Price interconnection of fuel and food markets: Evidence from biodiesel in the United States

The food versus fuel issue has attracted considerable attention with the rapid growth of global biofuel production. The previous literature examining the interconnectedness between biofuel, fossil fuel and agricultural markets employed futures and spot prices. However, food security needs to be discussed with non‐financial market prices, such as wholesale prices, since retail companies and households usually do not purchase products directly from financial markets, which encompass speculative activities, making them more volatile than non‐financial ones. This paper focuses on non‐financial markets in the United States to investigate the price‐interconnection of biodiesel, highway diesel, crude oil, and soybean, initially applying partial wavelet coherence and the Diebold‐Yilmaz connectedness index to price transmission research on biodiesel, highway diesel, crude oil, and soybean. Our main results are as follows: First, significant coherence between biodiesel and soybean, biodiesel and highway diesel, and highway diesel and crude oil is identified in the short and long term. Second, crude oil and biodiesel prices are found to be net transmitters, while soybean and highway diesel prices are net receivers. Finally, the crude oil market is identified as the source of spillovers among the four markets, and strongly influences the highway diesel market.

though the United States is the world's largest biodiesel producer as a single country, its biodiesel market has not yet been investigated in an appropriate manner for the following reasons. Kristoufek et al. (2012Kristoufek et al. ( , 2014 studied futures prices from the Chicago Board of Trade (CBOT) and the Intercontinental Exchange (ICE), together with the spot price of biodiesel in Germany, but did not, strictly speaking, concentrate on relationships with the biodiesel market in the United States. Furthermore, the existing studies analyzing the US markets concentrated on futures and spot prices (Kristoufek et al., 2012(Kristoufek et al., , 2014Vacha et al., 2013). Yet, future markets encompass many noncommercial traders (i.e., speculators) who can proliferate market volatilities by taking various positions (e.g., long and short positions with leveraged contracts and options). The heterogeneity of financial and non-financial markets could affect the conclusion of food versus fuel analyses since retailers and households usually do not purchase food or energy goods from financial markets.
Another issue is that advanced econometric techniques have been underutilized in the previous literature. Wavelet coherence has gained immense popularity, particularly in the areas of financial commodity markets and macroeconomics. The model enables us to continually identify the statistical significance of the time and frequency domains, degree of effects, and lead-lag relationship, which suits time-series price interconnection analyses and was initially applied to biofuel and related commodity markets by Vacha et al. (2013). Then, the partial wavelet coherence that controls the effects of third variables on pairwise connectedness was employed in the research field at hand by Kristoufek et al. (2016), who analyzed the futures price connections between the bioethanol, crude oil, gasoline, and corn markets. However, the partial wavelet model has not yet been applied to the price relationships between biodiesel and other relevant goods. Besides this, the Diebold-Yilmaz connectedness index (DYCI), which appeared a decade ago in novel work by Yilmaz (2012, 2014), is a popular method to simultaneously interrogate the correlations between various variables, making it possible to discover the pass-through source of variables' relationships.
This article fills the aforementioned knowledge gap by employing partial wavelet continuous transform and DYCI to explore the inter-relationships of biodiesel, highway diesel, soybean, and crude oil non-futures prices in the United States from April 2007 to December 2020. Specifically, our research applied the above two methods to investigate the price connection of biodiesel and relevant goods and visualize the dynamic variations of pairwise transmission effects between all variables. We also use the results generated from the two methods as robustness tests to verify the reliability of our findings.

| Literature review
The copious existing literature on price links between biofuels and food goods primarily focused on bioethanol price linkages with relevant commodities, while research on price spillovers from biodiesel and agricultural goods is relatively scarce. This section provides an overview of past analyses of price relationships between biodiesel and foods.
The following works do not use biodiesel prices but probe the connectivities of biodiesel-related commodity prices. Pal and Mitra (2017) used a quintile autoregressive distributed lag model to identify the connectivity between biodiesel and soybean prices from 2004 to 2014 and discovered that the prices have strong relationships in the long run. Peri and Baldi (2010) applied a threshold co-integration method to find the presence of asymmetric dynamic adjusting processes between the prices of rapeseed oil, sunflower oil, and fossil biodiesel. They found that rapeseed oil prices immediately adjust to the long-run equilibrium, which is decided by fossil diesel prices.
Several papers conducted research in this field in European regions. Hassouneh et al. (2012) concentrated on Spanish energy and food markets using cointegration and vector error correction (VEC) methods, asserting that the prices of biodiesel, sunflower oil, and crude oil are connected in the long term, although sunflower oil prices do not affect biodiesel prices in the short term. Abdelradi and Serra (2015) examined price volatility transmissions between biodiesel, rapeseed oil, and crude oil in European countries using GARCH models. Significant asymmetries were discovered in volatility spillovers between biodiesel and rapeseed oil prices. Cabrera and Schulz (2016), meanwhile, analyzed price and volatility risks due to links between Germany's energy and agricultural commodity prices by using VEC models. They found that prices move together and have an equilibrium in the long term, while there were positive correlations with persistent market shocks during most of the sample period. In research elsewhere, concerns about biodiesel being the cause of high agricultural commodity prices and price volatility were found to be unjustified. Serra and Gil (2012), for instance, analyzed the connectedness of diesel and crude oil prices in Spain, showing symmetric dependence, where extreme increases and decreases in the crude oil price were equally likely to be passed on to consumers. Busse et al. (2012) examined the interplay between diesel and biodiesel prices and between soy oil, rapeseed oil, and biodiesel prices in Germany. They showed the long-term relationships between biodiesel and diesel prices and between biodiesel, rapeseed oil, and soy oil prices. Yahya et al. (2022) investigated the dependence structure between oil, biodiesel and rapeseeds oil markets, using asymmetric copulas and cross-quantilogram approaches and found that crude oil price positively influences biodiesel prices. Janda and Kravec (2022) examined the price linkage dynamics of ethanol and biodiesel in Brazil, the US and the EU with the Johansen co-integration tests and a VEC model from 2003 to 2020 (the period was divided into four.). They uncovered that the co-movements and strength vary over time. The price transmissions between sugarcane-based ethanolrelated commodities in Brazil and biodiesel counterparts in the EU were more intensified in comparison with corn-based ethanol goods in the US.
Certain past research papers shed light on biodiesel markets in other regions. As an example, Kristoufek et al. (2012) applied a minimal spanning tree and hierarchical tree to investigate the interconnections between various fuel and agricultural commodities, considering futures prices from the Chicago Board of Trade and the Intercontinental Exchange, along with German biodiesel spot prices. They concluded that biodiesel prices are associated with energy prices but not with feedstock prices. Meanwhile, Vacha et al. (2013) used the wavelet coherence method on ethanol, biodiesel, gasoline, diesel, crude oil, corn, wheat, soybean, sugarcane, and rapeseed oil. They found that the biodiesel price correlated with the German diesel price during stable (noncrisis) periods at a low frequency, while the biodiesel price moved with soybean and rapeseed oil prices in the long run. The mutual dependency between biofuels and related goods was analyzed by Kristoufek et al. (2012), who found that biodiesel and German diesel were mutually dependent, bioethanol and biodiesel held mutual responsiveness with fossil fuels, and their connectedness was price-dependent. Furthermore, they discovered that biodiesel price was highly influenced by German diesel and soybean prices. In sum, most articles found that biodiesel prices correlate with feedstock prices in the long term, though the relationship is not always established in the short term. In addition, biodiesel price tends to move with fuel prices. Kristoufek et al. (2014) used the two-stage least-squares procedure to probe biodiesel market transmissions using German spot biodiesel and agricultural futures prices from the CBOT. In doing so, they discovered that the strength of spillovers increased during the global food crisis period around 2008. Gomez-Gonzalez et al. (2022) applied DYCI to examine realized volatility spillovers between different moments (e.g., skewness, kurtosis, and sign jump variations) of financial and commodity markets. Their results indicated that risk spillovers are time varying and sensitive to volatility measures. Moreover, they detected an increasing trend of interconnectedness between commodity markets and financial markets after the global financial crisis (2008)(2009).
Though state-of-the-art econometric techniques, such as wavelet transform and DYCI, have gained immense popularity in recent years, they have not yet been fully applied to price correlations in biofuel research on financial commodity markets and macroeconomics. Only two existing articles applied the wavelet coherence methodology to the biofuel and food price interdependency. Vacha et al. (2013) were the first to apply the technique to explore the interaction of biofuels with related goods. A partial wavelet model considering third variable effects on the pair was then employed by Kristoufek et al. (2016) concentrating on bioethanol markets. The DYCI has not yet been utilized in either bioethanol or biodiesel markets.
To conclude, to our knowledge, no publication has yet focused on non-futures price inter-connection analysis of biodiesel and related goods in the United States. In addition, no article has yet determined the long-and short-term impacts of those commodities on one another through the use of the DYCI method or the partial wavelet coherence framework.

| MATERIALS AND METHODS
In this section, we describe in detail the empirical methods applied in the present study. Methodologically, we focused on two mathematical tools: the multivariate wavelet technique introduced by Aguiar-Conraria and Soares (2014) and the time-domain connectedness measures proposed by Yilmaz (2012, 2014). Readers who are not interested in the technical details may skip to Section 4, where we interpret the empirical results of our analysis.

| Multivariate wavelet analyses
To examine the dynamic interaction and lead-lag relationship of soybean, biodiesel, highway diesel, and crude oil across the time and frequency domains, we employed multiple wavelet coherency, partial wavelet coherency, partial wavelet phase-difference, and partial wavelet gain.
First, following Aguiar-Conraria and Soares (2014), the continuous wavelet transform (CWT) of a time-series x(t) ∈ L 2 (ℝ) can be represented by Note that wavelet is the function of two variables W x ( , ), where is the scaling factor controlling the width of the wavelet, while is a translation parameter describing the location of the wavelet ( , ∈ ℝ and ≠ 0 ). The specific wavelet we use in this paper is the Morlet wavelet defined by M (t) = 1 1∕4 e i 0 t e −t 2 ∕2 . See [13] for a discussion of certain important properties of this wavelet. The window of function W x ( , ) becomes larger with a corresponding lower frequency for | | > 1. For simplicity, we omit ( , ) from the following formulae. Meanwhile, the window becomes narrower with a higher frequency for | | < 1. If the wavelet function is complex, the wavelet transform W x is also complex. Based on the CWT, we can derive the wavelet power spectrum (WPS) as The WPS gives us a measure of the variance distribution of univariate time-series.
Second, to investigate the dynamic correlation between fuel and food price returns, we must introduce a bivariate framework of wavelet analysis. For the bivariate case, the cross-wavelet transform of two time-series y(t) and x(t), denoted by W yx , is defined as W yx = W y W x , and its absolute value | | | W yx | | | is referred to as the cross-wavelet power (CWP). The CWP of two time-series reflects their covariance of time and frequency. Based on the CWP, we can derive the complex wavelet coherency yx , by where represents a smoothing operator in both scale and time. For simplicity, we introduce the notation yx to replace W yx and use y and x to denote , respectively. Hence, Equation (2) can simply . The absolute value of yx is called the wavelet coherency, which can be defined by The wavelet coherency is the ratio of cross-spectrum to the product of the spectrum of each series.
Given a complex wavelet may arise, Aguiar-Conraria and Soares (2014) also provide a phase difference tool to clarify the possible delays of the oscillations between two time-series. The wavelet-coherence phase difference is determined as follows.
where the smoothed real (ℜ) and imaginary (ℑ) should have been estimated in Equation (2). We identify the lead-lag relationship between y(t) and x(t) at each time and frequency by using the value of the phase difference. Specifically, a phase-difference of zero indicates that the two series move together: when yx ∈ 0, ∕ 2 , the two series move in-phase (positively related), with y(t) leading x(t). Nevertheless, when yx ∈ − ∕2, 0 , we say that x(t) leads y(t). On the other hand, the two series are in an anti-phase relationship (negatively related) when yx ∈ − , − ∕ 2 and yx ∈ ∕ 2, , where y(t) leads x(t) in the former and x(t) leads y(t) in the latter. Finally, we apply the technique of wavelet partial coherency, which helps identify the resulting wavelet coherency between two time-series y(t) and x(t) after eliminating the influence of the controlling variable z(t). Following Aguiar-Conraria and Soares (2014), the complex partial wavelet coherency between y and x, after controlling for z, is given by where yz and xz are defined similarly to yx , while and R yz and R xz are calculated in a similar way to R yx . The absolute value and angle of yx,z , will, respectively, be called the partial wavelet coherency and partial wavelet phasedifference between the series y and x, after controlling for z , and will be denoted by R yx,z and yx,z . To save space, we do not define R yx,z and yx,z , but these definitions are available from the authors on request. Furthermore, Mandler and Scharnagl (2014) generalized the concept of wavelet gain and defined a partial wavelet gain, which can be interpreted as a regression coefficient in the regression of y on x after controlling for other variables. Following [18], the partial wavelet gain G yx,z is indicated by The partial wavelet gain provides the magnitude of impact among variables at each time and frequency.

| DYCI measures
We also used a time-domain connectedness measure to examine the directional return connectedness and build a network among the prices of food, biofuels, and fossil fuels. Following Yilmaz (2012, 2014), we considered a four-variable VAR(k) system, defined as follows: where a 4 × 1 vector of the variables at time t is denoted by z t , including soybean, biodiesel, highway diesel, and crude oil. is a 4 × 1 vector of the constants. Φ 1 , ⋯ , Φ p expresses the coefficient matrices and u t stands for white noise with a covariance matrix ∑ . The model work with n × n matrix lag-polynomial Φ(L) = I n − Φ 1 L − ⋯ − Φ p L p with I n identity matrix can be written concisely as Φ(L)z t = u t . If the VAR model above follows a covariance stationary process, Equation (6) can be rewritten as an infinite moving average process z t = Ψ(L)u t , where Ψ(L) matrix of infinite lag polynomials can be calculated recursively from Φ(L) = [Ψ(L)] −1 and is key to understanding dynamics. To remove the relevance of the order from the results of the variance decomposition, we add the H-step-ahead forecast errors for H = 1, 2, … , as suggested by Koop et al. (1996) and Pesaran and Shin (1998). As such, the generalized variance decomposition becomes where the connectedness matrix g ij (H) is the contribution of the jth variable to the forecast error variance of the ith element at horizon h. ∑ marks the covariance matrix of errors in the VAR model. jj is the standard deviation of the innovation for the jth equation, while the 4 × 1 selection vector is represented by i , with the ith element equal to 1, and 0 otherwise.
The advantage of the DYCI approach is that it provides the direction of pairwise connectedness, which enables us to understand the spillovers between variables. We use ̃ g ij (H) to represent the directional connectedness from variable j to variable i, which is defined as For simplification, it is denoted as C i←j (H). Note that the own and cross-variable variance contribution sums to 1 under the generalized decomposition with Since one of our purposes is to estimate the total connectedness (TC), we calculate the pairwise connectedness and construct the TC as This shows the impact of connectedness across variables on the total forecast error variance. Moreover, it is important to investigate the spillover effect and identify which variables are transmitting a shock to others and which are receiving a shock from others. The directional connectedness (DC) to the variable i from all other variables j is given by In reverse, the directional connectedness from variable i to all other variables j is calculated as Based on Equations 10 and 11, we then obtain the net total connectedness (NC) from variables i to all other variables j as follows The NC indicates whether variables are net transmitters or receivers of shocks when the value is positive or negative.
Finally, to investigate specific variables' connections, it is also interesting to determine the net pairwise directional connectedness (NPDC) between two variables i and j, which can be set as This improves our understanding of the transmission mechanism between two specific variables.

| Materials
The primary objective of this current paper was to examine the price volatility pass-through mechanism between biodiesel, highway diesel, crude oil, and soybean. While our research focused on the link between biodiesel and soybean, we also included highway diesel and crude oil as substitutive goods and production factors of diesel, respectively. We collected monthly price data series for crude oil, biodiesel, highway diesel, and soybean in the United States to analyze their relationships in the sample period spanning April 2007 to December 2020. The producer prices of crude oil and soybean were from the US Energy Information Administration and Federal Reserve Economic Data, respectively. We obtained the wholesale price of biodiesel and the retail price of highway diesel from the USDA Economic Research Service.
All levels of prices were seasonally adjusted using the X-13-ARIMA method. The X-13-ARIMA (autoregressive integrated moving average), as developed by the US Census Bureau, is one of the most popular methods for seasonal adjustment. R t represents the monthly price returns, calculated as the logarithmic first difference of prices R t = ln p t − ln p t−1 . The price at time t is shown as p t . Table 1 reports the summary statistics and preliminary tests of price returns for biodiesel, soybean, highway diesel, and crude oil. In the table, positive mean values for biodiesel and soybean can be observed, while the mean values for highway diesel and crude oil are negative. On average, soybean data provided the highest returns, while the lowest yield was for the crude oil data. We can also see that the standard deviation value of crude oil was larger than that of the others, indicating greater volatility. Furthermore, all the variables were negatively skewed, indicating a high probability of negative returns. In terms of kurtosis, crude oil and highway diesel were leptokurtic, which suggests that the distributions of these two variables had fat tails.
The properties of data reveal several interesting features. First, over the sample period, the volatility has increased on average for the biodiesel and soybean market, while it has decreased for the fossil fuel (highway diesel and crude oil) market. Second, the higher standard deviation of crude oil shows that the volatility is higher in the crude oil market than in the other markets. This implies that crude oil price is more influenced by complex international factors and are difficult to predict. Third, the distributional properties of all the price returns exhibit the same characteristics. They are negatively skewed, nonnormal, and show high kurtosis. Specifically, the skewness and kurtosis values indicate serious downward volatility revisions in crude oil price returns during the sample period. Furthermore, all returns were non-normally distributed at a 1% significance level based on the Jarque-Bera (J-B) normality test, and the results of the augmented Dickey-Fuller (ADF) test (Dickey & Fuller, 1979) and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root test (Kwiatkowski et al., 1992) demonstrated that all price returns were stationary.
Finally, the results from the autoregressive conditional heteroscedasticity (ARCH)-Lagrange multiplier (LM) test (Engle, 1982) suggest that the ARCH effects in the return-series applied to all prices. On the other hand, the Breusch-Godfrey (B-G) LM test (Breusch, 1978;Godfrey, 1978) evidences that there was a serial correlation between soybean and crude oil. These results show there were potentially certain non-linearities in the series and highlight why we needed to carry out a wavelet-based analysis.
The monthly price returns of each variable are depicted in Figure 1a on the left, together with their wavelet power spectrum in Figure 1b on the right. The wavelet power spectrum indicates the magnitude of the variance for each return series and provides a first assessment of the behavior of each variable in the time and frequency domains. Here, all the returns are divided into two timescale sections: short-term variation (high frequency) of one to 4 months, and long-term variation (low frequency) of four to 8 months. Through 5000 Monte Carlo simulations applying phase-randomized surrogates, we exposed the critical values. For the cones of influence (COI), we used a black outline to point out the area influenced by edge effects.
Since Grinsted et al. (2004) argued that the results outside the COI region might be unreliable, we discuss the wavelet power spectra inside the COI in this study. First, looking at the soybean price in Figure 1b.  After identifying the volatility of all the price returns across the time and frequency domains, we further explored their movement together and lead-lag relationship, using multivariate wavelet analysis, as presented in the next section.

| Wavelet analysis
In this subsection, the time-varying interaction and leadlag relationship between biodiesel, soybean, highway diesel, and crude oil returns are analyzed by partial wavelet coherence combined with the partial phase-difference and partial wavelet gains. For definitions of the frequency band, significance level, COI, and color code used, refer to the wavelet power spectrum analysis. The results for the partial coherency between each pair of variables over F I G U R E 1 Price-return plot and wavelet power spectrum for soybean, biodiesel, highway diesel, and crude oil price returns. Notes: (a.1)-(a.4) convey the trend of each return. (b.1)-(b.4) are the wavelet power spectrum of each return. The black and gray contours represent the 5% and 10% significance levels, respectively. The strength of the wavelet power and the local volatility are shown by colors ranging from blue (lowest) to red (highest). The broken white line indicates the maximum local wavelet power spectrum for each variable. time for different frequency domains are presented in Figure 2a.1-f.1. The statistically significant coherence regions are represented by the red area with the black contour.
First, it is worth mentioning that the persistence of strong coherence regions at the 4-8-month frequency band can be identified in the pairs of biodiesel-soybean, biodiesel-highway diesel, and highway diesel-crude oil. This indicates a strong correlation exists among these three pairs of variables in the long-term scales. Specifically, regions with significantly high coherence can be observed running from 2014 to 2016 for biodieselsoybean in Figure 2a1. Moreover, according to the partial phase-difference in Figure 2a2, we can determine that the phase-differences are between 0 and ∕ 2, indicating that biodiesel positively leads soybean. In addition, the partial wavelet gains in Figure 2a3 show that the corresponding coefficient is close to 0.04 for the significant coherence. On the other hand, the partial coherence between biodiesel and highway diesel (Figure 2b1) is particularly strong between the beginning of 2012 and the middle of 2014. In this significant range, the phase difference in Figure 2b2 is between 0 and ∕ 2, implying a positive relationship between two variables, with biodiesel leading. Meanwhile, the partial wavelet gains in Figure 2b3 decrease from 0.1 to 0.09 in this period. Finally, the region of high partial coherence between highway diesel and crude oil in Figure 2f1 is observable across most of the sample. For the statistically significant region, the corresponding partial phase-differences in Figure 2f2 are between 0 and − ∕ 2, suggesting that crude oil positively leads highway diesel in the long term. Moreover, the partial wavelet gain (Figure 2f3) indicates that the coefficient of the period of coherence ranges from 0.09 to 0.1.
We turn our attention to the results for the high frequency (short-term), where some different patterns emerge. More specifically, according to Figure 2a1, the partial coherence between biodiesel and soybean becomes stronger after 2014, especially in 2015 and 2019-2020. Meanwhile, the corresponding partial phase-difference in Figure 2a2 shows that biodiesel positively relates to soybean (the phase-differences are between − ∕ 2 and ∕ 2 ), with soybean leading in 2015 and biodiesel leading in 2019-2020. Moreover, we find that many pairs of variables, such as biodiesel-highway diesel in Figure 2b1, biodieselcrude oil in Figure 2c1, and soybean-highway diesel in Figure 2d1, share a similar high-coherence region running from 2009 to 2014. The partial phase-difference exhibits the positive relationships between these three pairs of variables. Moreover, the results reveal that highway diesel leads biodiesel, biodiesel leads crude oil, and soybean leads highway diesel in significant regions. In addition, the outcome of soybean-crude oil in Figure 2e1 indicates that the region of high coherence extends from 2017 to 2020, and soybean positively leads crude oil. By contrast, the significantly high coherence between highway diesel and crude oil in Figure 2f1 can be identified throughout the sample period in the short term. Looking at the partial phase-difference, we can observe a positive correlation between these two variables, and crude oil typically leads highway diesel. Finally, it is interesting to note that the partial wavelet gains exhibit time variation and fluctuate dramatically across the sample period for all the variable pairs in the short term.
The wavelet analysis above provides notable evidence of significantly high coherence for biodiesel-soybean, biodiesel-highway diesel, and highway diesel-crude oil, as can be identified both in the short and long term. There are several key points to remark on from these empirical results. First, the renewable fuel biodiesel is strongly correlated with its major raw material soybean. The positive correlation between biodiesel and soybean suggests that the interactions uncovered between these two commodities are significant and point to common information shared by investors. Hence, from the food security point of view, policymakers need to be aware of the connections between food and fuel goods (i.e., soybean and biodiesel prices), particularly in the short run. Second, the positive correlation between biodiesel and highway diesel indicates that it is reasonable to presume these two commodities play a complementary role as fuel, with biodiesel blended into diesel most commonly at 5% but also at different concentrations in the US. Third, the significant connection between highway diesel and crude oil is also noteworthy, characterized by the oil market leading the highway market. This result is consistent with the findings of Ferrer et al. (2018), suggesting that crude oil prices are the primary driver of energy and that uncertainty in the oil market profoundly impacts the price of fuel.

| Dynamic connectedness analysis
The above wavelet analysis provides us with an overview of coherence and causality across the food and fuel markets but does not examine in detail the degree of connection or spillover effects among the returns of biodiesel, soybean, highway diesel, and crude oil. So, taking the DYCI approach, we further conducted a dynamic spillover analysis. Following Yilmaz (2012, 2014), we used a VAR model with a lag length of 1-4 months, a forecasting horizon from 1 to 16 months, and a rolling window from 1 to 36 months. The results show that the one-lag VAR model with a 10-step-ahead forecast horizon and 24-month rolling sample provides the best performance in terms of SBIC. For the final model specification, F I G U R E 2 Empirical results of partial wavelet analysis. Notes: Left: partial wavelet coherence. Center: partial phase-difference. Right: partial wavelet gain. The color code for coherence ranges from blue (low coherence) to red (high coherence). The black contour designates the 5% significance level.
a one-lag VAR model was applied to estimate the connectedness index in our analysis. First, Table 2 shows the static connectedness index across all variables. The off-diagonal elements capture shocks from (to) others, and diagonal elements present shocks of their own. We can see that the total connectedness index (TCI) in the system is 41.18% suggesting that the interdependence between food and fuel markets is significant. More importantly, the results of directional spillovers from all variables to one specific variable vary from 27.9% for soybean to 55.9% for highway diesel. This indicates that highway diesel is the most affected by shocks from others, while soybean is the least affected. Then, regarding the contribution to others, crude oil makes the highest total contribution (51.7%), and soybean is the lowest (23.1%). The results for the net spillover provide evidence that soybean and highway diesel are net recipients, while biodiesel and crude oil are net transmitters. Based on the connectedness index table, we also produced a net pairwise spillover matrix for all pairs. To visualize how the variables interact with one another, we plotted a graph of the network among them, as shown in Figure 3.
The directions of the arrows explain the net directional connectedness of variables and the bold line demonstrates a greater spillover than the other fine lines. It is clear from Figure 3 that crude oil is a net transmitter of shocks to the other three variables, while soybean is a net receiver of shocks from the other variables. Specifically, the results provide evidence that the degree of spillover from crude oil to highway diesel is larger than that "to others." Furthermore, our results indicate that biodiesel returns impact soybean and highway diesel. Moreover, we can determine that the magnitude of the return transmission from biodiesel to soybean is stronger than that of highway diesel. Overall, these findings suggest that biodiesel significantly affects soybean and/or highway diesel, while crude oil has a great influence on highway diesel.
Though the results of the static connectedness index provide an overall picture of the average spillovers among the variables, they do not illustrate the variations with time. So, to further investigate the net return spillover between each pair of variables, we conducted a time-varying connectedness that allowed us to move from the static perspective to a dynamic perspective. Figure 4 presents the net pairwise connections between all pairs in the period from May 2009 to December 2020. In general, as can be observed, these connections vary across different pairs of variables over time. First, when we look at the pairwise connectedness index of biodiesel-soybean, the net spillovers are nearly all positive, suggesting that shocks from the biodiesel market altered the soybean market for most of the period under study. It can then be noted that soybean became a net transmitter from mid-2015, lasting Note: The FEVD is based on a monthly VAR model for order 1. The return connectedness index is estimated by using and a 10-step-ahead forecast horizon and a 24-month rolling sample. "Net spillovers" are the difference between the "contributions to others" and the "from others." T A B L E 2 Total static connectedness index for biodiesel, soybean, highway diesel, and crude oil.

F I G U R E 3
The connectedness network for biodiesel, soybean, highway diesel, and crude oil. Notes: The arrow direction represents the net pairwise directional connectedness between variables. The weight of the lines explains the strength of pairwise directional connectedness from strongest (bold line) to weakest (fine line).
until the end of 2017. A possible explanation for this is that biodiesel price was significantly influenced by the shale oil boom from 2015 to 2017 due to the development of hydraulic fracking. Second, from the plots of dynamic net pairwise connectedness between biodiesel and highway diesel, biodiesel can be seen to have been a net transmitter, while highway diesel was a net recipient between 2009 and 2016. Subsequently, highway diesel became the net transmitter, while biodiesel was the net recipient, from 2017 to 2020. Third, in the case of highway diesel-crude oil, crude oil was clearly the net transmitter, and highway diesel was the net receiver during almost all examined periods. This result suggests crude oil played a leading role in transmitting shocks to highway diesel in our sample period. Finally, according to Figure 4, the net pairwise connections of biodiesel-crude oil, soybean-highway diesel, and biodiesel-crude oil shared certain features in common, such as the greater time-varying connectedness of these three pairs over the sample period when compared to the other three pairs. In pre-2012, the former variable in each pair was the primary transmitter and the latter the net receiver. However, the reverse transmission of shocks can be seen running from the latter to the former in almost all F I G U R E 4 Net pairwise returns spillovers among biodiesel, soybean, highway diesel, and crude oil. Notes: Dynamic net pairwise connectedness is estimated using the FEVD on 10-step-ahead forecasts and the 24-month rolling sample. Positive values indicate a net transmitter of return spillovers; negative values denote a net receiver. of the periods from 2013 to 2020. This interesting finding implies that a structural change or regime shift likely occurred in 2013, which might be partially attributed to the fluctuations in biodiesel prices that continued to decline until 2015 after peaking around 2011.

| Conclusions and policy implications
Our research utilized the partial wavelet coherence and DYCI approaches to analyze the connectedness of biodiesel and its related markets in the United States. Most of the results obtained from the experiments show consistency between the findings of the two methods. The primary outcomes are as follows. First, significantly high coherence for biodiesel-soybean, biodiesel-highway diesel, and highway diesel-crude oil was identified both in the short and long term. Second, crude oil and biodiesel prices were found to be net transmitters, while soybean and highway diesel prices were net receivers. Finally, the crude oil market was the causal source among the markets concerned, particularly affecting the highway diesel market.
There are several policy implications of the experimental results, as follows: 1. In our analysis, both the wavelet transform and DYCI method proved that the biodiesel price induces soybean price changes, which suggests that biodiesel production led by governmental policies reinforces a relationship between the prices of the two. Soybean is used not only as food but also as livestock feed. Since the United States is the second-largest soybean exporter to various nations such as China and Japan (Food and Agriculture Data, 2022), domestic meat prices in such importing countries could be affected by biofuel policies in the United States, as demonstrated by Guo and Tanaka (2020). The researchers proposed that the US farm-gate price of soybean significantly influences the international soybean price. Extended biodiesel production in the United States could thus exacerbate food price fluctuations in both domestic and global markets. 2. Another key finding was that the crude oil market is the source of spillovers among the four markets. More specifically, the direction of transmission from crude oil to biodiesel, highway diesel, or soybean was confirmed by the DYCI method. The crude oil price directly affects the soybean price, as well as indirectly altering it through the highway diesel and biodiesel prices. Accordingly, stabilizing crude oil prices leads to the steadiness of the other three markets. Recently, in July 2021, the Strategic Petroleum Reserve held by the US government reached 621 million barrels (US Energy Information Administration-EIA, 2021), equivalent to approximately 34 days of consumption in the country. The data from US EIA (2021) shows that the daily consumption of petroleum reached 18.19 million barrels. An increase in the petroleum emergency reserve would increase the country's buffer against shocks to the supply of highway diesel, biodiesel, and soybean brought about by price changes in the markets. 3. We discovered the connectivity between food and fuel markets. To alleviate the linkage, biodiesel should be produced from non-food materials such as jatropha curcas, as it quickly grows even in arid areas not suited for agriculture and can be harvested only 1 year after planting (Kazusa DNA Research Institute, 2022). On top of that, it can be cultured for over 50 years once it is planted. Therefore, cellulosic biofuel production should be encouraged by subsidizing its production and decreasing tax exemption for biodiesel production from edible materials.

| Future research
We will conclude by addressing future research topics. As indicated in the literature review, despite many economists having made price connections concerning the biofuel market, the underlying factors behind those connections have so far been left uninvestigated. For instance, it may be that the intensity of interconnection is affected by biofuel production levels. More specifically, it could be hypothesized that greater biofuel production leads to more intense relationships with producing factors such as corn, sugar, and soybean. Besides this, the costs associated with carbon dioxide emissions might influence the links at play. Looking ahead, research on such themes will support the policy-making process to facilitate the continuing provision of both food and fuel, and it will fill current knowledge gaps on the mechanisms of energy and food securities.

ACKNOWLEDGMENTS
This paper was written with financial support from the Japan Society for the Promotion of Science [grant number: 21 K05805]. The authors thank Ana-Maria Ignat for her proofreading, formatting, and miscellaneous research support.

CONFLICT OF INTEREST STATEMENT
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are openly available in Dryad at https://doi.org/10.5061/dryad.xd254 7dnc.

SUPPORTING INFORMATION
Additional supporting information can be found online in the Supporting Information section at the end of this article.