Changing value detrended cross correlation coefficient over time: Between crude oil and crop prices

Accepted Manuscript Changing value detrended cross correlation coefficient over time: Between crude oil and crop prices Subrata Kumar Mitra, Vaneet Bhatia, R.K. Jana, Parikshit Charan, Manojit Chattopadhyay

PII: DOI: Reference:

S0378-4371(18)30471-0 https://doi.org/10.1016/j.physa.2018.04.034 PHYSA 19466

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Physica A

Received date : 17 December 2016 Revised date : 7 November 2017 Please cite this article as: S.K. Mitra, V. Bhatia, R.K. Jana, P. Charan, M. Chattopadhyay, Changing value detrended cross correlation coefficient over time: Between crude oil and crop prices, Physica A (2018), https://doi.org/10.1016/j.physa.2018.04.034 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.

Your Reference: PHYSA 19466 Manuscript Number: PHYSA-162195

Title Page information Title: Changing Value Detrended Cross Correlation Coefficient Over Time: Between Crude Oil and Crop Prices

Keywords: crude oil; food crop prices; comovement; detrended cross correlation analysis

Order of Authors: Subrata Kumar Mitra, Vaneet Bhatia, R K Jana, Parikshit Charan, Manojit Chattopadhyay

Abstract: In this paper, we analyzed changing comovement relationship between prices of crude oil and food crops/food index using a detrended cross-correlation analysis (DCCA) approach proposed by Podobnik et al. (2008). In order to detect the changing nature of cross-correlation coefficient over time, we estimated the detrended cross-correlation coefficient on a rolling window and found that cross-correlations were negative before 2004 for most of the items but they increased considerably after the surge in crude oil price in 2007 that caused diversion of food grains for production of biofuels.

Authors and their affiliations: Sequence 1 2 3 4 5

Author Subrata Kumar Mitra Vaneet Bhatia R K Jana Parikshit Charan Manojit Chattopadhyay

Email ID [email protected] [email protected] [email protected] [email protected] [email protected]

Corresponding Author: Subrata Kumar Mitra Corresponding Author's Institution: Indian Institute of Management Raipur

Affiliation Indian Institute of Management Raipur, Sejbahar, Raipur 432015, India

Highlights 

Comovement between prices of crude oil and food crops/food index was analysed 



Used detrended cross‐correlation analysis (DCCA) approach 



Time varying relationship was captured using a rolling window approach 

Changing Value Detrended Cross Correlation Coefficient Over Time: Between Crude Oil and Crop Prices

1.

Introduction

Oil is the second most important contributor to energy consumption worldwide. It is one of the key drivers of the economy. The fluctuation in crude oil price acts as a barometer of the economy. As a result, crude oil price movements are a matter of concern for investors, economists, and policymakers. It has a direct influence on almost all production related activities. As a result, the fluctuation in crude oil price is an important topic of research (Wang, Chen & Jin, 2009).

It is well known that many nations meet the major portion of their oil requirement through imports. The increase in crude oil price provides an opportunity for many nations to focus more on producing biofuels, a major alternative source of energy (Nigam & Singh, 2011). Also, due to concerns related to environmental implications and depleting natural resources, biofuels have emerged as a sustainable alternative to crude oil. The demand for biofuels is expected to sustain in future due to obligatory blending requirements (OECD/FAO, 2017). Now, biofuels are made mostly from food crops such as sugarcane, soybean, wheat, maize, etc. So, under the scenario of increasing crude oil price, demands of these crops will be more. As a result, the prices of these crops will also increase.

The literature suggests that scholars have explored the relationship between crude oil price fluctuation and various commodities (Pindyck & Rotemberg, 1990; Cashin, McDermott, &

Scott, 1999; Lescaroux, 2009; de Nicola, De Pace, & Hernandez, 2016; Kang, McIver, & Yoon, 2017). Baffes (2007) and Mitchell (2008) have conducted similar studies for investigating the associations between prices of crude oil and other commodities. They observed that prices of both crude oil and food crops had increased significantly during the year 2007 to the middle of 2008. Liu (2014) analyzed the relationship between prices of crude oil and agricultural products including soybean, corn, wheat, and oat. Rezitis (2015) investigated the linkage between prices of crude oil and agricultural commodities and observed their effect on the exchange rate of US dollar.

We find from the literature that relationship between price fluctuation of crude oil and agricultural commodities have been less explored. Liu (2014) and Rezitis (2015) attempted to explore such relationships. However, their studies did not include maize and sugarcane which account for nearly 80% (53.91% from maize and 25.96% from sugarcane) of ethanol production (OECD/FAO, 2017). Therefore, these two food crops can be used as a substitution for analyzing the relationship between prices of crude oil and agricultural commodities. We included the major commodities that dictate the substitution of crude oil with biofuel and investigated the relationship between crude oil and agricultural commodities.

The purpose of this study is to analyze the dynamic relationship between prices of crude oil and major agricultural commodities namely, maize and sugarcane, used for biofuel production. More specifically, we attempt to investigate whether the cross-correlation between prices of agricultural commodity and crude oil are changing over time or not. We have employed time series data on monthly observations from February 1995 to November

2016 for this purpose. For checking the robustness of the obtained results, the relationship between the price of crude oil with other variables like wheat, rice, soybean and Food and Beverage (F&B) index was also investigated. F&B index was included for capturing the effect of a change in the price of crude oil on a broader category of food items.

To achieve the purpose of our study, we have used the Detrended Cross Correlation Analysis (DCCA) approach (Podobnik & Stanley, 2008). It has been used in a number of studies to detect cross-correlations because of its ability to compute robust estimates. However, we came across only one study that used DCCA to explore the co-movement between the price of crude oil and crop prices (Liu, 2014). The DCCA offers several advantages over other methods used in financial economics. First, it allows capturing the dynamic system for nonstationary time series data. Second, Stanley, Gabaix, Gopikrishnan, and Plerou (2007) noted that the correlations between variables change in time and DCCA helps to capture such dynamic relationship. Third, detrending approaches has the capability of uncovering crosscorrelation and power-law auto-correlations in the long range of the stochastic process. Podobnik, Horvatic, Petersen, and Stanley (2009) observed that ignoring periodic trends in a time series significantly affect the long-range correlations analysis. Finally, the DCCA is a robust method for investigating the cross-correlation between the two time series (Liu, 2014).

This study makes a couple of contributions to the related literature. First, we explore the relationship between prices of crude oil and food crops used for biofuel production as well as a broader set of agricultural commodities by considering the F&B index. Second, to capture the changing nature of DCCA correlation coefficient over time, we used moving windows of

different sizes, namely, 24, 36, 48, and 60 months. Further, the DCCA coefficient also depends on the time frame taken for measuring ρDCCA and therefore, different periods for estimating DCCA (n = 1 to 30 in steps of 1) were used.

The remainder of this paper is presented as follows: Section 2 presents a brief literature review.

Section 3 presents data sources, data description, and methodology. Section 4

presents the empirical analysis and important findings. Section 5 concludes the paper.

2.

Literature review

The research on exploring the relationship between crude oil and agricultural commodities prices can be segregated into three categories (Nazlioglu, Erdem, & Soytas, 2013). First, studies analyzing the impact of the price of crude oil on production costs of agricultural commodities. Second, the relationship between crude oil and biofuels prices. Third, the relationship between crude oil and agricultural commodities prices regarding investment fund activity. The present study belongs to the second category. Serra, Zilberman, Gil, and Goodwin (2011) explored the links between gasoline, oil, corn and ethanol prices in the US and reported a long run relationship between them. They also showed a strong association between energy and food prices. Hassouneh, Serra, Goodwin, and Gil (2012) analyzed the price transmissions between crude oil, biodiesel and sunflower oil in Spain. They reported an equilibrium and a long-run relationship between crude oil, sunflower, and biodiesel prices. Highlighting the debate regarding the use of agricultural production for biofuels, Abdelradi and Serra (2015) suggested a bi-directional and asymmetric relationship between food and biofuel prices.

Cabrera and Schulz (2016) reported a long run relationship between energy and agricultural commodity prices and suggested that such relationship preserves an equilibrium in Germany. de Nicola et al. (2016) suggested an increase in co-movement between crude oil and biofuel producing agricultural commodities. Pal and Mitra (2017) highlighted the relationship between soybean and diesel prices in the USA and reported a long run relationship between them that vary in quantiles. Kang, McIver, & Yoon (2017) provided an empirical evidence indicating the return and volatility spillovers between commodities. Alvarez-Ramirez, Alvarez, and Rodriguez (2008) detected a co-movement between prices of crude oil and agricultural commodities. Power and Turvey (2010) used wavelet analysis and found the presence of a significant long-range relationship in 14 commodity prices. Regarding methodology, the earlier literature indicates the use of traditional methods to analyze the relationship between prices of oil and agricultural commodities. For example, Serra et al. (2011) used the vector error correction model, Hassouneh et al. (2012) used parametric error correction and a multivariate local linear regression model, Abdelradi and Serra (2015) used multivariate GARCH model, and Cabrera and Schulz (2016) used multivariate multiplicative volatility model.

DCCA approach developed by Podobnik and Stanley (2008) has been used in a number of studies to detect co-movement of time series (Podobnik et al., 2009; Horvatic, Stanley, & Podobnik, 2011; Ma, Wei, Huang, & Zhao, 2013; El Alaoui & Benbachir, 2013; Liu, 2014; Guedes, Dionísio, Ferreira & Zebende, 2017). Liu (2014) utilized the DCCA to examine the behavior of crude oil and agricultural commodities prices and observed the existence of return

cross-correlations for soybean and corn but not for soybean and oat. He attributed that during 2006 to 2008, high oil price is the primary reason for the food crisis. Few papers also used cross correlation based approaches for establishing the relationship between the price of crude oil and other time series. Yang, Zhu, Wang, and Wang (2016) applied multifractal DCCA (MF-DCCA) approach to study the existence of cross-correlations between stock market of China and crude oil. It is found that DCCA approach has given better result compared to vector auto-regression (VAR) model. Hussain, Zebende, Bashir, and Donghong (2017) also used DCCA approach for investigating the co-movement behavior of exchange rate and oil price. They observed the evidence of negative and weak cross-correlation between them.

Although several studies have given evidence of positive correlation of prices between crude oil and commodities, the changing nature of the correlation was not well investigated in the prior studies. We, therefore, use the DCCA approach in this study.

3.

Data and Methodology

3.1 Data The period of this is from February 1995 to November 2016 and used monthly data of the crude

oil

price

from the

website

of U.S.

Energy

Information

Administration

(http://www.eia.gov/) and commodity price data from the website of International Monetary Fund (http://www.imf.org/). The descriptive analysis of data and the correlation coefficients between the series are produced in table 1. The time series plots of the price series are given in figure 1 Insert table 1 and figure 1 here

Initially, we investigated the stationarity of the data series using the Phillips Perron (PP) and augmented Dickey–Fuller (ADF) unit root tests, and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) LR test for all price series and food and beverage index at their level values. It can be seen from the Panel-A of Table 2 that for the obtained values, it is not possible to reject the null hypothesis of a unit root. However, using tests on their first differences, it can be rejected for all the series at 1% level of significance. A common practice followed in econometric analysis is to take differenced data before using in regression analysis. However, differencing of data takes away the long-term information contained in the data at their levels. In our analysis, there was no need in differencing data as DCCA approach can deal with nonstationary quite well (Horvatic et al., 2011; Podobnik & Stanley, 2008).

Insert table 2 here

3.2 Methodology In the study, we used the DCCA cross-correlation coefficient to measure the association of prices of crude oil and agricultural commodities used for producing biofuels. The DCCA approach was proposed by Podobnik, Jiang, Zhou, and Stanley (2011) that offers a statistical test for calculating the cross-correlation coefficient. Zebende (2011) proposed crosscorrelation coefficient for nonstationary time series. We denote the DCCA cross-correlation coefficient as ρDCCA(n). We can estimate the value of ρDCCA(n) for any two time series { X } and { Y } having size ‘n ’ . In the study, we used prices

of crude oil as series {X} and prices of agricultural commodities as series { Y } . The value of ρDCCA(n) can be estimated as follows:

where



,

,

, and





,

,

,



are detrended covariance and variance

functions of the two series, respectively, and ‘n’ is the size of the window. The value of ρDCCA(n) remains within the limit of 1 like the Pearson correlation coefficient. ρDCCA(n) = +1 indicates a very high positive and ρDCCA(n) = -1 indicates a very high negative cross-correlations, and ρDCCA(n) = 0 indicates the absence of cross-correlation. In the paper, we measure ρDCCA(n) between the crude oil price and major crops that are being used for biofuel production, namely, wheat, maize, rice, soybean and sugar and a Food and Beverage index as follows: 

Changes in the value of ρDCCA(n) coefficient for a different span of ‘n’.



Changes of cross-correlation coefficient over time using moving windows of 24, 36, 48,

and 60 months, respectively.

4.

Result

To start with the values of ρDCCA coefficient for individual food crop prices and a food index with crude oil prices were estimated and a line plot is presented in figure 2. The ρDCCA(n) values in the figure were plotted varying the value of ‘n’ from 5 to 260 in steps of 5, and it

gave a broader view of how the value of the coefficient changed from short to the long horizon.

Insert figure 2 here

The value of ρDCCA coefficient was low for the low value of ‘n’ but it increased and became close to +1 for the higher values of ‘n’. The value of the coefficient had always remained positive even in the short run but became very strong in the long run. Food and beverage Index shows very high cross correlation even for the lower value of ‘n’; it became more than 0.75 for n = 15 and increased more than 0.90 for n  115. Prices of wheat, maize, rice, and soybean exhibited high cross correlation (exceeding 0.50) from n = 15 onwards. However, ρDCCA coefficient for rice dropped to less than 0.50 for n = 20 to n = 80 and gradually exceeded 0.75 level for n  120. The ρDCCA coefficient for sugar was, however, remained lower compared to other food grain prices.

Although above analysis confirms the presence of high and positive cross correlated relationship, it does not reveal the changing nature of the strength of relationship over time. Hence, we used a moving window of different window sizes {24 months, 36 months, 48 months and 60 months} to estimate the value of ρDCCA coefficient for all variables taking a value of n = 24. Figure 3 displays time series plots of ρDCCA Coefficient value between various food crop prices/index and Crude oil with a window size of 60 months. Plots of other window sizes were not presented to save space. The x-axis represents time, and the first point in the x-axis (January 2000) presented ρDCCA value estimated for the period February 1995 to

January 2000. Subsequent plots of the ρDCCA coefficient are estimated on a similar moving window of 60 months.

Insert table 3 and figure 3 here

The values of the coefficient for n=24 months for the Food & Beverages Index, Wheat, Maize, Rice, Soybean, and Sugar with Crude oil on a rolling window size of 60 months is given in Table 3. It can be found that the ρDCCA coefficient for the Food & Beverages Index, Wheat, Maize, Rice, Soybean, and Sugar with Crude oil have changed with time. The value of ρDCCA Coefficient for all the above series was low before 2004. The value of the coefficient for Food & Beverage Index and wheat have increased considerably from 2004 onwards. Although the ρDCCA Coefficient values were negative in some cases prior to 2009, all food crops exhibited a positive relationship with Crude oil prices from 2009 onwards with the sole exception of sugar prices in 2016.

Global biodiesel production grew from less than one billion liters in the year 2000 to more than 25 billion liters by 2012. As the demand for agricultural crops for biofuel generation was very low prior to the year 2007, prices of those crops were largely depended on other socioeconomic factors rather on oil prices. Therefore, the cross-correlation coefficient of Oil and various crops was erratic and sometimes show a negative relationship. However, with a sharp increase in biofuel production from 2007 onwards, agricultural commodity and energy market prices have started showing a positive cross-correlation relationship. The figure clearly establishes the changing value of ρDCCA coefficient over time. The coefficient for most of the

items was negative prior the year 2004, but the strength of the relationship started rising thereafter as more food crops are increasingly being used for generation biofuels. The ρDCCA coefficient for soybean, wheat, and maize as well as food and beverage index started rising again in 2007 with the unprecedented rising trend of crude oil price. However, crude oil price fell drastically in the last quarter of 2015 and consequently, the value of ρDCCA coefficient for most of the food grains/index was also weakened. However, despite low crude oil prices in 2016, the value of ρDCCA coefficient started rising again in the latter part of the year.

ρDCCA plots in figure 3 were made taking a constant value of n = 24 as a test case. To get a broader picture on how the coefficient value changes with the changing value of ‘n’, we again estimated the value of ρDCCA coefficient for n = 6 to n = 30 using a step size of 1 and plotted the results as heat maps in figure 4. The analysis was carried out with different window sizes of 24 months, 36 months,48 months and 60 months and found that the values of ρDCCA have increased from the year 2007 for agricultural crops. The heatmaps showing the changing value of ρDCCA for 60 months is presented in figure 4. Heatmaps of other window sizes were not presented to save space. Darker shades (Red color in the soft copy) represent a high value of ρDCCA coefficient and white shade represent low values of ρDCCA coefficient. A close look at the figure reveals that the shades were lighter before the 100th period that corresponded the year of 2007 and shades became darker when oil price increased sharply in 2007. The darker shades continued to remain strong till the year 2015 and started fading in 2016.

Insert figure 4 here

5.

Conclusion

The sharp increase in biofuel production after the oil crisis in the year 2007 had caused the diversion of a significant quantity of agricultural crops for biofuel production. This has increased interdependence between the agricultural commodity and energy markets in recent years, and prices of agricultural crops have started showing a high co-movement relationship with crude oil prices. The focus of the study was to capture this changing nature of the strength of the relationship between the crude oil price and food grain prices/index by employing DCCA method and found that the value of ρDCCA coefficient turned from negative to positive territory from the year 2004 as oil prices started increasing. The ρDCCA(n) value of soybean, wheat, and maize started increasing again from 2007 in tandem with the sharp and unprecedented rise in crude oil prices. Thus, the rolling window approach helped us to analyze the change in the values of ρDCCA coefficient between food grain and crude oil prices. The scope of further analysis lies in using windows based on actual macroeconomic events. As the food crops are being converted into biofuels, the relationship between the price of food crops and the price of crude oil need to be carefully examined. When oil prices go up, diverting crops for generating biofuels would become more profitable, and biofuel producers would pay higher prices for their inputs. Historically, the food and energy prices were not much correlated, but now with the expansion of biofuel industry, the equations are bound to change. Many countries are net importers of both food and fuel, and consequently, high co-movement between food and fuel prices would

harm them. This would particularly cause serious implications on food shortages in poor countries.”

Acknowledgement We remain grateful to Professor Eugene Stanley, Main Editor of Physica A and the anonymous referees for their thorough and collegiate review. This has added considerable value to our work.

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Table 1: Descriptive Statistics of the variables used Crude F&B Statistic Oil Index Wheat Maize Rice Soybean Mean 54.08 123.01 179.10 155.49 368.61 313.77 Standard Deviation 34.82 35.00 65.91 66.45 153.58 121.34 Kurtosis -1.01 -1.15 -0.17 0.13 1.38 -0.75 Skewness 0.58 0.42 0.76 1.04 1.04 0.63 Minimum 9.82 75.83 88.55 75.06 162.10 158.31 Maximum 132.72 196.27 403.81 332.95 1015.21 622.91 Correlation coefficient with Crude Oil F&B Index Wheat Maize Rice Soybean Sugar

1.00 0.88 0.81 0.84 0.76 0.86 0.50

1.00 0.88 0.93 0.84 0.96 0.62

1.00 0.89 0.75 0.87 0.42

1.00 0.81 0.93 0.60

1.00 0.82 0.49

1.00 0.52

Sugar 23.59 4.53 3.57 1.96 17.08 40.37

1.00

Table 2: Result of unit root tests Panel A: Variables at their level values Intercept Variables ADF KPSS PP RBRTE -1.93 1.51 F&B Index -1.48 1.55 Wheat -2.29 0.95 Maize -1.84 1.15 Rice -2.12 1.07 Soybean -2.11 1.4 Sugar -2.46 0.67 1% level -3.45 0.73 5% level -2.87 0.46

-1.76 -1.29 -1.99 -1.86 -2.12 -1.84 -2.03 -3.45 -2.87

Panel B: Variables at their first difference Intercept Variables ADF KPSS PP RBRTE

Intercept and Trend ADF KPSS PP -2.37 0.17 -2.48 0.26 -2.44 0.17 -2.08 0.19 -2.46 0.21 -2.82 0.19 -2.91 0.12 -3.99 0.216 -3.42 0.146

Intercept and Trend ADF KPSS PP

-10.74***

0.1***

-10.68***

-10.74***

0.06***

-9.86***

0.08***

-9.92***

-9.84***

0.08***

Wheat

-12.84***

0.09***

-12.65***

-12.83***

0.07***

Maize

-12.71***

0.07***

-12.87***

-12.69***

Rice

-11.33***

0.05***

-9.49***

-11.31***

F&B Index

-1.97 0.61 -2.12 0.52 -1.76 0.71 -2.11 -3.99 -3.42

0.069** * 0.054** * 0.054** *

10.67*** -9.91*** 12.64*** 12.86*** -9.46***

11.00*** Sugar -8.01*** 0.05*** -11.67*** -8.00*** 0.04*** 11.66*** 1% level -3.45 0.73 -3.45 -3.99 0.216 -3.99 5% level -2.87 0.46 -2.87 -3.42 0.146 -3.42 The statistics are the pseudo-t ratios for ADF and PP tests and the LM statistics for KPSS test. ***, **, and * indicate statistical significance at 1, 5 and 10 percent, respectively. It can be seen that none of the series are stationary at their level values, but all the series became stationary at their first difference at 1% significance level for all the three different unit root tests. Soybean

-11.25***

0.05***

-11.02***

-11.25***

Table 3: ρDCCA coefficients for n = 24 months for a window size of 60 months. Month FB_Index Wheat Maize Rice Soybean Sugar Jan-00 0.31 0.35 0.23 -0.18 0.49 -0.51 Jan-01 0.06 0.22 0.14 -0.55 0.21 -0.80 Jan-02 -0.13 -0.05 -0.12 -0.71 0.06 -0.66 Jan-03 -0.31 -0.14 -0.44 -0.67 -0.02 -0.62 Jan-04 0.54 0.49 0.29 -0.18 0.61 0.10 Jan-05 0.73 0.38 0.46 0.64 0.53 -0.05 Jan-06 0.43 0.04 -0.24 0.89 -0.21 0.03 Jan-07 0.51 0.28 -0.14 0.82 -0.27 0.37 Jan-08 0.48 0.39 -0.08 0.77 -0.30 0.40 Jan-09 0.92 0.85 0.68 0.82 0.81 0.32 Jan-10 0.92 0.86 0.80 0.67 0.85 0.03 Jan-11 0.93 0.77 0.84 0.62 0.83 0.23 Jan-12 0.89 0.76 0.78 0.41 0.83 0.44 Jan-13 0.78 0.58 0.79 0.26 0.66 0.60 Jan-14 0.75 0.67 0.85 0.19 0.60 0.52 Jan-15 0.38 0.27 0.45 0.29 0.26 0.24 Jan-16 0.85 0.63 0.30 0.30 0.73 -0.02 Value ρDCCA(n=24 months) coefficients for Food & Beverages Index, Wheat, Maize, Rice, Soybean, and Sugar with Crude oil were estimated on a rolling window size of 60 months. The figures reveal that the values of ρDCCA Coefficient have changed with time.

Figu ure 1: Timee series plotss of the variiables used iin the studyy

Figu ure 2: ρDCCCA(n) coefficcients. The value of ρDCCCA Coefficiient of varioous food graains/index withh crude oil pprices were plotted p varyying size of n from 5 to 260 in stepp size of 5

Figure 3: Changee of ρDCCA(n n) coefficieents on a rrolling wind dow. Plots of ρDCCA Coefficient C value beetween variious food crrop prices/inndex and C Crude oil forr n = 24 moonths with a window size of 660 months. The plot deemonstrates that the vallues of ρDCCCA Coefficiennt have chaanged with time.

F&B Inddex

Wheat

Maize

Rice

Soybeann

Sugar

Figure 4: Heat maap is showin ng changingg value of ρDCCA Coeffficient of foood grain prices/iindex with crude c oil prrices. In thee figures, X--axis denotees rolling wiindow numbber. The first winndow is for 60 months from f Januarry 2000 to D December 20004, and theereafter the 60-month window w was movedd by one moonth. For exxample, the ssecond winddow was froom Februaryy 2000 to Januaryy 2005 and sso on. Y-axiis denoted chhanging vallues of n {6 months to 330 months} in monthlyy intervals. T The darker ppatches in thhese heatmaaps indicatee low and neegative com movement relationships betweeen Crude oiil and food pprices, wherreas white patches p reveeal the preseence of positivee and high coomovementt relationshiip.