- Email: [email protected]
Long-range dependence and market structure
Chaos, Solitons and Fractals 31 (2007) 995–1000 www.elsevier.com/locate/chaos
Long-range dependence and market structure Daniel O. Cajueiro a, Benjamin M. Tabak a
b,*
Universidade Cato´lica de Brası´lia—Mestrado em Economia de Empresas, SGAN 916, Mo´dulo B—Asa Norte, DF 70790-160, Brazil b Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900 Brasilia, Brazil Accepted 19 October 2005
Communicated by Prof. M.S. El Naschie
Abstract In this paper, we have found that although the Dow Jones Average Industrial Index does not possess long-range dependence in mean returns, individual stocks that form the index do. These results were obtained using the Local Whittle estimation procedure. Most stocks seem to be anti-persistent with Hurst exponents below 0.5, which is in line with mean reversion in the long run. Furthermore, open–open returns possess a stronger degree of anti-persistence than close–close returns due to market structure effects. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Long memory describes the dynamics of the correlation structure of time series at long lags. Testing for long memory in stock market returns has been in the research agenda for the past 20 years as it has many implications for portfolio and risk managers.1 If stock returns are said to possess long-range dependence, then one could build long-term forecasting models that could improve the portfolio management process. Furthermore, one could also forecast more precisely long-term volatilities, which is an essential input in both portfolio and risk management. Most studies have focused on stock market indices, which are aggregate indices. However, if a fractal structure exists in individual stock returns, itÕs presence may not be captured in aggregate indices. Therefore, studying individual stock returns may provide further insights in this debate. This study investigates the presence of long-range dependence in individual stocks from the Dow Jones for the US. This is one of the most important stock indices in the world and all stocks are highly liquid. Therefore, considerations of market liquidity should play no role in driving the results. Previous research has documented differences in open–open and close–close returns due to market structure for the US. Amihud and Mendelson [1] studied the New York Stock Exchange (NYSE) and compared the volatility of returns using open–open returns and close–close returns. The NYSE opens with a call auction, but trades as a specialist dealer market thereafter. Therefore, open–open returns should reflect the influence of the opening auction. Furthermore, *
1
Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045. E-mail address: [email protected] (B.M. Tabak). See [5–8].
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.077
996
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
close–close returns reflect the effect of the specialist dealer market. The authors in line with another study due to [13] concluded that NYSE open–open returns are more volatile than closing returns. These studies motivate testing for longrange dependence in both close–close and open–open returns and testing for significant differences. These differences could be attributed to market microstructure effects. [9] studied individual stocks from the Dow Jones Index and found that these stocks possess negative serial correlation. Furthermore, by comparing close–close and open–open returns the authors find that overreaction may account for the differences in the return behavior of opening and closing returns. This paper adds to the literature by testing whether Hurst exponents may help in assessing differences in market structure. We test for individual stocks that comprise the Dow Jones Industrial Average Index verifying whether Hurst exponents differ for close–close and open–open returns. In the next section, we present the method that is employed in the paper to estimate Hurst exponents. Section 3 presents the data, while in Section 4 we discuss the empirical results. Finally, Section 5 concludes the paper.
2. The Local Whittle estimator In this paper, the Local Whittle estimator is used to provide the Hurst exponent. A given market is said to have longrange dependence with persistent behavior if the Hurst exponent H > 0.5, with anti-persistent behavior if H < 0.5 and random walk behavior if H = 0.5. The Local Whittle estimator is a semi-parametric estimator, which only requires specifying the parametric form of the spectral density when the frequency k is close to zero, f ðkÞ GðH Þjkj12H ;
as k ! 0;
ð1Þ
when G(H) is a constant. The computation involves an additional parameter m, an integer less than N/2, where N is the size of the time series, and such that, as N ! 1, 1 m þ ! 0. m N
ð2Þ
This means that as N gets larger, m gets larger as well, although slower. For a spectral density of the form (1), the Whittle approximation of the Gaussian likelihood function is obtained by minimizing ! m 1 X Iðkj Þ 12H QðG; H Þ ¼ þ logðGk Þ ; ð3Þ j m j¼1 Gk12H j where kj = 2pj/N and I(kj) is periodogram. So this estimator sums the frequencies only up to 2pm/N. b Replacing above G by its estimate G, m X Iðkj Þ b ¼1 . G m j¼1 k12H j
One may define b H Þ 1 ¼ log RðH Þ ¼ Qð G;
ð4Þ
! m m 1 X IðkÞ 2H 1 X logðkj Þ . m j¼1 k12H m j¼1 j
ð5Þ
Robinson [12] showed that under certain technical assumptions, b ¼ arg min RðH Þ; H
ð6Þ
converges in probability to actual value H, i.e., b H Þ !d Normalð0; 1=4Þ. m1=2 ð H
ð7Þ
b converges to H. On the other hand, Therefore, the choice of m is quite important. The larger the value of m, the faster H if the series also presents short range behavior, then m should be small. In this paper, in order to ensure faster converb to H, the limiting value of m ¼ N 1 is used.2 gence of H 2
2 One should note that using this limiting value of m, larger frequencies are not considered. Therefore, the short range dependence phenomena should not be affecting our conclusions.
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
997
Table 1 Bloomberg codes Bloomberg code
Name
MMM UN equity AA UN equity MO UN equity AIG UN equity AXP UN equity BA UN equity CAT UN equity C UN equity KO UN equity DIS UN equity DD UN equity XOM UN equity GE UN equity GM UN equity HPQ UN equity HD UN equity HON UN equity IBM UN equity INTC UQ equity JNJ UN equity JPM UN equity MCD UN equity MRK UN equity MSFT UQ equity PFE UN equity PG UN equity SBC UN equity UTX UN equity VZ UN equity WMT UN equity INDU index
3M CO Alcoa Inc Altria Group Inc American International Group American Express Co Boeing Co Caterpillar Inc Citigroup Inc Coca-Cola Co/The The Walt Disney Co. Du Pont (EI) De Nemours Exxon Mobil Corp General Electric Co General Motors Corp Hewlett-Packard Co Home Depot Inc Honeywell International Inc Intl Business Machines Corp Intel Corp Johnson & Johnson Jpmorgan Chase & Co McdonaldÕs Corp Merck & Co. Inc. Microsoft Corp Pfizer Inc Procter & Gamble Co SBC Communications Inc United Technologies Corp Verizon Communications Inc Wal-Mart Stores Inc Dow Jones Indus. Avg
3. Data The series studied include 30 companies included in the Dow Jones Industrials Index. We also include the aggregate index for comparison purposes. All analysis is done on stock return series. The series were retrieved from the Bloomberg database. Table 1 presents the codes and the names of the companies that enter the sample. The New York Stock Exchange opens at 13:30 (GMT) and the first record of the INDU index for that day is registered at 13:35. The market closes at 20:00 (GMT) and the last record of the day is registered at 20:05. Overall, our sample period consists of approximately 3524 days. The sample starts in January 2, 1990 and ends in December 19, 2003.
4. Results Previous research has found that US stocks are predictable to some extent.3 Furthermore, technical analysis has been found to add value for portfolio management. For example, in a seminal paper [4] presented evidence suggesting that technical trading rules may benefit investors. Their analysis of the closing levels of the Dow Jones Industrial Average Index suggests that a variety of simple technical rules can be used to generate returns in excess of a buy and hold
3
See [10,11].
998
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
strategy. This research suggests that US stock returns possess serial correlation, which could be exploited in trading systems. We present evidence of long-term dependence for many stocks that comprise the Dow Jones Industrial Average Index. Table 2 presents Hurst exponents for close–close returns estimated using the Local Whittle methodology. Table 3 presents results for open–open returns. It is striking that in both cases we cannot reject the null of absence of longrange dependence for the aggregate index (Dow Jones Industrial Average Index—INDU) but this certainly is not true for many individual stocks. Most Hurst exponents are below 0.5, which suggests negative serial correlation (mean reversion in the long run), in line with the findings of [9]. We employ a nonparametric statistic to test for changes in the median of the distribution of Hurst exponents. The Kruskal–Wallis (KW) statistic, testing for difference in the median of Hurst exponents for open– open and close–close returns, is 17.19 with a p-value of 0.00, suggesting that we can reject with a 1% significance level the null of equality of medians. The medians for close–close and open–open returns are 0.48 and 0.46, respectively. This suggests that open–open returns possess a stronger degree of anti-persistence. The degree of anti-persistence is much stronger in open–open than in close–close returns. Approximately 50% of stocks present long-range dependence in close–close returns, while this figure increases to more than 83% for open–open returns. Our results are in line with [9] which finds that the tendency for overreaction in open–open returns is much stronger than for close–close returns. Furthermore, overreaction is the cause of serial correlation in opening returns.
Table 2 Hurst exponents for close–close stock returns are presented in the second column with associated standard errors in the third column, for m/2. The third and fourth columns presents a Z-statistic for the null H = 0.5 Code
H
Std. error
Z-Stat
p-Value
MMM AA MO AIG AXP BA CAT C KO DIS DD XOM GE GM HPQ HD HON IBM INTC JNJ JPM MCD MRK MSFT PFE PG SBC UTX VZ WMT INDU
0.46 0.49 0.49 0.50 0.47 0.50 0.50 0.50 0.49 0.48 0.47 0.39 0.47 0.48 0.47 0.50 0.50 0.48 0.49 0.48 0.52 0.49 0.50 0.48 0.49 0.47 0.47 0.48 0.45 0.46 0.49
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
3.72 0.50 0.54 0.29 2.14 0.08 0.37 0.14 0.51 1.76 2.90 9.01 2.26 2.06 2.31 0.23 0.07 1.72 1.24 1.70 1.30 0.52 0.20 1.28 0.92 2.14 2.71 1.94 4.41 3.38 0.90
0.00 0.31 0.29 0.61 0.02 0.47 0.35 0.45 0.30 0.04 0.00 0.00 0.01 0.02 0.01 0.59 0.47 0.04 0.11 0.04 0.10 0.30 0.58 0.10 0.18 0.02 0.00 0.03 0.00 0.00 0.18
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
999
Table 3 Hurst exponents for open–open stock returns are presented in the second column with associated standard errors in the third column, for m/2. The third and fourth columns presents a Z-statistic for the null H = 0.5 Code
H
Std. error
Z-Stat
p-Value
MMM AA MO AIG AXP BA CAT C KO DIS DD XOM GE GM HPQ HD HON IBM INTC JNJ JPM MCD MRK MSFT PFE PG SBC UTX VZ WMT INDU
0.44 0.48 0.48 0.50 0.45 0.49 0.47 0.45 0.46 0.46 0.43 0.39 0.44 0.48 0.46 0.47 0.48 0.46 0.45 0.43 0.47 0.46 0.47 0.45 0.44 0.44 0.46 0.47 0.45 0.45 0.49
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
4.71 1.46 2.00 0.35 3.93 1.15 2.15 4.09 3.48 3.80 6.07 9.13 4.83 1.87 3.62 2.67 1.85 3.66 4.46 5.64 2.68 3.81 2.56 4.01 5.02 4.88 3.26 2.41 4.03 4.35 0.60
0.00 0.07 0.02 0.64 0.00 0.12 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.03 0.00 0.00 0.00 1.00 0.00 0.01 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.28
5. Final considerations In this paper, we present evidence of mean reversion (anti-persistence, H < 0.5) for individual stock returns. Open– open returns present a stronger degree of anti-persistence than close–close returns. This is in line with recent research conducted by [9] that suggests that a greater tendency for overreaction in open–open returns is able to explain this phenomenon. Our results are important for a variety of reasons. In first place, one cannot reject the null of absence of long-range dependence for the Dow Jones Industrial Average Index. However, when focusing on individual stocks, long-range dependence is present in a huge proportion of them. This suggests that studies focusing on indices may present aggregation effects that mask results for long-range dependence. In second place, most stocks show Hurst exponents below 0.5, which suggests that previous results in the literature may be flawed, as they were searching for Hurst exponents greater than 0.5. Finally, most financial theories have to be reexamined in order to take into account these features the dynamics of asset prices.4
4
An important example would be option pricing theory, which assumes that asset prices follow Brownian motions (see [3]).
1000
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
References [1] Amihud Y, Mendelson H. Trading mechanisms and stock returns: an empirical investigation. Journal of Finance 1987;42:533–53. [3] Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 1973;81:637–54. [4] Brock W, Lakonishok J, LeBaron B. Simple technical trading rules and the stochastic properties of stock returns. Journal of Finance 1992;47:1731–64. [5] Cajueiro DO, Tabak BM. The HurstÕs exponent over time: testing the assertion that emerging markets are becoming more efficient. Physica A 2004;336:521–37. [6] Cajueiro DO, Tabak BM. Ranking efficiency for emerging-markets. Chaos, Solitons & Fractals 2004;22:349–52. [7] Cajueiro DO, Tabak BM. Evidence of long-range dependence in asian equity markets: the role of liquidity and market restrictions. Physica A 2004;342:656–64. [8] Cajueiro DO, Tabak BM. Ranking efficiency for emerging-markets II. Chaos, Solitons & Fractals 2005;23:671–5. [9] Chelley-Steeley P. Opening returns, noise and overreaction. Journal of Financial Research 2001;24:513–21. [10] Lo AW, Mackinlay AC. Stock market prices do not follow random walks: evidence from a simple specification test. Review of Financial Studies 1988;1:41–66. [11] Lo AW, Mackinlay AC. The size and power of the variance ratio test in finite samples: a Monte Carlo investigation. Journal of Econometrics 1989;40:203–38. [12] Robinson PM. Gaussian semiparametric estimation of long-range dependence. The Annals of Statistics 1995;23:1630–61. [13] Stoll H, Whaley R. Stock market structure and volatility. Review of Financial Studies 1990;3:37–71.
Long-range dependence and market structure Daniel O. Cajueiro a, Benjamin M. Tabak a
b,*
Universidade Cato´lica de Brası´lia—Mestrado em Economia de Empresas, SGAN 916, Mo´dulo B—Asa Norte, DF 70790-160, Brazil b Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900 Brasilia, Brazil Accepted 19 October 2005
Communicated by Prof. M.S. El Naschie
Abstract In this paper, we have found that although the Dow Jones Average Industrial Index does not possess long-range dependence in mean returns, individual stocks that form the index do. These results were obtained using the Local Whittle estimation procedure. Most stocks seem to be anti-persistent with Hurst exponents below 0.5, which is in line with mean reversion in the long run. Furthermore, open–open returns possess a stronger degree of anti-persistence than close–close returns due to market structure effects. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Long memory describes the dynamics of the correlation structure of time series at long lags. Testing for long memory in stock market returns has been in the research agenda for the past 20 years as it has many implications for portfolio and risk managers.1 If stock returns are said to possess long-range dependence, then one could build long-term forecasting models that could improve the portfolio management process. Furthermore, one could also forecast more precisely long-term volatilities, which is an essential input in both portfolio and risk management. Most studies have focused on stock market indices, which are aggregate indices. However, if a fractal structure exists in individual stock returns, itÕs presence may not be captured in aggregate indices. Therefore, studying individual stock returns may provide further insights in this debate. This study investigates the presence of long-range dependence in individual stocks from the Dow Jones for the US. This is one of the most important stock indices in the world and all stocks are highly liquid. Therefore, considerations of market liquidity should play no role in driving the results. Previous research has documented differences in open–open and close–close returns due to market structure for the US. Amihud and Mendelson [1] studied the New York Stock Exchange (NYSE) and compared the volatility of returns using open–open returns and close–close returns. The NYSE opens with a call auction, but trades as a specialist dealer market thereafter. Therefore, open–open returns should reflect the influence of the opening auction. Furthermore, *
1
Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045. E-mail address: [email protected] (B.M. Tabak). See [5–8].
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.077
996
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
close–close returns reflect the effect of the specialist dealer market. The authors in line with another study due to [13] concluded that NYSE open–open returns are more volatile than closing returns. These studies motivate testing for longrange dependence in both close–close and open–open returns and testing for significant differences. These differences could be attributed to market microstructure effects. [9] studied individual stocks from the Dow Jones Index and found that these stocks possess negative serial correlation. Furthermore, by comparing close–close and open–open returns the authors find that overreaction may account for the differences in the return behavior of opening and closing returns. This paper adds to the literature by testing whether Hurst exponents may help in assessing differences in market structure. We test for individual stocks that comprise the Dow Jones Industrial Average Index verifying whether Hurst exponents differ for close–close and open–open returns. In the next section, we present the method that is employed in the paper to estimate Hurst exponents. Section 3 presents the data, while in Section 4 we discuss the empirical results. Finally, Section 5 concludes the paper.
2. The Local Whittle estimator In this paper, the Local Whittle estimator is used to provide the Hurst exponent. A given market is said to have longrange dependence with persistent behavior if the Hurst exponent H > 0.5, with anti-persistent behavior if H < 0.5 and random walk behavior if H = 0.5. The Local Whittle estimator is a semi-parametric estimator, which only requires specifying the parametric form of the spectral density when the frequency k is close to zero, f ðkÞ GðH Þjkj12H ;
as k ! 0;
ð1Þ
when G(H) is a constant. The computation involves an additional parameter m, an integer less than N/2, where N is the size of the time series, and such that, as N ! 1, 1 m þ ! 0. m N
ð2Þ
This means that as N gets larger, m gets larger as well, although slower. For a spectral density of the form (1), the Whittle approximation of the Gaussian likelihood function is obtained by minimizing ! m 1 X Iðkj Þ 12H QðG; H Þ ¼ þ logðGk Þ ; ð3Þ j m j¼1 Gk12H j where kj = 2pj/N and I(kj) is periodogram. So this estimator sums the frequencies only up to 2pm/N. b Replacing above G by its estimate G, m X Iðkj Þ b ¼1 . G m j¼1 k12H j
One may define b H Þ 1 ¼ log RðH Þ ¼ Qð G;
ð4Þ
! m m 1 X IðkÞ 2H 1 X logðkj Þ . m j¼1 k12H m j¼1 j
ð5Þ
Robinson [12] showed that under certain technical assumptions, b ¼ arg min RðH Þ; H
ð6Þ
converges in probability to actual value H, i.e., b H Þ !d Normalð0; 1=4Þ. m1=2 ð H
ð7Þ
b converges to H. On the other hand, Therefore, the choice of m is quite important. The larger the value of m, the faster H if the series also presents short range behavior, then m should be small. In this paper, in order to ensure faster converb to H, the limiting value of m ¼ N 1 is used.2 gence of H 2
2 One should note that using this limiting value of m, larger frequencies are not considered. Therefore, the short range dependence phenomena should not be affecting our conclusions.
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
997
Table 1 Bloomberg codes Bloomberg code
Name
MMM UN equity AA UN equity MO UN equity AIG UN equity AXP UN equity BA UN equity CAT UN equity C UN equity KO UN equity DIS UN equity DD UN equity XOM UN equity GE UN equity GM UN equity HPQ UN equity HD UN equity HON UN equity IBM UN equity INTC UQ equity JNJ UN equity JPM UN equity MCD UN equity MRK UN equity MSFT UQ equity PFE UN equity PG UN equity SBC UN equity UTX UN equity VZ UN equity WMT UN equity INDU index
3M CO Alcoa Inc Altria Group Inc American International Group American Express Co Boeing Co Caterpillar Inc Citigroup Inc Coca-Cola Co/The The Walt Disney Co. Du Pont (EI) De Nemours Exxon Mobil Corp General Electric Co General Motors Corp Hewlett-Packard Co Home Depot Inc Honeywell International Inc Intl Business Machines Corp Intel Corp Johnson & Johnson Jpmorgan Chase & Co McdonaldÕs Corp Merck & Co. Inc. Microsoft Corp Pfizer Inc Procter & Gamble Co SBC Communications Inc United Technologies Corp Verizon Communications Inc Wal-Mart Stores Inc Dow Jones Indus. Avg
3. Data The series studied include 30 companies included in the Dow Jones Industrials Index. We also include the aggregate index for comparison purposes. All analysis is done on stock return series. The series were retrieved from the Bloomberg database. Table 1 presents the codes and the names of the companies that enter the sample. The New York Stock Exchange opens at 13:30 (GMT) and the first record of the INDU index for that day is registered at 13:35. The market closes at 20:00 (GMT) and the last record of the day is registered at 20:05. Overall, our sample period consists of approximately 3524 days. The sample starts in January 2, 1990 and ends in December 19, 2003.
4. Results Previous research has found that US stocks are predictable to some extent.3 Furthermore, technical analysis has been found to add value for portfolio management. For example, in a seminal paper [4] presented evidence suggesting that technical trading rules may benefit investors. Their analysis of the closing levels of the Dow Jones Industrial Average Index suggests that a variety of simple technical rules can be used to generate returns in excess of a buy and hold
3
See [10,11].
998
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
strategy. This research suggests that US stock returns possess serial correlation, which could be exploited in trading systems. We present evidence of long-term dependence for many stocks that comprise the Dow Jones Industrial Average Index. Table 2 presents Hurst exponents for close–close returns estimated using the Local Whittle methodology. Table 3 presents results for open–open returns. It is striking that in both cases we cannot reject the null of absence of longrange dependence for the aggregate index (Dow Jones Industrial Average Index—INDU) but this certainly is not true for many individual stocks. Most Hurst exponents are below 0.5, which suggests negative serial correlation (mean reversion in the long run), in line with the findings of [9]. We employ a nonparametric statistic to test for changes in the median of the distribution of Hurst exponents. The Kruskal–Wallis (KW) statistic, testing for difference in the median of Hurst exponents for open– open and close–close returns, is 17.19 with a p-value of 0.00, suggesting that we can reject with a 1% significance level the null of equality of medians. The medians for close–close and open–open returns are 0.48 and 0.46, respectively. This suggests that open–open returns possess a stronger degree of anti-persistence. The degree of anti-persistence is much stronger in open–open than in close–close returns. Approximately 50% of stocks present long-range dependence in close–close returns, while this figure increases to more than 83% for open–open returns. Our results are in line with [9] which finds that the tendency for overreaction in open–open returns is much stronger than for close–close returns. Furthermore, overreaction is the cause of serial correlation in opening returns.
Table 2 Hurst exponents for close–close stock returns are presented in the second column with associated standard errors in the third column, for m/2. The third and fourth columns presents a Z-statistic for the null H = 0.5 Code
H
Std. error
Z-Stat
p-Value
MMM AA MO AIG AXP BA CAT C KO DIS DD XOM GE GM HPQ HD HON IBM INTC JNJ JPM MCD MRK MSFT PFE PG SBC UTX VZ WMT INDU
0.46 0.49 0.49 0.50 0.47 0.50 0.50 0.50 0.49 0.48 0.47 0.39 0.47 0.48 0.47 0.50 0.50 0.48 0.49 0.48 0.52 0.49 0.50 0.48 0.49 0.47 0.47 0.48 0.45 0.46 0.49
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
3.72 0.50 0.54 0.29 2.14 0.08 0.37 0.14 0.51 1.76 2.90 9.01 2.26 2.06 2.31 0.23 0.07 1.72 1.24 1.70 1.30 0.52 0.20 1.28 0.92 2.14 2.71 1.94 4.41 3.38 0.90
0.00 0.31 0.29 0.61 0.02 0.47 0.35 0.45 0.30 0.04 0.00 0.00 0.01 0.02 0.01 0.59 0.47 0.04 0.11 0.04 0.10 0.30 0.58 0.10 0.18 0.02 0.00 0.03 0.00 0.00 0.18
D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
999
Table 3 Hurst exponents for open–open stock returns are presented in the second column with associated standard errors in the third column, for m/2. The third and fourth columns presents a Z-statistic for the null H = 0.5 Code
H
Std. error
Z-Stat
p-Value
MMM AA MO AIG AXP BA CAT C KO DIS DD XOM GE GM HPQ HD HON IBM INTC JNJ JPM MCD MRK MSFT PFE PG SBC UTX VZ WMT INDU
0.44 0.48 0.48 0.50 0.45 0.49 0.47 0.45 0.46 0.46 0.43 0.39 0.44 0.48 0.46 0.47 0.48 0.46 0.45 0.43 0.47 0.46 0.47 0.45 0.44 0.44 0.46 0.47 0.45 0.45 0.49
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
4.71 1.46 2.00 0.35 3.93 1.15 2.15 4.09 3.48 3.80 6.07 9.13 4.83 1.87 3.62 2.67 1.85 3.66 4.46 5.64 2.68 3.81 2.56 4.01 5.02 4.88 3.26 2.41 4.03 4.35 0.60
0.00 0.07 0.02 0.64 0.00 0.12 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.03 0.00 0.00 0.00 1.00 0.00 0.01 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.28
5. Final considerations In this paper, we present evidence of mean reversion (anti-persistence, H < 0.5) for individual stock returns. Open– open returns present a stronger degree of anti-persistence than close–close returns. This is in line with recent research conducted by [9] that suggests that a greater tendency for overreaction in open–open returns is able to explain this phenomenon. Our results are important for a variety of reasons. In first place, one cannot reject the null of absence of long-range dependence for the Dow Jones Industrial Average Index. However, when focusing on individual stocks, long-range dependence is present in a huge proportion of them. This suggests that studies focusing on indices may present aggregation effects that mask results for long-range dependence. In second place, most stocks show Hurst exponents below 0.5, which suggests that previous results in the literature may be flawed, as they were searching for Hurst exponents greater than 0.5. Finally, most financial theories have to be reexamined in order to take into account these features the dynamics of asset prices.4
4
An important example would be option pricing theory, which assumes that asset prices follow Brownian motions (see [3]).
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D.O. Cajueiro, B.M. Tabak / Chaos, Solitons and Fractals 31 (2007) 995–1000
References [1] Amihud Y, Mendelson H. Trading mechanisms and stock returns: an empirical investigation. Journal of Finance 1987;42:533–53. [3] Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 1973;81:637–54. [4] Brock W, Lakonishok J, LeBaron B. Simple technical trading rules and the stochastic properties of stock returns. Journal of Finance 1992;47:1731–64. [5] Cajueiro DO, Tabak BM. The HurstÕs exponent over time: testing the assertion that emerging markets are becoming more efficient. Physica A 2004;336:521–37. [6] Cajueiro DO, Tabak BM. Ranking efficiency for emerging-markets. Chaos, Solitons & Fractals 2004;22:349–52. [7] Cajueiro DO, Tabak BM. Evidence of long-range dependence in asian equity markets: the role of liquidity and market restrictions. Physica A 2004;342:656–64. [8] Cajueiro DO, Tabak BM. Ranking efficiency for emerging-markets II. Chaos, Solitons & Fractals 2005;23:671–5. [9] Chelley-Steeley P. Opening returns, noise and overreaction. Journal of Financial Research 2001;24:513–21. [10] Lo AW, Mackinlay AC. Stock market prices do not follow random walks: evidence from a simple specification test. Review of Financial Studies 1988;1:41–66. [11] Lo AW, Mackinlay AC. The size and power of the variance ratio test in finite samples: a Monte Carlo investigation. Journal of Econometrics 1989;40:203–38. [12] Robinson PM. Gaussian semiparametric estimation of long-range dependence. The Annals of Statistics 1995;23:1630–61. [13] Stoll H, Whaley R. Stock market structure and volatility. Review of Financial Studies 1990;3:37–71.