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In the present paper, Autoregressive Integrated Moving Average (ARIMA) models developed to forecast the prices of potato using time series data of eighteen years from 2002–2019. The best models selected by comparing Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), Mean Absolute Percent Error (MAPE), and Root Mean Square Error (RMSE). The study revealed that ARIMA (1,1,2), ARIMA (2,1,1)(0,0,2)_{[12]}, ARIMA (2,1,2), ARIMA (1,1,4)(0,0,1)_{[12]}, ARIMA (1,1,1)(0,1,2)_{[12]}, ARIMA (0,1,0) (0,1,1)[_{12}], and ARIMA (3,1,3) were the best fitted models for forecasting of price of potato for the states of Utter Pradesh, West Bengal, Madhya Pradesh, Gujarat, Punjab, Tripura and India respectively. The prices of potato in Utter Pradesh, West Bengal and India will be increasing with the first-quarter providing the highest price. The prices of potato in Madhya Pradesh and Tripura will be highest in the fourth quarter. In Punjab, the prices of potato will be increasing with the third-quarter. The forecast shows that market prices of potato in Utter Pradesh, West Bengal, Madhya Pradesh, Gujarat, Punjab, Tripura, and overall India would be ruling in the highest value of .1208 ₹/qt, 1812 ₹/qt, 1345 ₹/qt, 1712 ₹/qt, 1354 ₹/qt, 2636 ₹/ qt, and 1715 ₹/qt respectively for the year 2020.

The ARIMA models forecasted in 2020 showed the prices of potato in Utter Pradesh, West Bengal, and India will increase with the first-quarter providing the highest price.

Potato (

Under tropical and sub-tropical conditions, the losses due to poor handling and storage reported to be between 40–50 percent (

This study focuses on estimating the future price of potato in major producing states in India. To achieve this, we applied time series on the potato price data recorded throughout January 2002 to December 2019, which obtained from the AGMARKNET website as per the availability. The monthly wholesale price data of January 2002 to December 2018 was used for model predictions and from January 2019 to December 2019 was used for validation. By using R software, the data analyzed to fit the best model using an autoregressive integrated moving average (ARIMA) model. The selected best models were used to forecast the prices of potato in Utter Pradesh, West Bengal, Madhya Pradesh, Punjab, Gujarat, Tripura, and India up to December 2020.

ARIMA is one of the most traditional methods of nonstationary time series analysis. In contrast to the regression models, the ARIMA model allows time series to be explained by its past or lagged values and stochastic error terms. The models developed by this approach usually called ARIMA models because they use a combination of autoregressive (AR), integration (I)-referring to the reverse process of differencing to produce the forecast and moving average (MA). The ARIMA (p, d, q) model can be represented by the following general forecasting equation:

Where, _{t}_{1} … _{p}_{1}, … θ_{q} are the parameters of the MA model and the ε_{t}, ε_{t-1}, …, ε_{t-q} are the noise error terms. Where

The stationary check of time series data was performed, which revealed that the potato prices were non-stationary. We will have to test the differenced time series for stationarity (unit root problem) by the Augmented Dickey-Fuller test (ADF). The nonstationary time series data were made stationary by first-order differencing. After that, best fit ARIMA models were developed using the data from January 2002 to December 2018 and used to forecast the prices of potato for the year 2020. Candidate ARIMA models, were identified by finding the initial values for the orders of non-seasonal parameters

The best-fitted model was selected based on the low value of Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), Mean Absolute Percent Error (MAPE), and Root Mean Square Error (RMSE).

The fitted ARIMA model to be validate before using the forecasted results for broader use. The model was verified for accuracy by examining the residuals of the model using the autocorrelation function (ACF) and partial ACF (PACF). If the model shows random residuals, it indicates that the identified model is adequately predicts future prices and vice versa. The ACF and PACF residuals considered random, when all their ACFs were within limits. After satisfying the adequacy of the fitted model, it can be use for forecasting future prices.

Identification of the model was concerned with deciding the appropriate values (p, d, q) (P, D, Q). To check the stationarity of price series of potato, the Augmented Dickey-Fuller unit root tests used. The test confirmed that the data was non-stationary for without difference. In this case, differencing of lag 1 gave a significant result, so with differencing of lag 1(d =1) is stationary in respect to mean and variance. Thus, there is no need for further differencing the time series, and then the adopted difference order is d = 1 for the ARIMA (p, d, q). This test allows going further in the steps for the ARIMA model development, which are to find out the appropriate values of (p, d, q) (P, D, Q). It was done by observing the Auto Correlation Function (ACF) and the Partial Auto Correlation Function (PACF) values. The Auto Correlation Function helps in choosing the appropriate values for the ordering of moving average terms (MA) and the Partial AutoCorrelation Function for those autoregressive terms (AR). The autocorrelation function and the partial autocorrelation function for Utter Pradesh, West Bengal, Madhya Pradesh, Punjab, Gujarat, Tripura, and India were obtained and presented in

ACF and PACF graph of U.P.

ACF and PACF graph of W.B.

ACF and PACF graph of W.B.

ACF and PACF graph of Punjab

ACF and PACF graph of Gujarat

ACF and PACF graph of Tripura

ACF and PACF graph of Tripura

After the identification of the appropriate values of (p, d, q) (P, D, Q), the best-fitted models were selected based on the lowest values of RMSE, MAPE, AIC, and BIC. The model ARIMA (1,1,2), ARIMA (2,1,1) (0,0,2_{[}12_{]},_{[12]}, ARIMA (0,1,0)(0,1,1)_{[12]}, and ARIMA (3,1,3) were identified the best fitted models for Utter Pradesh, West Bengal, Madhya Pradesh, Gujarat, Punjab, Tripura, and India respectively. The results of these coefficients given in

The model validation checked using ACF and PACF plots and distribution of residuals by the Ljung-Box test plot, a statistical test that assesses whether any of a group of autocorrelations of a time series are different from zero which will influence the accuracy of the model. The potato price residual plots evaluate autocorrelations of the selected model. It was determined that most spikes fall within significance limits, which indicates that the model residuals (errors) are not autocorrelated. The fitted model is valid and can be used for making a forecast, and the residuals are normally and independently distributed. The result of Box Ljung Q statistics represented in

Residuals analysis of model ARIMA (1, 1, 2)

Residuals analysis of model ARIMA (2,1,1) (0,0,2)_{[12]}

Residuals analysis of model ARIMA (2,1,2)

Residuals analysis of model ARIMA (1,1,4)(0,0,1)_{[12]}

Residuals analysis of model ARIMA (1,1,1)(0,1,2)_{[12]}

Residuals analysis of model ARIMA (0,1,0)(0,1,1)_{[12]}

Residuals analysis of model ARIMA (3, 1, 3) Forecasting

The forecasted results of the prices of potato shown in

Forecasting prices of potato in India

In the study, ARIMA (1,1,2), ARIMA (2,1,1) (0,0,2)_{[12]}, ARIMA (2,1,2), ARIMA (1,1,4)(0,0,1)_{[12]}, ARIMA (1,1,1)(0,1,2)_{[12]}, ARIMA (0,1,0)(0,1,1)_{[12]}, and ARIMA (3,1,3) models were developed for forecasting price of potato for the states of Utter Pradesh, West Bengal, Madhya Pradesh, Gujarat, Punjab, Tripura, and India respectively. The models showed a good performance in the case of explaining variability in the data series and also, it's predicting ability.