Jute (Corchorus capsularis L.) known as the “golden fiber” is one of the most important commercial cash crops grown in India. It is one of the cheapest and strongest of all natural fibers and is considered the fiber of the future. After cotton, it is the most important fiber crop grown in India. India is the world’s largest producer of raw jute, accounting for more than half of global jute production. Among the states, West Bengal ranks first in area and production of jute in the country with a total area of 0.52 million hectares (78.49 per cent) and a total production of 7.75 million bales (82.49 per cent) with a 2701 kg/hectare productivity during 2022-23. (Directorate of Economics and Statistics, DA and FW). Agricultural prices play an important role in the whole national economy of India. Commodities price forecasts are critical to market participants who make production and marketing decisions and to policymakers who administer commodity programs and assess the market impacts of domestic or international events (Das et al., 2022; Sahu and Basak, 2024). Time series modelling is one such approach that collects and analyses past values to develop appropriate models that describe the inherent structure and characteristics of the series. Most of the research in time-series analysis assumes that the observations that are far apart in time are basically independent, but in many practical situations it is seen that many empirical economic series show that the distant observations are dependent; this is characterized as a long memory. This long memory component of the market cannot be adequately explained by systems that work with short memory parameters i.e., AR (p), MA(q), ARMA(p,q) and ARIMA (p,d,q). The autoregressive fractional integral moving average (ARFIMA) model has been widely employed for long memory time series forecasting in a divergent domain for several decades. Several studies have been conducted in different parts of India and abroad for modelling time series in the presence of long memory. Booth et al., (1982), Helms et al., (1984) and Sahu et al., (2023) applied the rescaled range (R/S) method to detect the existence of long memory in the prices data and provided evidence of long-memory behavior in financial markets. Granero et al., (2008) investigated the Hurst exponent for long memory processes in capital markets. Erfani and Samimi (2009) studied the long memory of the stock price index (TSIP) and established the ARFIMA and ARIMA models, concluding that the ARFIMA is a much better model in long memory. Paul (2017) worked in long memory on maximum and minimum temperature series and analysed them using the ARFIMA model and concluded that the ARFIMA model could be used successfully for modelling the temperature series. Similar works have been done by Liu et al., (2017), Safitri et al., (2019) and Sahu et al., (2024).
       
However, in this study, a wavelet domain based ARFIMA model is used for adequate modelling of the long memory features of weekly jute prices in the Cooch Behar market of West Bengal. Furthermore, we compared the forecasting abilities of the ARIMA and ARFIMA models.

Data description
 
In order to carry out our analysis, historical jute price data of Cooch Behar market has been taken from the Agricultural Marketing Information Network (https://agmarknet.gov.in) portal from January 2009 to December 2022 (672 weeks) with weekly data. In the present study, statistical analyses have been carried out using the powerful software “RStudio” (https://www.rstudio.com). The first 80% observations (i.e., from the 1^st week of 2009 to the 10^th week of 2020) are used for the model building purpose and the rest 20% observations (i.e., from the 11^th week of 2020 to the 48^th week of 2022) are used for model validation.
 
Test for stationarity
 
Testing data for stationarity is very important in research where the underlying variables are based on time. A time series is said to be stationary if its underlying generating process is based on a constant mean and constant variance, with its auto correlation function (ACF) essentially constant through time. If these conditions do not hold, the series is non-stationary. In order to check the data for stationarity, the Augmented Dickey-Fuller (ADF) test (Dickey and Fuller, 1979) and the Phillips-Perron (PP) unit root test (Phillips and Perron, 1988) have been used.
 
Long memory process
 
The concept of long memory, or long-range dependence, is an important one in time series analysis. A long memory feature occurs when the autocovariances for a stationary time series tend to zero like a power function but more slowly than an exponential decay. In this study, the Hurst exponent method (Hurst, 1951) has been used to test the long memory. The value of Hurst exponent (H) varies from 0 to 1, When 0.5 < H < 1, it implies that persistent long memory of a time series is strong. 
 
Autoregressive integrated moving average (ARIMA) model
 
The ARIMA model (Box and Jenkins, 1976) isa widely recognized statistical forecasting model that predicts future observations of a time series on the basis of some linear function of past values and white noise terms. The ARIMA (p, d, q) process is expressed by: 
 

  

 
Where, m is the mean of the series, yt is the Observed time series values at time t; φ_p (B) =  (1- φ₁B - φ₂B² - ... - φ_pB^p) and φ_q (B) = (1 - φ₁B - φ₂B² - ... - φ_qB^q)  are the autoregressive operator and moving average operator respectively in which φ_i (i = 1,2, ... ... , p) and φ_j (j = 1,2, ... ... ,q) are the autoregressive and moving average coefficients respectively; p and q are integers and often referred to as orders of autoregressive and moving average respectively; ε_t is the t^th white noise and B is the backshift operator (By_t = y_t-1). If the series is not stationary, the first difference or higher-order differences will produce a stationary time series.
 
Autoregressive fractionally integrated moving average (ARFIMA) model
 
One of the most challenging issues in statistical modelling is time series analysis and forecasting. In general, models for time series data can have many forms and represent different stochastic processes. In all linear time series forecasting approaches, one of the most important models is ARIMA, which has been widely used to forecast social, economic, agricultural, engineering and financial problems (Box and Jenkins, 1976). The ARIMA model can only capture the short-range dependence (SRD) property, but many practical agricultural datasets, principally commodity prices data, show the typical feature of a long memory process. Therefore, there is a need for a model that has the property of long memory. The autoregressive fractional integral moving average (ARFIMA) model meets this property.
       
The ARFIMA (p,d,q)  model (Granger and Joyeux,1980; Hosking,1981) is similar to that of the ARIMA (p,d,q)  model, except for the differencing value (d), which is given as follows:

  

 
There are three steps in the procedure of establishing an ARFIMA model. First, testing for long term memory in the time series and determining the fractional differencing parameter. Second, imposing fractional differencing on the series and obtaining an ARMA process. Third, determining the other two parameters of the ARFIMA model, namely  p and q. After fractional differencingis determined, we obtained the fractional differencing time series as (1-B)^dy_t with binomial expansion. The partial auto correlation function (PACF) is most useful for identifying the order (p) of an autoregressive model and the autocorrelation function (ACF) is useful for identifying the order (q) of moving average model.
 
Wavelet for long memory parameter estimation
 
The maximum overlap discrete wavelet transform (MODWT) is a modified DWT where the subsampling process is avoided, leading to a higher level of information in the resulting wavelet and scaling coefficients when compared to the DWT (Ghaemi et al., 2019). The MODWT determines scaling coefficient V_φ (m, n) and wavelet coefficient  W_ψ (m,n) (where scale parameter and translations parameter n) by applying low and high pass filters to the original dataset (Kumar et al., 2020).
       
For estimating the long memory parameter of ARFIMA model, the algorithm based on wavelet (Jensen, 1999) is followed. Let y_t be a mean zero fractionally differenced process with 0 < d < 1/2.  Using the autocovariance function of the I(d) process, Jensen (1999) found that as j → 0, the wavelet coefficients, W_ψ (m,n), associated with a mean zero  I(d) process with  0 < d < 1/2 are distributed N 90, σ²2^-2md), where σ²  is a finite constant. The wavelet coefficients from an I(d)process have a variance that is a function of the scaling parameter m, but is independent of the translation parameter, n. The correlation of the wavelet coefficients from an I(d) process decay exponentially over time and scale. Hence, define R(m) to be the wavelet coefficients’ variance at scale m, i.e., R(m) = σ²2^-2md. Taking the logarithmic transformation of R(j), we obtain the relationship R(m) = Inσ² - d In2^2m. Where ln R(m) is linearly related to ln 2^-2m by the fractional differencing parameter, d. Hence, the unknown d of a fractionally integrated series can be estimated by the ordinary least squares estimator d.
 
Information criteria and accuracy measures
 
In this paper, we used two widely applied criterion Akaike information criterion (AIC) and the Bayesian information criterion (BIC) to select the best model among a set of candidate models. Furthermore, the RMSE, MAE and MAPE are used as an accuracy measure to evaluate the performance of the models.

Primary statistical analysis
 
The weekly series of jute prices for the Cooch Behar market of West Bengalis shown in Fig 1. The Cooch Behar market series depicts an up-and-down pattern, with two sharp rises between 2013 and 2016 and another between 2020 and 2022.The descriptive statistics to summarize information from the weekly jute price data are listed in Table 1. As Table 1 shows, the series of jute prices in the Cooch Behar market has a mean of 4188, a standard deviation of 1518.44 and, 36.26% coefficient of variation, suggesting that it has been volatile. In addition, skewness and kurtosis statistics show that the price series is not normally distributed.




       
To begin with the implementation of ARIMA and ARFIMA models, the data series are divided into two sets: The training set and the testing set. First, the model is fitted using the training data with 538 observations (i.e., from the 1^st week of 2009 to the 10^th week of 2020) and then it is predicted over the validation period using a testing set with the last 134 observations (i.e., from the 11^th week of 2020 to the 48^th week of 2022).
 
Test for stationarity
 
The first step in applying ARIMA and ARFIMA models is to check whether the time series is stationary or not. In order to test for stationarity, we first conducted Augmented Dickey-Fuller and Phillips-perron unit root tests on the training dataset of the series. According to the results, the p-values (Test statistic) of the ADF and PP tests are 0.494(-2.20) and 0.251(-15.39), respectively, indicating that the time series under consideration is clearly non-stationary. The study, therefore, proceeded to find the stationary series.
 
Test for long memory and estimation
 
The presence of long memory in a time series (training set) was confirmed by investigating the autocorrelation function (ACF) plot, which shows that the correlations decay very slowly towards zero up to 250 lags (Fig 2), indicating the presence of long memory processes. Accordingly, the presence of long memory is tested as discussed in methodology and it is found that the R/S Hurst value (H = 0.873) is higher than 0.5, which firmly concludes the existence of the long memory characteristic of the jute prices. The models that consider the long memory property are very sensitive to the estimation of the long-memory parameter (i.e., the fractional differencing parameter) and for this reason, in this study, it has been estimated by using the wavelet-based ordinary least squares estimator (d_wavelet) andis found to be 0.327.


       
After determining the fractional differencing parameter we obtained the fractional and first-order differencing time series shown in Fig 3. For that, the stationary test results are shown in Table 2. The -values of the ADF and PP tests are less than 5%, which reveal the series has become stationary, which is also confirmed by Fig 3.

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Model Identification
 
To establish ARIMA and ARFIMA models, the values of, and must be determined. In the above section, we have identified the value of . Now in this section, we are going to find the optimal value of and which are order of autoregressive and moving average terms. We used the training set as in-sample data for the determination of the parameters and  of the ARIMA and ARFIMA models. First, we computed the values of autocorrelation and partial autocorrelation for fractionally differenced series and first-order differenced time series, as illustrated in Fig 4-5. On computation of ACF and PACF for each estimated difference parameter, it is observed that the decay rate of ACF has improved as compared to the decay of ACF in the actual training set (Fig 4). The orders of non-seasonal parameters  and (q) are obtained by looking for significant spikes in autocorrelation and partial autocorrelation functions.

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In the identification stage, we estimated different ARIMA and ARFIMA specifications with different combinations of  (AR terms) and  (MA terms), which are listed in Table 3 and selected the appropriate model from each method as having the minimum values of AIC and BIC values. Thus, the models selected for the training period are ARFIMA (1,0.327,1), ARFIMA (3,0.327,1) and ARIMA (3,1,0).


 
Validation and diagnostic checking
 
After appropriate ARFIMA and ARIMA models have been obtained, the next step is to see their ability to forecast the data. The model verification process is concerned with examining residuals obtained from fitted models to see if they contain any systematic pattern that could still be removed to improve the chosen models. This has been done through the Ljung-box diagnostic test and it is found that the -value of the Ljung-box test is more than 5% (Table 4), which means that the model residual meets the assumption of white noise residuals. The evaluation of forecasting performance has been done for the test set as an out of-sample period of 134 observations. Table 4 represents the results of the models based on the three different accuracy performance measures: RMSE, MAE and MAPE.


       
As shown in Table 4, comparing the validation results of all three models indicates that all are likely to perform well in the forecasting phase and, it is observed that the ARFIMA(3,0.327,1) model produces the lowest RMSE, MAE and MAPE, which are164.42, 105.66 and 1.70, respectively. It can be concluded that the wavelet method based ARFIMA (3,0.327,1) model is the most accurate compared to other models, where predictions indicate that there are narrow variations between the actual and predicted values of jute prices (Fig 6). The strength of the ARIMA model in forecasting jute prices in the Cooch Behar market is considerable, but Table 4 shows that the ARIMA does not perform well. That the most accurate model is conclude to forecast the weekly jute prices in the Cooch Behar market of West Bengal is the ARFIMA (3,0.327,1) model, which is given as:
 

(1 - 0.578B - 0.188B² - 0.210B³) (1 - B)^0.327 y_t = 1569.681 + (1 + 0.212B) ε_t

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The aim of this paper is to introduce an appropriate model for modeling and forecasting the weekly jute prices in the Cooch Behar market of West Bengal in the presence of long memory feature. The presence of long memory behavior is confirmed in jute price series using an ACF plot and Hurst rescaled range (R/S) analysis indicates that it would be better to develop and employ ARFIMA models. We considered the wavelet methods for estimating the fractional differencing parameter. ARIMA and ARFIMA models are fitted to the jute price series and the models selected are ARFIMA (1,0.327,1), ARFIMA (3,0.327,1) and ARIMA (3,1,0) on the basis of the minimum AIC and BIC values. A comparative study has been made between the forecasting performances of the ARIMA and ARFIMA models and it is found that the ARFIMA (3,0.327,1) model out performs the best fitted ARIMA (3,1,0) model in terms of the RMSE, MAE and MAPE. Hence, it is evident that long memory plays an important and dominant role in describing and modeling the jute prices. Finally, the ARFIMA (3,0.327,1) model is found to be the best optimal model to forecast the jute prices for the Cooch Behar market. The model demonstrated good performance in terms of explained variability and predicting power. The study has revealed that the ARFIMA model could be used successfully for modelling as well as forecasting, especially for data with the long memory (long-range dependency) property.

The authors declare that they have no conflict of interest.