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, Y_t is pricing, μ. is the mean of series, the ϕ₁ … ϕ_p are the parameters of the AR model, the θ₁, … 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 “p” stands for the order of the autoregressive process, ‘a’ is the order of the data stationary, and ‘q’ is the order of the moving average process. Seasonal ARIMA model is to be denoted by ARIMA (p, d, q) (P, D, Q), where P denotes the number of seasonal autoregressive components, Q denotes the number of seasonal moving average terms and D denotes the number of seasonal differences required to induce stationarity discussed Box et al. (1994). The steps, which were followed to define an ARIMA model as stated by Box and Jenkins:

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 p and q. They were obtained by looking for significant spikes in autocorrelation and partial autocorrelation functions.

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 Fig. 1-7.

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], 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 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 Table 1. A careful examination of ACF and PACF up to 24 lags of West Bengal, Gujarat, Punjab, and Tripura revealed the presence of seasonality.

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 Fig. 8-14. Out of sample MAPE, values were also present in the permissible range sown in Table 1.

The forecasted results of the prices of potato shown in Table 2. The forecasted prices of potato in the selected states were showed a random trend in the coming months. 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 price of potato fluctuated widely over the months. The prices of potato in Utter Pradesh, West Bengal, Gujarat, and India would be high in the period from January to May and then after more or less constant. The prices of potato in Madhya Pradesh and Tripura showed an increasing trend and pick from September to November. In Punjab, the forecasted prices of potato fluctuated widely over the months. We can also see the prices of potato in India would be high as compare to selected states except for Tripura.