Connection between climatic change and international food prices: evidence from robust long-range cross-correlation and variable-lag transfer entropy with sliding windows approach

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 well-established 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 time-varying version. Furthermore, variable-lag 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 long-term periods. Additionally, variable-lag transfer entropy confirms the causal role of the NAO index in influencing international food prices.

1 Introduction

The global COVID-19 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 2021-an 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 Niño and La Niña indices, and specific food products. For instance, [65] examines the effects of El Niño and La Niña events on corn and soybean prices, showcasing increased volatility during El Niño in the Spring-Summer 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 Niña and El Niño events. [40] establishes a positive correlation between wheat export prices and La Niña events, highlighting their consistent short and long-term impacts. [80] studies the dynamics of the fishmeal-soybean flour price ratio, demonstrating significant impacts of the bi-monthly multivariate El Niño 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 Niño index, while noting a negative relationship with La Niña 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 index’s impact on food prices in various regions and countries. [44] establishes the NAO index’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 large-scale 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 variable-lag transfer entropy causality test. Theoretical and empirical implications are discussed in Sect. 6. Finally, our conclusions are presented in Sect. 7.

Table 1 Summary of previous research

2 Methodology

2.1 Robust bivariate Hurst exponent

The robust bHe was introduced by [20]. This robust bHe is constructed using the detrended cross-correlation function and is specifically designed to analyze the existence of significant cross-correlations between two stochastic processes, referred to as X and Y. In the realm of signal processing and analysis, the cross-correlation method has emerged as a versatile and indispensable tool with myriad applications. This research sets out to explore the diverse practical uses of cross-correlation 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 method’s effectiveness in characterizing complex surface structures [84]. Additionally, the study delves into the application of cross-correlation in estimating the channel impulse response in wireless communication channels, highlighting its crucial role in enhancing communication system performance [75]. In the realm of multi-sensor signal processing, the cross-correlation 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 cross-correlation as a non-destructive testing method for detecting leaks in buried pipes within industrial and household facilities, underscoring its significance in real-world problem-solving [27].

Assumes that both processes, X and Y, have zero means and exhibit long-range temporal autocorrelation characterized by power-law auto-correlations, as described by the following equations:

$$ \mathbb{E} \bigl[X(\ell )X(\ell +s) \bigr]\sim s^{2H_{X}-2} \quad \text{and}\quad \mathbb{E} \bigl[Y(\ell )Y(\ell +s) \bigr]\sim s^{2H_{Y}-2}, $$

(1)

where $H_{X}$ and $H_{Y}$ represent the Hurst exponents, falling within the range of $[0.5,1[$. The power-law cross-correlations are defined as:

$$ \mathbb{E} \bigl[X(\ell )Y(\ell +s) \bigr]\sim As^{2\gamma -2} \quad \text{and} \quad \mathbb{E} \bigl[Y(\ell )X(\ell +s) \bigr]\sim Bs^{2\gamma -2}, \quad \text{where } (A,B)\in \mathbb{R}^{2}_{+}. $$

(2)

The robust bHe computation involves the following steps:

• 1 Profile construction: we divide each time series into non-overlapping boxes of size s, where s is defined as $N/s$, and $\ell _{v}=(v-1)s$. The profiles in the v-th 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 Cross-correlation calculation: the cross-correlation 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 q-th order cross-correlation function: the q-th order cross-correlation 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 long-range cross-correlation properties of processes X and Y. A value of $\gamma _{X,Y}(2)$ around 0.5 indicates no cross-correlation, while $\gamma _{X,Y}(2)>0.5$ suggests a positive power-law cross-correlation, and $\gamma _{X,Y}(2)<0.5$ indicates anti-correlation.

In situations where the processes X and Y exhibit cross-correlations 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]:

$$ \mathbb{E} \bigl[F_{X,Y}(2,s)^{2} \bigr]= \frac{\sigma _{X}\sigma _{Y} \rho _{XY}}{2}s^{H_{X}+H_{Y}}\mathcal{O}(1)+ \frac{\sigma _{X}\sigma _{Y} \rho _{XY}}{2}s. $$

(7)

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 cross-correlation 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 cross-correlation),

• Alternative Hypothesis: $H_{X}+H_{Y}>1$ (indicating a positive power-law cross-correlation).

For each value of N, the critical value, denoted as $H_{(X,Y)}^{c}$, is computed as:

$$ H_{(X,Y)}^{c}=\mu _{H_{X,Y}}+Z_{\alpha} \sigma _{H_{X,Y}}. $$

(8)

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 p-value of $H_{X,Y}^{o}$ is calculated as follows:

$$ \text{p-v} \bigl(H_{(X,Y)}^{o} \bigr)=\frac {1}{N_{rep}} \sum_{\ell =1}^{N_{rep}} \mathbf{1}_{\{H_{X,Y}\leq (\widehat{H}_{X,Y})_{\ell}\}}. $$

(9)

In this equation, $(\widehat{H}_{X,Y})_{\ell}$ represents the estimated value of the robust bHe for the ℓ-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 $\text{p-v}(H_{X,Y}^{o})<0.05$. In either case, we conclude that the positive power-law cross-correlation 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 $1 \leq T \leq N-h$, 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 $\{X_{i}\}_{i=T,\dots ,h+T-1}$ and $\{Y_{i}\}_{i=T,\dots ,h+T-1}$.

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

• 3 We compute the cross-correlation ${f_{v}}^{X,Y}(s)$ for each box using Equation (4) for $v=1,\dots ,N^{*}_{s}$, and the second order cross-correlation 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 1–4 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 −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.

Figure 1

Studied time series

Table 2 Descriptive statistics of studied time series

According to Table 2 (Panel B), based on the results of the ADF test, all time series are stationary, as all p-values are lower than 0.05. However, according to the results of the JB test, the NAO index is normally distributed, as the p-value is higher than 0.05. In contrast, all other time series are not normally distributed, as the associated p-values 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 Dickey-Fuller (ADF) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests can be sensitive to the influence of outliers [32, 63], we mitigate this issue by utilizing the unit root test based on Breitung’s variance ratio [11]. Additionally, to assess the normality distribution of the studied time series, we employ the robust Jarque-Bera (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 non-stationarity in favor of the alternative hypothesis of stationarity, as all p-values 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 Jarque-Bera (JB) test, with corresponding p-values (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 σ, within a subset of data series with a length of N. This test facilitates the distinction between random processes characterized by a power-law spectrum and deterministic low-dimensional chaotic signals. The application of the variance growth test in [23] demonstrates the stochastic nature of certain financial time series.

Figure 2

Studied time series after removing outliers observations

Figure 3

Autocorrelation functions of studied time series without outliers and of associated fGn process

Table 3 Results of stationarity and normality tests, and the Hurst exponent (H) of the studied time series without outliers

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 low-dimensional chaotic data, the variance reaches a plateau for N values exceeding the Poincaré 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 σ 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 power-law distribution.

Figure 4

Results of correlation dimension method for studied time series without outliers

Figure 5

Results of variance growth test for studied time series without outliers

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 self-similar stochastic models used to capture both anti-persistent 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 anti-persistence inherent in the process [79].

4 Cross-correlation analysis using robust bHe

We adopt the sliding window approach to assess the long-range cross-correlation 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 short-term scale, and from 150 to 200 days as a long-term scale. The time-varying long-range cross-correlation is determined as the average of $H_{X,Y}(T,n)$ for all n, expressed as:

$$ H(T)_{X,Y}(2)=\frac {1}{4} \sum_{n=\{50,100,150,200\}}H_{X,Y}(T,n). $$

(10)

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 power-law cross-correlation. The contour plots illustrating sliding windows long-range cross-correlation and the corresponding time-varying long-range cross-correlation 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 power-law cross-correlation 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 long-range cross-correlation 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 6

Sliding window and time-varying robust bHe for NAO and return of corn time series

Figure 7

Sliding window and time-varying robust bHe for NAO and return of Oats time series

Figure 8

Sliding window and time-varying robust bHe for NAO and return of Soybean time series

Figure 9

Sliding window and time-varying robust bHe for NAO and return of Wheat time series

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 long-term scales across all studied time periods. In Fig. 6b, the time-varying long-range cross-correlation coefficient surpasses the critical value of 1.297 for all time periods, except during mid-September 2021 and mid-December 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 cross-correlation between the NAO index and the return of corn prices could be linked to the NAO index’s association with other climatic change indices affecting corn price or yield fluctuations [42, 59, 61]. Moreover, previous research has highlighted the NAO index’s impact on water quality [70, 73] and rainfall patterns [71], with the well-established importance of irrigation and rainfall on corn productivity [30, 68, 89]. Hence, our findings regarding the cross-correlation between the NAO index and corn prices may be associated with the NAO index’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 short-term scale ($50 \leq n \leq 100$). Furthermore, for the long-term scale ($150 \leq n \leq 200$), $H_{X,Y}$ consistently exceeds 1.9 across all studied time periods. The corresponding long-range cross-correlation coefficient, depicted in Fig. 7b, displays time-varying behavior throughout all studied time periods. Additionally, $H_{X,Y}$ remains above 1.297 for the majority of the studied time periods, except during mid-November 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 long-range cross-correlation between the NAO index and the return of soybean prices, revealing that $H_{X,Y}$ consistently exceeds 1.297 for both short- and long-term scales. Additionally, the evolution of the long-range cross-correlation coefficient over time, as depicted in Fig. 8b, is characterized by a non-constant, time-varying 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 long-term scales. Furthermore, in Fig. 9b, the time-varying long-range cross-correlation coefficient remains above 1.297, except during mid-November 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 variable-lag transfer entropy

In this section, we explore the causal relationship between the NAO index and the return of international food prices by employing a variable-lag transfer entropy (VLTE) causality test [2] with a sliding windows approach. We use the same sliding windows and scales for the power-law cross-correlation 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 short-term 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 long-term 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 short-term scale throughout the entire studied time period. Additionally, for the long-term 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 short-term scale and almost the entire long-term 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.

Figure 10

Time-scale VLTE ratio for different pairs of time series

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 power-law cross-correlation coefficient and the variable-lag 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 power-law cross-correlation coefficient. This novel metric enhances our ability to analyse and understand the cross-correlation between the NAO index and international food prices. Secondly, our introduction of a color map of the power-law cross-correlation coefficient, obtained through a sliding window framework, enables us to infer the time-varying robust power-law cross-correlation coefficient. This visualization approach provides a nuanced understanding of how the cross-correlation 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 cross-correlation methods in various works [43, 87].

6.2 Empirical implications

The global food crisis, intensified by the COVID-19 pandemic, has disrupted food prices across nearly all countries. Numerous studies indicate a negative impact on food prices during the pandemic compared to the pre-pandemic period [4, 8]. The ongoing effects of the COVID-19 pandemic, combined with the Russian-Ukrainian 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 power-law cross-correlation 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 index’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^∘C and striving for 1.5^∘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 plant-based 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, resource-efficient, varied, and nutrient-rich farming systems. This involves producing more varied and nutrient-dense 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 COVID-19 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 power-law cross-correlation coefficient with a sliding windows approach for analysis. This approach is utilized to generate a color map and the time-varying version of the power-law cross-correlation coefficient. Our key finding is that, for both short- and long-term analyses, the power-law cross-correlation 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 power-law cross-correlation. In simpler terms, substantial increases in the NAO index precede significant increases in each international food price. These findings are further validated through a variable-lag 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 (AR-NAO) or the time-varying AR-NAO model can be employed for this purpose. For multivariate models, the vector autoregressive fractionally integrated moving average (VARFIMA) model is suggested to capture both long-range and short-range 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 power-law cross-correlation 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 power-law cross-correlation 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 other-a crucial aspect for preventive actions. To address this limitation, future research will explore the robust power-law cross-correlation 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 power-law cross-correlation between time series in both time and frequency.