest in measuring the volatility of stock market indices. While there are several ways to estimate the volatility of a stock market index series, the ARCH model suggested in an article by Engle (1982) is a widely used approach with many advantages.2 The advantage of the ARCH approach is that it can be used to estimate a series that measures conditional, or predictable, volatility. Engle and Susmel (1993) employ a variant of this procedure to test for common ARCH features of weekly return data for 18 stock market return series between January 1980 and January 1990. Their testing procedure seeks common factors in two-by-two market comparisons. Building on this work, the present chapter uses daily data for 21 national markets, including those studied by Engle and Susmel, from 12/31/85 to 9/30/93. A common generic ARCH model is initially fitted to each market index series and the resulting estimated conditional heteroskedasticity series are related in a 7 order VAR model, using regional groupings to test for dynamic linkages. VAR models are fitted for the complete period and the 1/1/88 - 9/30/93 period, which excludes the October 1987 world wide stock market crash. The chapter first offers a brief discussion of related literature. Next, the data and ARCH-class models are discussed. The estimated conditional heteroskedasticity series are compared graphically and VAR models are estimated to study whether the conditional volatility series are related in the sense of Granger (1969) across markets. Finally, more extensive models of volatility are presented.

6.1 Related Literature

Transmission of Conditional Volatility

Our work is motivated by an interest in the integration of international stock markets. Literature on this subject has been summarized by Roll (1989). This literature has two major threads, one focusing on modeling level effects and the other on modeling second-moment or volatility effects. An example of the former approach is described by Eun and Shim (1989), who estimate a VAR model to study dynamic patterns of daily stock market returns in nine countries from 1980 to 1985. An example of the latter approach is Engle and Susmel (1993), who use the ARCH approach to investigate whether international stock markets have the same volatility process. Their study examines 18 major markets and seeks to isolate common factors, using a multivariate MARCH-1 test to relate volatility in one market to that in another. In addition to studying volatility, they present correlations between returns in three time groupings: European markets, Far Eastern markets, and North American markets. The work of Hamao, Masulis and Ng (1990) combines both approaches. Their study examines the correlations of both price changes and volatility, using daily and intraday stock-price activity for the three-year period 1 April 1985 to 31 March 1988. Because their study was confined to three centers, Tokyo, London and New York, it was possible to include lagged returns and jointly estimated squared residuals from one center in ARCH / GARCH models of other centers. If our study had been confined to three centers, we might have used this promising approach.

Because our study groups as many as 14 national markets in one grouping, the joint estimation strategy of Hamao, Masulis and Ng (1990) is not tractable. We cannot jointly estimate 14 ARCH / GARCH models. An advantage of our simpler, country by country, ARCH / GARCH estimation strategy is that if a dynamic pattern were found between the conditional volatility series of two centers, it would suggest that market participants are overlooking a trading opportunity. Why do market participants not simply reestimate ARCH / GARCH models for each center every day? They could then make forecasts of changes in the expected volatility, which could be used in the VAR model to make forecasts of conditional volatility in other centers. Of course, our procedure implicitly assumes that the conditional volatility process in each center can be modeled as if it were an independent stochastic process. We think this assumption is a reasonable one and it has been made by all ARCH / GARCH studies that are not bivariate. We next examine whether the separately estimated conditional volatilities in any center have cross-market influences.

Our estimated conditional volatility measures contain less information than estimates obtained from a bivariate (multivariate) approach that includes information from other centers. If Granger causality is not found between centers, there is no assurance that the same finding would be obtained if we used a heterokedasticy measure estimated conditional on more information. On the other hand, a finding of Granger causality between our measures of conditional heteroskedasticity indicates a relationship would be found with the more general measure. At the very least, such a finding suggests a market participant could use this information to make predictions of volatility.

Transmission of Conditional Volatility

Recent related studies on volatility transmission include Chan, Karolyi and Stulz (1992), who estimate a bivariate GARCH in the mean process to study foreign influences on the risk premium for U. S. assets, Karolyi (1995), who uses a multivariate GARCH process to model the international transmission of stock returns and volatility between the United States and Canada, and Lin, Engle and Ito (1994), who use a two-stage GARCH approach to model bidirectional cross-market effects. In these papers, ARCH-class models are extended in order to study volatility dynamics across markets. Lin, Engle and Ito (1994, p. 510) note "whether the volatility is correlated across markets is important in examining the speed of market adjustment to new information." A closely related topic is the dynamics of information transmission between markets. Cheung and Kwan (1992) studied this topic and found public information originating in the United States affects foreign markets, although the opposite may not be true.

There is particularly intense interest in determing the extent to which volatility is predictable and whether it is time-varying. While these questions have been studied by Lin, Engle and Ito (1994) in a two-by-two market analysis, our study presents a multicountry approach. In addressing this question, we must consider what measure of volatility is appropriate? While Bennett and Kelleher (1988) and others have measured volatility, using the standard deviations of daily percent changes (total volatility), our work follows other ARCH-class studies in using conditional or predictable volatility. Good arguments can be made for both approaches. Studies using the total volatility approach do not have the signal loss associated with the conditional volatility approach and might find evidence for the dynamic transmission of volatility, which might not be found with the other approach. For market participants, however, such an approach should have less practical value, since total volatility is known only after the fact, while a measure of expected (conditional) volatility can be obtained from the forecasts of an ARCH-class model. In a model using expected volatility, if the volatility in one market is found to Granger (1969) cause the volatility of another market, then, in theory, it would be possible for a market participant to use this relationship to predict conditional volatility in the caused market. Such a use would not be possible with the total volatility approach.

In summary, our work extends bivariate GARCH research and allows us to consider more than a two-by-two national market comparison of conditional volatility. Because it is not presently econometrically tractable to jointly estimate a GARCH model with more than two markets, we first fit a common ARCH-class model to each of the 21 markets. We then relate the resulting conditional heteroskedasticity series to one another, using VAR type models to test for Granger (1969) causality effects. This procedure is discussed in the next section.

6.2 Data and Model

Transmission of Conditional Volatility

Daily data on the stock market indices for 21 national markets have been obtained from the Financial Times for the period 12/31/85 - 9/30/93. Means, standard errors, skewness and kurtosis are listed in Table 6.1. The series listed include 17 of the 18 markets examined by Engle and Susmel (1993) in their study of weekly data. While Singapore is not a part of our study, Mexico, New Zealand, Ireland and South Africa have been included. The 21 national markets have been grouped by region or approximate time zone to avoid the difficulty of attempting to interpret possibly spurious dynamic relationships. That difficulty might arise if some markets open substantially later than other markets. The regional groupings of this study, Europe, Far East and North America, are the same as those used by Engle and Susmel (1993) in their study of stock market indices from weekly data. While many researchers have transformed the data and studied returns (the percent change in the market index), this study estimates ARCH models with a trend variable on stock index levels. This choice has been made because of the high frequency of the data used and because the level is what is traded in the market.3

Table 6.1 indicates that 10 of 18 series with significant skewness have negative skewness and 18 of 21 series have significant kurtosis. Using weekly return data, Engle and Susmel (1993) find negatively significant skewness for all European markets, except those of Austria, Belgium, and Denmark. In our study 11 out of 14 European markets have significant skewness, with 7 of 11 having negative skewness.4 Unit root test results, using the augmented Phillips and Perron (1988) test, are reported in Table 6.2. Tests for lags up to 9 are performed, although only results for lag 2 are reported. Column PP(2) reports significant unit roots tests for all series in a test without a trend (the 5% and 1% critical values are -2.86 and -3.43, respectively). Column PPT(2) reports significant unit roots for detrended data for all series (the 5% and 1% critical values are -3.41 and -3.96, respectively). Since unit roots have been found for all raw data series, it is imperative that the residuals of the estimated ARCH-class models be tested for unit roots and that trend terms be included in the estimated models. If the residuals of these ARCH-class models are found not to have unit roots, then the interpretation is that the estimated model has filtered out the nonstationarity, or in the case of ARCH-class models of one series, the series is cointegrated with its lags (Hamilton, 1994 pp. 651-653). This result is obtained and the test statistics for the residuals are listed in the PP'(2) and PPT'(2) columns. Before discussing these results in detail, we will discuss the ARCH class models estimated.

Transmission of Conditional Volatility

Table 6.1 Descriptive Data on Stock Market Index Series

───────────────────────────────────────────────────────────────────────────

MarketMeanS. E.Skewness Kurtosis

European Markets

Austria141.126 52.286.73069*-.24549*

Belgium126.776 22.054-1.257* .98463*

Germany101.836 17.466.07290-.98726*

Spain135.453 26.324-1.240*1.19108*

France 125.100 28.074-.3230*-1.0309*

Italy 80.833 14.135-.0068-.59138*

Netherlands127.591 25.367-.0062-.82080*

Switzerland 94.495 14.331 .6966* .28238*

United Kingdom149.021 28.832-.6562*-.39903*

Ireland139.976 29.649-.6070*-.10091

Denmark183.648 60.635-.1846*-1.5507*

Norway160.054 44.014 .3511*-.64874*

Sweden150.929 38.791-.3300*-1.0332*

South Africa163.802 47.363 .1884*-.74403*

Far Eastern Markets

Australia129.577 23.031-.9116*-.0578

Hong Kong144.649 61.4521.0122*-.01811

Japan136.495 34.564-.2410*-.47678*

New Zealand 65.541 19.5791.0003*1.2365*

North American Markets

Mexico626.070582.622 .7389*-1.0994*

United States135.475 27.813 .2289*-1.1224*

Canada125.516 15.387-.6258*-.27282*

─────────────────────────────────────────────────────────────────────────────

Levels of significance < .05 for skewness and kurtosis are marked with a *. All data are daily from 12/31/85 to 9/30/93.

Transmission of Conditional Volatility

Table 6.2 Stationarity Tests on Stock Market Index Series

────────────────────────────────────────────────────────────────────

MarketPP(2) PP'(2)PPT(2)PPT'(2)

European Markets

Austria-1.43 -35.71-1.34-36.24

Belgium-2.99 -42.11-3.11-42.14

Germany-1.83 -38.41-2.50-38.43

Spain-3.43 -34.40-2.92-34.42

France -1.97 -38.57-2.93-38.80

Italy-2.75 -30.81-3.79-31.15

Netherlands-0.88 -26.75-3.13-26.75

Switzerland-0.82 -37.74-1.73-37.75

United Kingdom-2.14 -25.91-3.38-25.94

Ireland-2.52 -35.84-2.69-35.88

Denmark-1.28 -36.71-1.30-37.43

Norway-1.65 -28.30-1.56-28.31

Sweden-2.07 -33.90-2.40-34.58

South Africa-1.95 -37.85-2.35-37.88

Far Eastern Markets

Australia-2.65 -42.60-2.57-42.63

Hong Kong 0.63 -40.83-1.20-40.89

Japan-2.44 -32.27-2.33-32.27

New Zealand-1.46 -36.71-2.88-36.91

North American Markets

Mexico 0.47 -32.57-1.62-35.41

United States-0.93 -40.21-3.13-40.23

Canada-2.33 -19.63-1.92-19.67

PP(2) = Phillips-Perron test with 2 lags on raw series.

PP'(2)= Phillips-Perron test with 2 lags on ARCH model

residual.

PPT(2) = Phillips-Perron test with a trend and 2 lags on

raw series.

PPT'(2)= Phillips-Perron test with a trend and 2 lags on ARCH model residual.

5% and 1% critical values are -2.86 and -3.43, respectively, for tests without trend terms and -3.41 and -3.96 for tests with trend terms. Augmented Dickey-Fuller tests have also been run but have not been reported, since the findings are similar to those from the augmented Phillips-Perron tests. Only lag 2 results are reported. Tests up to lag 9 for models with and without trends have been examined with no change in the basic finding. Coefficients for the ARCH model estimated are reported in Table 6.3. For data sources, see text.

Transmission of Conditional Volatility

Table 6.3 Basic ARCH Models from 12/31/85-9/30/93

β0 β1 β2τ µα0 α1 L

European Markets

1 Austria .0593 1.065 -.0646 .000019 -.15840 2.4659 .57057 -2321.5

(.310) (23.41) (-1.42) (.184)(-3.345) (74.19)(15.19)

2 Belgium .7500 1.021 -0.0271 -.000098-.03039 1.3944 .18353 -1500.09

(2.53) (3.97) (-.106) (-1.16) (-.116) (47.54) (7.94)

3 Germany .7588 .1546 .8365 -.0002 -.85688 1.5568 .19615 -1622.10

(1.97) (.827) (4.89) (-1.47) (-4.82) (57.35) (12.31)

4 Spain .4811 1.1859 -.1890 .00002 .05971 2.1316 .2916 -2016.8

(2.28) (8.54) (-1.36) (.241) (.417) (48.42) (15.38)

5 France .6988 1.0416 -.0489 -.0003 -.09386 1.7497 .23246 -1776.3

(3.17) (8.51) (-.401) (-2.46) (-.789) (37.46) (10.76)

6 Italy .9405 .9557 .0361 .00022 -.2489 1.0201 .2511 -1249.7

(3.82) (11.99) (.456) (3.83) (-3.98) (35.82) (9.45)

7 Netherlands 1.0956 .2972 .6913 -.00049 -.7180 1.1558 .2759 -1390.6

(2.46) (1.23) (2.87) (-2.24) (-3.11) (34.14) (11.18)

8 Switzerland .0810 1.7686 -.7697 -.00003 .74419 1.04014 .1647 -1229.4

(1.19) (10.31) (-4.51) (-1.13) (4.038) (40.57) (12.38)

9 U. K. .8573 1.1381 -.1454 -.00029 .048042 2.2195 .17283 -1970.1

(2.59) (6.61) (-.850) (-2.25) (.279) (47.61) (8.40)

10 Ireland .4676 1.3082 -.3115 -.00004 .18153 2.7267 .2767 -2246.2

(2.32) (13.19) (-3.19) (-.484) (1.68) (47.07) (15.46)

11 Denmark .1954 .9625 .0365 -.00009 -.1789 3.3080 .2911 -2448.5

(.878) (8.06) (.307) (-.713) (-1.56) (53.24) (14.57)

12 Norway .0727 1.3961 -.3960 .00003 .2197 3.7151 .4304 -2680.0

(.480) (18.94) (-5.37) (.415) (2.77) (30.70) (17.82)

13 Sweden .6239 1.1495 -.1551 -.0003 -.0249 2.6322 .4525 -2354.2

(3.47) (14.02) (-1.90) (-3.11) (-.289) (44.01) (15.91)

14 S. Africa .5209 1.3542 -.3587 -.0002 .3170 8.9728 .1659 -3365.1

(1.66) (4.37) (-1.16) (-1.63) (.983) (49.23) (7.50)

Far Eastern Markets

1 Australia .5248.8852 .1124 .000147 -.26229 1.5225 .53227 -1851.6

(2.29) (9.471)(1.21) (2.08)(-3.00) (28.79)(37.22)

2 New Zealand .2118 1.861-.8636 .000057.73351 .57665.58673 -890.9

(6.66) (129.7)(-61.1) (4.48) (24.7) (35.7)(26.8)

3 Hong Kong .0011 1.9712 -.9722 -.00013 .82677 2.6168 1.0481 -2547.7

(.074) (371.0) (-183.3) (-9.23) (88.6) (43.62) (22.14)

4 Japan .6194 .8613 .1367 .00020 -.2963 2.6791 .3373 -2278.8

(2.20) (9.24) (1.46) (2.07) (-3.55) (32.27) (14.31)

North American Markets

1 Mexico .1162 1.2273 -.2284 -.0010 -.2489 37.33 .16395 -5660.4

(.233) (129.8) (-24.27) (-1.65) (-35.99) (56.67) (31.97)

2 U. S. .5503 .9976 -.0027 -.0002 -.0765 1.1961 .2030 -1357.0

(1.99) (4.23) (-.011) (-1.40) (-.333) (48.19) (22.68)

3 Canada .4260 .9564 .0405 .00003 -.3117 .5234 .3050 -613.8

(2.27) (17.48) (.75) (.69) (-6.79) (35.98) (20.40)

The ARCH model estimated maximizes Li = -.5(log(vit) + (eit)²/vit), where

eit = xit - βi0 - βi1xit-1 - βi2xit-2 + τit + µieit-1. vit = αi1(eit-1)² . xit = stock market index for the ith market for the tth period. t scores in are given parenthetically ( ).

Transmission of Conditional Volatility

If we define xit as the stock market index for the ith market in the tth period, then the ARCH model suggested by Engle (1982) maximizes Li,

where

Li = -.5(log(vit) + (eit)²/vit) (6.2-1)

eit = xit - βi0 - βi1xit-1 - βi2xit-2 + τit + µieit-1(6.2-2)

and

vit = αi0 + αi1(eit-1)². (6.2-3)

The above model contains a trend term τi, a moving average term µi and first- and second-order autoregressive terms, βi1 and βi2. The series eit is the residual of the means equation, while vit is the estimated conditional heteroskedasticity (or volatility) for the ith series. If αi1 is significant, it indicates that the conditional volatility of the ith market is predictable. The results for estimating equations (6.2-1), (6.2-2) and (6.2-3) for the 21 markets in the period 12/31/85 - 9/30/93 are given in Table 6.3 and the stationarity tests on the residual series are given in Table 6.2. Since all values for PP'(2) and PPT'(2) are below the critical values listed above, we find that the estimated ARCH model has filtered the market index series such that the residual is stationary. This finding gives us confidence to proceed with tests of the dynamic relationships between the estimated conditional heteroskedasticity (or volatility) measures vit. There is evidence of variance persistence or predictability, since αi1 in equation (6.2-3) is significant for all markets with all t scores greater than 7.50. The dynamic properties of vit for i=1,...,21 are of special interest, since if we could show that vit-k Granger (1969) causes vjt for i j and k > 0, then it would be possible to make predictions on volatility in the jth market based on volatility in the ith market. Because αi1 has already been found to be significant, predictions of volatility in the jth market could be made.5 Before moving to formal Granger tests, we will inspect the plots of the estimated vit for all markets, which are given in Figures 6.1, 6.2 and 6.3.

Transmission of Conditional Volatility

Conditional or predicted volatility data for all 21 markets are plotted in compact 3 by 3 format, with the markets of United Kingdom, United States and Germany displayed in the first row of each graph for comparison. First, note that except for Germany, Austria and to some extent Mexico, there is substantial volatility in all markets in the fall of 1987 during the world-wide stock market crash. In examining regional markets such as North America, we are struck by the similarity between the United States and Canada (see Figure 6.1), in contrast with Mexico, which shows substantial volatility in the latter third of the sample (1991-1993) (see Figure 6.3). In the Far East, Australia and New Zealand are the most similar (see Figure 6.1), while in Europe, Germany and France are quite similar (see Figure 6.2). Next we systematically test the dynamic relationships between these conditional volatility measures.

Transmission of Conditional Volatility

6.3 Transmission of Volatility Across National Markets

To test for Granger (1969) causality of conditional heteroskedasticity or volatility across markets within a region, VAR models of order 7 were estimated for the 14 markets in the European time zone, four markets in the Far East and three markets in North America. The lag length of 7 has been selected to remove all autocorrelation and cross correlation among the residuals and to extend 7 trading days. To save space, the estimated VAR coefficients have not been reported. Granger causality significance tests are reported in Tables 6.4 and 6.5. The reported statistic Si,j gives the significance level of a test of whether conditional volatility in the jth country Granger (1969) causes conditional volatility in the ith country. Table 6.4 reports test results from 12/31/85 - 9/30/93, while Table 6.5 reports test results from 1/1/88 - 9/30/93. The subperiod has been selected to remove the effect of the worldwide stock market crash in the fall of 1987. In the analysis that follows, the results for these two periods are contrasted.

Transmission of Conditional Volatility

Looking first at the 3 by 3 market model estimated for North America, we observe Mexico has no effect on Canada or the United States, since S3,1 and S2,1 are .096 and .0008 in the complete period and .40 and .15 in the subperiod respectively. In the complete period, the United States maps to Canada (S3,2= 1.000) and Canada maps to the United States (S2,3 = .9993). The feedback from Canada to the United States appears to be caused by the 1987 crash, since in the subperiod the effect occurs only from the United States (S3,2 = .994) to Canada rather than from Canada to the United States (S2,3 = .163). This finding is similar to what was found by Cheung and Kwan (1992).

In the Far East, in a four-market model, the effect of the 1987 crash appears as feedback which is not observed in the subperiod. Looking first at Japan in the subperiod, we find S4,j is not significant for any case in which j 4, indicating that conditional volatility in Japan is not influenced by conditional volatility of stockmarkets in Australia, New Zealand or Hong Kong. In the complete period, S4,j is significant for j=1,...,4. The same effect is observed for Hong Kong in which the other three markets have significant effects in the complete period but none in the subperiod. In the subperiod, Australia affects New Zealand (S2,1=1.00), with no feedback (S1,2=.72), while in the full period there is a two-way relationship between Hong Kong and Australia (S1,3 = 1.00, S3,1 = .9999) and between Hong Kong and Japan (S3,4=.992, S4,3= 1.000). In the complete period, one-way relationships are found for New Zealand to Australia (S1,2 = .998), New Zealand to Japan (S4,2=.998) and Australia to Japan (S4,1 = .9999).