- Email: [email protected]
ρx,y between open-close stock markets
Physica A 534 (2019) 122152
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
ρx,y between open-close stock markets L.S. Almeida da Silva a , E.F. Guedes b , Paulo Ferreira c,d,e , Andreia Dionísio e , ∗ G.F. Zebende f , a
Study of Complex Systems Group, State University of Feira de Santana, Bahia, Brazil Department of Statistics, Federal University of Bahia, Bahia, Brazil Research Center for Endogenous Resource Valorization, Portalegre, Portugal d Instituto Politécnico de Portalegre, Portugal e CEFAGE-UE, IIFA, Universidade de Évora, Largo dos Colegiais 2, 7000 Évora, Portugal f State University of Feira de Santana, Bahia, Brazil b c
highlights • We propose to study the DCCA cross-correlation coefficient between open-close stock market index. • This coefficient was applied in some global index. • Patterns and differences are observed depending on the time scale.
article
info
Article history: Received 19 April 2019 Received in revised form 29 June 2019 Available online 31 July 2019 Keywords: Stock market DFA DCCA Detrended cross-correlation coefficient
a b s t r a c t In this paper we propose to study the auto and the cross-correlation between openclose stock market indexes. Our choice took into account the main indexes around the world: North America (Dow Jones and NASDAQ — USA; IPC — MEXICO), South America (BOVESPA — Brazil; MERVAL — Argentina), Asia (Nikkei_225 — Japan; SSE — China; Hang Seng — Hong Kong), Europe (IBEX_35 — Spain; CAC_40 — French; DAX — German; FTSE_100 — England). Thus, for the opening and closing stock market index and its respective return, we applied the DFA method for measuring auto-correlation, as well as, the cross-correlation between these signals with DCCA cross-correlation coefficient. Our results show that for auto-correlation there is long-range (power-law) auto-correlations with: αDFA > 1.0 for the original index and αDFA ≃ 0.5 for the return. For crosscorrelations we found perfect DCCA cross-correlation between open and close indexes at long time-scale, but in short time-scale there are differences between the stock markets. From the point of view of the return, DCCA cross-correlation coefficient between open and close values showing lower DCCA cross-correlation levels. In this case the open returns are not related with the close returns, evidencing the time independence of these observations in this time-scale, for some stock markets. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Despite the great importance of public opinion on the issues of economic globalization, its origins and consequences continue to be studied. The financial stock market index is the main way to study an economy or its relations globally. The ∗ Corresponding author. E-mail address: [email protected] (G.F. Zebende). https://doi.org/10.1016/j.physa.2019.122152 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
reasons that influence the indexes are extremely diverse, may be from the volume of domestic and foreign investment to how much the economy of a given country is coupled with the rest of the world [1]. In this scenario of globalization it is possible to imagine that the great economies are very important for the global financial equilibrium [2,3]. Following the proposal initiated by Almeida da Silva in your masters degree [4], this paper aimed at the study the time series of the economic indexes of the main stock markets. Our main objective will be to study each stock market with the opening and closing index, day by day. In the light of this objective, we will consider the application of the DFA method [5] and the DCCA cross-correlation coefficient ρx,y [6] in these time series of the stock market. To provide continuity, the paper is organized as follows: Section 2 present the classic descriptive statistic, Section 3 describe the DFA methodology, Section 4 present the results and discussions, while Section 5 draws the conclusions. 2. Data and descriptive statistics We chose some of the most important indexes in the world, including both developed and emerging markets. It turns possible to identify if results are different considering stock market development. Our data was retrieved in Yahoo Finance and comprises the time sample between 12/01/2010 and 03/15/2019. The selected indexes are:
• (^BVSP):The benchmark index of about 60 stocks that are traded on the São Paulo (Brazil), with 2050 points. • (^DJI): Stock market index that indicates the value of 30 large, publicly owned companies based in the United States, with 2085 points.
• (^FCHI): The CAC 40 index represents a capitalization-weighted measure of the 40 most significant stocks among the 100 largest market caps on the Euronext Paris, with 2117 points.
• (^GDAXI): DAX (Deutscher Aktienindex) is a blue chip stock market index consisting of the 30 major German companies trading on the Frankfurt Stock Exchange, with 2098 points.
• (^HSI): Hang Seng Index is a free float-adjusted market-capitalization-weighted stock-market index in Hong Kong, with 2039 points.
• (^IBEX): The IBEX 35 (Spanish Exchange Index) is the benchmark stock market index of the Madrid, with 2119 points. • (^IXIC): NASDAQ Composite is a stock market index of the common stocks and similar securities listed on the • • • •
NASDAQ stock market, with 2085 points. (^MERV): MERVAL Index is the most important index of the Buenos Aires Stock Exchange, with 2008 points. (^MXX): Mexican Stock Exchange, is the stock exchanges in Mexico, with 2073 points. (^N225): Nikkei Stock Average, is a stock market index for the Tokyo Stock Exchange, with 2035 points. (^000001.SS): SSE index is a stock market index of all stocks that are traded at the Shanghai Stock Exchange, with 2016 points.
Initially, taking into account the mean and the standard deviation the descriptive statistics is presented in Table 1. As we can see, although the mean values give an idea of how much the index is worth in the time period, these values differ from each other (each index is based on its own criterion) and that they usually have a very high standard deviation around their mean value (see Table 1). In this context, the return is normally used for the analysis of economic and financial time series, that is, for the original stock market index St at the time t, the return will be:
( rt ≡ ln
St
)
St −1
(1)
Consequently, with rt , each time series is normalized (with mean around zero), facilitating in this way the comparison between the most diverse indexes. The next step was to describe the descriptive statistics with the third (Skewness) and fourth (Kurtosis) moments of the stock market indexes and their respective returns, see Fig. 1. From this figure, directly is observed that the time series are non-stationary, because the value of Skewness differ from 0 and the Kurtosis differs from 3 (normal distribution). For all-cases the Shapiro–Francia (p-valor) normality test are less than 0.0001, and this also shows that the distributions are different from normal distribution. Specifically in the case of index analysis, we can see that the values of Skewness and Kurtosis are the same for the stock market opening (□) and the stock market closing ( ), with positive Skewness and Kurtosis less than 3 (platykurtic distribution). For the return (Eq. (1)), in the opening ( ) and in the closing ( ) stock market, we have the mean equal to zero with standard deviation less than 0.01. Also, we can identify Kurtosis great than 3 (leptokurtic distribution) and negative Skewness, with a noticeable difference between the opening and closing of the stock market (see Fig. 1(b)). But, descriptive statistic is poor on the aspect of correlations analysis. Therefore, in order to give a better view on the auto and cross-correlations, we briefly describe the statistical method for this type of analysis in the next section. 3. DCCA auto and cross-correlation analysis Before we introduce the DCCA cross-correlation coefficient, we start with the DFA method [5] for auto-correlation analysis. DFA method is based in the fluctuation function, FDFA , defined from a time series {xi } with i = 1, 2, 3, . . . , N (length).
L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
3
Table 1 Descriptive statistics with the mean value and the standard deviation (open/close index). Mean Open ^000001.SS
sd Close
Open
Close
2 777.26
2 780.68
572.56
573.63
^BVSP
60 925.50
60 940.02
12 121.67
12 153.43
^DJI
17 496.89
17 501.78
4 295.37
4 293.20
4 379.10
4 378.80
706.69
706.80
^FCHI ^GDAXI ^HSI
4 379.10
4 378.80
706.69
706.80
23 675.67
23 661.03
3 165.54
3 160.97
^IBEX
9 444.33
9 439.89
1 143.55
1 143.63
^IXIC
4 686.05
4 686.83
1 582.92
1 582.64
^MERV
12 521.22
12 533.23
10 093.08
10 103.92
^MXX
43 189.48
43 188.96
4 302.67
4 301.16
^N225
15 845.57
15 843.27
4 687.35
4 686.92
Fig. 1. Descriptive statistics for all indexes. Skewness and Kurtosis are presented here for (a) Open-Close index, and (b) Open-Close return.
Continuing with the methodology, the integrated time series is calculated: Xk =
k ∑
(xi − ⟨x⟩) ∴ cumulative sum
(2)
i=1
where, ⟨x⟩ represent the mean and k = 1, 2, 3, . . . , N. Xk is divided into box windows of length n, and the local trend of each box, Xk,n , is obtained by a least-squares fit of each series. The fluctuation function is calculated by:
N 1 ∑ FDFA (n) = √ (Xi − Xi,n )2 N
(3)
i=1
Therefore, if there is power-law auto-correlations, then: FDFA (n) ∼ nαDFA
(4)
Exponent αDFA quantifies the empirical strength of the long-range (power-law) auto-correlations of the signal, a selfaffinity parameter [7] (see Table 2 for αDFA properties). The advantage of DFA over other methods lies in allowing the
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L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152 Table 2 αDFA exponent and properties. Exponent
Type of signal
αDFA αDFA αDFA αDFA αDFA αDFA
Anti-persistent Uncorrelated, white noise Long-range correlated persistent 1/f noise Non-stationary Brownian motion
< 0.5 ⋍ 0.5 > 0.5 ⋍ 1.0 > 1.0 ⋍ 3/2
detection of long-range auto-correlations embedded in seemingly non-stationary time series, and also avoiding spurious detection of apparent long-range auto-correlations, which are an artifact of non-stationarity. It is possible to generalize the DFA method to measure cross-correlation, and this generalization will be introduced briefly as DCCA method [8]. Therefore, considering two time series, {xi } and {yi }, with 1 = 1, 2, . . . ., N, and the integrated time series as Eq. (2), we obtain two new time series, {Xk } and {Yk }. Thus, if we have overlapping boxes, these two integrated time series, {Xk } and {Yk }, are divided into (N − n) boxes of equal length n, with 4 ≤ n ≤ N4 . The local trend of each box, Xk,n and Yk,n , are obtained by a least-squares fit of each series, and the covariance function of the residuals in each box are calculate the by: 2 fDCCA (n, i) =
i+n ∑
1 (n + 1)
(Xk − Xk,n )(Yk − Yk,n )
(5)
k=i
Finally, the average over all (N − n) overlapping boxes is calculated to obtain the detrended covariance function: 2 FDCCA (n) =
1 (N − n)
N −n ∑
2 fDCCA (n, i)
(6)
i=1
If power-law are present in DCCA cross-correlation function, then: 2 FDCCA (n) ∼ n2λ
(7)
The λ exponent quantifies long-range (power-law) cross-correlations and also identifies seasonality [9,10], but λ does not quantify the level of cross-correlations. It is possible to quantify the level of cross-correlation with DFA and DCCA methods using the DCCA cross-correlation coefficient [6]. ρx,y (n) is defined as the ratio between the detrended covariance 2 (Eq. (6)) and the detrended variance function FDFA (Eq. (3)): function FDCCA
ρx,y (n) =
2 (n) FDCCA
FDFAx (n) FDFAy (n)
(8)
It is observed that, ρx,y (n) ranges from −1 ≤ ρx,y ≤ 1, and: i. ρx,y = 1 means a perfect DCCA cross-correlation; ii. ρx,y = 0 there is no DCCA cross-correlation; iii. ρx,y = −1 means a perfectly anti DCCA cross-correlation. The advantage of this detrended cross-correlation coefficient lies in measuring cross-correlations between two nonstationary time series at different time scales [11–27], among others papers. 4. Results and discussions Initially, we used the data containing the index of each stock market at the opening and at its closing. Our choice was based on the main world indexes, located in the North America, South America, Asia and Europe. Also, besides the values of the indexes, we did analysis for the value of the return, Eq. (1). The results for auto and DCCA cross-correlation analysis are presented respectively right below. 4.1. DFA exponents DFA analysis show that, FDFA (n) ∼ nαDFA , for the stock market index and the returns, see Fig. 2. The values with αDFA exponent (linear adjustment) for each stock market are presented in Table 3. The αDFA exponents values are larger than the unit for the stock market indexes, which indicates non-stationarity (as expected) of the time series. For the return, the values are relatively close to 0.5. The indexes closer to 0.5 (coined with the efficiency [28]) are ^N225 (result common to other studies), ^HSI, ^BVSP and ^MERV. The indexes farthest from the efficiency level are ^IXIC, ^DJI and ^MXX. Most of the indexes show anti-persistence, only China is the exception.
L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
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Fig. 2. FDFA as a function of time scale n for all indexes: (a) Index Open, (b) Index Close, (c) Return Open, and (d) Return Close. Table 3 DFA exponent for index and return. The values of the linear adjustments for αDFA always had R2 > 0.99. Index
Return
DFA exponent
1 2 3 4 5 6 7 8 9 10 11
^BVSP ^DJI ^FCHI ^GDAXI ^HSI ^IBEX ^IXIC ^MERV ^MXX ^N225 ^000001.SS
DFA exponent
Open
Close
Open
Close
1.40 1.43 1.42 1.41 1.42 1.40 1.44 1.42 1.32 1.44 1.47
1.40 1.42 1.37 1.41 1.42 1.40 1.44 1.42 1.32 1.44 1.48
0.48 0.41 0.43 0.46 0.48 0.45 0.39 0.48 0.41 0.48 0.53
0.48 0.41 0.43 0.46 0.48 0.44 0.40 0.48 0.41 0.47 0.54
4.2. DCCA cross-correlation coefficient Fig. 3 present the results for ρx,y between the Open and Close stock market indexes. We observed that there is a positive DCCA cross-correlation between opening and closing indexes, and more, ρx,y for n ≥ 66 tends to a perfect DCCA cross-correlation with fast convergence. It means that a change in the opening/closing price today is fully internalized after 66 days for all indexes. In small time scales, there are differences between the values of ρx,y , but for all cases ρx,y > 0.5. Considering the return indexes, Fig. 4 show the results for ρx,y between the Open and the Close values. In this case, the convergence to perfect DCCA cross-correlation occurs only for long time-scales, which means less memory in the signs. In an economic point of view, the results about the original indexes are expected, since it is normal to have high similarities between the open and the close values. In what refers to returns, the results are quite interesting. Our results point to the existence of low DCCA cross-correlations between open and closing returns in some indexes, namely ^BVSP, ^DJI, ^MMX and ^MERV. In these cases, since the correlation is null, we may conclude that the open return does not influence the close return of the same day. For example, for ^HSI in the short time-scale, the DCCA cross-correlation is around 0.5. Fig. 5 show the smaller time-scale, n = 4, and in this case it is possible to identify differences between the values of ρx,y depending on the stock market.
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L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
Fig. 3. ρx,y as a function of time scale n for DCCA cross-correlation between Open and Close stock market index. Vertical line show n = 66 days.
Fig. 4. ρx,y as a function of time scale n for DCCA cross-correlation between the return Open-Close values.
Fig. 5. ρx,y with n = 4 for all indexes (a) and respective return (b).
L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
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5. Conclusions We found respectively for the stock market indexes and the return long-range (power-law) DFA auto-correlation. In these cases αDFA > 1.0 for the original index (identifying that the original time series of the index are non-stationary) and for the return αDFA ≃ 0.5, with predominance for anti-persistent behavior depending on the stock market. Also, between open and close values, the indexes and the return present higher level of DCCA cross-correlation for long time-scale. Therefore, for short time-scale, the results show different pattern in the stock markets. In the case of the stock market indexes, the transition for perfect DCCA cross-correlation is found for n > 66 days, whereas for the returns only for large time-scales. Lastly, by the specialized literature, price of opening stock markets are more commonly used by investors with the intention of short-term financial returns. However, as show this paper, there is no difference between opening and closing prices for decision-making with long time-scales, n > 66 days (for prices) or n > 365 days (for returns). In this sense, the velocity of spread of the information of the cross-correlation between open and close stock market can be an important parameter for investors and regulation authorities. Acknowledgments Paulo Ferreira and Andreia Dionísio gratefully acknowledge financial support by National Funds of the FCT: Portuguese Foundation for Science and Technology within the project UID/ECO/03182/2019. Everaldo Guedes thanks the FAPESB (Fundação de Amparo à Pesquisa do Estado da Bahia), Brazil (Grant BOL 0976/2016) and Gilney Zebende thanks the CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil (Grant 304362/2017-4). References [1] K.C. Knorr, A. Preda (Eds.), The Sociology of Financial Markets, Oxford University Press, 2006. [2] E.E. 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Donghong, Oil price and exchange rate co-movements in Asian countries: Detrended cross-correlation approach, Physica A 465 (1) (2017) 338–346. [20] B. Podobnik, Z.-Q. Jiang, W.-X. Zhou, H.E. Stanley, Statistical tests for power-law cross-correlated processes, Phys. Rev. E 84 (2011) 066118. [21] X. Jin, Time-varying return-volatility relation in international stock markets, Int. Rev. Econ. Finance 51 (2017) 157–173. [22] P. Ferreira, Assessing the relationship between dependence and volume in stock markets: A dynamic analysis, Physica A 516 (2019) 90–97. [23] E. Guedes, A. Dionísio, P. Ferreira, G. Zebende, DCCA Cross-correlation in blue-chips companies: A view of the 2008 financial crisis in the Eurozone, Physica A 479 (2017) 38–47. [24] X.-Y. Qian, Y.-M. Liu, Z.-Q. Jiang, B. Podobnik, W.-X. Zhou, H.E. Stanley, Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces, Phys. Rev. E 91 (2015) 062816. [25] B. Podobnik, D. Horvatic, A.M. Petersen, H.E. Stanley, Cross-correlation between volume change and price change, Proc. Natl. Acad. Sci. 106 (52) (2009) 22079–22084. [26] B. Podobnik, I. Grosse, D. Horvatić, S. Ilic, C.P. Ivanov, E.H. Stanley, Quantifying cross-correlations using local and global detrending approaches, Eur. Phys. J. B 71 (2) (2009) 243–250. [27] D. Horvatic, H.E. Stanley, B. Podobnik, Detrended cross-correlation analysis for non-stationary time series with periodic trends, Eur. Phys. Lett. 94 (1) (2011) 18007. [28] P. Ferreira, A. Dionisio, E.F. Guedes, G.F. Zebende, A sliding windows approach to analyse the evolution of bank shares in the European Union, Physica A 490 (2018) 1355–1367.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
ρx,y between open-close stock markets L.S. Almeida da Silva a , E.F. Guedes b , Paulo Ferreira c,d,e , Andreia Dionísio e , ∗ G.F. Zebende f , a
Study of Complex Systems Group, State University of Feira de Santana, Bahia, Brazil Department of Statistics, Federal University of Bahia, Bahia, Brazil Research Center for Endogenous Resource Valorization, Portalegre, Portugal d Instituto Politécnico de Portalegre, Portugal e CEFAGE-UE, IIFA, Universidade de Évora, Largo dos Colegiais 2, 7000 Évora, Portugal f State University of Feira de Santana, Bahia, Brazil b c
highlights • We propose to study the DCCA cross-correlation coefficient between open-close stock market index. • This coefficient was applied in some global index. • Patterns and differences are observed depending on the time scale.
article
info
Article history: Received 19 April 2019 Received in revised form 29 June 2019 Available online 31 July 2019 Keywords: Stock market DFA DCCA Detrended cross-correlation coefficient
a b s t r a c t In this paper we propose to study the auto and the cross-correlation between openclose stock market indexes. Our choice took into account the main indexes around the world: North America (Dow Jones and NASDAQ — USA; IPC — MEXICO), South America (BOVESPA — Brazil; MERVAL — Argentina), Asia (Nikkei_225 — Japan; SSE — China; Hang Seng — Hong Kong), Europe (IBEX_35 — Spain; CAC_40 — French; DAX — German; FTSE_100 — England). Thus, for the opening and closing stock market index and its respective return, we applied the DFA method for measuring auto-correlation, as well as, the cross-correlation between these signals with DCCA cross-correlation coefficient. Our results show that for auto-correlation there is long-range (power-law) auto-correlations with: αDFA > 1.0 for the original index and αDFA ≃ 0.5 for the return. For crosscorrelations we found perfect DCCA cross-correlation between open and close indexes at long time-scale, but in short time-scale there are differences between the stock markets. From the point of view of the return, DCCA cross-correlation coefficient between open and close values showing lower DCCA cross-correlation levels. In this case the open returns are not related with the close returns, evidencing the time independence of these observations in this time-scale, for some stock markets. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Despite the great importance of public opinion on the issues of economic globalization, its origins and consequences continue to be studied. The financial stock market index is the main way to study an economy or its relations globally. The ∗ Corresponding author. E-mail address: [email protected] (G.F. Zebende). https://doi.org/10.1016/j.physa.2019.122152 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152
reasons that influence the indexes are extremely diverse, may be from the volume of domestic and foreign investment to how much the economy of a given country is coupled with the rest of the world [1]. In this scenario of globalization it is possible to imagine that the great economies are very important for the global financial equilibrium [2,3]. Following the proposal initiated by Almeida da Silva in your masters degree [4], this paper aimed at the study the time series of the economic indexes of the main stock markets. Our main objective will be to study each stock market with the opening and closing index, day by day. In the light of this objective, we will consider the application of the DFA method [5] and the DCCA cross-correlation coefficient ρx,y [6] in these time series of the stock market. To provide continuity, the paper is organized as follows: Section 2 present the classic descriptive statistic, Section 3 describe the DFA methodology, Section 4 present the results and discussions, while Section 5 draws the conclusions. 2. Data and descriptive statistics We chose some of the most important indexes in the world, including both developed and emerging markets. It turns possible to identify if results are different considering stock market development. Our data was retrieved in Yahoo Finance and comprises the time sample between 12/01/2010 and 03/15/2019. The selected indexes are:
• (^BVSP):The benchmark index of about 60 stocks that are traded on the São Paulo (Brazil), with 2050 points. • (^DJI): Stock market index that indicates the value of 30 large, publicly owned companies based in the United States, with 2085 points.
• (^FCHI): The CAC 40 index represents a capitalization-weighted measure of the 40 most significant stocks among the 100 largest market caps on the Euronext Paris, with 2117 points.
• (^GDAXI): DAX (Deutscher Aktienindex) is a blue chip stock market index consisting of the 30 major German companies trading on the Frankfurt Stock Exchange, with 2098 points.
• (^HSI): Hang Seng Index is a free float-adjusted market-capitalization-weighted stock-market index in Hong Kong, with 2039 points.
• (^IBEX): The IBEX 35 (Spanish Exchange Index) is the benchmark stock market index of the Madrid, with 2119 points. • (^IXIC): NASDAQ Composite is a stock market index of the common stocks and similar securities listed on the • • • •
NASDAQ stock market, with 2085 points. (^MERV): MERVAL Index is the most important index of the Buenos Aires Stock Exchange, with 2008 points. (^MXX): Mexican Stock Exchange, is the stock exchanges in Mexico, with 2073 points. (^N225): Nikkei Stock Average, is a stock market index for the Tokyo Stock Exchange, with 2035 points. (^000001.SS): SSE index is a stock market index of all stocks that are traded at the Shanghai Stock Exchange, with 2016 points.
Initially, taking into account the mean and the standard deviation the descriptive statistics is presented in Table 1. As we can see, although the mean values give an idea of how much the index is worth in the time period, these values differ from each other (each index is based on its own criterion) and that they usually have a very high standard deviation around their mean value (see Table 1). In this context, the return is normally used for the analysis of economic and financial time series, that is, for the original stock market index St at the time t, the return will be:
( rt ≡ ln
St
)
St −1
(1)
Consequently, with rt , each time series is normalized (with mean around zero), facilitating in this way the comparison between the most diverse indexes. The next step was to describe the descriptive statistics with the third (Skewness) and fourth (Kurtosis) moments of the stock market indexes and their respective returns, see Fig. 1. From this figure, directly is observed that the time series are non-stationary, because the value of Skewness differ from 0 and the Kurtosis differs from 3 (normal distribution). For all-cases the Shapiro–Francia (p-valor) normality test are less than 0.0001, and this also shows that the distributions are different from normal distribution. Specifically in the case of index analysis, we can see that the values of Skewness and Kurtosis are the same for the stock market opening (□) and the stock market closing ( ), with positive Skewness and Kurtosis less than 3 (platykurtic distribution). For the return (Eq. (1)), in the opening ( ) and in the closing ( ) stock market, we have the mean equal to zero with standard deviation less than 0.01. Also, we can identify Kurtosis great than 3 (leptokurtic distribution) and negative Skewness, with a noticeable difference between the opening and closing of the stock market (see Fig. 1(b)). But, descriptive statistic is poor on the aspect of correlations analysis. Therefore, in order to give a better view on the auto and cross-correlations, we briefly describe the statistical method for this type of analysis in the next section. 3. DCCA auto and cross-correlation analysis Before we introduce the DCCA cross-correlation coefficient, we start with the DFA method [5] for auto-correlation analysis. DFA method is based in the fluctuation function, FDFA , defined from a time series {xi } with i = 1, 2, 3, . . . , N (length).
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Table 1 Descriptive statistics with the mean value and the standard deviation (open/close index). Mean Open ^000001.SS
sd Close
Open
Close
2 777.26
2 780.68
572.56
573.63
^BVSP
60 925.50
60 940.02
12 121.67
12 153.43
^DJI
17 496.89
17 501.78
4 295.37
4 293.20
4 379.10
4 378.80
706.69
706.80
^FCHI ^GDAXI ^HSI
4 379.10
4 378.80
706.69
706.80
23 675.67
23 661.03
3 165.54
3 160.97
^IBEX
9 444.33
9 439.89
1 143.55
1 143.63
^IXIC
4 686.05
4 686.83
1 582.92
1 582.64
^MERV
12 521.22
12 533.23
10 093.08
10 103.92
^MXX
43 189.48
43 188.96
4 302.67
4 301.16
^N225
15 845.57
15 843.27
4 687.35
4 686.92
Fig. 1. Descriptive statistics for all indexes. Skewness and Kurtosis are presented here for (a) Open-Close index, and (b) Open-Close return.
Continuing with the methodology, the integrated time series is calculated: Xk =
k ∑
(xi − ⟨x⟩) ∴ cumulative sum
(2)
i=1
where, ⟨x⟩ represent the mean and k = 1, 2, 3, . . . , N. Xk is divided into box windows of length n, and the local trend of each box, Xk,n , is obtained by a least-squares fit of each series. The fluctuation function is calculated by:
N 1 ∑ FDFA (n) = √ (Xi − Xi,n )2 N
(3)
i=1
Therefore, if there is power-law auto-correlations, then: FDFA (n) ∼ nαDFA
(4)
Exponent αDFA quantifies the empirical strength of the long-range (power-law) auto-correlations of the signal, a selfaffinity parameter [7] (see Table 2 for αDFA properties). The advantage of DFA over other methods lies in allowing the
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L.S.A. da Silva, E.F. Guedes, P. Ferreira et al. / Physica A 534 (2019) 122152 Table 2 αDFA exponent and properties. Exponent
Type of signal
αDFA αDFA αDFA αDFA αDFA αDFA
Anti-persistent Uncorrelated, white noise Long-range correlated persistent 1/f noise Non-stationary Brownian motion
< 0.5 ⋍ 0.5 > 0.5 ⋍ 1.0 > 1.0 ⋍ 3/2
detection of long-range auto-correlations embedded in seemingly non-stationary time series, and also avoiding spurious detection of apparent long-range auto-correlations, which are an artifact of non-stationarity. It is possible to generalize the DFA method to measure cross-correlation, and this generalization will be introduced briefly as DCCA method [8]. Therefore, considering two time series, {xi } and {yi }, with 1 = 1, 2, . . . ., N, and the integrated time series as Eq. (2), we obtain two new time series, {Xk } and {Yk }. Thus, if we have overlapping boxes, these two integrated time series, {Xk } and {Yk }, are divided into (N − n) boxes of equal length n, with 4 ≤ n ≤ N4 . The local trend of each box, Xk,n and Yk,n , are obtained by a least-squares fit of each series, and the covariance function of the residuals in each box are calculate the by: 2 fDCCA (n, i) =
i+n ∑
1 (n + 1)
(Xk − Xk,n )(Yk − Yk,n )
(5)
k=i
Finally, the average over all (N − n) overlapping boxes is calculated to obtain the detrended covariance function: 2 FDCCA (n) =
1 (N − n)
N −n ∑
2 fDCCA (n, i)
(6)
i=1
If power-law are present in DCCA cross-correlation function, then: 2 FDCCA (n) ∼ n2λ
(7)
The λ exponent quantifies long-range (power-law) cross-correlations and also identifies seasonality [9,10], but λ does not quantify the level of cross-correlations. It is possible to quantify the level of cross-correlation with DFA and DCCA methods using the DCCA cross-correlation coefficient [6]. ρx,y (n) is defined as the ratio between the detrended covariance 2 (Eq. (6)) and the detrended variance function FDFA (Eq. (3)): function FDCCA
ρx,y (n) =
2 (n) FDCCA
FDFAx (n) FDFAy (n)
(8)
It is observed that, ρx,y (n) ranges from −1 ≤ ρx,y ≤ 1, and: i. ρx,y = 1 means a perfect DCCA cross-correlation; ii. ρx,y = 0 there is no DCCA cross-correlation; iii. ρx,y = −1 means a perfectly anti DCCA cross-correlation. The advantage of this detrended cross-correlation coefficient lies in measuring cross-correlations between two nonstationary time series at different time scales [11–27], among others papers. 4. Results and discussions Initially, we used the data containing the index of each stock market at the opening and at its closing. Our choice was based on the main world indexes, located in the North America, South America, Asia and Europe. Also, besides the values of the indexes, we did analysis for the value of the return, Eq. (1). The results for auto and DCCA cross-correlation analysis are presented respectively right below. 4.1. DFA exponents DFA analysis show that, FDFA (n) ∼ nαDFA , for the stock market index and the returns, see Fig. 2. The values with αDFA exponent (linear adjustment) for each stock market are presented in Table 3. The αDFA exponents values are larger than the unit for the stock market indexes, which indicates non-stationarity (as expected) of the time series. For the return, the values are relatively close to 0.5. The indexes closer to 0.5 (coined with the efficiency [28]) are ^N225 (result common to other studies), ^HSI, ^BVSP and ^MERV. The indexes farthest from the efficiency level are ^IXIC, ^DJI and ^MXX. Most of the indexes show anti-persistence, only China is the exception.
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Fig. 2. FDFA as a function of time scale n for all indexes: (a) Index Open, (b) Index Close, (c) Return Open, and (d) Return Close. Table 3 DFA exponent for index and return. The values of the linear adjustments for αDFA always had R2 > 0.99. Index
Return
DFA exponent
1 2 3 4 5 6 7 8 9 10 11
^BVSP ^DJI ^FCHI ^GDAXI ^HSI ^IBEX ^IXIC ^MERV ^MXX ^N225 ^000001.SS
DFA exponent
Open
Close
Open
Close
1.40 1.43 1.42 1.41 1.42 1.40 1.44 1.42 1.32 1.44 1.47
1.40 1.42 1.37 1.41 1.42 1.40 1.44 1.42 1.32 1.44 1.48
0.48 0.41 0.43 0.46 0.48 0.45 0.39 0.48 0.41 0.48 0.53
0.48 0.41 0.43 0.46 0.48 0.44 0.40 0.48 0.41 0.47 0.54
4.2. DCCA cross-correlation coefficient Fig. 3 present the results for ρx,y between the Open and Close stock market indexes. We observed that there is a positive DCCA cross-correlation between opening and closing indexes, and more, ρx,y for n ≥ 66 tends to a perfect DCCA cross-correlation with fast convergence. It means that a change in the opening/closing price today is fully internalized after 66 days for all indexes. In small time scales, there are differences between the values of ρx,y , but for all cases ρx,y > 0.5. Considering the return indexes, Fig. 4 show the results for ρx,y between the Open and the Close values. In this case, the convergence to perfect DCCA cross-correlation occurs only for long time-scales, which means less memory in the signs. In an economic point of view, the results about the original indexes are expected, since it is normal to have high similarities between the open and the close values. In what refers to returns, the results are quite interesting. Our results point to the existence of low DCCA cross-correlations between open and closing returns in some indexes, namely ^BVSP, ^DJI, ^MMX and ^MERV. In these cases, since the correlation is null, we may conclude that the open return does not influence the close return of the same day. For example, for ^HSI in the short time-scale, the DCCA cross-correlation is around 0.5. Fig. 5 show the smaller time-scale, n = 4, and in this case it is possible to identify differences between the values of ρx,y depending on the stock market.
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Fig. 3. ρx,y as a function of time scale n for DCCA cross-correlation between Open and Close stock market index. Vertical line show n = 66 days.
Fig. 4. ρx,y as a function of time scale n for DCCA cross-correlation between the return Open-Close values.
Fig. 5. ρx,y with n = 4 for all indexes (a) and respective return (b).
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5. Conclusions We found respectively for the stock market indexes and the return long-range (power-law) DFA auto-correlation. In these cases αDFA > 1.0 for the original index (identifying that the original time series of the index are non-stationary) and for the return αDFA ≃ 0.5, with predominance for anti-persistent behavior depending on the stock market. Also, between open and close values, the indexes and the return present higher level of DCCA cross-correlation for long time-scale. Therefore, for short time-scale, the results show different pattern in the stock markets. In the case of the stock market indexes, the transition for perfect DCCA cross-correlation is found for n > 66 days, whereas for the returns only for large time-scales. Lastly, by the specialized literature, price of opening stock markets are more commonly used by investors with the intention of short-term financial returns. 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