Cross-correlation between interest rates and commodity prices

Accepted Manuscript Cross-correlation between interest rates and commodity prices Qing Wang, Yiming Hu PII: DOI: Reference:

S0378-4371(15)00159-4 http://dx.doi.org/10.1016/j.physa.2015.02.053 PHYSA 15935

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Physica A

Received date: 25 November 2014 Revised date: 28 January 2015 Please cite this article as: Q. Wang, Y. Hu, Cross-correlation between interest rates and commodity prices, Physica A (2015), http://dx.doi.org/10.1016/j.physa.2015.02.053 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

>Investigate cross-correlations between interest rate and agricultural commodity markets >The cross-correlations are all significant and persistent. >We find strong multifractality in both auto-correlations and cross-correlations. >The time-variation property of cross-correlations is also revealed.

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Cross-correlation between interest rates and commodity prices

Qing Wang, Yiming Hu Institute of Chinese Financial Studies of Southwestern University of Finance and Economics

Date of this revision: Jan, 2015

Abstract In this paper, we investigate cross-correlations between interest rate and agricultural commodity markets. Based on a statistic of Podobnik et al. (2009), we find that the cross-correlations are all significant. Using the MF-DFA and MF-DXA methods,

we

find

strong

multifractality

in

both

auto-correlations

and

cross-correlations. Moreover, the cross-correlations are persistent. Finally, based on the technique of rolling window, the time-variation property of cross-correlations is also revealed.

Keywords: Interest rate; Cross-correlations; MF-DXA;Time variation



Corresponding authors. E-mail address: [email protected]. 1

1. Introduction Interest rate and commodity price can affect each other in the economic sense. On one hand, interest rate is one of the major determinants of agricultural commodity prices. According to the standard cost of carry theory, higher interest rate will improve the requirements for the future benefit from commodity storage, leading to decreases in spot prices [1]. On the other hand, the usefulness of commodity prices in formulating monetary policy has also been well documented in the literature [2-4]. The reason is that commodity markets are informationally efficient, respond quickly to general economic conditions and thereby can provide instantaneous information about the state of economy. Additionally, many commodities are also the important inputs of industry production. Therefore, increases in commodity prices can result in higher manufactured goods and finally cause the general inflation [5]. In the literature, the relations between commodity price and the interest rate are always investigated based on the vector autoregressive (VAR) model or the vector error correction (VEC) one [6-10]. Both VAR and VEC assume the linear joint dynamics as they are actually the standard regressions with the lagged variables. According to the arguments in the econophysics studies, the linkages among economic variables are intrinsically nonlinear and change over time [11-19]. Thus, it is less appropriate to capture their joint dynamics using linear specifications. Unfortunately, nonlinearity in the relations between commodity price and the interest rate has not been addressed in the literature. Moreover, it is well known that due to business cycles and some extreme events, the relations among different economic 2

variables are not constant, but change over time. The existing studies have not documented the time variations in the linkages between agricultural commodity price the interest rate. Our focus is to fill these gaps. We start from fresh perspectives in investigating the relationships between commodity price and the interest rate. First, we employ a simple but very powerful statistical test of Podobnik et al. [20] to qualitatively examine whether their cross-correlations are significantly. Second, we use a method borrowed from statistical physics to investigate the long-range cross-correlations. This multifractal detrended cross-correlation analysis (MF-DXA) [21], a multifractal generalization of the seminal DXA firstly developed by Podobnik and Stanley [22], has been considered as a powerful tool in analyzing nonlinear cross-correlations in nonstationary time series. Recently, MF-DXA is also widely applied to finance researches [23-25]. Third, we use a rolling window method to investigate the evolution of cross-correlations over time. We use the daily data of effective Federal Funds rate and four major agricultural commodity prices (corn, soybean, wheat and rice). We find that the cross-correlations are significant at 10% significance level for greater lag orders. The results based on MF-DXA indicate that the cross-correlations are persistent and multifractal. The degrees of persistence and multifractality differ depending on the type of commodity. The evidence based on rolling window method reveals the significant time-varying property in cross-correlations. The remainder of this paper is organized as follows. Section 2 provides the 3

description of methodologies used in current paper. Section 3 contains data. Section 4 lists the main empirical findings. In Section 5, we perform some meaningful discussions. The last section concludes the paper.

2. Methodology Let us briefly introduce the MF-DXA method [21]. Assume that there are two series x(i) and y(i) (i = 1,2,… ,N), where N is the equal length of these two series. Step 1. Construct the profile i

i

t=1

t=1

X (i )  （x(t )- x）, Y (i )  （y (t )- y）

(1)

where, x and y denote the average of the two whole time series x(i) and y(i). Step 2. The profiles X(i) and Y(i) are divided into Ns = [N/s] non-overlapping windows (or segments) of equal length s. Since the length N is not always a multiple of the considered time scale s. In order to not discard the Section of series, the same procedure is repeated starting from the opposite end of each profile. Thus, 2Ns non-overlapping windows are obtained together. Step 3. The local trends Xν(i) and Yν(i) for each segment ν (ν = 1, 2, 3,

. . ., 2Ns) are

evaluated by least squares fits of the data, then the detrended covariance is determined by

1t 2 v v F ( s, v)   X ((v  1) s  i)  X (i)  Y ((v  1) s  i)  Y (i) s i=1

(2)

for each segment ν, ν = 1, 2, …, Ns and

1t 2 v v F (s, v)   X ( N  (v  N s)s  i)  X (i)  Y ( N  (v  N s)s  i)  Y (i) s i=1 4

(3)

for each segment ν, ν=Ns+1, Ns+1, …, 2Ns. Then the trends Xν(i) and Yν(i) denote the fitting polynomial with order m in each segment ν (conventionally called MF-DXA-m.[26] ). Step 4. qth-order the fluctuation function as follows. F q ( s)  [

1 2N s

2 Ns

 [ F (s, )] 2

 =1

1/ q q /2

]

(4)

If q  0, then  1 2 Ns   2 F 0( s)  exp  ln[ F (s, )]      4 N s  =1

(5)

When q = 2, MF-DXA is the standard of the DXA. Step 5. Analyze the scaling behavior of the fluctuations by observing log-log plots Fq(s) verse s for each values of q. If the two series are long-range cross-correlated, Fq(s) will increase for values of s, we can obtain a power-law expression F q( s) ~ s H xy

(q)

(6)

This can be presented as follows

log F q(s)  H xy(q) log( s)  log A

(7)

The scaling exponent Hxy(q) is known as the generalized cross-correlation exponent, describing the power-low relationship. Especially, if the two time series are equal, MF-DXA is the MF-DFA. when the scaling exponent Hxy(q) is independent q, the cross-correlation between two series is monofractal. If the scaling exponent Hxy(q) is dependent on q, the cross-correlation between two series is multifractal. Moreover, if the scaling exponent Hxy(q) > 0.5, the cross-correlations between the return fluctuations of the two series related to q are long-range persistent. If the scaling 5

exponent Hxy(q) < 0.5, the cross-correlations between the return fluctuations of the two series related to q are anti-persistent. If Hxy(q) = 0.5, there are no cross-correlations between the two series. Furthermore, for positive q, Hxy(q) describes the scaling behavior of the segments with large fluctuations. On the contrary, for negative q, Hxy(q) describes the scaling behavior of the segments with small fluctuations. According to the multifractal formalism, the Renyi exponent xy(q) can be used to characterize the multifractal nature

 xy (q)  q H xy (q)  1

(8)

If the Renyi exponent xy(q) is liner of the q, we can conclude that the two correlated series is monofractal, otherwise, it is multifractal. The multifractal spectra fxy(α) describes the singularity content of the time series one can obtain through the Legendre transform:

 xy(q)  H xy(q)  q H 'xy(q)

(9)

f xy ( )  q( xy  H xy(q))  1

(10)

In order to measure the time-varying degree of multifractality, the financial risk measure ΔH was proposed by [27-29], as the following

H  H max (q)  H min(q)

(11)

ΔH is equal to the difference of the Hmax(q) and Hmin(q). The greater are ΔH, the stronger is the degree of multifractality, therefore the greater is the stock risk, and vice versa.

6

3. Data description In this paper, we choose US interest rate (Effective Federal Funds Rate) and four agricultural commodities, corn (CBOT Corn Futures), soybean (CBOT Soybean Futures), wheat (CBOT Wheat Futures) and rice (CBOT Rice Futures). Our data is collected from the Datastream, covering the period from 1 January 2000 to 31 September 2014. We collect the data from Datastream. In addition, the trade days are not the same, so we use the STATA software to match the day, which has a trading of the interest rate and agricultural commodities future markets in that day. After removing the different days, we get the four pairs observations (interest rate and corn, 3707 observations; interest rate and soybean, 3711 observations; interest rate and wheat, 3712 observations; interest rate and rice, 3711 observations) in our empirical study. Agricultural commodities returns are computed by the formula, Rt =100*(log(Pt)– log(Pt-1)), where Pt is the closing price index at time t. In addition, we use the first order differences of interest rate to investigate the cross-correlations with the agricultural commodities price. Fig. 1 plots agricultural commodity prices and interest rate. Table 1 shows the descriptive statistics of interest rate and agricultural commodities return. All of the interest rate and agricultural commodities prices return are significantly skewed and leptokurtic at 1% level1, suggesting that they are the fat-tail distributed. The Jarque-Bera statistic further shows that the null hypothesis of normality is rejected at the 1% level of significance. The Ljung–Box statistic for serial 1

We use the econometric software, WINRATS (V 8.00, 32 bit) to test the null hypothesis, “Skewness = 0” and “ Kurtosis = 3”. 7

correlation shows that the null hypotheses of no autocorrelation up to the 20th order are rejected. Furthermore, the Augmented Dickey–Fuller support the rejection of the null hypothesis of a unit root at the 1% significance level, implying that the all of agricultural commodities prices are stationary and can be modeled directly without further transforms. Table 1 shows that the interest rate series is not stationary, which apply the empirical evidence to analyze between the first order differences of interest rate and agricultural commodities prices in our paper.

(Insert Table 1 about here)

(Insert Fig. 1 about here)

4. Empirical results 4.1cross-correlations test In order to quantify the cross-correlation between interest rate and the commodity markets, we introduce a new cross-correlation statistic proposed by Podobnik et al. [22] in analogy to the Ljung-Box test [32]. The cross-correlation statistic between two series {x(i)}, {y(i)}, which have the same length N, functions as follow N

Ci 

 x k y k i

k i 1 N

N

x y

k 1

2 k

k 1

(12)

2 k

Then the cross-correlation test statistic m

2

Ci i 1 N  i

Q cc (m)  N 2 

(13)

which is approximately 2(m) distributed with m degrees of freedom. If there are no 8

cross-correlations between two time series, the cross-correlation test agrees well with the 2(m) distribution. If the cross-correlations test exceeds the critical value of the

2(m) distribution, then the cross-correlations are significant at a special significance level. As a comparison, we also describe the critical value of the 2(m) distribution at the 10% level of significance for the degrees of freedom varying from 1 to 1000. According to equations (11) - (12), we can obtain the cross-correlation statistic (logarithmic form) Qcc(m) for the interest rate and the commodity prices return (e.g., corn, soybean, wheat and rice). From Fig.2, the cross-correlation statistic Qcc(m) for the four pairs (interest rate and corn, interest rate and soybean, interest rate and wheat, interest rate and rice) are always larger than (or close to) the critical values for the

2(m) distribution at the 10% level of significance, which suggests there are existed long-range cross-correlations. (Insert Fig. 2 about here)

Furthermore, to verify the results as above, we also apply another new method proposed by Podobnik et al. [33], the function as below:

 DCCA 

2

F DCCA(n) F DFA1 {n F} DFA

(14) n2 { }

Then, The value of  DCCA ranges between 1  DCCA  1 .If  DCCA equal to zero, which the two series have no cross-correlation, and it splits the level of cross-correlation between positive and the negative case. We calculate the values of

 DCCA based on different values of window size n (n=16, 32, 64, 128, 256) as Table 2 and draw the conclusions that are consistent with the cross-correlation test as above. (Insert Table 2 about here) 9

4.2 MF-DFA analysis In this paper, following previous works [14, 26-28], we set the time scale range to 10 < s < N/4, where N is the length of the return time series. It is well known that when H(q) is constant for all q, the time series are mono-fractal. Otherwise, the series are multifractal. In Table 3, we clearly find that four agriculture commodity prices return are multi-fractal, which the empirical results apply further evidence that the agriculture commodity stock exist a strong multi-fractal features as the same conclusions as the Ref. [11, 24]. However, to the best of my knowledge, there are few empirical studies of the multi-fractality in the interest rate market. Therefore, in our analysis, we use the MF-DFA method to investigate the auto-correlations of the US interest rate, which the empirical results show that the first order differences of interest rate is multi-fractal. Furthermore, when q = 2, the generalized Hurst exponent H(q) is exactly the Hurst exponent. From Table 3, we obtain that the difference of interest rate and wheat return series are less than 0.5. That means the both them present the anti-persistence. And the corn, soybean and rice present the persistence. (Insert Table 3 about here)

4.3 MF-DXA analysis Following the MF-DFA, we also choose the time scale s between 10 and N/4 (N is the length of each time series). Moreover, according to equations (12) - (13), we can only test for the presence of cross-correlation qualitatively. Thus, to present the cross-correlations quantitatively, we use to the MF-DXA method proposed by Zhou [21] to estimate a quantitative cross-correlation exponent. 10

It is well known that if the cross-correlation exponent Hxy(q) varies with different values of q, the two correlated series is multifractal; otherwise, it is monofractal. In Table 4, we discover that the cross-correlation exponent Hxy(q) between the first order differences of interest rate and agricultural commodities prices decreases with q varying from -10 to 10, implying that between the first order differences of interest rate and four agricultural commodities prices exist strong multi-fractal features during the sample period. Moreover, for q = 2, the bivariate Hurst exponent 0 < Hxy(2) < 1 has similar properties and interpretation as a univariate Hurst exponent [34]. when q = 2, the cross-correlation exponent Hxy(2) between the interest rate and corn price return is 0.52952, implying that the cross-correlation of between them are persistent in the whole sample. In addition, we can find that the Hxy(2) between the between the interest rate and other commodity prices return have also larger than 0.5 (interest rate and soybean, H(2) = 0.53752; interest rate and wheat, H(2) = 0.50920; interest rate and rice, H(2) = 0.55403), which mean that the two correlated series are persistent or long-range dependence. (Insert Table 4 about here)

5. Discussion

5.1 The strength of multifractality

In order to measure the time-varying degree of multifractality, the financial risk 11

measure ΔH was proposed by [27-29], so we follow the Eq. (11) to get the ΔH. Table 5 shows that the strength of multifractality of the rice is larger than others commodity markets (e.g., soybean, wheat and rice), which imply that the risk (volatility) of the rice is bigger than the corn, soybean and wheat markets. (Insert Table 5 about here)

Moreover, we use the MF-DXA method to get the ΔH. From Table 6, we find that cross-correlation between interest rate and corn price return is not only significant multifractal, but also is bigger than others in the degree of multifractality. Therefore, we can show that the change of interest rate has a larger influence on corn price than other commodity prices (for example, soybean, wheat and rice), reinforcing the stylized fact of multifractality. (Insert Table 6 about here)

5.2 Rolling window and the market efficiency

The method of rolling windows began to be widely used to investigate many topics on the financial markets after the influential work in Cajueiro and Tabak [31], such as market efficient [36-37] and risk management [38-39]. For different research purpose,the length of rolling windows is not fixed. Ref. [35, 39] used the length of several years to analyze the evolution of long term correlation, whereas Wang at al. [13] only used 250 data points as the rolling windows. Grech and Mazur [37] argued that if the length of rolling window is too large, thecalculated exponent may lose its

12

locality. Grech and Mazur [37] also argued that the local exponent at a given time t depends on the time-window length. Furthermore, when rather short time series are used, the universal multifractal hypothesis might be misleading [37]. Thus, for different purposes, the selection of the length of rolling window should be careful. In this paper, we take the same measure of Wang et al. [13], choosing 250 days as the length of rolling windows to investigate the dynamics of short-term cross-correlations (see in Fig.3). (Insert Fig. 3 about here)

Moreover, when q = 2, the bivariate Hrust exponent 0 < Hxy(2) < 1 has similar properties and interpretation as a univariate Hrust exponent [34]. For a financial market, if the Hurst exponent of the asset prices or returns is more close to 0.5, the market will be more close to weak-form efficiency. Therefore, we introduce the market efficiency index [40-41] as EI  | H  0.5 |

(15)

Based on Eq. (15), we can get the smaller the EI value of a market is, the higher the efficiency of the market could be. Fig. 3 shows the dynamics of the value of EI with the fixed window (= 250) using the method of rolling window. On the one hand, we find that the four commodity markets are not efficiency in sample period. On other hand, the cross-correlations between the interest rate and commodity markets are time-varying.

13

5.3. A binomial measure from P-model

Zhou [21] observed that for two time series constructed by a binomial measure from the p-model, the following relationship exists:

H xy(q)  [ H xx(q)  H yy(q)] / 2

(16)

The generalized Hurst exponents Hxx(q) (and Hyy(q)) of the interest rate (corn, soybean, wheat and rice) estimated by MF-DFA. If Hxx(q) (and Hyy(q)) is a constant, the time series (e.g., commodity markets) is mono-fractal; otherwise, it is multi-fractal. From Fig. 4, we can find that Hxx(q) and Hyy(q) vary with q, providing the further empirical evidences that the interest rate and commodity markets are multi-fractal. For each q, the right side is not equal to the left side for Eq. (16). (Insert Fig. 4 about here)

6. Conclusion

In this paper, we investigate the cross-correlations between the interest rate and the commodity markets using the multifractal detrended cross-correlation analysis method. Some interesting conclusions are as below. First, according to the cross-correlation statistic, the empirical results show the cross-correlations between the interest rate and the commodity markets are significant. Of course, we not only use the qualitative analysis of the cross-correlation test, but also take the quantitative analysis of the MF-DXA to confirm the cross-correlation between them. Thus, our evidence suggests that shocks to the real interest rate 14

contribute significantly to movements in commodity prices. Second, dependent on the MF-DFA and MF-DXA method, the empirical results can evident that the interest rate and the commodity markets all exhibit the multifractal features, and the cross-correlations between them are also the multifractality. In addition, the strength of multifractality of the rice is larger than others commodity markets, which imply that the risk of the rice is bigger than the corn, soybean and wheat markets. Furthermore, cross-correlation between interest rate and corn commodity market is bigger than others in the degree of multifractality. Finally, we employ the technique of the rolling window to investigate the dynamic of the scaling exponent Hxy(q) and find that the cross-correlations between the interest rate and commodity markets are time-varying. Furthermore, we explore the relationship between the cross-correlation exponents Hxy(q) and the average scaling exponents [Hxx(q) +Hyy(q)]/2]. Clearly, we provide the empirical evidence that the linear regression models (for example, VAR and VEC) cannot be used to describe the dynamics of cross-correlations between the interest rate and the commodity markets.

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20

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soybean

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Fig. 1. Interest rate and agricultural commodity prices.

5000

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3

critical value interest rate and corn interest rate and soybean interest rate and white interest rate and rice

2.5

2

Qcc(m)

1.5

1

0.5

0

-0.5

-1

0

0.5

1

1.5

2

2.5

m

Fig. 2. The cross-correlation statistic between the interest rate and the commodity prices return (e.g., corn, soybean, wheat and rice).

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2500

2000

2500

1500

2000

2500

1500

2000

2500

n

interest rate and soybean 0.7 0.6

H

0.5 0.4 0.3 0.2

0

500

1000

1500

n

interest rate and wheat 0.7 0.6

H

0.5 0.4 0.3 0.2

0

500

1000

n

interest rate and rice 0.7 0.6

H

0.5 0.4 0.3 0.2

0

500

1000

n

Fig. 3. Time-varying EI with the rolling window. The rolling window length is 250.

interest rate and corn

interest rate and soybean

0.8

0.8

Hxx-interest rate Hyy-corn Hxy [Hxx+Hyy]/2

0.7

Hxx-interest rate Hyy-soybean Hxy [Hxx+Hyy]/2

0.7

0.5

0.5

Hq

0.6

Hq

0.6

0.4

0.4

0.3

0.3

0.2

0.2

0.1 -10

-8

-6

-4

-2

0

2

4

6

8

0.1 -10

10

-8

-6

-4

-2

q

0

2

4

6

8

10

q

interest rate and wheat

interest rate and rice

0.8

0.8

Hxx-interest rate Hyy-wheat Hxy [Hxx+Hyy]/2

0.7

Hxx-interest rate Hyy-rice Hxy [Hxx+Hyy]/2

0.7

0.5

0.5

Hq

0.6

Hq

0.6

0.4

0.4

0.3

0.3

0.2

0.2

0.1 -10

-8

-6

-4

-2

0

q

2

4

6

8

10

0.1 -10

-8

-6

-4

-2

0

2

4

6

8

10

q

Fig. 4. The relationship of Hxx(q), Hyy(q), Hxy(q), [Hxx(q) + Hyy(q)]/2 and different q with varying from −10 to 10.

Table 1 Descriptive statistics Mean

Max

Min

S.D

Ske

Kur(excess) ***

J-B

ADF

Q(20)

-1.3920

107273.1910***

2.0433

7.0300

0.040000

2.1168

0.7537

corn

0.0126

12.7571

-26.8620

1.9111

-0.6575***

12.5805***

24706.2521***

-59.0786***

28.4320**

soybean

0.0187

96.6359

-98.3147

3.6026

-0.6774***

434.4353***

29214736.1285***

33.4218***

541.3210***

wheat

0.0177

8.7943

-10.8104

2.0681

0.1004***

2.0311***

644.1130***

-62.1094***

31.1850***

1.8056

***

***

30.1580***

0.0196

28.0816

-24.4535

0.4454

28.1796

***

684.0745

***

interest rate

rice

-0.8806

***

122908.6775

***

-57.1571

Note： „„Max‟‟, “Min”, „„S.D‟‟, „„Ske‟‟, „„Kur‟‟, “J-B” and ADF denote Maximum, Minimum, Stv. Dev, Skewness, Kurtosis, Jarque-Bera statistic [30] and the Augmented Dickey–Fuller statistic [31] respectively. Q(i) denotes the value of Ljung-Box-Pierce Q statistics with i lags. ***

denote 1% significance levels;

**

denote 5% significance levels;

*

denote 10% significance levels;

Table 2 The value of

 DCCA when for a given window size n

n

16

32

64

128

256

interest rate and corn

0.17226

0.18899

0.19444

0.35021

0.46099

interest rate and soybean

0.12123

0.167754

0.23887

0.41106

0.51343

interest rate and wheat

0.18364

0.20942

0.30571

0.34887

0.47120

interest rate and rice

0.21894

0.21189

0.30021

0.27062

0.45795

Table 3 Generalized Hurst exponents of the interest rate and four commodity prices return with q varying from -10 to 10 (the MF-DFA) q

interest rate

corn

soybean

wheat

rice

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

0.66239 0.65662 0.65002 0.64243 0.63365 0.62348 0.61173 0.59855 0.58541 0.57612 0.53568 0.51101 0.41751 0.32377 0.25585 0.21014 0.17864 0.15596 0.13895 0.12576 0.11523

0.73421 0.72473 0.71342 0.69987 0.68365 0.66458 0.64314 0.62088 0.59988 0.58117 0.56349 0.54862 0.53317 0.51761 0.50210 0.48724 0.47367 0.46168 0.45130 0.44236 0.43467

0.64116 0.63402 0.62601 0.61720 0.60786 0.59845 0.58965 0.58216 0.57656 0.57317 0.56989 0.56709 0.55578 0.53476 0.50800 0.48156 0.45868 0.43989 0.42462 0.41215 0.40184

0.51604 0.50969 0.50269 0.49510 0.48719 0.47949 0.47299 0.46907 0.46917 0.47406 0.48416 0.49197 0.49742 0.49642 0.48929 0.47843 0.46640 0.45480 0.44429 0.43504 0.42698

0.67120 0.66516 0.65831 0.65056 0.64194 0.63260 0.62306 0.61403 0.60547 0.59556 0.57545 0.55844 0.52484 0.48147 0.43668 0.39853 0.36881 0.34610 0.32855 0.31473 0.30362

Table 4 Cross-correlation exponents for the interest rate and the four commodity prices return with q varying from -10 to 10 (MF-DXA) q

interest rate and corn

interest rate and soybean

interest rate and wheat

interest rate and rice

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

0.70008 0.69133 0.68103 0.66889 0.65462 0.63824 0.62046 0.60321 0.58931 0.58001 0.56768 0.55738 0.52952 0.48754 0.44109 0.40036 0.36834 0.34385 0.32500 0.31022 0.29840

0.61519 0.61089 0.60631 0.60151 0.59662 0.59196 0.58812 0.58603 0.58641 0.58763 0.57694 0.56774 0.53752 0.49715 0.45539 0.41900 0.38982 0.36699 0.34906 0.33476 0.32316

0.66763 0.65839 0.64729 0.63392 0.61790 0.59922 0.57895 0.55988 0.54572 0.53785 0.53086 0.52489 0.50920 0.48692 0.46312 0.44181 0.42419 0.40990 0.39827 0.38868 0.38066

0.70604 0.69922 0.69140 0.68244 0.67224 0.66083 0.64853 0.63604 0.62431 0.61348 0.59637 0.58259 0.55403 0.51657 0.47699 0.44185 0.41332 0.39079 0.37296 0.35868 0.34706

Table 5 The commodity stock markets of strength of multifractality ΔH

corn 0.29954

soybean 0.23932

wheat 0.08906

Notes: ΔH is calculated by the H(-10) minus H(10) using the MF-DFA method.

rice 0.36758

Table 6 The commodity stock markets of strength of multifractality ΔH

corn 0.40168

soybean 0.29203

wheat 0.28697

Notes: ΔH is calculated by the H(-10) minus H(10) using the MF-DXA method.

rice 0.35898