Long-memory dynamics and weather-adjusted forecasting of greengram prices: A comparative analysis of ARFIMA, ARIMAX and ARFIMAX models

Introduction Background and Context

Agricultural commodity prices represent one of the most critical factors influencing farm income, food security, and market stability in developing economies like India. Among agricultural commodities, pulse crops occupy a pivotal position in Indian agriculture due to their nutritional richness, soil conservation properties, and export value. Greengram (Vigna radiata L.), popularly known as mung bean, is one of India's most important pulse crops, contributing significantly to domestic protein requirements and international trade (Hosking, 1981) [4] . Globally, greengram production has shown consistent growth, with total production reaching approximately 8 million metric tonnes annually, with India commanding a leading position. At the national level, India is the largest producer, consumer, and exporter of greengram globally, accounting for approximately 60-70% of world production (Hosking, 1981). According to recent agricultural statistics, greengram cultivation in India spans approximately 4.5 million hectares with an annual production of 2.5-3.0 million metric tonnes, maintaining a global ranking of the number one producer. Within India, greengram cultivation is concentrated across several states, with significant contributions from Maharashtra, Madhya Pradesh, Andhra Pradesh, and Telangana. Telangana, post-reorganization in 2014, has emerged as one of India's major greengram-producing states, accounting for approximately 8-12% of national production (Department of Agriculture and Cooperation, 2023) [2] . Within Telangana, Khammam district stands as one of the most important greengram cultivation centers, with an average area under cultivation of 85,000-95,000 hectares, annual production of 120,000-140,000 metric tonnes, and average productivity ranging from 1.4-1.6 tonnes per hectare. Khammam Market, functioning as a primary agricultural market under the Telangana Marketing Board, serves as a crucial price-discovery center for greengram in the region. Despite its agricultural importance, greengram prices exhibit substantial volatility, influenced by multiple factors including supply-demand dynamics, weather conditions, global market trends, and macroeconomic policies. Price unpredictability creates significant risks for farmers (income uncertainty), traders (profit volatility), and consumers (food security concerns). This volatility necessitates robust forecasting frameworks to support informed decision-making.

Statistical Models in Agricultural Price Forecasting

Traditional time-series models such as ARIMA have been widely applied to agricultural commodity price forecasting (Box & Jenkins, 1976). However, agricultural price series often exhibit long-memory characteristics where past shocks have persistent, long-lasting effects on future prices, which conventional ARIMA models may inadequately capture. The ARFIMA framework explicitly models long-memory dynamics through the fractional integration parameter (d), where 0 < d < 0.5 indicates mean reversion with slow decay of autocorrelation (Granger & Joyeux, 1980). Incorporating exogenous weather variables into time-series models recognizes that agricultural prices are inherently climate-dependent.

Study Objectives

The primary objectives of this research are to: (1) examine longmemory dynamics in greengram prices using fractional integration techniques; (2) quantify the influence of meteorological variables on price fluctuations; (3) develop and compare three advanced time-series forecasting models; (4) identify the superior forecasting model based on statistical criteria. Weather Data: Monthly meteorological observations, including minimum temperature (°C), maximum temperature (°C), rainfall (mm), and relative humidity (%), sourced from NASA POWER (https://power.larc.nasa.gov/data-access-viewer/) database for Khammam district coordinates. Data Processing: Price data were log-transformed to stabilize variance and enable interpretation as percentage changes. Weather variables were standardized for comparability across scales (Lütkepohl, 2005) [5] . The analysis has been carried out in R version 4.4.2.

Materials and Methods

Stationarity Testing

Two complementary unit root tests were applied to assess stationarity:

Augmented Dickey-Fuller (ADF) Test: Tests null hypothesis of unit root (non-stationarity) versus the alternative of stationarity. Test statistics follow Dickey-Fuller distribution.

Phillips-Perron (PP) Test:

Non-parametric test robust to heteroskedasticity and serial correlation in residuals. Test that assesses whether a time series is stationary by testing the null hypothesis of a unit root. (Granger & Joyeux, 1980) where: Time series decomposition (Figure 2) reveals three components:

Model Specifications and Equations ARFIMAX Model Equation

(1) Trend: upward from April 2022 (₹5,900) to September 2023 (₹7,150), then declining to May 2025 (₹6,500); (2) Seasonal: regular fluctuations with amplitude ±₹400-500, corresponding to agricultural calendar; (3) Irregular: relatively small random shocks compared to trend and seasonal components. the presence of a unit root in the price series at standard significance levels (α = 0.05). This non-stationarity justifies the use of differencing methods in ARIMAX or fractional differencing in ARFIMA/ARFIMAX frameworks.

Model Estimation Results

The ARFIMAX (0, d = 0.4647, 0) model with exogenous weather variables was estimated using Maximum Likelihood Estimation. The fractional differencing parameter d = 0.4647 was highly significant (p < 0.001), confirming long-memory dynamics in greengram prices. Among weather variables, relative humidity showed the strongest effect (β = -0.0528, p = 0.065), indicating that increased humidity reduces prices through supply-side mechanisms. Maximum temperature showed negative association (β = -0.0325), while rainfall and minimum temperature effects were weak and non-significant.

Model Fit: Log-likelihood = 111.108, AIC = -208.217, BIC = -196.754

The ARFIMAX model substantially outperformed ARIMAX (AIC difference: 103.46 points) and was comparable to pure ARFIMA (AIC difference: 3.74 points), suggesting that weather variables add practical interpretability at minimal cost to model fit. Economic Interpretation: Long-memory in prices reflects the structural nature of agricultural markets 1. Supply Rigidity: Greengram production requires planning decisions 3-6 months ahead. Supply shocks cannot be immediately reversed, causing price adjustments to persist. 2. Market Concentration: Regional markets influence central markets, creating slow information transmission.

Inventory Effects

Applications and Implications

For Farmers: Forecasts provide basis for harvest timing, storage management, and contracting decisions. Long-memory understanding indicates that recent prices persist current low prices may not quickly reverse.

For Traders: Weather-adjusted forecasts improve decisionmaking in physical commodity and derivatives trading. Longmemory dynamics create momentum-based opportunities.

For Policymakers: Long-memory findings suggest that shortterm price support policies may be insufficient; longer-horizon interventions may be necessary.

For Researchers: Study demonstrates practical applicability of ARFIMAX frameworks in Indian agricultural markets, suggesting similar approaches for other pulse crops.

Final Conclusion

This research successfully advances greengram price forecasting methodology by integrating long-memory modeling with weather variables. The ARFIMAX (0, d=0.465, 0) framework provides a theoretically sound, empirically validated, and practically applicable tool for stakeholders across standard ARIMA should encourage broader adoption of long-memory modeling in Indian agricultural commodity markets.

References

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