In this section, we present our methodology, following a three-step approach to evaluate the efficacy of energy-based indices in forecasting agricultural commodity prices.

In the first step, we create a composite index derived from a set of energy-related variables. These variables represent different energy products such as Barley (BAP), Sunflower (SUN), Durum Wheat (DWW), and also Maize (MPP) and Fossil Energy Consumption (FSE), or any other relevant indicator that are deemed to have an influence on agricultural commodity prices. Let $x_t$ be the index by normalizing these variables and taking their average growth rate over time. Specifically, the index is calculated as the average of the normalized values of selected energy products. We denote the energy variables as $E_1,E_2,\cdots ,E_n$, where n is the number of the products used to construct the index.

$\text{index}=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left(\frac{E_{i,t}}{E_{i,t-1}}\right)$ (1)

where $E_{i,t}$ denotes the value of the i-th energy variable at time t, and $E_{i,t-1}$ is the value of the i-th energy variable at the previous time period t − 1. The index is computed as the average of the normalized values of the energy variables. In what follows, we calculate the growth rate of our index using log-difference transformation, to achieve stationarity on our data $x_t=\ln \left(inde{x}_t\right)-\ln \left(inde{x}_{t-1}\right)$.

Let y be our dependent variable, where in our case is chosen among Barley (BAP), Sunflower (SUN), Durum Wheat (DWW), Maize (MPP) and also Fossil Energy Consumption (FSE). Respectively, we calculate the growth rate of the dependent variable as: $ldiff\left(y_t\right)=\ln \left(y_t\right)-\ln \left(y_{t-1}\right)$.

In the second step we compute the benchmark models for our dependent variable (which is also expressed as a growth rate) and these models are: a first order autoregressive model, a first order delay model (using only the lag of the index) and a first order autoregressive delay model (using the first lag of the dependent variable and of the index together);

${\hat{y}}_{t+1|t}^{ar}={\hat{\rho }}_0+{\hat{\rho }}_1{y}_t$ (2)

for the autoregressive model

${\hat{y}}_{t+1|t}^{delay}={\hat{\beta }}_0+{\hat{\beta }}_1{x}_t$ (3)

for the delay model

${\hat{y}}_{t+1|t}^{ar-delay}={\hat{\rho }}_0+{\hat{\rho }}_1{y}_t+{\hat{\beta }}_1{x}_t$ (4)

for the autoregressive-delay model

${\hat{y}}_{t+1|t}^{adl}={\hat{y}}_{t+1|t}^j+\gamma \cdot {\widehat{ϵ}}_{t|t-1}^{adl}$ (5)

for the adaptive learning method

In the third step we estimate these models by recursive least squares and quantile regression (at the median) and forecast one-year ahead (we start with 4 or 6 years for least squares and quantile regression respectively and increase our sample), then we record the results and find the best performing model forecast which we pass on to the adaptive learning method.

Our data stem from the official Eurostat database, specifically from the Agriculture, Forestry, Fisheries, Energy & Environment sections, covering the period from 1990 to 2023. The dataset pertains exclusively to Greece, ensuring a geographically focused analysis. Our study also employs annual time series data, with all agricultural commodity values reported in euros, consistent with Eurostat’s default currency framework. A crucial characteristic of our data is the dynamic interplay between energy and agricultural commodity markets. This relationship brings significant implications for market participants and stakeholders ( Cabrera & Schulz, 2016 ), demonstrating that understanding volatility linkages between these sectors is essential for decision-makers to develop effective strategies in volatile market conditions without sustainable and economic consistency ( Han et al., 2015 ). In particular, several important time series of agricultural commodities are available only at low frequency, such as annual data, which presents both analytical challenges and opportunities for further analysis. This incident reduces the available information for building accurate forecasting. To address this challenge while also maintaining analytical consistency, we employ the idea of adaptive learning to such short time series in the context of annual growth rates for Greek agricultural commodities.

The analysis of the existing study relies on a wide range of variables, starting with the dependent variables, including the following: BAP, SUN, DWW and MPP prices, all measured in euros per 100 kg (Pp100 kg) and also FSE, measured in thousand tonnes of oil equivalent (TTOE). BAP is the fifth most widely cultivated cereal crop globally, known for its resilience under diverse environmental conditions and its economic viability due to lower input requirements and simpler agronomic management ( Verstegen et al., 2014 ; Farooq, 2015 ). However, its cultivation area is going to get reduced in the upcoming years, making accurate forecasting essential, particularly given its strategic agricultural role ( Parviz, 2018 ; Sharafi & Nahvinia, 2024 ). MPP is the most widely produced crop globally and serves as a vital energy source, particularly in developing countries, where it plays a key role in food and agriculture ecosystems ( Zelingher & Makowski, 2022 ). Despite its global importance, maize prices are highly sensitive to production shocks in some regions, underscoring the need for reliable forecasting, especially as some countries depend heavily on this agricultural commodity and its prices ( Zelingher et al., 2020 ). On the other hand, SUN is a key variable in this analysis due to its role as a substitute commodity influenced by biofuel markets ( Paris, 2018 ). Also, DWW is considered as a crop of strategic importance in Greece, contributing to 1.3 million metric tons of production annually, reflecting its central role in the country’s agricultural output and Mediterranean agricultural markets and systems ( Toscano et al., 2014 ). Last but not least, FSE is critical for rational energy consumption strategic planning in the electricity sector, as power generation remains heavily reliant on fossil fuels and future consumption is expected to increase steadily ( Sun, He, & Chang, 2015 ).

To improve forecasting accuracy, we create energy related indices, including fuel oil (FOL), hydro power (HYR), natural gas (NGS), renewables and biofuels (RNW), wind power (WND), lignite (LGN), other oleaginous products (OLE), oil and petroleum products (OPP), pumped hydro power (PHY), refinery gas (RFG), solid fossil fuels (SFF), and total energy consumption (TTL), which all serve as leading indicators for agricultural price fluctuations ( Groen & Pesenti, 2011 ). These indices are analyzed through two estimation methods the QR and OLS methods, using the specific γ values² of 0.35, 0.40, and 0.50. Furthermore, in order to determine and select the most accurate forecasting model, we examine the performance of all indices across our dataset. To select the most accurate forecasting model, we created a ratio index which is calculated by dividing the adaptive learning value with each of the performance metric values. Our forecasting framework employs standard performance measures (MAE, MAPE, RMSE) in line with contemporary advances in adaptive learning applications for volatile markets, allowing for systemic evaluation of model performance across various specifications. We define the metrics as follows:

$\begin{array}{l}\text{MSE}\left(m,n_1\right)\stackrel{\text{def}}{=}{n}_1^{-1}\underset{t=n_0+1}{\overset{n}{{\displaystyle \sum }}}{\widehat{ϵ}}_t^{m,2}\\ \text{MAE}\left(m,n_1\right)\stackrel{\text{def}}{=}{n}_1^{-1}\underset{t=n_0+1}{\overset{n}{{\displaystyle \sum }}}\left|{\widehat{ϵ}}_t^m\right|\\ \text{MAPE}\left(m,n_1\right)\stackrel{\text{def}}{=}{n}_1^{-1}\underset{t=n_0+1}{\overset{n}{{\displaystyle \sum }}}\left|\frac{{\widehat{ϵ}}_t^m}{y_t}\right|\times 100\end{array}$ (6)