3.1. Wavelet analysis

Wavelet analysis is used to decompose a signal into its main components, enabling the possibility to focus on specific frequencies (Percival et al., Reference Percival, Wang and Overland2004). In contrast to the Fourier transform, wavelet transform combines information on both time and frequency domains, allowing us to track timewise particular frequencies (Mensi et al., Reference Mensi, Tiwari, Bouri, Roubaud and Al-Yahyaee2017). A wavelet transform is based on the mathematical operation of convolution, which specifies that the integral of the product of two functions, one of which is reversed and shifted, produces a third function that has similar features as the shifted and reversed function (Torrence and Compo, Reference Torrence and Compo1998). The wavelet transform is based on two specific functions: (1) the father wavelet, ϕ(t), and (2) the mother wavelet, Ψ(t). A series of wavelets called daughter wavelets, Ψ _u,s (t), can be built by simply scaling and translating (shifting) Ψ(t):

$${{\rm{\Psi }}_{u,s}}\left( t \right) = {1 \over {\sqrt s }}{\rm{\Psi }}({{t - u} \over s}),$$ (1)

where ${1 \over {\sqrt s }}$ is a normalization factor ensuring unit variance of the wavelet (i.e., ||Ψ _u,s (t)||² = 1), and u and s are the location and scaling parameters, respectively (Crowley, Reference Crowley2005). The scaling parameter controls for the length of the wavelet and is related to the frequency of the input signal such that a larger (lower) value implies the wavelet will correlate with the low (high) frequencies contained in the time series. The term u determines the location of the wavelet in the time domain. A number of wavelets have been developed to capture specific frequency characteristics of time series, and these include the Daubechies, Haar, Morlet, and Mexican hat. There are two types of wavelet transforms that are widely used in the literature: (1) the discrete wavelet transform (DWT) and (2) the continuous wavelet transform (CWT) (Crowley, Reference Crowley2005). The DWT is suitable for data compression and noise reduction, whereas the CWT is useful for smooth extraction of frequencies. Mother wavelets have to satisfy two main conditions: (1) zero mean (i.e., ${\rm{}}\mathop \int \noilimits_{ - \infty }^\infty {\rm{\Psi }}\left( t \right)dt = 0$ ) and (2) unit energy (localized in time or space; i.e., $\mathop \int \nolimits_{ - \infty }^\infty {{\rm{\Psi }}^2}\left( t \right)dt = 1$ ). In addition, wavelets have to satisfy the admissibility condition, which guarantees a reconstruction of the original time series from its wavelet transform using the inverse transform (Shalini and Prasanna, Reference Shalini and Prasanna2016). For this article, we use the DWT, given the flexibility it offers in denoising time series (Crowley, Reference Crowley2005), but also because it produces a minimum number of coefficients necessary to reconstruct a series. Given its parsimonious nature, the DWT has wide applications (Moya-Martínez et al., Reference Moya-Martínez, Ferrer-Lapeæa and Escribano-Sotos2015). To minimize the boundary effects at the extremities of time series when applying the DWT, we use the periodic decomposition often done in similar studies. Based on the DWT, any time series can be described as a linear combination of father and mother wavelets (Mensi et al., Reference Mensi, Tiwari, Bouri, Roubaud and Al-Yahyaee2017):

$${{X}}\left( t \right) = \mathop \sum \limits {_ks_{j,k}}{{\rm{\Phi }}_{j,k}}\left( t \right) + \mathop \sum \limits {_kd_{j,k}}{{\rm{\Psi }}_{j,k}}(t) + \ldots + \mathop \sum \limits {_kd_{1,k}}{{\rm{\Psi }}_{1,k}}(t),$$ (2)

where j represents the multiresolution, or scale level, and k depicts the number of coefficients at each scale level. Further, s_j,k and d_j,k are the scaling (or smooth) and detail (or wavelet) coefficients, respectively, and can be expressed as

$${s_{j,k}} = \mathop \vint \nolimits X(t){{\rm{\Phi }}_{j,k}}\left( t \right)dt,\,\rm and$$ (3)

$${d_{j,k}} = \mathop \vint \nolimits X(t){{\rm{\Psi }}_{j,k}}(t)dt,{\rm{ for \it{\ j} }} = {\rm{ 1}},{\rm{ 2}},{\rm{ }} \ldots ,{\it{ j}}.$$ (4)

The detail coefficient d_j,k captures the high frequencies contained in the input signal, or time series, and the scale coefficient s_j,k captures the smooth part, or the long-term trend, of the input function (Moya-Martínez et al., Reference Moya-Martínez, Ferrer-Lapeæa and Escribano-Sotos2015). The original input function X(t) can be reconstructed as a linear combination of the calculated coefficients (Mensi et al., Reference Mensi, Tiwari, Bouri, Roubaud and Al-Yahyaee2017):

$${{X}}\left( t \right) = {S_j}(t) + {D_j}(t) + {D_{j - 1}}(t) + \ldots + {D_1}(t),$$ (5)

with the smooth, or approximation, components of the time series represented by S _j = ∑ _k S_j,k ϕ _j,k (t) and the details components of the series specified as D_j = ∑ _k d_j,k Ψ _j,k (t). In practice, a wavelet with some desired properties is chosen and convoluted with a time series to extract the various frequencies that are contained in the series. It is then possible to rebuild the series by excluding, for example, certain frequencies. The reconstruction of a time series using the DWT approach most often relies on Mallat’s pyramid algorithm (Mallat, Reference Mallat1989), which consists of applying a series of low-pass and high-pass filters. Explicitly, a time series, X(t), is convolved with high-pass and low-pass filters to extract the detail, D ₁(t), and approximation, S ₁(t), components of the series. Then S ₁(t) becomes the input for the subsequent iteration phase to derive D ₂(t) and S ₂(t). This iterative process is repeated until the desired decomposition level j is achieved (Crowley, Reference Crowley2005).

Denoising a series is one of the most common applications of wavelet analysis and involves selecting a threshold value λ, which is then used to filter the derived wavelet coefficients. There are two types of thresholding: (1) hard thresholding, where wavelet coefficients with the absolute value less than the threshold are set to zero, and (2) soft thresholding, where the absolute values of the wavelet coefficients above λ are shrunk (Haven, Liu, and Shen, Reference Haven, Liu and Shen2012). We define λ according to Donoho (Reference Donoho1995) such that $\itlambda = \sqrt {2{\sigma ^2}} {\rm{log}}(N)$ , where N represents the length of the signal, and σ stands for the variance of the noise, which is estimated by computing the variance of the wavelet coefficients derived from the first decomposition level.Footnote ⁷ Therefore, the threshold level increases with the volatility of the time series. The denoised time series is then constructed by substituting the detailed wavelet coefficients derived through the DWT with the “thresholded” wavelet coefficients.

In this article, we use the Daubechies “extremal phase wavelets” (Daubechies, Reference Daubechies1992), as previous studies have shown that the Daubechies extremal phase wavelets are appropriate for financial data, and implement the DWT to denoise the cash and staple food index series. Typically, denoising the price indices implies removing those wavelet coefficients that do not contribute significantly to the signal. In terms of our index series, noise may represent short-term speculative behavior, scalping, herd behavior, outliers, or irrational price movements (Gardebroek and Hernandez, Reference Gardebroek and Hernandez2013). Daily observations such as futures prices typically contain a lot of noise, which does not necessarily contribute to the underlying movement in prices.

3.2. GARCH approach

As highlighted in the literature review, the MGARCH model is widely applied in the analysis of integration between markets. In this article, we study the volatility spillover between cash crop futures price index and staple food futures price index at an international level. The basis for constructing the indices is discussed later in Section 3.3. Our approach assumes that the variance-covariance matrix follows a BEKK-GARCH specification. The bivariate BEKK-GARCH model is expressed as

$$A(L){r_t} = {\varepsilon _t},$$ (6)

and

${\varepsilon _t}|{\omega _{t - 1}} \sim N(0,{H_t})$

where A(L) is a polynomial matrix in the lag operator L, r_t is a 2 × 1 daily return vector at time t, and ε_t is a 2 × 1vector of random errors representing the shocks, or innovations, at time t. H_t is a 2 × 2 conditional variance-covariance matrix, given market information ω _t–1 available at time t − 1. Equation (6) represents the mean conditional equation and describes the impact of own and lagged shocks as well as lagged innovations in other markets on the conditional mean of a variable at time t. The order of the system can be selected on the basis of a standard information criterion (e.g., Akaike information criterion [AIC], Schwarz information criterion [SIC]).

With respect to the form that H_t can take, it generally depends on the number of variables and the objective of the research. Often, when the number of variables is large, a less flexible MGARCH specification is chosen. This is because model convergence during the estimation process is difficult to achieve if the number of variables is larger than three and, in particular, when exogenous variables are included (El Hedi Arouri et al., Reference El Hedi Arouri, Lahiani and Nguyen2015). Convergence issues with higher model dimension can be limited by restrictive specifications such as the diagonal BEKK-GARCH and the scalar BEKK-GARCH models. These parsimonious specifications reduce the computational complexity and facilitate model solution. The list of more flexible GARCH specifications is fairly exhaustive, and we only mention here the most commonly used models, which comprise the full BEKK-GARCH model, introduced by Engle and Kroner (Reference Engle and Kroner1995); the constant conditional correlation–GARCH model, specified by Bollerslev (Reference Bollerslev1990); the DCC-GARCH model of Engle (Reference Engle2002); and the vector autoregressive (VAR)–GARCH introduced by Ling and McAleer (Reference Ling and McAleer2003).

Based on the model proposed by Engle and Kroner (Reference Engle and Kroner1995), the conditional variance-covariance matrix of the BEKK-GARCH specification can be expressed as

$${H_t} = C_0^{\rm{'}}{C_0} + A_{11}^{\rm{'}}{\varepsilon _{t - 1}}\varepsilon _{t - 1}^{\rm{'}}{A_{11}} + G_{11}^{\rm{'}}{H_{t - 1}}{G_{11}},$$ (7)

or in matrix form as

$${H_t} = C_0^{\rm{'}}{C_0} + {\left[ {\matrix{ {{a_{11}}} \ {{a_{12}}} \cr {{a_{21}}} \ {{a_{22}}} \cr } } \right]^{\rm{'}}}\left[ {\matrix{ {\varepsilon _{1,t - 1\ }^2} \hskip 25pt{{\varepsilon _{1,t - 1,}}{\varepsilon _{2,t - 1}}} \cr \hskip -11pt{{\varepsilon _{1,t - 1,}} {\varepsilon _{2,\ t - 1}}} \hskip 14pt{\varepsilon _{2,t - 1}^2} \cr } } \right]\left[ {\matrix{ {{a_{11}}} \ {{a_{12}}} \cr {{a_{21}}} \ {{a_{22}}} \cr } } \right]{\rm{}} + {\rm{}}{\left[ {\matrix{ {{g_{11}}} \ {{g_{12}}} \cr {{g_{21}}} \ {{g_{22}}} \cr } } \right]^{\rm{'}}}{H_{t - 1}}\left[ {\matrix{ {{g_{11}}} \ {{g_{12}}} \cr {{g_{21}}} \ {{g_{22}}} \cr } } \right],$$ (8)

where $C_0^{\rm{'}}{C_0}$ represents the decomposition of the intercept matrix, with C ₀ restricted to be a lower triangular matrix. The unrestricted n × n matrices A and G contain the own autoregressive conditional heteroskedasticity (ARCH) and cross-market ARCH effects and the own GARCH and cross-market GARCH effects, respectively. With this specification, it is possible to trace the effect of innovations and volatility in one market and how they transmit to other markets. These estimates are contained in matrices A and G. Expanding equation (8) yields the variance-covariance equations:

$${h_{11,t}} = c_{11}^2 + a_{11}^2\varepsilon _{1,t - 1}^2 + 2{a_{11}}{a_{21}}{\varepsilon _{1,t - 1}}{\varepsilon _{2,t - 1}} + a_{21}^2\varepsilon _{2,t - 1}^2 + g_{11}^2{h_{11,t - 1}} + 2{g_{11}}{g_{21}}{h_{12,t - 1}} + {\rm{}}g_{21}^2{h_{22,t - 1}},$$ (9)

$${h_{12,t}} = {c_{11}}{c_{21}} + {a_{11}}{a_{12}}\varepsilon _{1,t - 1}^2 + \left( {{a_{21}}{a_{12}} + {a_{11}}{a_{22}}} \right){\varepsilon _{1,t - 1}}{\varepsilon _{2,t - 1}} + {a_{21}}{a_{22}}\varepsilon _{2,t - 1}^2 + {\rm{}}{g_{11}}{g_{12}}{h_{11,t - 1}} + \left( {{g_{21}}{g_{12}} + {g_{11}}{g_{22}}} \right){h_{12,t - 1}} + {g_{21}}{g_{22}}{h_{22,t - 1}},$$ (10)

$${h_{22,t}} = c_{21}^2 + c_{22}^2 + a_{12}^2\varepsilon _{1,t - 1}^2 + 2{a_{12}}{a_{22}}{\varepsilon _{1,t - 1}}{\varepsilon _{2,t - 1}} + a_{22}^2\varepsilon _{2,t - 1}^2 + g_{12}^2{h_{11,t - 1}} + {\rm{}}2{g_{12}}{g_{22}}{h_{12,t - 1}} + g_{22}^2{h_{22,t - 1}}.$$ (11)

With the assumption that error terms follow a multivariate standard normal distribution, the BEKK-GARCH models are estimated by maximizing the log-likelihood function using the Berndt, Hall, Hall, and Hausman (BHHH) algorithm. The conditional log-likelihood function L for a sample of T observations is

$$L(\theta )\, = \,\sum\nolimits_{t\, = \,1}^T {{l_t}} (\theta ),\,{\rm{with}}$$ (12)

$$l\left( \theta \right) = - {\rm{log}}2\pi - {1 \over 2}{\rm{log}}\left| {{H_t}\left( \theta \right)} \right| - {1 \over 2}\varepsilon _t^{\rm{'}}(\theta )H_t^{ - 1}(\theta ){\varepsilon _t}(\theta ),$$

where θ represents the vector of all the parameters to be estimated.

3.3. Data

Two price indices are produced to capture movements in cash crop and staple futures prices. The cash crop futures index is constructed by taking a weighted average of the daily closing futures prices realized at the ICE for sugar no. 11 (raw sugar; SB) futures, cocoa (CC) futures, coffee “C” (KC) futures, and cotton no. 2 (CT) futures. We first normalize the prices and use the daily traded volumes (number of contracts traded) as weights to derive the daily futures price index. We follow a similar procedure for the staple food futures prices, where we use the daily closing futures prices realized at the CBOT for corn (C1) futures, soybeans (SB1) futures, and wheat (W1) futures and use the respective traded volumes as weights. For both indices, daily futures prices and volume data are sourced from Bloomberg and cover the period of January 3, 1990, to August 30, 2016. All futures prices are historical first generic price series, and expiring active futures contracts are rolled to the next deferred contract after the last trading day of the front month.Footnote ⁸

The choice of the commodities included in the two indices is based on a preanalysis that involves identifying the top exported cash crops and the top imported staple foods by the LIFDCs group.Footnote ⁹ We then select those crops for which an international futures contract exists. On this basis, coffee, cocoa, cotton, and sugar futures prices are selected to represent the group of cash crops, and wheat, corn, and soybeans futures prices are selected to characterize the staple food group. As with similar studies, the analysis is undertaken using the returns of the index series by taking the differences in the logarithm of two consecutive price indices. The choice of transforming the index series is determined by the fact that the cash crop and staple food indices are integrated at different orders. Both the augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) tests fail to reject the presence of unit root for the staple food index, whereas both tests reject the null hypothesis of nonstationarity for the cash crop index. Given that the indices are integrated at different orders—that is, staples being I(1) and cash crops I(0)—a vector error correction model (VECM) framework is not suitable in our case. Hence, a VAR is specified in returns in order to get consistent estimates. Figure 1 shows the daily movements of both cash and staple food price indices. The graph highlights the extent of the volatility that underpins both markets.Footnote ¹⁰

Figure 1. Daily movements of staple foods and cash crop price indices (2010 = 1).

4.1. Descriptive statistics

The descriptive statistics of the price index return series are reported in Table 1. The statistics show that the food price index has the largest daily return and the lowest standard deviation, in comparison with the cash crop index. Overall, the series are asymmetric, with a small positive skewness, and have large kurtosis coefficients. The Jarque-Bera test statistics rejects the null hypothesis of normality for both price return indices. The ARCH test for heteroskedasticity points to the presence of the ARCH effect in both index series. Also, the Ljung-Box test for autocorrelation evidences the presence of autocorrelation. These results corroborate the use of an MGARCH model in assessing the volatility integration between cash crop and staple food markets. They are also in-line with the underlying characteristics of commodity price movements, notably volatility clustering, as described by Deaton and Laroque (Reference Deaton and Laroque1992). With respect to the stationarity of the series, ADF and the PP tests reject the null hypothesis of nonstationarity at the 1% level of significance. The unconditional correlation between cash and staple food price index using the Pearson coefficient is estimated to be 0.74 at the 5% level of significance.

Table 1. Descriptive statistics of the price index returns

Notes: Q(14) refers to the Ljung-Box test for autocorrelation of order 14. ARCH(14) is the Engle (Reference Engle1982) test for conditional heteroskedasticity of order 14, and the Jarque-Bera test is used to test for normality. The augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) methods are used to test for nonstationarity of the index series. ARCH, autoregressive conditional heteroskedasticity.

4.2. Results of the wavelet analysis

As discussed in Section 3, we use the Daubechies “extremal phase wavelets” to decompose the series into approximated and detailed series. The Daubechies family of wavelets is used in finance and economics because of their desirable properties of ortho-normality, asymmetry, and higher number of vanishing moments (Daubechies, Reference Daubechies1992). Figure 2 illustrates the decomposition exercise based on a multiresolution analysis (MRA) at various scales for both food and cash crop price indices. A maximum scale level j of 12 is selected, which is standard in the literature, because previous studies have shown that moderate filters are appropriate for financial data (Gençay, Selçuk, and Whitcher, Reference Gençay, Selçuk and Whitcher2001, Reference Gençay, Selçcuk and Whitcher2005; In and Kim, Reference In and Kim2013). Note that, according to Nyquist’s rule, half of the sample can be eliminated at each successive scale level. For illustrative purposes, we present three detailed series and one approximation series derived from the calculated values of the wavelet transform coefficients. A wavelet coefficient can be interpreted as the difference between two adjacent averages for a certain scale (Percival et al., Reference Percival, Wang and Overland2004). Practically, it shows how the average of a particular series changes when considering various scales (e.g., 2 days, 20 days, or 360 days). Analyzing the change in the average of price series at different scales helps detect any possible trends, discontinuities, or abrupt changes in the series. In Figure 2, the highest scale level (frequency) component d1 corresponds to a time-scale (frequency) of 2¹ = 2 days (daily effects), while d5 accounts for variations in a time-scale (frequency) of 2⁵ = 32 days. The coarser, or smoother, part of the series (S7) captures the trend.

Figure 2. Wavelet decomposition results at selected scales for cash crop and staple food series. Note: “foodi” stands for food price index, and “cashi” represents the cash crop price index.

The MRA in Figure 2 suggests that the variations in the series are relatively heterogeneous across scales and time, with high fluctuations evidenced at finer-scale resolutions for both cash and staple food series. Some localized features are interesting to point out. For example, a stretch of high volatility toward the end of the sample is revealed for the staple food index, whereas there are marked variances at the start and toward the end of the sample for the cash crop price index, underpinned by large fluctuations in the value of the wavelet coefficients. The smooth series for the cash and staple food series (S7 in Figure 2) highlight the upward trend underlining both series up to their respective peak. Hence, the MRA suggests that the fluctuations in the decomposed series are heterogeneous across time and scales, implying that we can gain additional insights into the dynamics of the indices by considering their relationship at various scale levels. After obtaining the wavelet transform values, we proceed by denoising the series, as described in Section 3. Then, the series are reconstructed by adding to the trend, selected frequencies, or detail series, as described in equation (5). The selected scales (frequencies) are (1) low frequency (d = 9), (2) medium frequency (d = 5), and (3) high frequency (d = 1),Footnote ¹¹ as illustrated in Figures 3 and 4.

Figure 3. Reconstructed staple food price series at selected scales. Note: “foodi” stands for food price index.

Figure 4. Reconstructed cash crop price series at selected scales. Note: “cashi” represents the cash crop price index.

It is also possible to quantify how much each scale contributes to the overall variability of the index series through scale-based variance decomposition, as in Percival et al. (Reference Percival, Wang and Overland2004). The wavelet variance decomposition indicates that the largest contribution to the sample variance is accounted for by variations at the largest scale (long-run fluctuations) for both the cash crop and staple food indices. Hence, long-run variations have more weights on the series than short-run fluctuations.

In general, the results of the wavelet analysis show that the index series have common trends and volatility patterns, although differences in volatility prevail at specific scales. This suggests that the level of interdependence and the dynamics of volatility between cash crop and staple food price indices may vary depending on the considered timescale. As a result, we apply a BEKK-GARCH model using the denoised series at various time-frequency domains to account for the heterogeneity in the variance dynamics. This is described in the next section.

4.3. GARCH model

Using the staple food and the cash crop return indices, we estimate four bivariate VAR-BEKK-GARCH models. The first model uses the original series, whereas the other three models are applied to the denoised series but for different scale frequencies: low frequency (model 2), medium frequency (model 3), and high frequency (model 4). The VAR specification describes the conditional mean of the model, and the GARCH component explores the volatility interactions. We apply the AIC and SIC information criteria to identify the optimal lag order of the VAR system and run univariate GARCH for both index series to which we apply the same information criteria to examine the lag order for the GARCH component. The information criteria selects VAR(3) and GARCH(1,1) as the optimal specification. A comparative estimation of the log-likelihood values derived from other alternative lag specifications confirms the data are best characterized by a GARCH(1,1) specification.

The estimation results are reported in Table 2. The ARCH terms (a_1i, a_2i) indicate whether the conditional volatility is driven by lagged innovations, and the GARCH estimates (g_1i, g_2i) show if the current conditional volatility is influenced by its lagged values, reflecting volatility persistence. In general, estimation results for the four pairwise bivariate VAR(3)-BEKK-GARCH(1,1) models reveal some similar patterns with respect to the estimated ARCH and GARCH coefficients. First, the coefficients are found to be statistically significant for most of the pairwise estimations. Second, the estimated values for the ARCH coefficients are generally lower than the GARCH estimates, implying that lagged shocks do not affect current conditional variance as much as lagged volatility values. The diagnostic tests carried out on the standardized residuals and squared standardized residuals show a significant reduction in ARCH effects and autocorrelation depicted in the return series (see Table 1), indicating that the estimated models are sufficiently flexible to describe the volatility dynamics between staples and crop returns.

Table 2. Estimates of VAR(3)-BEKK-GARCH(1,1) for staple food and cash crop price indices at various time-frequency domains

Notes: A bivariate model VAR(3)–full-BEKK-GARCH(1,1) model is estimated for each model from January 2, 1990, to August 28, 2016. The information criteria AIC (Akaike information criterion) and SIC (Schwarz information criterion) were used to select the optimal lag order for the VAR model and the GARCH specification. Model 1: original series; model 2: low frequency; model 3: medium frequency; and model 4: high frequency. LB and LB² are the Ljung-Box Q-statistic for standardized and standardized square residuals, respectively. P values reported in parentheses. Asterisk (*) stands for significant at the standard 5% level. Stationarity condition tests show that the estimated full BEKK-GARCH model is stationary. The estimates of matrix A (ARCH effects) and G (GARCH effects) shown in Table 2 are reported as expressed in equation (8). Note that we only show results for conditional variances. Estimated results for the conditional correlations are presented in Figure 5. ARCH, autoregressive conditional heteroskedasticity; BEKK, Baba, Engle, Kraft and Kroner; GARCH, generalized autoregressive conditional heteroskedasticity; VAR, vector autoregressive.

Table 2 also reports estimations for the mean price return equations. Results indicate that, generally, the own autoregressive parameters for both staple food and cash crop return indices are found to be statistically significant, implying short-term predictability. Results also show that some cross-market returns parameters are found to be positive and statistically significant, but their number is much less than in the case of own mean spillover estimates. Also, we note that the information transmission flows mostly from the staple food to the cash markets, as shown by the number of significant coefficients capturing the effect of changes in staple food crop returns on cash crop returns. This result may in fact reflect the relatively greater liquidity in the staple food futures markets relative to cash crop futures markets.

The diagonal elements of matrix A (see equation 7), which captures own shocks, and the diagonal elements of matrix G, associated with own GARCH effect, are significant for most of the estimated models. That is, own news and past volatility movements affect the current conditional variance values. Also, a general assessment shows that the off-diagonal elements of matrix A and matrix G are for most cases significant, but with some degree of variations, reflecting asymmetries in the dynamics. In terms of model 1 (i.e., original series), results are generally in-line with those obtained with the other models. For the staple food and cash crop equations, own ARCH and own GARCH terms are highly significant. In absolute terms, estimates of the ARCH coefficients are generally found to be much smaller than those obtained for the GARCH component, implying larger effects of past conditional variances than lagged innovations on current conditional variances.

For the low frequency model (i.e., long run), results indicate that the current conditional variance for cash crop return indices depends on their own ARCH and own GARCH terms, meaning that market volatility of cash crops can generally be predicted on the basis of past shocks and past variance. However, in contrast to model 1, the own ARCH effect is found to be larger than the own GARCH effect, suggesting that unexpected shocks play a much more important role in driving variability of staple returns at low frequencies. Likewise, the own ARCH estimate for staples and cash crop equations are found to be greater than the own GARCH effects for the medium-frequency model. In the case of the high-frequency model, the own GARCH effect is larger than the own ARCH effect for both the staple food and cash crop equations, in-line with the outcome obtained with model 1. That is, at high frequencies, the conditional variances of cash crop and staple food returns are influenced by their respective past variances more so than unexpected news.

We now turn our attention to volatility transmission between staple food and cash crops, which is captured by the cross-estimates of ARCH and GARCH terms. Overall, there is significant volatility transmission between staple foods and cash crops as evidenced by the number of significant cross-effects terms estimated for the various pairwise systems. We note that the cross-market GARCH estimates are generally much larger than those of the cross-market ARCH effects. This is an indication that the conditional volatility of cash crop (staple food) markets is largely influenced by periods of volatility in the staple food (cash crop) markets rather than by the effects of lagged price return innovations in the staple food (cash crop) markets. Specifically, the GARCH cross-market effects are all statistically significant, with the exception of model 1, where past volatility in the cash market is statistically insignificant in the staple food market, and the medium-frequency model, where the past volatility in the food market is statistically insignificant in the cash market. On the other hand, the cross-market ARCH effects are all statistically significant, with the exception of model 1, where past innovations in the staples market do not show a statistically significant influence on the volatility of cash crop returns. Overall, the results show that the absolute values of the estimated cross-market GARCH and ARCH estimates are generally higher and statistically significant in the low-frequency case than for the other frequency models, suggesting that the level of volatility interdependence between cash crop and staple returns is much stronger at lower frequencies. Further, the low-frequency model yields the largest Pearson correlation estimates, reflecting a tighter interdependence in the long run. The fact that the conditional correlations are larger at lower frequencies may suggest that external factors common to both markets, such as macroeconomic variables and world energy prices, explain the larger correlation in the long run. In the short run, commodity-specific factors (e.g., supply shocks affecting sugar crops) dominate movements in prices, a feature that underlines the lower conditional correlation between staples and cash crops. These results are also corroborated by the estimated conditional correlations, which indicate that the correlation at lower frequency is mostly positive and increasing in periods of high commodity prices (see Figure 5). Figure 5 also shows that as the frequency increases from low to high, the conditional correlation between staple and cash crop markets weakens. As mentioned, weaker volatility integration may be attributed to the influence of commodity-specific factors rather than common factors across staples and cash crops.

Figure 5. Estimated conditional correlation between the cash crop price index and the staple food price index at various time-frequency domains.

Estimation results also show that the cross-market values associated with the staple foods are generally larger than those relevant to cash crops. This means that information coming from the food markets influences cash crop markets to a larger extent than in the opposite direction, which could reflect the effect of greater liquidity underlying the staple food futures. The implication for LIFDCs is that market information relevant to staple foods affects ultimately the variability of cash crop earnings. Despite the bidirectional nature of the relationship, both the own GARCH and own ARCH effects are found mostly larger in magnitude than the cross effects, highlighting the dominant role of intrinsic market factors.

Figure 5 shows the estimated conditional correlations between staple food and cash crop return series at various timescales calculated following equation (10). The estimated values exhibit high volatility throughout the sample period, with values ranging between −0.5 and 0.5, notably for the medium- and high-frequency scales. In the case of the low-frequency model, conditional correlations fluctuate between 0.5 and 1, with occasional and abrupt changes mostly toward the negative values and periods of upward or downward trends.

Relatively high conditional correlation values associated with low-frequency scale implies that cash crop sales are a good hedge against increases in staple food import bills and can contribute to limiting current account instability in the long run, more so than in the short term. The extent to which export earnings offset current account deficits because of import bills depends on the elasticity of cash crop markets. The smaller the elasticity, the larger the increase in export earnings resulting from higher prices. What do these results mean for a country like Burundi, which relies on cash crop exports and imports of staple foods? Strong and positive conditional correlation between cash crop and staple food markets means that the government can evaluate more accurately its financial needs in the face of current account imbalances because of import bills by taking into consideration the fact that revenues from cash crop exports can reduce funding requirements and, hence, borrowing costs. Second, the government can also use price information relevant to international staple foods in the design and planning of investment strategies for the cash crop subsector, given the linkages between both commodity subsectors. For example, information on staple food price prospects can be utilized to strengthen the robustness of national cash crop price projections.