Chaotic dynamics in pacific rim capital markets.

Pandey, Vivek K. ; Kohers, Theodor ; Kohers, Gerald 等

INTRODUCTION

The informational efficiency of financial markets has been an age old, yet intriguing topic of debate among finance theorists and practitioners. Random-walk tests employed in the past have established the weak-form efficiency of financial markets in most major industrialized nations. However, recent advances in the study of nonlinear deterministic systems have uncovered chaotic processes that can generate data series that may appear random to linear science. These discoveries have sparked a renewed rigor in the examination of capital market efficiency. Moreover, recent deliberations about viewing economies as evolutionary dynamical processes lend credence to the hypothesis that aggregate stock market behavior may be driven by a "vision of the future" (Grabbe 1996) and hence may embody an underlying deterministic mechanism. In light of these recent developments, investigations of underlying chaotic deterministic mechanisms in stock market aggregates has taken on an increased significance.

Grabbe (1996) presents the possibility of self-organization of human societies, and thus by implication of the economy, with a shared image or a vision of the future. At the singular level, this vision might be subconscious or nonexistent, but at the aggregate level such a vision might be discernible. In international stock markets, a large volume of the trading occurs while traders are speculating. They may not afford the luxury of acting late on any relevant news. Very often, the trader must anticipate other traders' moves and try to preempt such moves. As such, each trader must not just act on his or her expectations but rather act on anticipation of other traders' moves who themselves are trying to anticipate the first's and everyone else's moves and so on. Evolutionary dynamics provide a solution in the form of spontaneous order involving dynamic feedback at a higher, or aggregate, level. In the international stock markets context, what appears to be competition amongst traders and institutions at the lower level, where expectations are generated, functions as co-ordination at the higher (global) level. Hence it is likely that even in face of rational expectations, stock market aggregates, such as country market indexes used in this study, may be generated by some form of complex deterministic mechanism. As such market aggregates may not be priced efficiently in the traditional sense.

The subject of informational efficiency of U.S. financial markets continues to receive much attention in the literature (for examples, see Atkins and Dyl (1990); Ball and Kothari (1989)). More recent studies have begun the task of employing chaos theory in testing the efficiency of financial markets (e.g., Brock et al. (1987, 1991); Scheinkman and LeBaron (1989); Hsieh (1989, 1991, 1993, 1995); Kohers et al. (1997); Pandey et al. (1998) and Willey (1992)).

On the international level, several significant developments have created an increased interest in the efficiency of international markets. For example, relatively recent developments in financial market deregulation, the gradual lifting of restrictions on capital movements, the relaxation of exchange controls, major progress in computer technology and telecommunications, as well as a significant increase in the cross-listings of multinational company stocks have all led to a substantial rise in global stock market activities. Furthermore, the improvements in communication and computer technology not only have made the flow of international information cheaper and more reliable, but also have lowered the cost of international financial transactions. In addition, greater coordination in trade and capital flows policies among the industrialized nations may have contributed to more similar economic conditions and developments in these countries, which would be reflected in their respective stock markets. Largely as a result of these developments, many experts suggest that, especially in recent years, stock markets have moved toward a far greater degree of global integration, which has led to a renewed interest in the efficiency of foreign financial markets

In examining the pricing efficiency of stock markets, the vast majority of research has relied on linear modeling techniques which have serious limitations in detecting multi-dimensional patterns. Using recently developed methodology, this study intends to broaden the limited scope of previous research on the subject. This approach employs powerful statistical tools to detect low dimensional deterministic chaos in the stock markets of the U.S. and four major Pacific Rim countries. More specifically, in testing the efficiency of the national stock indexes representing the United States, Australia, Hong Kong, Japan, and Singapore, this paper employs tests for nonlinear dynamics, a process which, in comparison to traditional linear models, is capable of detecting more complex patterns that otherwise appear to be random. Each country's stock market index is examined individually to determine if the time series is generated by some form of deterministic chaos. Country indexes exhibiting low-dimensional deterministic chaos may contain some informational inefficiency (in the weak form sense); thus, it may be possible to use nonlinear dynamics to predict future stock returns.

The remainder of this paper is organized as follows. Section II reviews the relevant literature related to research dealing with efficiency and nonlinear dynamics in the major international markets. The methodology, including a brief description of the nonlinear dynamics approach, time frame, and data selection for the six major stock markets is explained in Section III. The findings are discussed in Section IV. The paper concludes with a summary in Section V.

RELATED LITERATURE

Most research on global market efficiency has dealt with the systematic movements of stock prices, the lead-lag relationship among market indexes, and the benefits of diversification (e.g., Cochran et al. (1993), Maldonado and Saunders (1981) and Panton et al. (1976)).

Studies employing tests based on nonlinear dynamics have just begun to surface in the literature. Brock et al. (1987, 1991), Hsieh (1989, 1991, 1993, 1995), Kohers et al. (1997), Pandey et al. (1998), Scheinkman and LeBaron (1989) and Willey (1992), have found a preponderance of evidence that residuals of whitened stock index returns are not IID (independently and identically distributed). The evidence is mixed as to whether this rejection of IID results from nonlinear dependencies or nonstationarity of data series.

Very few studies have examined international stock markets for the existence of chaotic dynamics. Frank et al. (1988), found some evidence of chaotic determinism in international markets. Mercado-Mendez and Willey (1992) generated evidence of chaotic processes in the Japanese Nikkei index. However, they did not find evidence of chaos in the Financial Times of London Industrial Index and the Dow Jones Industrial Average returns. Sewell et al. (1993) document nonlinear dependencies in the stock markets of Hong Kong, Korea, Japan, Singapore, and Taiwan, while Errunza et al. (1994) identify similar nonlinear dependencies in the markets of Germany, Japan, and the emerging markets of Argentina, Brazil, Chile, India, and Mexico. Omran (1997) reports finding strong evidence of the existence of nonlinear dependancy in the U.K. stock markets. Pandey et al. (1998) find preponderance of evidence of nonlinear dynamics in the aggregate stock market indexes of U.K., Switzerland, France, Italy and the U.S. However, they were unable to conclude that low-dimensional deterministic systems were the driving mechanism behind any of the examined indexes. In summary, the existing evidence on the presence of chaotic processes in international stock markets is highly sketchy and inconclusive.

Previous studies have provided useful information on various aspects of efficiency of financial markets and their respective degrees of linkage to each other. However, very little evidence exists on the nonlinear dynamics of the major global stock markets. Tests for chaotic determinism, in contrast to most traditional linear-form tests, are capable of detecting more complex patterns which otherwise appear to be random. Thus, by utilizing a statistical methodology specifically designed to detect low-dimensional deterministic chaos in the major Pacific Rim stock markets, this study fills an important void in the existing literature.

DATA AND METHODOLOGY

This study examines the stock markets of four major Pacific Rim countries along with the U.S. equity market. Specifically, the sample used in this research consists of the weekly national stock indexes of the following countries: Australia, Hong Kong, Japan, Singapore, and the United States. These indexes, representing market-weighted price averages, were retrieved from Morgan Stanley Capital International Perspective (MSCI) of Geneva, Switzerland. The indexes represent stock markets worldwide for which data was available on a consistent and reliable basis. The combined market capitalization of the companies that comprise the indexes represents approximately 60 percent of the aggregate market value of the various national stock exchanges. Since these national indexes are constructed on the basis of the same design principles and are adjusted by the same formulas, they are fully comparable with one another. The Morgan Stanley Capital International indexes are considered performance measurement benchmarks for global stock markets and are accepted benchmarks used by global portfolio managers as well as researchers (e.g., Cochran et al. (1993)). Each one of the country indexes is composed of stocks that broadly represent the stock compositions in the different countries.

Attempting to detect systematic patterns in the movements of the various global stock market indexes by using a common currency would introduce a serious bias. Specifically, any pattern detected using a common currency could be attributable to: (a) movements in the stock market, (b) movements in foreign exchange rates, and (c) any combination of the two. To avoid the possibility that any detected systematic pattern is due to foreign exchange rate developments, the various national stock markets are measured in terms of their respective local currencies.

The period examined in this study extends from February 23, 1978 through March 27, 1997. To avoid biases arising from possible structural shifts from regime changes and other shifts in market dynamics, the overall time frame is also subdivided into two subperiods of approximately equal length, that is February 23, 1978--September 10, 1987, and September 17--March 27, 1997.

Testing for Nonlinear Dynamics:

Prior to proceeding with their examination for nonlinear determinism, each index returns series is filtered for linear correlations using autoregressive models of order p denoted AR (p) of the form:

[[gamma].sub.t] = [[PHI].sub.0] + [[summation].sub.i=1,p] [[phi].sub.i] [[gamma].sub.t-1] + [[omega].sub.t]

where [[omega].sub.t] is a random error term uncorrelated over time, while [omega] = ([[omega].sub.n]) is the vector of autoregressive parameters. The lags (or order 'p') used in the autoregressions for the appropriate model are determined via the Akaike Information Criterion (AIC) (see Akaike (1974)).

In examining the efficiency of financial markets, the first step lies in testing for the randomness of security or portfolio returns. Such an approach was adopted in earlier studies of market efficiency using linear statistical theory and very general nonparametric procedures. Examinations of chaotic dynamics have revealed that deterministic processes of a nonlinear nature can generate variates that appear random and remain undetected by linear statistics. Hence, this study employs tests that have recently evolved from statistical advances in chaotic dynamics. One of the more popular statistical procedures that has evolved from recent progress in nonlinear dynamics is the BDS statistic, developed by Brock et al. [1987], which tests whether a data series is independently and identically distributed (IID).

The BDS statistic, which can be denoted as [[W.sub.m,T].([epsilon]) is given by

[[W.sub.m,T].([epsilon]) = [square root of] T [[C.sub.m,T].sup.([epsilon]) - [C.sub.1,T].sup.([epsilon])]/[[sigma].sub.m,T].sup.([epsilon])

where: T = the number of observations,

[epsilon] = a distance measure,

m = the number of embedding dimensions,

C = the Grassberger and Procaccia correlation integral, and

[[sigma].sup.2] = a variance estimate of C.

For more details about the development of the BDS statistic, see Appendix A. Simulations in Brock et al. demonstrate that the BDS statistic has a limiting normal distribution under the null hypothesis of independent and identical distribution (IID) when the data series consists of more than five hundred observations. The use of the BDS statistic to test for independent and identical distribution of pre-whitened data has become a widely used and recognized process (e.g., Brorsen and Yang (1994), Hsieh (1991, 1993), Kohers et al. (1997), Pandey et al. (1998), Sewell et al. (1992), Willie (1992)). After data has been pre-whitened and nonstationarity is ruled out, the rejection of the null of IID by the BDS statistic points towards the existence of some form of nonlinear dynamics.

Rejection of the null hypothesis of IID by the BDS is not conclusive evidence of the presence of chaotic dynamics. Structural shifts in the data series can be a significant contributor to the rejection of the null. To avoid biases arising from structural shifts from regime changes and other shifts in market dynamics, the sample period of February 1978--December 1996 is subdivided into two subperiods (i.e., February 23, 1978--September 10, 1987 and September 17, 1987--March 27, 1997) which are examined individually for the violation of the IID assumption.

Furthermore, in order to ascertain whether the data series are, indeed, a result of chaotic processes, two other tests are performed. The Modified Rescaled Range (R/S) analysis is a powerful indicator of long-term persistence of a series where the influence of a set of past returns on a set of future returns is effectively captured. In addition, the three moments test effectively distinguishes deterministic (mean) nonlinearities from nonlinearities of variance, the latter which could result from a stochastic rather than deterministic influence.

The R/S statistic, which was developed by Hurst (1951), has been used in several studies for the purpose of detecting long term dependencies in time series data. Over the years, a number of modifications and refinements have been made to the classical R/S statistic (e.g., see Lo (1991)). According to Lo (1991), one of the drawbacks of the classical R/S statistic is that it detects short-term as well as long-term dependency, but does so without distinguishing between them. Thus, if a time series were to have strong short-term dependencies, the R/S statistic may be biased towards an indication that long-range dependence also exists.

The Rescaled Range analysis is based on the simple hypothesis that any IID data would show an increase in standardized ranges which are proportional to increase in sample sizes as samples of increasing subperiod lengths are considered. If increases in standardized ranges are less than (more than) proportional to increasing sample sizes, then the data is persistent (antipersistent) and not IID. Both the R/S statistic and the Modified R/S statistic have been utilized by researchers as a measurement for detecting long range dependencies in time series data.

The classical R/S statistic is defined as:

[Q.sub.n] [equivalent to] 1/[s.sub.n] [Max 1<=k<=n [[summation].sub.j=1,k] ([x.sub.j]- [bar.x])- Min 1<=k<=n [[summation].sub.j=1,k]([x.sub.j]- [bar.n])

where: n = the length of the returns series being examined,

k = length of contiguous subperiods contained within n,

[x.sub.j] = observation within subperiod of length k, and

[S.sub.n] = standard deviation of the sample series.

Hence the R/S analysis gives one a fair idea of whether the examined time series shows any temporal persistence (antipersistence). To control for possible short-term dependence in time series data, Lo (1991) incorporates sample autocovariances in the construction of the Modified R/S statistic. See appendix B for more details on the development of the R/S and the Modified R/S analysis. The R/S analysis shows robustness even in face of small sample sizes.

Hsieh (1989, 1991) and Brock et al. (1991) developed the three moments test to specifically capture mean nonlinearity in a given series. Briefly stated, this test uses the concept that mean nonlinearity implies additive autoregressive dependence, whereas variance nonlinearity implies multiplicative autoregressive dependence. Using this notion and exploiting its implications, Hsieh (1989, 1991) constructed a test that examines the third order moments of a given series. Additive dependencies will lead to some of these third order moments being correlated. By its construction, this test will not detect variance nonlinearities. ****page 134 The third order sample correlation coefficients are computed as:

[r.sub.(xxx)(i,j)] = [1/T [summation] [x.sub.t] [x.sub.t-i] [x.sub.t-j]] / [1/T [summation] [x.sub.t.sup.2]].sup.1,5]

where: [r.sub.(xxx)](i,j) = the third order sample correlation coefficient of xt with xt-1 and xt-j, and

T = the length of the data series being examined.

Hsieh (1991) developed the estimates of the asymptotic variance and covariance for the combined effect of these third order sample correlation coefficients which can be used to construct a [chi square] statistic to test for the significance of the joint influence of the [r.sub.(xxx)](i,j)'s for specific values of j, such that 1<= i <= j. If the [chi square] statistics for relatively low values of j are significant, this outcome would be a strong indicator of the presence of mean nonlinearity in the examined series. As chaotic determinism is a form of mean-nonlinearity, the three moments test provides strong evidence of the presence of chaos.

The Grassberger and Procaccia (1983) correlation dimension test is a graphical measure of identifying a chaotic attractor in a chaotic series. The GP correlation dimension test utilizes the correlation integral, [C.sub.m,T].sup.([epsilon]), (also used in the development of the BDS statistic) to compute probabilities of data points in clustering sequences (see Appendix A for a more detailed description). In addition, to obtain evidence of chaotic dynamics, a graphical examination of the slope of log [C.sub.m,T].sup.([epsilon]) over log (g) for small values of g is obtained. This value, [C.sub.m,T] (g)/log ([epsilon]), is the point estimate of the Grassberger and Procaccia correlation dimension v. When plotted against m, the number of embedding dimensions, the point estimate of v (i.e., [v.sub.m]) will converge to a constant beyond a certain m if the data series is indeed chaotic. Appendix A provides a more detailed description of the development of the GP correlation dimension v.

The intuition that lies behind the search for convergence of the GP correlation dimension, ?m, into a single value ? as m, the dimensionality, increases is as follows. A two-dimensional relationship can be characterized as a line which retains its two-dimensional form whether viewed in a map of two, three, or more dimensions. Similarly, an m dimensional relationship will retain its m dimensional form when examined in m or more dimensions. A truly random series, however, retains no form in any dimensionality and fills up the entire space. The GP correlation dimension is a measure of fractional dimensionality, and will remain constant when examined in a space-map of dimensions higher than the identified GP dimension for a deterministic series. It is important to note here that the GP correlation dimension is a relative measure of the fractional dimensionality of a series and does not actually identify the absolute dimensionality of the examined series.

Hence, first the examined index returns are filtered for linear influences using autoregressive filters. The lag lengths for these AR processes are determined by using the Akaike Information Criteria (AIC) (see Akaike (1974)). Next, each of these series is tested for the null of IID using the BDS statistic. The above process is then repeated for subperiods of equal length to rule out the possibility that any observed rejection of the IID assumption in the previous step was due to nonstationarity of the data series. Furthermore, an examination of the rescaled ranges will reveal if the rejection of IID, in case it is observed, results from temporal dependencies. Finally, the three moments test and the GP correlation dimension test is used to determine if the observed nonlinearity is, indeed, mean nonlinearity to provide conclusive evidence of the existence of chaotic determinism.

RESULTS

Aside from its ability to detect nonlinear relationships, the BDS statistic, by its very design, is also sensitive to linear processes. Since this study concerns itself with the detection of nonlinear dynamics in stock return series, autocorrelations were filtered out using autoregressive models. The appropriate lag length for each series was determined using the Akaike Information Criterion (AIC). Table 1 shows the appropriate lag lengths used for each country's index returns examined.

Table 2 lists the BDS statistics for each data series for dimensions m = 2, ..., 10 and the distance measure [epsilon] = 0.5 [sigma], 0.75 [sigma], and 1.00 F. The BDS statistic has an intuitive explanation. For example, a positive BDS statistic indicates that the probability of any two m histories, ([x.sub.t], [x.sub.t-1], ..., [x.sub.t-m+1]) and ([x.sub.s], [x.sub.s-1], ..., [x.sub.s-m+1]), being close together is higher than what would be expected in truly random data. In other words, some clustering is occurring too frequently in an m-dimensional space. Thus, some patterns of stock return movements are taking place more frequently than is possible with truly random data.

An examination of the results in Table 2 reveals that most BDS statistics are significantly positive. For each country index, the BDS statistics are computed for [epsilon] = 1.00 [sigma], 0.75 [sigma], and 0.50 [sigma]. A lower [epsilon] value represents a more stringent criteria since points in the m-dimensional space must be clustered closer together to qualify as being "close" in terms of the BDS statistic. Hence, [epsilon] = 0.50 [sigma] reflects the most stringent test, while [epsilon] = 1.00 [sigma] is the most relaxed criterion used in this analysis.

In this study, the values of m examined go only as high as 10. Two reasons dictate the choice of 10 as the highest dimension analyzed. First, with m = 10, only 99 non-overlapping 10 history points exist in each return series. Examining a higher dimensionality would severely restrict the confidence in the computed BDS statistic. Second, the interest of this study lies only in detecting low dimensional deterministic chaos. High dimensional chaos is, for all practical purposes, as good as randomness (see, e.g., Brock et al. (1987)).

One must remember that the BDS statistic only reveals whether or not the data series examined is different from a random identically and independently distributed (IID) series. The results in Table 2 represent a summary rejection of the null hypothesis of IID for all the stock index series examined. However, it is possible that the stock index series examined reject the IID hypothesis because of nonstationarity in the data. Exogenous influences such as regime changes or regulatory reforms, among others, could impact the stock returns series in such a way that they give the appearance of not being random (although they are truly random in stable times) over the 19 year period under investigation in this study.

Tables 3 and 4 provide the BDS statistics for the same 5 country stock index series for the subperiods February 23, 1978 through September 10, 1987 and September 17, 1987 through March 23, 1997, respectively. An examination of the BDS statistics for the subperiods reveals a different picture from that which emerged for the overall period. Due to the small sample size (i.e., 498 observations per country index for each subperiod), one cannot expect the BDS statistic of IID data to be asymptotic standard normal. Brock et al. (1991) provide results of simulations with 1,000 repetitions for a small sample of 500 observations. These simulation results are adopted in this study as benchmarks for evaluating the significance of the computed BDS statistics for the two subperiods, as reported in Tables 3 and 4.

An examination of Table 3 reveals that the filtered returns for the U.S. index is IID over subperiod #1. This observation is consistent with those presented in Table 4 for subperiod #2. However, earlier, Table 2 had revealed that the U.S. stock index provided a rejection of the null of IID for the overall period. Such inconsistency suggests that the returns data for U.S. is plagued with nonstationarity. Consequently, the filtered returns for this index can not be examined for long-term nonlinear dependencies.

A majority of the BDS statistics computed over subperiods #1 and #2 (Tables 3 and 4) for the index returns of Australia, Hong Kong, Japan, and Singapore do provide summary rejection of the null hypothesis of IID. These results confirm the observations made from the examination of Table 2. Hence, one may conclude that the filtered index returns of Australia, Hong Kong, Japan, and Singapore are not significantly plagued by nonstationarity to impede further analysis. The next step in the examination is to attempt to determine if the rejection of the IID hypothesis is the result of any latent memory effects or chaotic dynamics present in these returns.

Table 5 reports the results for the weekly returns during different time periods for the index representing Australia, Hong Kong, Japan, Singapore, and the U. S. The table contains the classical R/S ([V.sub.n]) and four Modified R/S statistics ([V.sub.n](q)). The four Modified R/S statistics are computed with q-values of 12, 24, 36, and 48 weeks. The "%-Bias" is also reported. The %-Bias is the estimated bias of the classical R/S statistic ([V.sub.n]) and is computed as (([V.sub.n])/[V.sub.n](q)-1)x100.

A total of 996 weekly returns are available for each Australia, Hong Kong, Japan, and the U. S., and 986 weekly returns for Singapore. These returns are divided up into subperiods of two equal lengths. Table 5 indicates that in order to be statistically significant at the 5% level, the R/S and Modified R/S values should fall within the interval of 0.809 and 1.862. For Australia, Hong Kong, Japan, and the U.S., the results reveal that at the 5 percent level of significance, no long-term dependency is present in the total data set or any of the two subperiods. However, the Singapore index does provide indication of the presence of some long-term dependancy during subperiod #1. This persistence is significantly affected by smaller q values of 12 and 24 with bias of 6.5% and 11.6% respectively, which gives an indication of potential short-term dependencies. However, the R/S statistics for the same index during the second subperiod does not indicate any persistence. Hence the R/S statistic fails to provide a strong indication of long-term dependancy for the entire data set analyzed for any index.

The results of the three moments test are presented in Table 6. This table shows the [chi square] statistics for a combined test of the significance of all examined three moment correlations [r.sub.(xxx)](i,j) up to a certain lag length. Where 1 <= i <= j <= 5, the [chi square] statistic has 15 degrees of freedom. On the other hand, when all three moment correlations are examined up to a lag length of 10 (i.e., 1 <= i <= j <= 10), the [chi square] statistic has 55 degrees of freedom. As one may observe from Table 5, the [chi square](15) statistics for the Japanese and the Singapore index returns are significant. These results suggest that these two index returns may be influenced by low-dimensional chaos. The [chi square](55) statistics reveal that the Singapore index returns may have high-dimensional and hence more complex chaotic patterns.

After identification by the three moments test as possibly driven by chaotic deterministic processes, the Japanese and the Singapore index returns are subjected the graphical procedure of the determination of the Grassberger and Procaccia (1983) correlation dimension. Convergence of the point estimates of the GP correlation dimensions to a single number would provide evidence of the existence of a chaotic attractor which, in turn, would give conclusive evidence of the existence of a deterministic process. As observed from Figure 1, the point estimates of the GP correlation dimension for the Japanese index returns remain well below 2.5 even when the number of embedded dimensions rises up to twenty. Data limitations prevent the examination of higher order of embedded dimensions, hence one is unable to obtain evidence of the existence of a chaotic attractor. When the same returns series is randomly scrambled (and thereby destroying the temporal order), one observes that the point estimates of the GP correlation dimension rises rather rapidly with embedding dimensions. This provides us with an indication that there might be some chaotic influences present in the observed returns series. Similar observations can be obtained for the Singapore index returns where the point estimates of GP correlation dimension remains below 3.5 even when the embedding dimensions rises to twenty.

To sum up the findings of this study, the filtered index returns of all examined countries exhibit non-IID behavior. However, the index returns of U.S. is afflicted by nonstationarity of data series. As such, this index could not be examined further. The rescaled range analysis provides no strong indication of long-term dependency in the returns of any of the examined indexes. However, short-term dependencies may be prevalent in Singapore index returns. Finally, the three moments test revealed that the Japanese and the Singapore index returns appear to be influenced by a low-dimensional chaotic process. The Singapore index returns may be driven by a relatively more complex, high-dimensional, chaotic patterns. The GP correlation dimension analysis provides further evidence to support the above observations. However, in the absence of identification of a chaotic attractor, very possibly an artifact of data limitations, one is unable to obtain conclusive proof of the existence of deterministic chaos.

SUMMARY AND CONCLUSIONS

This study examined the weekly returns for five country stock indexes (i.e., Australia, Hong Kong, Japan, Singapore, and the United States). To briefly restate the methodology, the tools used in this paper are autoregressive filters to remove linear influences, the BDS statistics to check for the violation of the IID assumption, the rescaled range analysis to attain an idea of long-term temporal persistence, Hsieh's three moments test to obtain an indication of chaotic determinism, and the Grassberger and Procaccia correlation dimension analysis performed in search of conclusive evidence of chaotic determinsm. The results of this research provide evidence to suggest that some form of nonlinear influence abounds in the index returns of Australia, Japan, Hong Kong, and Singapore. Although nonstationarity of data precluded the investigators from drawing conclusions about nonlinear influence in the index returns of the U.S., it is probable that during stable periods, the U.S. index may also be influenced by a nonlinear process. The Japanese index returns do appear to be afflicted by a low-dimensional chaotic process. In contrast, the Singapore index returns seem to be affected by a more complex, higher-dimensional chaotic form.

The above findings have important implications concerning the predictability of these index returns. The nonlinear influence that appears to be prevalent in the Australian and Hong Kong index returns are not deterministic. As such, the weak-form efficiency of these markets is not compromised by these findings. Although data limitations hamper the search for conclusive evidence of the existence of chaotic deterministic patterns in these indexes, the Japanese and Singapore stock indexes do provide strong indication of being driven by a low-dimensional deterministic pattern. These results imply that potential predictability exists in this index returns. On the other hand, a chaotic process need not necessarily imply the existence of exploitable profit opportunities, either due to their complexity which may defy the identification of parameter values of the process itself or for the simple fact that their high sensitivity to initial conditions may magnify minute errors in forecasts to a point beyond the needed accuracy required for profit taking.

Several recent studies have examined the influence of conditional heteroskedasticity on stock returns. In one of these studies, Errunza et al. (1994) uncovered significant ARCH effects in the Japanese stock market. However, it must be pointed out that discovering the influence of stochastic nonlinearity, such as ARCH effects, does not imply predictability of stock returns over an extended period of time. By isolating the effects of mean nonlinearities, the tests employed in this study investigate the possibility of predictability of the returns series examined. The evidence of chaotic nonlinearities presented in this research suggests that the returns in the Japanese and Singapore stock markets are potentially predictable. Consequently, the results presented here lend credence to evolutionary dynamic model of aggregate stock price behavior. As such, these results may pose a challenge to the existence of informational efficiency in these stock markets and may give investors a renewed incentive to attempt to identify specific return patterns in the hope that the chaotic price generating process can yield economically significant profit opportunities.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

APPENDIX A

BDS STATISTIC AND GRASSBERGER AND PROCACCIA CORRELATION DIMENSION

An efficient way to test for chaos is to consider a measure called the correlation integral examined by Grassberger and Procaccia (1983), which is calculated as

[C.sub.m]([epsilon]) = [lim.sub.T-0] {(t,s), 0<t, s<T : [parallel][x.sup.m.sub.t] - [x.sup.m.sub.s][parallel]<[epsilon]} + [T.sup.2]

where: [x.sup.m.sup.t] is an m history point embedded in m dimensional space such that [x.sup.m.sub.t] = {[x.sub.t-m+1], ..., [x.sub.t]}, and [parallel][parallel] is the sup- or max- norm.

In simpler terms, the correlation integral, [C.sub.m]([epsilon]), is defined as the fraction of pairs ([x.sub.m.sup.t] [x.sub.s.sup.m]

which are close to each other in the sense that [max.sub.i=0], ... m-1] {[parallel][x.sub.s-i]-[x.sub.t-i] [parallel]} < [epsilon]

For finite sequences such as {[x.sub.t]} a scalar sequence of T observations, the Grassberger and Procaccia correlation integral may be computed as:

[C.sub.m]([epsilon]) = 2 / [T.sub.m] ([T.sub.m-1]) [[summation].sub.t<s] [I.sub.[epsilon]] ([x.sub.t.sup.m],

where: [T.sub.m] = T-(m-1),

[x.sup.m.sub.t] = Dimensional vector of m histories, i.e., ([x.sub.p][X.sub.t+1, ..., [x.sub.t+m-1]), [I.sub.[epsilon]] ([x.sup.m.sub.t], [x.sup.m.sub.s] = m) = an indicator function that equals 1 when [parallel][x.sub.t.sup.m] - [x.sup.m.sub.s][parallel] <[epsilon] and 0 otherwise. [parallel][parallel] = the sup-norm.

Brock, et al. (1987) provide a statistical test for nonlinearity using the correlation integral. They demonstrate that, under the null hypothesis that {[x.sub.[epsilon]} is independently and identically distributed (IID), with nondegenerate cumulative distribution F, for a fixed m and [epsilon]

[C.sub.m,T]([epsilon]) [right arrow] C([epsilon]).sup.m] with probability equal to 1 as T [right arrow][infinity]

where: C([epsilon]) = [integral] [F (z + [epsilon])--F (z-[epsilon]) ] dF(z)

Also, [C.sub.m,T]([epsilon])[right arrow] C[([epsilon]).sup.m] has an asymptotic normal distribution with mean zero and a known variance [[sigma].sup.2.sub.m]([epsilon]). A consistent estimate of C[([epsilon]).sup.m] is provided by [C.sub.1,T][([epsilon]).sup.m]. Then the BDS statistic, which can be denoted as

[W.sub.m,T.sup.([epsilon]) = [square root] T [[C.sub.m,T.sup.([epsilon])] - [C.sub.1,T]([epsilon])]/ [[sigma].sub.mT].sup.([epsilon]) has a limiting standard normal distribution under the null hypothesis of IID.

Furthermore, in order to ascertain whether the data series are, indeed, a result of chaotic processes, the Grassberger and Procaccia (1983) correlation dimension v is examined, i.e.,

v = [lim.sub.m[right arrow][infinity] [v.sub.m] where [v.sub.m] = [lim[epsilon][right arrow]0] log [C.sub.m,T.sup.([epsilon])] + log [epsilon]

A graphical examination of log log [[C.sub.m,T.sup.([epsilon])/([epsilon]) / log [epsilon] with respect to m, for small values of [epsilon], reveals its slope. Since [v.sub.m] is the point estimate of v, an examination of the above-mentioned graph will be illustrative. If [v.sub.m] does not increase with m, the data are consistent with chaotic processes (see Hsieh (1991)). For truly random processes, the value of v is [infinity].

APPENDIX B

THE RESCALED RANGE AND THE MODIFIED RESCALED RANGE ANALYSIS

The R/S statistic, which was developed by Hurst (1951), has been used in several studies for the purpose of detecting long term dependencies in time series data. Over the years, a number of modifications and refinements have been made to the classical R/S statistic (e.g., see Mandelbrot (1972, 1975), Mandelbrot and Taqqu (1979), and Lo (1991)). According to Lo (1991), one of the drawbacks of the classical R/S statistic is that it detects short range as well as long range dependency, but does so without distinguishing between them. Thus, if a time series were to have strong short range dependencies, the R/S statistic may be biased towards an indication that long range dependence also exists.

Both the R/S statistic and the Modified R/S statistic have been utilized by researchers as a measurement for detecting long range dependencies in time series data. In these studies, the classical R/S statistic is computed along with the Modified R/S for comparison purposes (for example, see Lo (1991) and Pan et al. (1996)).

The classical R/S statistic is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where: n = the length of the returns series being examined,

k = length of contiguous subperiods contained within n,

[x.sub.j] = observation within subperiod of length k, and

[s.sub.n] [equivalent to] [1/n [[summation].sub.j] [([x.sub.j]- [bar.n]).sup.2]].sup.1/2]

The Modified R/S statistic differs from the classical R/S in that, in addition to the variance, it incorporates the autocovariance of X in the denominator (see equations below). (For a more detailed description of the Modified R/S statistic, see Lo (1991).)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[omega].sub.j](q) [equivalent to] 1 - j/(q+1), q<n

The adjusted variance/covariance function, which is used to scale the ranges for the Modified R/S statistic may then be denoted as:

[[sigma].sup.2.sub.n](q) [equivalent to] [[sigma].sup.2.sub.x] + 2 [[summation].sub.j=1,q][[omega].sub.j](q)[[gamma].sub.j]

When properly normalized, as the sample size n increases without bound, the rescaled range converges in distribution to a well-defined random variable V, that is,

1/[square root of]n [Q.sub.n] [??] V

The cumulative distribution function of V takes on the following form: [F.sub.v](v) = 1 + 2 [[summation].sub.k=1,[infinity] (1-[4k.sup.2][v.sup.2])[e.sup.-2(kv)(kv)]

Fractiles of this cumulative distribution function are used as benchmarks for the evaluation of R/S statistics presented in Table 5.

REFERENCES

Akaike, H. (1974). A new look at statistical model identification, IEEE Transactions on Automatic Control, AC-19 (6), December, 716-723.

Atkins, A. B. & E. A. Dyl. (1990). Price reversals, bid-ask spreads, and market efficiency, Journal of Financial and Quantitative Analysis, 25(4), December, 535-547.

Ball, R. & S. P. Kothari. (1989). Nonstationary expected returns: implications for tests of market efficiency and serial correlation in returns, Journal of Financial Economics, 25(1), November, 51-74.

Booth, G. & F. Kaen. (1979). Gold and silver spot prices and market information efficiency, Financial Review, 14, 21-26.

Booth, G., F. Kaen & P. Koveos. (1982). R/s analysis of foreign exchange rates under two international monetary regimes, Journal of Monetary Economics, 10, 407-415.

Brock, W., W. Dechert, & J. Scheinkman. (1987). A test for independence based on the correlation dimension, Working Paper, University of Wisconsin at Madison, University of Houston, and University of Chicago.

Brock, W., D. Hsieh,& B. LeBaron. (1991). Nonlinear Dynamics, Chaos, and Instability, Cambridge, MA. and London: MIT Press.

Brorsen, B. W. & S. R. Yang. (1994). Nonlinear dynamics and the distribution of daily stock index returns, Journal of Financial Research, 17(2), Summer, 187-203.

Cochran, S. J., R. H. DeFina, & L. O. Mills (1993). International evidence on the predictability of stock returns, Financial Review, 28(2), May, 159-180.

Errunza, V., K. Hogan, Jr., O. Kini, & P. Padmanabhan. (1994). Conditional heteroskedasticity and global stock return distributions, Financial Review, 29(3), August, 293-317.

Frank, M., R. Gensay, & T. Stegnos. (1988). International chaos? European Economic Review, 32, 1569-1584.

Grabbe, J. Orlin. (1996). International Financial Markets, Englewood Cliffs, NJ: Prentice Hall.

Greene, M. & B. Fielitz. (1977). Long-term dependence in common stock returns, Journal of Financial Economics, 4, 339-349.

Helms, B., F. Kaen & R. Rosenman. (1984). Memory in commodity futures contracts, Journal of Futures Markets, 4, 559-567.

Hsieh, D. (1995). Nonlinear dynamics in financial markets: evidence and implications, Financial Analysts Journal, 51(4), 55-62.

Hsieh, D.(1993). Implications of nonlinear dynamics for financial risk management, Journal of Financial and Quantitative Analysis, 28(1), March, 41-64.

Hsieh, D. (1991). Chaos and nonlinear dynamics: Application to financial markets, Journal of Finance, 5, December, 1839-1877.

Hsieh, D. (1989). Testing for nonlinear dependence in daily foreign exchange rates, Journal of Business, 62(3), 339-368.

Hurst, H. E. (1951). Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 116, 770-799.

Kohers, T., V. Pandey & G. Kohers. (1997). Using nonlinear dynamics to test for market efficiency among the major us stock exchanges, The Quarterly Review of Economics and Finance, 37(2), Summer, 523-545.

Kamarotou, H. & J. O'Hanlon. (1989). Informational efficiency in the U.K., U.S., Canadian and Japanese equity markets: A note, Journal of Business Finance and Accounting, 16(2), Spring, 183-192.

Lo, A. (1991). Long-term memory in stock market prices, Econometrica, 59, 1279-1313.

Lo, A. & C. MacKinlay. (1988). Stock market prices do not follow random walks: Evidence from a simple specification test, Review of Financial Studies, 1, 41-66.

Maldonado, R. & A. Saunders. (1981). International portfolio diversification and the inter-temporal stability of international stock market relationships, 1957-78, Financial Management, Autumn, 54-63.

Mandelbrot, B. (1972). Statistical methodology for non-periodic cycles: From the covariance to r/s analysis, Annals of Economic and Social Measurement, 1, 259-290.

Mandelbrot, B.(1975). Limit theorems on the self-normalized range for weakly and strongly dependent processes, Z. Wahrscheinlichkeitsheorie verw. Gebiete, 31, 271-285.

Mandelbrot, B. & M. Taqqu. (1979). Robust r/s analysis of long-run serial correlation, Bulletin of the International Statistical Institute, 48(2), 59-104.

Mercado-Mendez, J. & T. Willey. (1992). Testing for nonlinear dynamics in international stock indices, Working Paper, Central Missouri State University.

Omran, M. F. (1997). Nonlinear dependence and conditional heteroskedasticity in stock returns: UK evidence, Applied Economics Letters, 4, 647-650.

Pan, M. S., A. Lin & H. Bastin. (1996). An examination of the short-term and long-term behavior of foreign exchange rates, Financial Review, 31(3), August, 603-622.

Pandey, V., T. Kohers & G. Kohers. (1998). Deterministic nonlinearity in major european stock markets and the U.S., The Financial Review, 33(1), February, 45-63.

Peters, E. (1991). Chaos and Order in Capital Markets: a New View of Cycles, Prices, and Market Volatility, New York: John Wiley and Sons, Inc.

Scheinkman, J. &B. LeBaron. (1989). Nonlinear dynamics and stock returns, Journal of Business, 62, 311-337.

Schollhammer, H. & O. Sand. (1985). The interdependence among the stock markets of major european countries and the united states: An empirical investigation of interrelationships among national stock price movements, Management International Review, 25(1), 17-26.

Sewell, S. P., S. R. Stansell, I. Lee, &M. S. Pan. (1993). Nonlinearities in emerging foreign capital markets, Journal of Business Finance and Accounting, 20(2), January, 237-248.

Watson, J. (1980). The stationarity of inter-country correlation coefficients: A note, Journal of Business Finance and Accounting, Summer, 297-303.

Willey, T.(1992). Testing for nonlinear dependence in daily stock indices, Journal of Economics and Business, 44(1), February, 63-76.

Vivek K. Pandey, The University of Texas at Tyler

Theodor Kohers, Mississippi State University

Gerald Kohers, Sam Houston State University

Table 1

Autoregression Lags Used to Filter Returns on the Stock Markets Analyzed

Country Stock Market Index Autoregressive Model Used to Filter

Australia AR(2)
Hong Kong AR(4)
Japan None
Singapore AR(3)
United States AR(8)

NOTE: AR = Autoregressive model with (x) lags. Lags are determined via
the Akaike Information Criterion.

Table 2
BDS Statistics for Filtered Returns for Pacific Rim Stock Markets
February 23, 1978-March 27, 1997

 Country Stock Market Index
 [epsilon]/
m [sigma] Australia Hong Kong

2 0.50 3.1300 4.2612
3 0.50 3.9393 4.4120
4 0.50 3.9458 4.8466
5 0.50 4.3411 5.4869
6 0.50 4.7038 6.3719
7 0.50 4.9278 7.4772
8 0.50 4.6986 9.1204
9 0.50 5.1504 11.7740
10 0.50 6.6158 12.3040
2 0.75 3.3827 4.1080
3 0.75 4.4093 4.7207
4 0.75 4.7301 5.6394
5 0.75 5.1405 6.6573
6 0.75 5.6880 7.8477
7 0.75 5.9919 8.9359
8 0.75 6.3742 10.3150
9 0.75 6.5526 11.6410
10 0.75 6.8094 13.0380
2 1.00 3.7934 4.6746
3 1.00 4.9044 5.5573
4 1.00 5.2481 6.5684
5 1.00 5.7544 7.4684
6 1.00 6.2884 8.6117
7 1.00 6.7790 9.6940
8 1.00 7.3358 10.9670
9 1.00 7.7916 12.3480
10 1.00 8.3783 13.7990

 Country Stock Market Index

m Japan Singapore U.S.

2 6.3591 3.7293 2.1248
3 8.1539 4.1775 3.0271
4 9.2897 5.0646 4.2399
5 10.4570 5.8226 5.6239
6 12.4680 6.5147 6.1754
7 14.7380 8.0701 6.2936
8 18.3800 9.5185 6.6431
9 22.5310 11.2850 7.1003
10 26.0310 13.7310 9.0366
2 6.6160 4.5920 2.3416
3 8.1922 5.3964 2.8909
4 9.5154 6.2227 3.8348
5 10.3700 7.0751 5.2128
6 11.6140 7.7191 6.1264
7 12.8740 8.2799 6.9666
8 14.3770 8.3429 7.9091
9 16.2970 8.7380 9.8188
10 18.4270 9.1412 11.2010
2 7.1931 4.9974 2.5043
3 8.7851 5.9354 3.2574
4 10.1000 6.6505 4.0146
5 10.8600 7.3793 4.8927
6 11.7270 7.8480 5.6911
7 12.5360 8.2784 6.2898
8 13.5080 8.5116 6.7730
9 14.6410 9.0591 7.6165
10 15.9120 9.5270 8.2419

NOTE: m = embedding dimension. n = 996 (Singapore: 986)

Except where noted with *, all BDS statistics are significant at
the .05 level.

Table 3
BDS Statistics for Filtered Returns for Pacific Rim Stock Markets
Subperiod 1: February 23, 1978-September 10, 1987

 Country Stock Market Index:
 [epsilon]/
m [sigma] Australia Hong Kong

2 0.50 1.6248 * 3.3690
3 0.50 1.9220 * 2.9262
4 0.50 1.7018 * 2.9165
5 0.50 2.7599 * 3.0983
2 0.75 1.9666 2.7595
3 0.75 2.6705 2.5084
4 0.75 2.9543 2.8256
5 0.75 3.4698 3.3116
2 1.00 1.8505 2.9929
3 1.00 2.4450 2.8254
4 1.00 2.5536 3.2596
5 1.00 3.0288 3.7194

 Country Stock Market Index:

m Japan Singapore U.S.

2 6.6144 2.3694 1.4527 *
3 7.7209 2.4796 1.6049 *
4 7.2947 3.2108 1.3523 *
5 7.8438 3.8152 0.6865 *
2 6.7248 2.7465 1.0832 *
3 7.7200 2.9738 1.4503 *
4 7.7984 3.3302 0.9546 *
5 8.3228 4.0173 1.0256 *
2 6.7643 2.8576 1.5277 *
3 8.0128 3.1586 1.6310 *
4 8.4112 3.5085 1.4306 *
5 8.7399 3.8625 1.4486 *

NOTE: m = embedding dimension. n = 498.

Except where noted with *, all BDS statistics are significant.

For specifics, see the following tabulated values generated by
Brock et al. (1987)

With 500 observations Significance Level: 5%

 [epsilon]/ [epsilon]/
m [sigma] = 0.50 [sigma] = 1.00

2 1.98 1.81
3 2.25 1.83
4 2.42 1.87
5 2.98 1.94

With 500 observations Significance Level: 1%

 [epsilon]/ [epsilon]/
m [sigma] = 0.50 [sigma] = 1.00

2 3.00 2.61
3 3.30 2.73
4 3.64 2.87
5 4.45 2.99

Table 4
BDS Statistics for Filtered Returns for Pacific Rim Stock Markets
Subperiod 2: September 17, 1987-March 27, 1997

 Country Stock Market Index:
 [epsilon]/
m [sigma] Australia Hong Kong

2 0.50 2.5267 2.1594
3 0.50 3.7034 2.6478
4 0.50 4.0210 2.7717
5 0.50 4.0621 2.8602
2 0.75 2.8316 2.5120
3 0.75 4.0600 3.3908
4 0.75 4.3563 4.1647
5 0.75 4.4807 4.5986
2 1.00 3.5453 3.3170
3 1.00 4.6936 4.5005
4 1.00 4.8809 5.2931
5 1.00 4.8281 5.8935

 Country Stock Market Index:

m Japan Singapore U.S.

2 2.7775 2.9679 1.6618 *
3 2.7492 3.7833 2.1253 *
4 2.4594 4.1501 2.9296
5 2.3935 4.2360 2.6362
2 3.2993 3.7671 0.0136 *
3 3.3511 4.7843 0.7012 *
4 3.7124 5.5072 1.9988 *
5 3.8649 5.8612 2.8289
2 4.3092 4.1822 0.3434 *
3 4.6357 5.0306 1.0603 *
4 5.2381 5.5164 2.2423
5 5.4630 5.9359 3.0077

NOTE: m = embedding dimension. n = 498 (Singapore: 488).

Except where noted with *, all BDS statistics are significant.

For specifics, see the following tabulated values generated
by Brock et al. (1987)

With 500 observations Significance Level: 5%

 [epsilon]/ [epsilon]/
m [sigma] = 0.50 [sigma] = 1.00

2 1.98 1.81
3 2.25 1.83
4 2.42 1.87
5 2.98 1.94

With 500 observations Significance Level: 1%

 [epsilon]/ [epsilon]/
m [sigma] = 0.50 [sigma] = 1.00

2 3.00 2.61
3 3.30 2.73
4 3.64 2.87
5 4.45 2.99

Table 5
Classical R/S and Modified R/S (q = 12, 24, 36, and 48) Statistics
with %/Bias ((Vn/Vn(q)-1) x 100)

Sample Size Vn Vn(12) %-Bias

 Australia

All 1.11 1.18 -6.1%
Subperiod 1 1.48 1.62 -8.6%
Subperiod 2 0.92 0.98 -6.2%

 Hong Kong

All 1.11 1.18 -6.1%
Subperiod 1 1.48 1.62 -8.6%
Subperiod 2 0.92 0.98 -6.2%

 Japan

All 1.69 1.61 5.6%
Subperiod 1 1.42 1.43 -1.0%
Subperiod 2 1.32 1.30 2.0%

 Singapore

All 1.21 1.15 5.2%
Subperiod 1 2.07 * 1.95 * 6.5%
Subperiod 2 1.21 1.17 3.0%

 U.S.

All 1.15 1.18 -1.8%
Subperiod 1 1.03 1.21 -14.3%
Subperiod 2 0.86 0.80 7.6%

 Fractiles of the Distribution FV(v)

P(V<v) .025 .050 .100
v 0.809 0.861 0.927

Sample Size Vn(24) %-Bias Vn(36)

 Australia

All 1.22 -9.2% 1.25
Subperiod 1 1.59 -6.7% 1.47
Subperiod 2 1.01 -9.4% 1.11

 Hong Kong

All 1.22 -9.2% 1.25
Subperiod 1 1.59 -6.7% 1.47
Subperiod 2 1.01 -9.4% 1.11

 Japan

All 1.54 10.1% 1.50
Subperiod 1 1.42 -0.4% 1.46
Subperiod 2 1.30 2.1% 1.29

 Singapore

All 1.17 3.8% 1.21
Subperiod 1 1.86 * 11.6% 1.79
Subperiod 2 1.22 -1.4% 1.33

 U.S.

All 1.24 -7.0% 1.34
Subperiod 1 1.32 -21.5% 1.48
Subperiod 2 0.85 1.1% 0.88

 Fractiles of the Distribution FV(v)

P(V<v) .200 .800 .900
v 1.018 1.473 1.620

Sample Size %-Bias Vn(48) %-Bias

 Australia

All -10.8% 1.26 -11.6%
Subperiod 1 0.9% 1.38 7.7%
Subperiod 2 -17.5% 1.21 -24.3%

 Hong Kong

All -10.8% 1.26 -11.6%
Subperiod 1 0.9% 1.38 7.7%
Subperiod 2 -17.5% 1.21 -24.3%

 Japan

All 13.1% 1.46 16.3%
Subperiod 1 -2.9% 1.43 -0.5%
Subperiod 2 2.5% 1.29 2.4%

 Singapore

All 0.1% 1.25 -2.8%
Subperiod 1 15.9% 1.74 19.3%
Subperiod 2 -9.1% 1.40 -14.0%

 U.S.

All -13.8% 1.44 -19.6%
Subperiod 1 -29.9% 1.56 -33.9%
Subperiod 2 -2.1% 0.90 -4.3%

 Fractiles of the Distribution FV(v)

P(V<v) .950 .975 .995
v 1.747 1.862 2.098

Table 6
Chi-Square Statistics for the Joint Influence of Three Moment
Correlations for the Filtered Index Returns

Lags Statistic Australia Hong Kong

1<=i<=j<=5 [chi square] (15) 2.61 26.00
1<=i<=j<=10 [chi square] (55) 25.93 17.30

Lags Japan Singapore U.S.

1<=i<=j<=5 49.74 * 52.07 * 20.69
1<=i<=j<=10 21.63 123.22 * 18.70

* Significant at the 1% level for a right-tailed test