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Connection between climatic change and international food prices: evidence from robust longrange crosscorrelation and variablelag transfer entropy with sliding windows approach
EPJ Data Science volumeÂ 13, ArticleÂ number:Â 56 (2024)
Abstract
As nations progress, the impact of climate change on food prices becomes increasingly substantial. While the influence of climate change on the yields of major agricultural products is widely recognized, its specific effect on food prices remains uncertain. This study delves into the impact of the North Atlantic Oscillation (NAO) index, a wellestablished climate indicator, on global food prices. To accomplish this, a robust bivariate Hurst exponent (robust bHe) is applied. The study employs a sliding windows approach across various time scales to produce a color map of this coefficient, presenting a timevarying version. Furthermore, variablelag transfer entropy with a sliding windows approach is utilized to discern causal relationships between the NAO index and international food prices. The findings reveal that significant increases in the NAO index are correlated with noteworthy upswings in various international food prices over both short and longterm periods. Additionally, variablelag transfer entropy confirms the causal role of the NAO index in influencing international food prices.
1 Introduction
The global COVID19 pandemic has given rise to significant challenges, particularly in the domains of poverty and hunger. Projections suggest a potential surge of 75 million individuals into extreme poverty by the end of 2022, accompanied by sustained elevated levels of global hunger [25, 82, 86]. Against this backdrop, our study aims to illuminate a specific aspect of the intricate interplay between climate dynamics and global food prices.
Before the pandemic, the number of people contending with hunger rose to 828 million in 2021an increase of approximately 46 million from 2020 and a staggering 150 million from 2019 [1, 31]. The global food crisis has been exacerbated by nations imposing food trade restrictions to fortify domestic supplies and mitigate price fluctuations [53, 58]. Beyond trade dynamics, shifts in global temperatures, alterations in precipitation patterns, and the heightened frequency of extreme weather events have further impacted global food production [17, 28]. Since 1961, climate change has resulted in a 21% decline in global agricultural productivity, posing a threat to economic development, particularly in developing nations [86].
In this intricate global context, our study focuses on unraveling the specific impact of the North Atlantic Oscillation (NAO) index on international food prices. The NAO index, indicative of the redistribution of atmospheric mass between the Arctic or subarctic regions and the subtropical regions of the Atlantic, significantly influences global crop production [45, 67]. Furthermore, it plays a crucial role in shaping winter conditions in Asia, Europe, and the United States, subsequently influencing agricultural production [74, 83, 85].
While numerous studies have investigated the impact of the NAO index on agricultural production, our current review reveals a gap in the exploration of its direct influence on international food prices. However, research has delved into the correlation between other climate indices, such as the El NiÃ±o and La NiÃ±a indices, and specific food products. For instance, [65] examines the effects of El NiÃ±o and La NiÃ±a events on corn and soybean prices, showcasing increased volatility during El NiÃ±o in the SpringSummer phase and differing impacts on soybean prices across seasons. Similarly, [81] examines monthly spot prices of wheat from various regions, indicating price trends post La NiÃ±a and El NiÃ±o events. [40] establishes a positive correlation between wheat export prices and La NiÃ±a events, highlighting their consistent short and longterm impacts. [80] studies the dynamics of the fishmealsoybean flour price ratio, demonstrating significant impacts of the bimonthly multivariate El NiÃ±o Southern Oscillation (ENSO) on the price ratio for up to a year following ENSO shocks. Additionally, [41] finds a positive relationship between rice prices and the El NiÃ±o index, while noting a negative relationship with La NiÃ±a shocks.
While a direct study exploring the potential relationship between the NAO index and international food prices remains absent, several research studies have explored the NAO indexâ€™s impact on food prices in various regions and countries. [44] establishes the NAO indexâ€™s influence on soybean production in Australia and Europe, while [72] provides evidence linking the NAO index with soybean production in Italy. Furthermore, [3] demonstrates the global aggregated variability in corn, soybean, and wheat output, attributed respectively to the ENSO index, the Indian Ocean Dipole, tropical Atlantic variability, and the NAO index. Moreover, [60] reveals a strong correlation between largescale changes in the NAO index and soybean yield variability across different continents. Studies like [93] and [12] investigate the correlation between the NAO index and corn, rice, wheat, barley, oats, and potatoes, highlighting varying relationships based on crop type and seasonal changes.
The summary of previous research, Table 1, offers a concise overview of the relationship between different climatic change indices and food prices, detailing the methodologies employed in these studies. The structure of this article is outlined as follows: Sect. 1 is dedicated to the introduction. In Sect. 2, we introduce the methodologies used in this study. Section 3 provides a description of the sample data along with descriptive statistics. The results of the robust bivariate Hurst exponent are outlined in Sect. 4. Following that, Sect. 5 presents the results of the variablelag transfer entropy causality test. Theoretical and empirical implications are discussed in Sect. 6. Finally, our conclusions are presented in Sect. 7.
2 Methodology
2.1 Robust bivariate Hurst exponent
The robust bHe was introduced by [20]. This robust bHe is constructed using the detrended crosscorrelation function and is specifically designed to analyze the existence of significant crosscorrelations between two stochastic processes, referred to as X and Y. In the realm of signal processing and analysis, the crosscorrelation method has emerged as a versatile and indispensable tool with myriad applications. This research sets out to explore the diverse practical uses of crosscorrelation functions, extending across various domains. One notable application focuses on the assessment of similarity between 2D surface profiles and 3D real surface topography images, showcasing the methodâ€™s effectiveness in characterizing complex surface structures [84]. Additionally, the study delves into the application of crosscorrelation in estimating the channel impulse response in wireless communication channels, highlighting its crucial role in enhancing communication system performance [75]. In the realm of multisensor signal processing, the crosscorrelation function proves to be instrumental, particularly in accurate time delay estimation [92]. Furthermore, the method is applied to evaluate objective multiple linear regression thresholds, offering a robust approach in the low and middle frequencies [90]. Notably, the article also explores the utility of crosscorrelation as a nondestructive testing method for detecting leaks in buried pipes within industrial and household facilities, underscoring its significance in realworld problemsolving [27].
Assumes that both processes, X and Y, have zero means and exhibit longrange temporal autocorrelation characterized by powerlaw autocorrelations, as described by the following equations:
where \(H_{X}\) and \(H_{Y}\) represent the Hurst exponents, falling within the range of \([0.5,1[\). The powerlaw crosscorrelations are defined as:
The robust bHe computation involves the following steps:

1
Profile construction: we divide each time series into nonoverlapping boxes of size s, where s is defined as \(N/s\), and \(\ell _{v}=(v1)s\). The profiles in the vth box are constructed as follows:
$$ X_{v}(k)=\sum_{j=1}^{k}X( \ell _{v}+j) \quad \text{and}\quad Y_{v}(k)=\sum _{j=1}^{k}Y(\ell _{v}+j), \quad \text{for } k=1,\dots ,s. $$(3) 
2
Trend estimation: we estimate the local trends of \(X_{v}(k)\) and \(Y_{v}(k)\), denoted as \(\widetilde{X_{v}}(k)\) and \(\widetilde{Y_{v}}(k)\), respectively.

3
Crosscorrelation calculation: the crosscorrelation for each box is computed using the following equation:
$$ f_{v}^{X,Y}(s) = \frac{1}{s} \sum_{k=1}^{s} \bigl(X_{v}(k) \widetilde{X_{v}}(k) \bigr) \bigl(Y_{v}(k) \widetilde{Y_{v}}(k) \bigr), \quad \text{for } v=1,\dots ,N_{s}. $$(4) 
4
qth order crosscorrelation function: the qth order crosscorrelation function is determined as follows:
$$ F_{X,Y}(q,s)=\textstyle\begin{cases} (\frac{1}{N_{s}}\sum_{v=1}^{N_{s}}(f_{v}^{X,Y}(s))^{ \frac {q}{2}} )^{\frac {1}{q}}, & \text{when } q\neq 0; \\ \exp (\frac{1}{2N_{s}}\sum_{v=1}^{N_{s}}\log (f_{v}^{X,Y}(s)) ), & \text{when } q=0. \end{cases} $$(5) 
5
Scaling relation: for sufficiently large s, we expect a scaling relation as follows:
$$ F_{X,Y}(q,s)\sim s^{\gamma _{X,Y}(q)}, $$(6)where the Hurst exponent \(\gamma _{X,Y}(q)\) characterizes the longrange crosscorrelation properties of processes X and Y. A value of \(\gamma _{X,Y}(2)\) around 0.5 indicates no crosscorrelation, while \(\gamma _{X,Y}(2)>0.5\) suggests a positive powerlaw crosscorrelation, and \(\gamma _{X,Y}(2)<0.5\) indicates anticorrelation.
In situations where the processes X and Y exhibit crosscorrelations and are affected by outlier observations, we establish the robust bHe, denoted as \(H_{X}+H_{Y}\). This is determined by examining the behavior of \(\mathbb{E}[F_{X,Y}(2,s)^{2}]\), particularly for large values of s, as presented in the following equation [20]:
Here, \(\sigma _{X}\) represents the standard deviation of \(X(1)\), \(\sigma _{Y}\) is the standard deviation of \(Y(1)\), and \(\rho _{XY}\) indicates the correlation between \(X(1)\) and \(Y(1)\).
2.2 Statistical significance of robust bHe
Assessing the statistical significance of the robust bHe is crucial in understanding the significance of crosscorrelation between processes. To gauge this significance, we employ a hypothesis testing approach, comparing the observed \(H_{(X,Y)}^{o}\) to critical values (\(H_{(X,Y)}^{c}\)) at various confidence levels. This approach aligns with the methodology proposed by [66]. The assessment unfolds through the following steps:

1
Generation of simulated time series: initially, we generate pairs of independent and identically distributed (i.i.d.) time series with \(H_{X}=H_{Y}=0.5\). These time series are drawn from a Gaussian distribution.

2
Computation of robust bHe: we apply the methodology developed by [20] to compute the robust bHe for each pair of i.i.d. time series.

3
Replication: this process of generating i.i.d. time series and estimating the robust bHe is repeated for a large number of iterations, typically \(N_{\text{rep}}=10{,}000\) times.

4
Probability density function computation: the probability density function of the robust bHe is computed, offering insights into the distribution of the robust bHe values.
Following this, we proceed to test the null hypothesis against the alternative hypothesis:

Null Hypothesis: \(H_{X}+H_{Y}=1\) (indicating no crosscorrelation),

Alternative Hypothesis: \(H_{X}+H_{Y}>1\) (indicating a positive powerlaw crosscorrelation).
For each value of N, the critical value, denoted as \(H_{(X,Y)}^{c}\), is computed as:
Here, \(\mu _{H_{X,Y}}\) represents the mean and \(\sigma _{H_{X,Y}}\) represents the standard deviation of the robust bHe computed over 10,000 iterations. The term \(Z_{\alpha}\) corresponds to the quantile from the standard normal distribution corresponding to the chosen confidence level of \(1\alpha \). Additionally, the pvalue of \(H_{X,Y}^{o}\) is calculated as follows:
In this equation, \((\widehat{H}_{X,Y})_{\ell}\) represents the estimated value of the robust bHe for the â„“th simulated bivariate i.i.d time series, and \(\mathbf{1}_{\{H_{X,Y}\leq (\widehat{H}_{X,Y})_{\ell}\}}\) equals 1 if \(H_{X,Y}\leq (\widehat{H}_{X,Y})_{\ell}\) and 0 otherwise. We proceed to reject the initial hypothesis under two conditions: when \(H_{X,Y}^{o}>H_{(X,Y)}^{c}\) or when \(\text{pv}(H_{X,Y}^{o})<0.05\). In either case, we conclude that the positive powerlaw crosscorrelation is statistically significant.
2.3 Sliding windows approach of bHe
Let N the length of the initial time series, h the size of window where \(1000 \leq h \leq N/2\), n the time scale and T the time period where \(1 \leq T \leq Nh\), then we obtain the color map of \(H_{X,Y}(T,n)\) using sliding windows framework as follows:

1
We fixe \(h=N/2\) and, for fixed T in \([1,h+1]\), we consider the pairs of time series \(\{X_{i}\}_{i=T,\dots ,h+T1}\) and \(\{Y_{i}\}_{i=T,\dots ,h+T1}\).

2
For fixed n, each time series is covered with \(N^{*}_{s}=[(hn)/s]\) non overlapping boxes of size s. We compute the profiles in the vth box \([\ell _{v}+1,\ell _{v}+s]\), where \(\ell _{v}=(v1)s\) using Equation (3).

3
We compute the crosscorrelation \({f_{v}}^{X,Y}(s)\) for each box using Equation (4) for \(v=1,\dots ,N^{*}_{s}\), and the second order crosscorrelation function \(F(T,n)_{X,Y}(2,s)\) by Equation (5) as:
$$ F(T,n)_{X,Y}(2,s)^{2}= \frac{1}{N^{*}_{s}}\sum _{v=1}^{N^{*}_{s}}f_{v}^{X,Y}(s). $$ 
4
We obtain \(H_{X,Y}(T,n)\) by the liner regression of \(\log (F(T,n)_{X,Y}(2,s)^{2})\) on \(\log (s)\).

5
We repeat steps 1â€“4 for \(T=1,\dots ,h+1\) and for \(n=50,100,150,200\).
3 Data and descriptive statistics
The time series under analysis include the North Atlantic Oscillation (NAO) index, obtained from https://www.cpc.ncep.noaa.gov/products/precip/CWlink/pna/nao.shtml, and the international prices of corn, soybean, oats, and wheat in U.S. Dollars per bushel, sourced from the Bloomberg Terminal. These time series span from January 06, 2020, to May 18, 2022, resulting in a daily time series comprising 600 observations. Except for the NAO index, we examine the fluctuation of each time series by analyzing its returns, given by \(x_{t+1}x_{t}\). The visual representation of the studied time series is depicted in Fig. 1, where the red points denote the detected outlier observations. Descriptive statistics for the studied time series are presented in Table 2. Upon careful examination of the result presented in Table 2 (Panel A), distinct patterns within the time series become evident. Specifically, the NAO index stands out with a negative mean of âˆ’1.839, indicating an average below zero. It is accompanied by a significant variance denoted by a standard deviation of 110.543. This underscores the substantial variability associated with the NAO index. In contrast, the mean values for the remaining time series are 0.0001 for corn and equal to 0.001 for soybean, wheat, and oats, respectively. These are accompanied by variances of 0.084, 0.16, 0.127, and 0.081. These findings suggest relatively stable average returns across these commodities, presenting a notable contrast to the more fluctuating nature of the NAO index.
According to Table 2 (Panel B), based on the results of the ADF test, all time series are stationary, as all pvalues are lower than 0.05. However, according to the results of the JB test, the NAO index is normally distributed, as the pvalue is higher than 0.05. In contrast, all other time series are not normally distributed, as the associated pvalues of the JB test are lower than 0.05. Given that the returns of international food prices are not distributed according to the normal distribution, we computed the correlation between the NAO index and each studied international food price using the Spearman coefficient. The results of the Spearman coefficient are given in the last column of Table 2, indicating no significant correlation between the NAO index and the other studied time series. The robust bHe, as proposed in [20], necessitates the utilization of fractional Gaussian noise (fGn) processes contaminated with additive outlier observations. In our approach to compute this exponent, we first identify and remove outliers from the studied time series. Subsequently, we evaluate the normality, stationarity, and stochastic nature of the time series without outliers. If the refined time series demonstrates these characteristics, we consider it to be indicative of a fGn process, laying the groundwork for the computation of the robust bHe.
To identify outlier observations in the examined time series, we employ the extreme value theory (EVT) test proposed by [46]. The outcomes of this test, indicating the percentage of outliers in each time series, are presented in Table 2 (Panel A). Recognizing that the augmented DickeyFuller (ADF) and KwiatkowskiPhillipsSchmidtShin (KPSS) tests can be sensitive to the influence of outliers [32, 63], we mitigate this issue by utilizing the unit root test based on Breitungâ€™s variance ratio [11]. Additionally, to assess the normality distribution of the studied time series, we employ the robust JarqueBera (robust JB) test [34]. The results of these tests are detailed in Table 2 (Panel A).
The outcomes of the outlier detection test reveal that the NAO index, the returns of oats, corn, soybean, and wheat time series contain 3.5%, 3.291%, 3.041%, 2.083%, and 1.958% outliers, respectively. Furthermore, the robust stationarity test rejects the initial hypothesis of nonstationarity in favor of the alternative hypothesis of stationarity, as all pvalues are below 1%. Thus, the studied time series are deemed stationary. The absence of normality in the distribution of the studied time series is affirmed by the robust JarqueBera (JB) test, with corresponding pvalues (given in parentheses) being less than 1%, indicating the rejection of the initial hypothesis of normality.
The outcomes of the stationarity and normality tests for the studied time series without outliers are presented in Table 3, and these time series are visually represented in Fig. 2. The ADF and JB test results in Table 3 indicate that, subsequent to the removal of outliers, the examined time series demonstrate both stationarity and a Gaussian distribution. Furthermore, the estimated values of the Hurst exponent (H), determined through the corrected rescaled range (R/S) approach [47, 88], and by augmenting 0.5 to the estimator value of the fractional difference parameter d obtained via the exact local Whittle estimator [76], consistently surpass 0.5. This suggests a prevailing characteristic of long memory within the scrutinized time series, given that values greater than 0.5 signify enduring correlations and interdependence among observations. The persistence observed in these time series, impervious to the influence of outliers, further emphasizes the robust manifestation of long memory phenomena. This attribute, signifying a sustained influence of prior observations on future values, deepens our comprehension of the underlying dynamics and provides valuable insights into the nature of the scrutinized data [10, 39]. To ascertain if these time series, post outlier removal, adhere to a fGn process, we conduct a graphical comparison of their autocorrelation functions with that of a fGn process using the estimated Hurst exponent H. The autocorrelation functions align closely, as depicted in Fig. 3. Additionally, the stochastic nature of the time series without outliers is examined using the robust correlation dimension estimator proposed in [18]. This estimator is based on the Gaussian kernel correlation integral in [19, 24, 91], and the analysis includes the variance growth test introduced in [37]. The variance growth test evaluates the appropriateness of fitting a given time series to \(1/f^{\alpha}\) stochastic noise by computing the Root Mean Square (RMS) deviation, denoted as Ïƒ, within a subset of data series with a length of N. This test facilitates the distinction between random processes characterized by a powerlaw spectrum and deterministic lowdimensional chaotic signals. The application of the variance growth test in [23] demonstrates the stochastic nature of certain financial time series.
The authors in [37] specifically established that for stochastic \(1/f^{\alpha}\) colored noise, the variance scales as \(\sigma \sim N^{\alpha 1}\), thus continuing to grow indefinitely as a function of N. In contrast, for lowdimensional chaotic data, the variance reaches a plateau for N values exceeding the PoincarÃ© return time [37]. Consistent with the methodology outlined in [37], we partition each time series of length T into subsets with varying lengths. This involves exploring all subsets using a length N sliding window where \(N = 2, \dots , T\). Subsequently, we calculate all the Ïƒ values for these subsets of length N.
The results obtained from the application of the correlation dimension are illustrated in Fig. 4. Whereas, the results obtained from implementing the variance growth test are illustrated in Fig. 5, where \(\log (\sigma )\) is plotted against the logarithm of the subset lengths. The results depicted in Fig. 4 indicate an increase in correlation dimension with the embedding dimension (m), confirming the stochastic nature of the studied time series after removing outliers. This observation is further supported by the dynamic confidence interval of the estimated correlation dimension. These results are also validated by the growth variance test results shown in Fig. 5, where the values of RMS deviation continue to grow with increasing length. This suggests that the latter series can be effectively modeled by random noise processes with a powerlaw distribution.
In summary, based on these findings, we conclude that the studied time series can be characterized as fGn processes contaminated by outliers. Herein, we recall that fGn processes are selfsimilar stochastic models used to capture both antipersistent and persistent dependencies in various fields, such as hydrology, finance, and climate science. fGn is a generalization of the Gaussian random walk, and its distinctive feature lies in the Hurst exponent, which quantifies the degree of persistence or antipersistence inherent in the process [79].
4 Crosscorrelation analysis using robust bHe
We adopt the sliding window approach to assess the longrange crosscorrelation between the NAO index and the returns of various international food prices for the time period spanning March 16, 2021, to May 18, 2022, with time scales (n) set at 50, 100, 150, and 200 days. Specifically, we classify n from 50 to 100 days as a shortterm scale, and from 150 to 200 days as a longterm scale. The timevarying longrange crosscorrelation is determined as the average of \(H_{X,Y}(T,n)\) for all n, expressed as:
Additionally, we establish 1.297 as the statistical critical value for \(H_{X,Y}\) at a confidence level of 95%, obtained from a simulation study (the critical value for \(H_{X,Y}\) for a confidence level of 95% is obtained using the procedure in Sect. 2.2). Consequently, each \(H_{X,Y}\geq 1.297\) is deemed statistically significant, indicating a positive powerlaw crosscorrelation. The contour plots illustrating sliding windows longrange crosscorrelation and the corresponding timevarying longrange crosscorrelation coefficients (with the dashed red line representing \(H(T)_{X,Y}=1.297\)) for different pairs are provided in Figs. 6, 7, 8, and 9. The key finding in this section emphasizes a noteworthy observation: there is a substantial positive powerlaw crosscorrelation across various time scales (n) and all time periods (T) between the fluctuations in the NAO index and those in the studied international food prices. This implies a persistent longrange crosscorrelation between the NAO index and the returns of different international food prices. In practical terms, this signifies that a significant alteration in the NAO index is consistently followed by a substantial change in the returns of wheat, soybean, corn, and oats prices, respectively.
Figure 6a depicts the color map of \(H_{X,Y}\) for the NAO index and the return of corn prices. Notably, \(H_{X,Y}\) consistently exceeds 1.9 for both short and longterm scales across all studied time periods. In Fig. 6b, the timevarying longrange crosscorrelation coefficient surpasses the critical value of 1.297 for all time periods, except during midSeptember 2021 and midDecember 2021. This outcome indicates that substantial changes in the NAO index are followed by significant alterations in the return of corn prices, suggesting an influential role of the NAO index in influencing corn price fluctuations.
To the best of our knowledge, there is no existing scientific research confirming a direct connection between the fluctuations in the NAO index and the return of corn prices. This novel finding may be attributed, in part, to the impact of the NAO index on corn yield, as suggested by previous studies [44, 57, 93], which subsequently influences corn prices. Additionally, the crosscorrelation between the NAO index and the return of corn prices could be linked to the NAO indexâ€™s association with other climatic change indices affecting corn price or yield fluctuations [42, 59, 61]. Moreover, previous research has highlighted the NAO indexâ€™s impact on water quality [70, 73] and rainfall patterns [71], with the wellestablished importance of irrigation and rainfall on corn productivity [30, 68, 89]. Hence, our findings regarding the crosscorrelation between the NAO index and corn prices may be associated with the NAO indexâ€™s influence on corn production, particularly its sensitivity to water quality for irrigation and rainfall conditions.
Examining Fig. 7a, which illustrates the color map of \(H_{X,Y}\) for the NAO index and the return of oats prices, reveals that \(H_{X,Y}\) surpasses 1.7 for the shortterm scale (\(50 \leq n \leq 100\)). Furthermore, for the longterm scale (\(150 \leq n \leq 200\)), \(H_{X,Y}\) consistently exceeds 1.9 across all studied time periods. The corresponding longrange crosscorrelation coefficient, depicted in Fig. 7b, displays timevarying behavior throughout all studied time periods. Additionally, \(H_{X,Y}\) remains above 1.297 for the majority of the studied time periods, except during midNovember 2021 and the beginning of April 2022, where \(H_{X,Y}\) falls below 1.297.
This outcome underscores that significant fluctuations in the NAO index correspond to notable changes in the return of oats prices. Similar to the previous observation, there is a scarcity of existing research exploring the relationship between the NAO index and fluctuations in oats prices. Our interpretation leans towards a potential association with the correlation between the NAO index and oats yield [13, 51], or the broader connection between the NAO index and certain climatic changes impacting oats prices [38, 55]. Furthermore, building on the link between the NAO index and water or rainfall, we posit that the relationship between the NAO index and oats prices can be explained by the influence of water and rainfall on oats yield [49, 52].
Figure 8a presents the color map illustrating the longrange crosscorrelation between the NAO index and the return of soybean prices, revealing that \(H_{X,Y}\) consistently exceeds 1.297 for both short and longterm scales. Additionally, the evolution of the longrange crosscorrelation coefficient over time, as depicted in Fig. 8b, is characterized by a nonconstant, timevarying pattern. Furthermore, \(H_{X,Y}\) remains above 1.297, except for the middle of November 2021. This outcome leads us to infer that substantial changes in the NAO index are closely followed by significant fluctuations in soybean prices. Importantly, our finding aligns with existing research that highlights a significant relationship between the NAO index and the fluctuation of soybean prices [44, 60].
Figure 9a illustrates the color map of \(H_{X,Y}\) for the NAO index and the return of wheat prices, revealing that \(H_{X,Y}\) consistently exceeds 1.8 for all time periods and across both short and longterm scales. Furthermore, in Fig. 9b, the timevarying longrange crosscorrelation coefficient remains above 1.297, except during midNovember 2021 and the beginning of January 2022. These findings suggest a compelling connection: substantial changes in the NAO index are accompanied by notable fluctuations in wheat prices. Importantly, this observation aligns with various other studies that have demonstrated a significant relationship between the NAO index and the fluctuation of wheat prices [35, 50, 77].
5 Causality analysis using variablelag transfer entropy
In this section, we explore the causal relationship between the NAO index and the return of international food prices by employing a variablelag transfer entropy (VLTE) causality test [2] with a sliding windows approach. We use the same sliding windows and scales for the powerlaw crosscorrelation coefficient analysis. [6] established complete equivalence between the Granger causality test and the transfer entropy approach for Gaussian time series. Furthermore, [26] endorsed the use of the transfer entropy approach and the nonlinear Granger causality test after a comprehensive comparison with ten causality methodologies and tests. On a different note, the effectiveness of the transfer entropy method in the presence of outlier observations in the studied time series is demonstrated in [29]. Moreover, transfer entropy has found practical applications in various domains [21, 22]. We employ the VLTE method, capable of inferring a causal relationship of Granger or transfer entropy where a cause impacts an effect with arbitrary dynamic delays. According to [2], the VLTE method involves computing the transfer entropy from X to Y, denoted as \(\mathcal{T}_{X \to Y}\), and from Y to X, denoted as \(\mathcal{T}_{Y \to X}\). If the VLTE ratio, defined as \(\text{VLTE ratio} = \mathcal{T}_{X \to Y}/\mathcal{T}_{Y \to X}\), is greater than 1, then we conclude that the variable X transfer entropy causes the variable Y. The results of the VLTE method for our studied time series are presented in Fig. 10a, Fig. 10b, Fig. 10c, and Fig. 10d. In Fig. 10a, the VLTE ratio surpasses 1 for the shortterm scale across the entire studied time period. Consequently, we deduce that the NAO index transfer entropy causes the international price of corn. Moving on to Fig. 10b, the color map depicting the VLTE ratio for pairs of the NAO index and the return of corn prices reveals values exceeding 1 for both short and longterm scales, implying that the NAO index transfer entropy causes the fluctuation of the international price of oats. Figure 10c illustrates the VLTE ratio for pairs of the NAO index and the return of the international price of soybean. Here, the VLTE ratio exceeds 1 for the shortterm scale throughout the entire studied time period. Additionally, for the longterm scale, the ratio remains greater than 1, except for the time interval from the beginning of the studied period to the start of November 2021. This suggests that the NAO index transfer entropy causes the fluctuation of the international price of soybean. Examining Fig. 10d, the VLTE ratio is greater than 1 for the shortterm scale and almost the entire longterm scale (\(90 \leq n \leq 180\)), specifically for the time interval from the beginning of the studied period to the end of November 2021. This outcome indicates that the NAO index transfer entropy causes the fluctuation of the international price of wheat. The results obtained through the VLTE causality test with the sliding windows approach corroborate those obtained using the robust bHe.
6 Implications for theory and practice
6.1 Theoretical implications
The primary objective of this article is to explore the potential impact of the NAO index on international food prices across various time scales. To accomplish this, we utilized a robust powerlaw crosscorrelation coefficient and the variablelag transfer entropy with a sliding window approach designed to effectively capture the influence of the NAO index on international food prices in both time and scales. This study makes a significant contribution to the existing literature on the impact of climate change on food prices.
The theoretical implications of our study emerge in the following ways. Firstly, we extend the current literature by introducing an innovative, robust powerlaw crosscorrelation coefficient. This novel metric enhances our ability to analyse and understand the crosscorrelation between the NAO index and international food prices. Secondly, our introduction of a color map of the powerlaw crosscorrelation coefficient, obtained through a sliding window framework, enables us to infer the timevarying robust powerlaw crosscorrelation coefficient. This visualization approach provides a nuanced understanding of how the crosscorrelation varies over time, offering valuable insights into the dynamics of the relationship.
Significantly, this study introduces a novel approach from both theoretical and empirical perspectives. To the best of our knowledge, no academic researcher has previously presented this innovative methodology. This contribution is noteworthy, particularly in light of the prevalence of crosscorrelation methods in various works [43, 87].
6.2 Empirical implications
The global food crisis, intensified by the COVID19 pandemic, has disrupted food prices across nearly all countries. Numerous studies indicate a negative impact on food prices during the pandemic compared to the prepandemic period [4, 8]. The ongoing effects of the COVID19 pandemic, combined with the RussianUkrainian conflict, have contributed to soaring food costs in both domestic and international markets [9]. Consequently, escalating food prices could potentially become the new norm, posing endemic and widespread hazards to global food security.
Our study reveals a positive powerlaw crosscorrelation and thus, an information flow between the fluctuations of the NAO index and those of international prices for oats, soybean, corn, and wheat, respectively, in both short and long terms. This is achieved through the application of a robust bHe and the VLTE causality method. Initially, we establish the statistical critical value for the bHe. Subsequently, we create a color map by computing the robust bHe across different time scales using a sliding window approach. Additionally, we use the same windows to obtain the color map for the VLTE measure. Our results indicate that substantial fluctuations in the NAO index are consistently followed by corresponding fluctuations in the prices of the studied international foods. Moreover, we demonstrate that the NAO index transfer entropy causes the fluctuations in international food prices. Consequently, the NAO index can be considered an explanatory variable for international food prices, heightening uncertainty in global food markets.
Despite the NAO index increasing uncertainty in international food markets, our study finds that the NAO index and international food prices are fundamentally intertwined within a single modeling framework. Thus, we demonstrate that the NAO index is an explanatory factor for the fluctuation of international food prices. To mitigate food market uncertainty, actions can be taken to address the fluctuation of the NAO index. Notably, given the NAO indexâ€™s correlation with temperature, reducing temperature can help control the NAO index and, subsequently, international food prices [5, 15]. The Paris Agreement, signed in 2015 to prevent severe climate change, is considered in this objective, aiming to keep global warming well below 2^{âˆ˜}C and striving for 1.5^{âˆ˜}C.
Various strategies, such as decarbonization techniques and technologies like nuclear power, renewable energy, and the use of alternative fuels, serve as traditional mitigation measures [7, 14]. Additionally, emerging technologies known as negative emissions technologies, including bioenergy carbon capture and storage, biochar, enhanced weathering, direct air carbon capture and storage, ocean fertilization, ocean alkalinity enhancement, soil carbon sequestration, afforestation, and reforestation, hold promise for capturing and sequestering carbon dioxide from the atmosphere [36, 54, 64, 69].
Considering that emissions from agriculture may become the main source of emissions worldwide by the middle of the century [16, 56], it is crucial to apply various strategies to mitigate emissions from the food industry. This includes a shift to more plantbased diets [78], reducing food waste [33], and improving crop yields and farming practices [48, 62].
Our findings suggest that changes in the NAO index can propagate to changes in the international price of food in both short and long terms. A significant increase in the NAO index is likely to lead to further increments in international food prices. Policymakers can implement different strategies to ensure food security. To secure food and nutrition security in the face of a warming climate, governments, private companies, and international partners must collaborate to work toward more productive, resourceefficient, varied, and nutrientrich farming systems. This involves producing more varied and nutrientdense food with less water and fertilizer for a growing population while simultaneously reducing land use change and greenhouse gas emissions. Other temporary measures that governments can consider include import levies or price subsidies with clear sunset provisions for basic food commodities. Additionally, governments should assist food production, refrain from stockpiling, and utilize food reserves when available to improve the food supply.
7 Conclusion
Even before the onset of the COVID19 pandemic, global food insecurity had been on the rise due to various factors such as increasing food prices, declining wages, disrupted supply chains, and the impacts of climate change. This study delves into the influence of fluctuations in the NAO index on the corresponding fluctuations in the international prices of wheat, corn, soybean, and oats. Wheat and corn, along with rice, constitute a significant portion of the human diet, while oats play a crucial role in the production of healthy food products. Soybeans, being a major source of both human and animal protein, are closely linked to global meat consumption and are expected to see increased demand.
After identifying outliers through robust statistical tests in our time series data, we employ a novel robust powerlaw crosscorrelation coefficient with a sliding windows approach for analysis. This approach is utilized to generate a color map and the timevarying version of the powerlaw crosscorrelation coefficient. Our key finding is that, for both short and longterm analyses, the powerlaw crosscorrelation coefficient for pairs of time series (NAO index, wheat), (NAO index, soybean), (NAO index, oats), and (NAO index, corn) surpasses the critical value of 1.297 at a 95% confidence level. This indicates a statistically significant impact of NAO index fluctuations on various international food prices, revealing a significant positive powerlaw crosscorrelation. In simpler terms, substantial increases in the NAO index precede significant increases in each international food price. These findings are further validated through a variablelag transfer entropy causality test.
Practically, our results can inform policymakers in the development of public policies aimed at mitigating the effects of the NAO index on international food prices. Moreover, predicting international food prices based on the evolution of the NAO index could yield satisfactory results. Univariate models like the autoregressive model (AR) with the NAO index as an exogenous variable (ARNAO) or the timevarying ARNAO model can be employed for this purpose. For multivariate models, the vector autoregressive fractionally integrated moving average (VARFIMA) model is suggested to capture both longrange and shortrange dependence dynamics between the NAO index and international food prices, facilitating forecasting. Different scenarios for the future evolution of various food prices can be simulated using these proposed econometric models.
Our proposed method, based on the robust powerlaw crosscorrelation coefficient, addresses outliers caused by occasional events, such as those in climate change time series or international food prices. Additionally, it allows for the selection of the required time scale. However, while proving the existence of a powerlaw crosscorrelation between two processes, our method does not provide the specific time interval when a large increment in one process is followed by a large increment in the othera crucial aspect for preventive actions. To address this limitation, future research will explore the robust powerlaw crosscorrelation coefficient with time interval localization. Additionally, combining our proposed method with decomposition techniques like wavelet decomposition or empirical mode decomposition will be considered to test powerlaw crosscorrelation between time series in both time and frequency.
Availability of data and materials
Data is available upon reasonable request from the corresponding author.
The link to NAO index: https://www.cpc.ncep.noaa.gov/products/precip/CWlink/pna/nao.shtml.
International prices of corn, soybean, oats, and wheat is obtained from the Bloomberg Terminal.
Code availability
The source code developed in this research is available upon request. Interested parties may contact the corresponding author for access to the code
Abbreviations
 NAO:

North Atlantic Oscillation
 bHe:

bivariate Hurst exponent
 ENSO:

El NiÃ±o Southern Oscillation
 fGn:

fractional Gaussian noise
 VLTE:

variablelag transfer entropy
 ADF:

augmented DickeyFuller
 KPSS:

KwiatkowskiPhillipsSchmidtShin
 JB:

JarqueBera
References
(WHO) WHO (2022) Un report: global hunger numbers rose to as many as 828 million in 2021. https://www.who.int/news/item/06072022unreportglobalhungernumbersrosetoasmanyas828millionin2021. Accessed December 25 2023
Amornbunchornvej C, Zheleva E, BergerWolf T (2021) Variablelag granger causality and transfer entropy for time series analysis. ACM Trans Knowl Discov Data 15:1â€“30
Anderson WB, Seager R, Baethgen W, Cane M, You L (2019) Synchronous crop failures and climateforced production variability. Sci Adv 5(7):eaaw1976
Bairagi S, Mishra AK, Mottaleb KA (2022) Impacts of the covid19 pandemic on food prices: evidence from storable and perishable commodities in India. PLoS ONE 17(3):1â€“15
Bandara JS, Cai Y (2014) The impact of climate change on food crop productivity, food prices and food security in south Asia. Adv Econ Anal Policy 44(4):451â€“465
Barnett L, Barrett AB, Seth AK (2009) Granger causality and transfer entropy are equivalent for Gaussian variables. Phys Rev Lett 103:238701
Bataille C, Ahman M, Neuhoff K, Nilsson LJ, Fischedick M, Lechtenbohmer S, SolanoRodriquez B, DenisRyan A, Stiebert S, Waisman H, Sartor O, Rahbar S (2018) A review of technology and policy deep decarbonization pathway options for making energyintensive industry production consistent with the Paris agreement. J Clean Prod 187:960â€“973
Beckman J, Baquedano F, Countryman A (2021) The impacts of COVID19 on GDP, food prices, and food security. Q Open 1(1):qoab005
Behnassi M, Haiba ME (2022) Implications of the RussiaUkraine war for global food security. Nat Hum Behav 6:754â€“755
Beran J (1994) Statistics for long memory processes. Monographs on statistics and applied probability, volÂ 61. Chapman & Hall, New York
Breitung J (2002) Nonparametric tests for unit roots and cointegration. J Econom 108(02):343â€“363
Brown I (2012) Influence of seasonal weather and climate variability on crop yields in Scotland. Int J Biometeorol 57:605â€“614
Brown I (2013) Influence of seasonal weather and climate variability on crop yields in Scotland. Int J Biometeorol 57:605â€“614
Bustreo C, Giuliani U, Maggio D, Zollino G (2019) How fusion power can contribute to a fully decarbonized European power mix after 2050. Fusion Eng Des 146:2189â€“2193
Chen B, Villoria NB (2019) Climate shocks, food price stability and international trade: evidence from 76 maize markets in 27 netimporting countries. Environ Res Lett 14(1):014007
Crippa M, Solazzo E, Guizzardi D, MonfortiFerrario F, Tubiello FN, Leip A (2021) Food systems are responsible for a third of global anthropogenic ghg emissions. Nat Food 2:198â€“209
CWorldwide (2022) How climate change increases hunger and why weâ€™re all at risk. Tech. rep., concern worldwide
Dhifaoui Z (2016) Robust to noise and outliers estimator of correlation dimension. Chaos Solitons Fractals 93:169â€“174. https://www.sciencedirect.com/science/article/pii/S0960077916303198. https://doi.org/10.1016/j.chaos.2016.10.017
Dhifaoui Z (2018) Statistical moments of Gaussian kernel correlation sum and weighted least square estimator of correlation dimension and noise level. J Stat Plan Inference 193:55â€“69. https://www.sciencedirect.com/science/article/pii/S0378375817301374. https://doi.org/10.1016/j.jspi.2017.08.001
Dhifaoui Z (2022) Robustness of detrended crosscorrelation analysis method under outliers observations. Fluct Noise Lett 21(04):2250039
Dhifaoui Z, Khalfaoui R, Abedin MZ, Shi B (2022) Quantifying information transfer among clean energy, carbon, oil, and precious metals: a novel transfer entropybased approach. Finance Res Lett 49:103138. https://www.sciencedirect.com/science/article/pii/S1544612322003610. https://doi.org/10.1016/j.frl.2022.103138
Dhifaoui Z, Khalfaoui R, Ben Jabeur S, Abedin MZ (2023) Exploring the effect of climate risk on agricultural and food stock prices: fresh evidence from emdbased variablelag transfer entropy analysis. J Environ Manag 326:116789. https://www.sciencedirect.com/science/article/pii/S0301479722023623. https://doi.org/10.1016/j.jenvman.2022.116789
Dhifaoui Z, Kortas H, Benammou S (2014) Correlation dimension of fractional Gaussian noise: new evidence from wavelets. Int J Bifurc Chaos 24(04):1450041. https://doi.org/10.1142/S0218127414500412
Diks C (1996) Estimating invariants of noisy attractors. Phys Rev E 53:R4263â€“R4266. https://link.aps.org/doi/10.1103/PhysRevE.53.R4263. https://doi.org/10.1103/PhysRevE.53.R4263
Division UNS (2022) End poverty in all its forms everywhere. https://unstats.un.org/sdgs/report/2022/goal01/#:~:text=Now%2C%20rising%20inflation%20and%20the,compared%20with%20pre%2Dpandemic%20projections. Accessed December 25 2023
Edinburgh T, Eglen SJ, Ercole A (2021) Causality indices for bivariate time series data: a comparative review of performance. Chaos 31:083111
Elandalibe K, Jbari A, Bourouhou A (2015) Application of crosscorrelation technique for multi leakage detection. In: 2015 third World Conference on Complex Systems (WCCS), ppÂ 1â€“4. https://doi.org/10.1109/ICoCS.2015.7483243
EPA (2022) Climate impacts on agriculture and food supply. Tech. rep., United States Environmental Protection Agency
Falkowski M, Domanski PD (2020) Impact of outliers on determining relationships between variables in largescale industrial processes using transfer entropy. In: 2020 7th International Conference on Control, Decision and Information Technologies (CoDIT), volÂ 1, ppÂ 807â€“812. https://doi.org/10.1109/CoDIT49905.2020.9263965
Florio E, Mercau J, Jobbagy E, Nosetto M (2014) Interactive effects of watertable depth, rainfall variation, and sowing date on maize production in the western pampas. Agric Water Manag 146:75â€“83
Food, (FAO) AO (2022) Chap.Â 2 food security and nutrition around the world, the state of food security and nutrition in the world 2022. https://www.fao.org/3/cc0639en/online/sofi2022/foodsecuritynutritionindicators.html. Accessed December 25 2023
Franses PH, Haldrup N (1994) The effects of additive outliers on tests for unit roots and cointegration. J Bus Econ Stat 12(4):471â€“478
Fund WW (2022) Fight climate change by preventing food waste. Tech. rep
Gel YR, Gastwirth JL (2008) A robust modification of the JarqueBera test of normality. Econ Lett 99(01):30â€“32
Gimeno L, Ribera P, Iglesias R, de la Torre Ramos L, GarcaHerrera R, Hernandez E (2002) Identification of empirical relationships between indices of enso and nao and agricultural yields in Spain. Clim Res 21:165â€“172
Goglio P, Williams A, BaltaOzkan N, Harris N, Williamson P, Huisingh D, Zhang Z, Tavoni M (2020) Advances and challenges of life cycle assessment (lca) of greenhouse gas removal technologies to fight climate changes. J Clean Prod 244:118896
Goode B, Cary JR, Doxas I, Horton W (2001) Differentiating between colored random noise and deterministic chaos with the root mean squared deviation. J Geophys Res 106(A10):21277â€“21288. https://doi.org/10.1029/2000JA000167
Gordeev RV, Pyzhev AI, Zander EV (2022) Does climate change influence Russian agriculture? Evidence from panel data analysis. Sustainability 14(2):718
Granger CWJ, Joyeux R (1980) An introduction to longmemory time series models and fractional differencing. J Time Ser Anal 1(1):15â€“29. https://doi.org/10.1111/j.14679892.1980.tb00297.x
Gutierrez L (2017) Impacts of el ni osouthern oscillation on the wheat market: a global dynamic analysis. PLoS ONE 12(6):1â€“22
Hasudungan P, Irham I, Utami AW (2021) The impact of el ni o southern oscillation and covid19 on the rice price dynamics in Indonesia: the vector error correction model approach. IOP Conf Ser Earth Environ Sci 883(1):012061
Hatfield JL, Dold C (2018) Climate change impacts on corn phenology and productivity. In: Amanullah FS (ed) Corn. IntechOpen, Rijeka. Chap.Â 6
He LY, Chen SP (2011) A new approach to quantify powerlaw crosscorrelation and its application to commodity markets. Physica A 390(21):3806â€“3814. https://doi.org/10.1016/j.physa.2011.06.013. https://www.sciencedirect.com/science/article/pii/S0378437111004602
Heino M, Guillaume JHA, MÃ¼ller C, Iizumi T, Kummu M (2020) A multimodel analysis of teleconnected crop yield variability in a range of cropping systems. Earth Syst Dyn 11(1):113â€“128
Heino M, Puma MJ, Ward PJ, Gerten D, Heck V, Siebert S, Kummu M (2018) Twothirds of global cropland area impacted by climate oscillations. Nat Commun 9:1257
HolesovskÃ½ J, Campulova M, MichÃ¡lek J (2018) Semiparametric outlier detection in nonstationary times series: case study for atmospheric pollution in brno, Czech Republic. Atmos Pollut Res 9:27â€“36
Hurst HE (1951) Longterm storage capacity of reservoirs. Trans Am Soc Civ Eng 116(1):770â€“799
Islam SM, Gaihre YK, Islam MR, Ahmed MN, Akter M, Singh U, Sander BO (2022) Mitigating greenhouse gas emissions from irrigated rice cultivation through improved fertilizer and water management. J Environ Manag 307:114520. https://www.sciencedirect.com/science/article/pii/S0301479722000937. https://doi.org/10.1016/j.jenvman.2022.114520
Jia H, Zhang T, Yin X, Shang M, Chen F, Lei Y, Chu Q (2019) Impact of climate change on the water requirements of oat in northeast and North China. Water 11(1):91
Kettlewell P, Sothern R, Koukkari W (1999) U.k. Wheat quality and economic value are dependent on the North Atlantic oscillation. J Cereal Sci 29(3):205â€“209
Kim MK, McCarl BA (2005) The agricultural value of information on the North Atlantic oscillation: yield and economic effects. Clim Change 71:117â€“139
Klink K, Wiersma JJ, Crawford CJ, Stuthman DD (2014) Impacts of temperature and precipitation variability in the northern plains of the United States and Canada on the productivity of spring barley and oat. Int J Climatol 34(8):2805â€“2818
Kristalina G, SebastiÃ¡n Sosa BR (2022) Global food crisis demands support for people, open trade, bigger local harvests. Tech. rep., IMF
Lawrence MG, Schafer S, Muri H, Scott V, Oschlies A, Vaughan NE, Boucher O, Schmidt H, Haywood J, Scheffran J (2018) Evaluating climate geoengineering proposals in the context of the Paris agreement temperature goals. Sci Rep 9:3734
Ljungqvist FC, Thejll P, Christiansen B, Seim A, Hartl C, Esper J (2022) The significance of climate variability on early modern European grain prices. Cliometrica 16(1):29â€“77
Lynch J, Cain M, Frame D, Pierrehumbert R (2021) Agricultureâ€™s contribution to climate change and role in mitigation is distinct from predominantly fossil co2emitting sectors. Front Sustain Food Syst 4:518039. https://doi.org/10.3389/fsufs.2020.518039
Malone R, Meek D, Hatfield J, Mann M, Jaquis R, Ma L (2009) Quasibiennial corn yield cycles in Iowa. Agric For Meteorol 149(6):1087â€“1094
Mari EP, Axel VT (2022) Trade restrictions are inflaming the worst food crisis in a decade. Tech. rep., World Bank
Moschini G, Ji Y, Lee S (2021) Corn yields and climate change: the innovation challenge. Center for Agricultural and Rural Development (CARD) publications aprwinter20211, Center for Agricultural and Rural Development (CARD) at Iowa State University. https://ideas.repec.org/p/ias/cpaper/aprwinter20211.html
Najafi E, Pal I, Khanbilvardi R (2019) Climate drives variability and joint variability of global crop yields. Sci Total Environ 662:361â€“372
Noah SD, Thomas WH, Martin S, Monika V (2012) Response of corn markets to climate volatility under alternative energy futures. Nat Clim Change 2:514â€“518
Northrup DL, Basso B, Wang MQ, Morgan CLS, Benfey PN (2021) Novel technologies for emission reduction complement conservation agriculture to achieve negative emissions from rowcrop production. Proc Natl Acad Sci 118(28):e2022666118. https://doi.org/10.1073/pnas.2022666118
Otero J, Smith J (2005) The kpss test with outliers. Comput Econ 26:59â€“67
Palmer C (2019) Mitigating climate change will depend on negative emissions technologies. Engineering 5(6):982â€“984
Peri M (2017) Climate variability and the volatility of global maize and soybean prices. Food Secur 9:673â€“683
Podobnik B, Jiang ZQ, Zhou WX, Stanley HE (2011) Statistical tests for powerlaw crosscorrelated processes. Phys Rev E 84:066118
Porter JR, Semenov MA (2005) Crop responses to climatic variation. Philos Trans R Soc Lond B, Biol Sci 360(1463):2021â€“2035
Ren X, Jia Z, Chen X (2008) Rainfall concentration for increasing corn production under semiarid climate. Agric Water Manag 95(12):1293â€“1302
Ricke KL, Millar RJ, MacMartin DG (2017) Constraints on global temperature target overshoot. Sci Rep 7:14743
Rust W, Bloomfield JP, Cuthbert M, Corstanje R, Holman I (2022) The importance of nonstationary multiannual periodicities in the North Atlantic oscillation index for forecasting water resource drought. Hydrol Earth Syst Sci 26(9):2449â€“2467
Rust W, Bloomfield JP, Cuthbert MO, Corstanje R, Holman IP (2021) Nonstationary control of the nao on European rainfall and its implications for water resource management. Hydrol Process 35(3):e14099
Salinger MJ, Verdi L, Dalla Marta A, Dalu G, Baldi M, Messeri G, Vallorani R, Morabito M, Crisci A, Altobelli F et al. (2022) Linking maize yields in veneto Italy, to largescale atmospheric variability, circulation regimes and weather types. J Agric Sci 160(6):423â€“439
Sarafanov A (2009) On the effect of the North Atlantic oscillation on temperature and salinity of the subpolar North Atlantic intermediate and deep waters. ICES J Mar Sci 66(7):1448â€“1454
Shah H, Hellegers P, Siderius C (2021) Climate risk to agriculture: a synthesis to define different types of critical moments. Clim Risk Manag 34:100378. https://www.sciencedirect.com/science/article/pii/S2212096321001078. https://doi.org/10.1016/j.crm.2021.100378
Sharif Z, ShaÃ¡meri AZ (2007) The application of cross correlation technique for estimating impulse response and frequency response of wireless communication channel. In: 2007 5th student conference on research and development, ppÂ 1â€“5. https://doi.org/10.1109/SCORED.2007.4451386
Shimotsu K, Phillips PCB (2005) Exact local Whittle estimation of fractional integration. Ann Stat 33(4):1890â€“1933. http://www.jstor.org/stable/3448627
Shmelev SE, Salnikov V, Turulina G, Polyakova S, Tazhibayeva T, Schnitzler T, Shmeleva IA (2021) Climate change and food security: the impact of some key variables on wheat yield in Kazakhstan. Sustainability 13(15):8583
Shukla P, Skea J, Slade R, Khourdajie AA, van Diemen R, McCollum D, Pathak M, Some S, Vyas P, Fradera R, Belkacemi M, Hasija A, Lisboa G, Luz S, Malley J (2022) Climate change 2022: mitigation of climate change. Contribution of working group iii to the sixth assessment report of the intergovernmental panel on climate change. Contribution of working group iii to the sixth assessment report of the intergovernmental panel on climate change. Tech. rep., Cambridge University Press, Cambridge, UK and New York, NY, USA
Sorbye SH, Rue H (2016) Fractional gaussian noise: prior specification and model comparison. arXiv:1611.06399
Ubilava D (2014) El niÃ±o southern oscillation and the fishmeal soya bean meal price ratio: regimedependent dynamics revisited. Eur Rev Agric Econ 41(04):583â€“604
Ubilava D (2017) The enso effect and asymmetries in wheat price dynamics. World Dev 96:490â€“502
USAID (2022) Response to global food security crisis: fact sheet. Tech. rep
Visbeck MH, Hurrell JW, Polvani L, Cullen HM (2001) The North Atlantic oscillation: past, present, and future. Proc Natl Acad Sci 98(23):12867â€“12877
Vorburger T, Song JF, Chu W, Ma L, Bui S, Zheng A, Renegar T (2011) Applications of crosscorrelation functions. Wear 271(3):529â€“533. The 12th International Conference on Metrology and Properties of Engineering Surfaces. https://www.sciencedirect.com/science/article/pii/S0043164810001407. https://doi.org/10.1016/j.wear.2010.03.030
Wang G, You L (2004) Delayed impact of the North Atlantic oscillation on biosphere productivity in Asia. Geophys Res Lett 31(12):L12210
WBank (2022) Trade and food security in a climate changeimpacted world. Tech. rep
Wei YL, Yu ZG, Zou HL, Anh V (2017) Multifractal temporally weighted detrended crosscorrelation analysis to quantify powerlaw crosscorrelation and its application to stock markets. Chaos, Interdiscip J Nonlinear Sci 27(6):063111
Weron R (2002) Estimating longrange dependence: finite sample properties and confidence intervals. Phys A, Stat Mech Appl 312(1):285â€“299. https://www.sciencedirect.com/science/article/pii/S0378437102009615. https://doi.org/10.1016/S03784371(02)009615
Xu R, Li Y, Guan K, Zhao L, Peng B, Miao C, Fu B (2021) Divergent responses of maize yield to precipitation in the United States. Environ Res Lett 17(1):014016
Xu ZM, Devel E, Vinck B, Van Cauwenberge P (1995) Application of crosscorrelation function in the evaluation of objective MLR thresholds in the low and middle frequencies. Scand Audiol 24(4):231â€“236
Yu D, Small M, Harrison RG, Diks C (2000) Efficient implementation of the Gaussian kernel algorithm in estimating invariants and noise level from noisy time series data. Phys Rev E 61:3750â€“3756. https://link.aps.org/doi/10.1103/PhysRevE.61.3750. https://doi.org/10.1103/PhysRevE.61.3750
Zhang L, Wu X (2006) On the application of cross correlation function to subsample discrete time delay estimation. Digit Signal Process 16(6):682â€“694. https://doi.org/10.1016/j.dsp.2006.08.009. https://www.sciencedirect.com/science/article/pii/S1051200406001230
Zhou M, Wang H, Yang S, Fan K (2012) Influence of springtime North Atlantic oscillation on crops yields in northeast China. Clim Dyn 41:3317â€“3324
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Dhifaoui, Z. Connection between climatic change and international food prices: evidence from robust longrange crosscorrelation and variablelag transfer entropy with sliding windows approach. EPJ Data Sci. 13, 56 (2024). https://doi.org/10.1140/epjds/s13688024004821
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DOI: https://doi.org/10.1140/epjds/s13688024004821