Time series modeling of Indian non-basmati rice exports: A comparative approach

Introduction

Rice (Oryza sativa L.) is one of the world's most important cereal crops, providing food security and livelihoods to billions of people, particularly in Asia and Africa. Among the various types, non-basmati rice constitutes the largest share of global rice production, consumption and trade. The term "non-basmati rice" refers to all rice varieties other than basmati and is primarily consumed as a staple food due to its short-to medium-grain length, non-aromatic nature, affordability and adaptability to diverse agro-climatic conditions. India, the world's largest rice exporter, predominantly produces non-basmati varieties such as Swarna, IR-64, MTU-1010 and Sona Masuri, owing to their high yield potential, stability, and suitability for large-scale cultivation. Major export destinations during 2023-24 included Benin, Guinea, Togo, Côte d'Ivoire and Vietnam, while total non-basmati rice exports reached 11.12 million tonnes in FY 2023-24 (APEDA). Agriculture contributes approximately 15 per cent to India's GDP and provides livelihoods to over half of the population, with agricultural exports accounting for nearly 10 per cent of total exports (Charishma and Singh, 2024) [2] . Non-basmati rice alone accounts for more than 80 per cent of India's total rice exports in several years, driven by strong demand from African and Asian markets due to its affordability and availability (DGCI&S, 2024) [4] . To forecast the future performance of non-basmati rice exports, this study employs time-series models such as ARIMA, SARIMA and Holt-Winters exponential smoothing, which effectively capture trend and seasonal patterns in export data. Comparing the forecasting performance of these models enables the identification of the most reliable approach, offering valuable insights for policymakers, traders and industry stakeholders to support evidence-based decision-making and strengthen India's position in the global rice market.

Materials and Methods

Monthly data with respect to Non-Basmati Rice exports of India for period from 2015-2016 to 2024-25 was collected from Agricultural and Processed Food Products Export Development Authority (APEDA), Government of India. As the study is based on time-series data, the dataset was initially screened to ensure that no outliers were present.

Test for outliers

Outlier detection can be done in several ways viz Descriptive statistics, Boxplot method, Statistical tests like Grubbs' Test, Dixon's Q Test etc. In the present study to identify the existence of outliers the statistic Z is calculated as difference between absolute value of the observation and mean divided by the Standard Deviation (SD) as shown in Equation (1).

Where, SD is the Standard deviation of variable X and n indicates the number of observations. If Z is greater than 1.96, then it is implied that outlier exists in the present sample (Dasyam et al., 2015) [3] .

ARIMA Model

ARIMA model was explained in Box-Jenkins (BJ) methodology. George P Box and Gwilym Jenkins (Box-Jenkins, 1970) introduced the approach in their book "Time Series Analysis: Forecasting and Control". This is why it is often referred to as the Box-Jenkins approach. (Agarwal, 2019) [1] . It (p,d,q) is a combination of Auto Regressive (AR) and Moving Average (MA) with an order of integration or differencing (d), where p and q are the order of autocorrelation and moving average respectively (Gujarati et al. 2012) [5] .

Auto Regressive (AR):

The autoregressive model is denoted by AR, which predicts the future values based on its own past observations of order p, as follows:

Where, c is constant term, Ø p is the p-th autoregressive parameter and et is the error term at time t.

Moving Average (MA):

The moving average (MA) model implies that the present value of a time series is formed as a linear function of the current and previous q error terms, as follows:

where c is constant term, θ q is the q-th moving average parameter and e t-k is the error term at time t-k.

ARIMA in general form is as follows

where Δ denotes difference operator like

Here, Z t-1 , …, Z t-p are values of past series with lag 1, …, p respectively. Box-Jenkins methodology consist of four steps as follows

 Identification of model

The Box-Jenkins methodology employs Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots to specify ARIMA parameters (p, d, q), necessitating time series data that exhibit stationarity characterized by stable mean, variance and autocorrelations over time. Stationarity is evaluated through the Augmented Dickey-Fuller (ADF) test alongside visual analysis of ACF and PACF patterns; a slow, persistent ACF decay suggests non-stationarity, whereas abrupt cutoffs or rapid attenuation confirm stationarity. For non-stationary series, differencing is repeatedly applied until the ACF displays quick decay and consistent variance, with the required differencing order defining parameter d.

 Parameter Estimation

The objective of parameter estimation is to derive reliable estimates for the ARMA model coefficients based on the predetermined orders p and q.

 Diagnostic Checking Following parameter estimation for a tentatively selected ARIMA model, diagnostic checks are essential to validate its adequacy. Analysis of the residuals' ACF and PACF plots helps assess model fitness, confirmed by achieving minimal MAPE, RMSE and MAE values.

 Forecasting

Parameter estimation equips the ARIMA model to serve as a predictive tool for forecasting future time series values. ARIMA modeling's widespread adoption stems from its proven forecasting efficacy, yielding more dependable short-term predictions compared to conventional econometric approaches.

SARIMA

Seasonality in time series analysis refers to recurring fluctuations that follow a consistent pattern over a specific number of time intervals, denoted as S. For instance, in quarterly observations, S = 4, representing four periods per year, while in monthly observations, S = 12. Similar to how regular differencing is used to handle non-stationarity in a series, seasonal differencing is applied when the data exhibit seasonal non-stationarity. The seasonal autoregressive (SAR) and seasonal moving average (SMA) components are parameters that define the seasonal ARIMA model. In this framework, the seasonal AR and MA terms forecast the variable 𝑥𝑡 using lag values that are multiples of the seasonal period S. The seasonal ARIMA model is represented by ARIMA (p, d, q) (P, D, Q) where p is the number of autoregressive terms, q is the number of moving average terms, d is the order of differencing to induce stationarity, P is the number of seasonal Autoregressive components, Q is the number of seasonal moving average terms and D is the number of seasonal differences needed to make the series stationarity.

The seasonal ARIMA model has the following expression:

∇𝑑= (1 -𝐵) 𝑑 denotes the number regular differences and ∇𝑠𝐷= (1 -𝐵𝑠) 𝐷 denotes number of seasonal differences.

Where, ∅(𝐵) is stationary Autoregressive operator, 𝜃(𝐵) is a stationary moving average operator, 𝜀𝑡 is a white noise (Brockwell and Davis, 1996).

QS test for Seasonality

The QS test is a seasonal lag-based variation of the Ljung-Box test that solely takes into account positive autocorrelation.

The current approach still assumes that statistics are distributed as a χ2.

Results and Discussion

Before the actual analysis, the quantity series consisting of 120 observations was divided into two subsets: a training set and a testing set. The training set, used for model building and estimation, contained 108 observations, while the remaining 12 observations corresponding to the most recent months were reserved for the testing set to evaluate model performance. Later, the dataset was examined for outliers using Equation (1). The results indicated that no extreme observations were present in the data. The mean export quantity was estimated at 898.103 MT and the standard deviation was 451.241 MT, indicating considerable variability in export quantities. These descriptive statistics were presented in Table 1. At first, stationary of Non-Basmati Rice exports in India from 2015-16 to 2024-25 was tested by time series plots and ADF (Augmented Dickey-Fuller) test. The time series plot clearly indicated that the data was non-stationary because of prominent fluctuation trend as shown in Figure 1. ADF test for unit root also confirmed that the data was nonstationary and it became stationary at first difference (Table 2). It was also be confirmed by observed difference before and after differentiation of ACF and PACF plots. Among the all the models, ARIMA model (2, 1, 3) was found to be best fit than any other because of the minimum values of the BIC, AIC, MAPE and MAE values. To further investigate the seasonality indicated by large values in the data, which raises doubts about its presence, a QS test had also been conducted. From the test, it was found that the QS test for the level series was significant, which confirms seasonality of exports as shown in Table 4. www.extensionjournal.com Hence, Seasonal time series models such as SARIMA and Winter smoothing models were also tried. Among the all the models, SARIMA model (2, 1, 3) (0, 1, 1) 12 was found to be best fit than any other SARIMA, as because of the minimum values of the AIC, MAPE and MAE values. The parameters were estimated for the best selected model i.e., SARIMA (2,1,3) (0, 1, 1) 12 as mentioned in Table 5. Similar results were obtained for the various studies of Insaad and Sribunrueang, 2025 [6] ; Aquije et. al. 2025 [8] ; Valipour, 2015 [7] . Later, Holt-Winter exponential smoothing was also developed to the selected data set. Similarly, parameters of Holt-Winter were estimated as level (α) = 0.819, trend (β) = 0.01 and seasonality (γ) = 0.99 respectively were estimated as per table 7.

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