Persistence and Long-range Dependence in the Emerging Stock Market of Kuwait

Persistence and Long-range Dependence in the Emerging Stock Market of Kuwait

Assaf, A


A major issue in financial economics is the behavior of stock returns over long horizons. This study provides empirical evidence of the long-range dependence in the emerging stock market returns and volatility of Kuwait. We test for long memory in the daily returns, absolute returns, squared returns and modified log-squared returns. The measures of long-term persistence employed are the modified rescaled range statistic R/S proposed by Lo (1991), the rescaled variance V/S statistic developed by Giraitis et al. (2003), and the semiparametric Gaussian estimator of Robinson (1995). We also examine the volatility process using the FIGARCH (Fractionally Integrated GARCH) model of Baillie et al. (1996). Significant long memory is established as a feature of both the returns and volatility. This evidence supports the proposition that the emerging stock market of Kuwait has an underlying fractal structure and is consistent with the evidence reported by studies on developed markets. Our results should be useful to regulators, practitioners and derivative market participants, whose success depends on the ability to forecast stock price movements.

Keywords: Long Memory, R/S, V/S, Semiparametric Gaussian Estimator, Emerging Markets, Kuwaiti Stock Exchange

JEL classification: G1; G12; G14; G15


It is commonly observed that asset returns, whilst approximately uncorrelated, are temporally dependent. In particular, the autocorrelation functions of various volatility measures – absolute and squared returns – decay at a very slow mean-reverting hyperbolic rate. This feature is labeled a “long memory” or “long-range dependence”. The long memory property describes the high-order structure of a series. If a series exhibits long memory, there is persistent dependence even between distant observations. Such series are characterized by distinct but nonperiodic cyclical patterns. The presence of long memory dynamics causes nonlinear dependence in the moments of the distribution and hence a potentially predictable component in the series dynamics. Mandelbrot (1977) characterizes such processes as having fractal dimensions. On the other hand, the short memory, or short term dependence, property describes the low-order correlation structure of a series. For short memory series, correlations among observations at long lags become negligible.

Given the implications of long memory for the theory and practice of financial economics, considerable attention has been directed to the analysis of long memory in asset returns. Long-range dependence has been found in the returns of a variety of assets classes. Long memory analysis has been conducted for exchange rate returns (e.g., Cheung 1993, Cheung and Lai 1993, 1995, and Fisher et al. 1997), index and commodity futures returns (e.g., Helms, Kaen, and Rosenman, 1984, Milona, Koveos, and Booth, 1985, Kao and Ma, 1992, Eldridge, Bernhardt, and Mulvey, 1993, Fang, Lai, and Lai, 1994, and Corazza, Malliaris, and Nardelli, 1997), and European, Asian and American equity market returns (e.g., Greene and Fielitz, 1977, Lo, 1991, Nawrocki, 1995, Cheung, Lai, and Lai, 1993, and Jacobsen, 1996).

Despite the extensive research into the empirical aspects of this relation in the well-developed financial markets, less is known about it in emerging securities markets. Emerging markets are typically much smaller, less liquid, and more volatile than well known world financial markets (e.g., Domowitz, Glen, and Madhavan, 1998). There is also some evidence that emerging markets may be less informationally efficient. This could be due to several factors such as poor-quality (low precision) information, high trading costs, and/or less competition due to international investment barriers. Further, the industrial organization found in emerging economies is often quite different from that in developed economies. These conditions may contribute to a different structure of asset price determination and dynamics in emerging financial markets1.

In this study we investigate the presence of fractional dynamics in the returns and volatility of the emerging stock market of Kuwait2. The focus on the Kuwaiti stock market is appropriate for a number of reasons. First, Kuwait is one of the countries in the Gulf Cooperation Council (GCC) countries and is becoming an increasingly important component of the regional economy, with its equity market becoming an integral segment of the regional GCC stock markets3. Understanding the behavior of this market is thus an important undertaking. Second, this market allows comparison of developed markets with maturing markets to determine if the returns-generating process and presence or absence of fractional dynamics depends on the degree of market development4. Third, the presence of long-memory dynamics in stock market returns would provide evidence against the weak form of market efficiency as it implies non-linear dependence in the first moment of the distribution and hence a potentially predictable component in the series dynamics. Fourth, the presence of fractal structure in this market may reflect fractal dynamics in the underlying economy which, in turn, would be of value in modeling business cycles.

A further application for the long memory lies in the volatility of financial time series. Ding, Granger, and Engle (1993) examine the long memory properties of several transformations of the absolute returns of daily S&P 500 data series and found considerable evidence of long memory in volatility. The work of Ding et al. (1993) has motivated a variety of studies on long memory property in volatility. Based on the evidence of strong persistent autocorrelation in the squared and absolute value of returns, one might be interested in applying long memory tests to the squared observations or other pertinent variables. We analyze long-memory in the volatility by considering the series of absolute returns, squared returns, and modified log-squared returns as proxies of the volatility measures. We test for long memory by using the modified rescaled range statistic (R/S) developed by Lo (1991), the rescaled variance (V/S) statistic proposed by Giraitis, Kokoszka Leipus and Teyssiere (2003) and the semiparametric Gaussian estimator of Robinson (1995). We further estimate an ARFIMA (p,d,q) model for returns and volatility and examine the volatility process using the FIGARCH (Fractionally Integrated GARCH) model of Baillie et al. (1996). Significant long memory is conclusively demonstrated in both the returns and volatility. The evidence is invariant to the methods used in either testing or estimating long memory components. Our findings support the proposition that returns and volatility in the emerging stock market of Kuwait have an underlying fractal structure, and disputes the hypothesis of market efficiency. Therefore, the Kuwaiti stock market, even with its different institutions and information flows than the developed market, presents similar fractal structure to the preponderance of studies employing other developed markets. The implication is that differences in institutions and information flows in this market are not that important enough to affect the valuati on process of equity securities and produce similar results to those occurring in developed markets.

The paper is organized as follows. Section 2 provides the empirical methodology and describes the tests and estimators employed. Section 3 presents the empirical results. Section 4 contains a summary of our findings and concluding remarks.

Empirical Methodology

Table 1 summarizes the statistical properties of the returns: we show the first four moments, the autocorrelation coefficient at lag one and the Ljung and Box test statistic for autocorrelation in returns and squared returns. First, the higher variability of the Kuwaiti stock market returns is visible. Considering the autocorrelation of returns, at lag one the index has a value of 0.093 and is significant at the 5% level. We also observe two stylized facts for return series which has universal validity, as documented in the survey by Pagan (1996). The first stylized fact is nonnormality of the unconditional distribution of returns in the form of leptokurtosis. This phenomenon has been termed fat tails. The second stylized fact is that the volatility of returns is time-varying. This dependence is indicated by the significant Ljung-Box Q(20) test statistics showing strong autocorrelation in squared returns. Thus, this time-dependence in volatility shows that a specification which omits the dynamics in variance neglects an important characteristic of the time series.

Table 1 also includes the results of KPSS tests proposed by Kwiatkowski et al. (1992) for the null hypothesis of short memory against long-memory alternatives. We consider two tests, denoted by Const and Trend based on a regression on a constant, and on a constant and time trend, respectively. As the table shows, the trend-stationarity null hypothesis is strongly rejected for return series. Therefore, the return series cannot be characterized as I(0) processes, which suggests that a fractionally differenced process may be an appropriate representation for these series.

The results from the R/S statistic are reported in Table 2. R/S is sensitive to the order of truncation q and there is no statistical criteria for choosing q in the framework of this statistic. Andrew’s (1991) rule gives mixed results. If q is too small, this statistic does not account for the autocorrelation of the process, while if q is too large, it accounts for any form of autocorrelation and the power of this test tends to its size. For that reason, Teverovsky, Taqqu and Willinger (1999) suggest to use this statistic with other tests. Since there is no data driven guidance for the choice of this parameter, we consider different values for q=0,2,4,6,8,10,15,25,50. At the 5% significance level, the null hypothesis of a short-memory process is rejected if the modified R/S statistic does not fall within the confidence interval [0.809, 1.862]7.

For returns, the null hypothesis is rejected for all lag orders, except for q=25 and 50. The result illustrates the issue of the choice of the bandwidth parameter q. We reject the null hypothesis of no long-memory up to lag 15. However, when q=25, and 50, this null hypothesis is accepted, as the power of this test is too low for these levels of truncation orders. The results for the volatility measures differ from those of the returns series. For absolute and squared returns, in only one case, where q=50, can the null hypothesis of no long memory not be rejected at the 5% significance level. For modified log-squared returns, long memory is present at all lags.

In Table 3, we report the V/S statistic results. We consider the same lag order as in the case of R/S statistic. There is strong, robust evidence that the null hypothesis is strongly rejected for returns and volatility series, and the presence of long memory is similar in value across all measures and for each lag order. Those results fall in line with those reported by R/S analysis. The combined evidence from both statistics is a good representation of the data generating process, and suggests that a fractionally differenced process may be appropriate representation for these series.

We further estimate the long memory parameter (d) in returns and volatility, by applying Robinson’s (1995b) estimator. The following set of bandwidth parameters for this estimator is used: m=500,400,300,350,250, and 150. Table 4 reports our results. Long memory is found in the returns, absolute returns, squared returns and modified log-squared returns. The estimated d values fall between 0 and 0.5, a property of the fractionally integrated processes, in their ability to capture long memory in returns and volatility with a slower rate of decay in their autocorrelation function than the exponential decay of standard autoregressive moving average ARMA (d=0) process.8 This evidence of long-range dependence is qualitatively the same across different choices of the bandwidth employed and volatility measures. However, for squared returns, long memory parameter estimates are lower than those of absolute and modified log-squared returns.

Further Analysis

The ML estimates are broadly consistent with the semi-parametric Gaussian estimates of Robinson (1995b) and more supportive of long memory. We have obtained d estimates that are significant for all series, with long memory effects far more pronounced in volatility than in returns series. For absolute returns, there is a stronger evidence of long memory than that for squared returns. The values of d lie in the interval from 0 to 1/2, typically around 0.3, indicating positive dependence between distant observations for returns and volatility.

Where Γilde;(.) is the gamma function and v is the degrees of freedom parameter. A second point concerns the minimum number of observations required to estimate the FIGARCH model. This number is related to the order of the expansion of the fractional filter (1-L)^sup d^. Because of the positive value of d, it is advisable to use a sufficiently high truncation lag order. In this respect, we chose a truncation order equal to 1000.

In order to assess the relevance of the FIGARCH specification, we also estimate a GARCH model. The estimation results are included in Table 6. Unsurprisingly, the Student’s-t distribution is found to outperform the normal. Simple likelihood ratio tests point out that the degree of freedom v needs to be included in the estimation procedure. As a whole, Table 6 suggests that the FIGARCH specification is supported by the data. Indeed, in all cases, the parameter d is highly significantly different both from 0 and 1, rejecting the validity of both the GARCH and the IGARCH specifications. Hence, there is strong support for the hyperbolic decay and persistence as opposed to the conventional exponential decay associated with the stable GARCH (1,1). Finally, a sequence of diagnostic statistics is provided and fails to detect any need to further complicate the model. These tests are skewness (b^sub 3^) and kurtosis (b^sub 4^) values as well as the Box-Pierce statistics of the residuals (Q(20)) and the squared residuals (Q^sub 2^(20)) at lag equal to 20. In general, the estimations carried out with assumed conditional Gaussian errors exhibit kurtosis, which tends to motivate further the use of a Student’s-t distribution. As a whole, our MA(1) FIGARCH (l,d,l) model and Student’s-t distribution seems a satisfying representation to our data.


One of the important questions in studies of asset return and volatility has been how long the effects of shocks persist. This is particularly important for emerging financial markets. In this article, the modified R/S statistic of Lo (1991), the V/S statistic of Giraitis et al. (2003) and the semiparametric Gaussian estimator of Robinson (1995) are applied to investigate the long memory properties in return and volatility of the emerging stock market of Kuwait. We further investigate the long memory in volatility by estimating the fractional parameter d within an ARFIMA (p,d,q) model and the FIGARCH model of Baillie et al. (1996). Significant long memory is found, not only in returns, but in absolute returns, squared returns and modified log-squared returns.

Some immediate benefits emerge from the results of this study. First, our results show that the Kuwaiti Stock Market had an underlying fractal structure, and disputes the hypothesis of market efficiency. Second, our results have an important bearing on the pricing of equity derivatives. While options in the Gulf region are still limited to over-the-counter trading, the market appetite for the risk-return profile of the Gulf-based derivatives is strong. As trading in equity derivatives strengthens, it is important that the pricing models used include an assumption about long-range dependence. Third, our results would be useful for investors employing asset allocation strategies and may demonstrate that, despite their infancy, equity prices in Kuwait exhibit characteristics akin with more mature markets. We conclude that stock market returns and volatility in Kuwait, even with its institutions and information flows than the developed market, present similar return-generating process and fall in tandem with those patterns observed in the more mature stock markets of the developed countries.

1 For recent research on emerging markets and discussions of some of the differences between emerging and developed markets, see Errunza (1994); and Harvey (1995).

2 For a presentation of the institutional, organizational and structural aspects of the Kuwaiti Stock Market, see Al-Loughani, N. and Chappell, D. (2001).

3 Research on these markets has focused on the issue of efficiency as well as on their integration with international markets. Butler and Malaikah (1992) examine individual stock returns in both the Kuwaiti and Saudi Arabian markets over the second half of the 1980s and conclude with market inefficiency in both markets. Darrat and Hakim (1997) examine price linkages among three Arab stock markets (Amman, Cairo and Casablanca) and their integration with international markets, and find that theses markets are integrated within the region but not at the international level. Darrat and Pennathur (2002) studied economic and financial integration among the countries in the Arab Maghreb region (Algerian, Morrocco, and Tunisia) and found that they share a robust relation binding together their financial and economic policies. Abraham et al. (2002) examine the random walk properties of three Gulf stock markets – Kuwait, Saudi Arabia, and Bahrain – after correcting for infrequent trading. They cannot reject the random walk hypothesis for the Saudi and Bahrain markets, however, the Kuwaiti market fails to follow a random walk even after the correction. Two other studies examined the efficiency of the Kuwaiti stock market: Al-Loughani (1995) and Al-Loughani and Moosa (1997) and concluded with stationarity but not random walk.

4 The Kuwaiti stock market is constrained by a number of factors: there is a lack of market makers; a significant part of economic activity remains under government control; short selling is illegal and there is no facility for securities lending and borrowing; information disclosure requirements are lax; and derivatives are not available.

5 We provide a brief description of the estimator and the reader is referred to Robinson (1995a) and Robinson (1995b) for further explanation.

6 Data provided by the National Bank of Kuwait.

7 The fractiles are given in Lo (1991).

8 Some figures were made for the periodogram and the spectrum of absolute returns series. Both the spectrum and the periodogram have a pole at the zero frequency, which is a signature of the presence of long-memory in the volatility series. This long-range dependence is illustrated by plotting of the autocorrelation function of the series of absolute returns up to lag 150. There is a significant autocorrelation observed even at a long lag. Figures could be made available upon request.


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