The time series data on monthly prices of onion required for the study was collected from the registers maintained by the respective market committees, National horticulture board database (Anonymous 2018) and NHRDF (National Horticulture Research and Development Foundation). The data related to monthly modal prices (₹/qtl) for the period from January 2003 to December 2017 (15 years) was used for time series analysis and for price forecasting from January to June 2018.

To analyse all the four components of a time series viz., trend, seasonal, cyclical and irregular fluctuations, a multiplicative model of the following type was used as elucidated in Areef et al. (2019),

Monthly data Y_t = T _t × S _t × C_t × I_t w

where,

In order to estimate the seasonal variation, the twelve month centered moving average method was used which gives us the periodic changes without seasonality. To estimate the seasonal index, a 12 month centered moving average was calculated as follows:

This is sequential manner for each points of time t. In this fashion, a 12 month centered moving average removes a large part of fluctuation due to the seasonal effects so that what remains is mainly attributable to other sources viz., long term effects T_t,, cyclical effect C_t and the irregular variation I_t w which is due to random causes is also minimized by the process of smoothing out effect. Thus, this affords a means of not only estimating TC effect but also estimating seasonal components. In the next step of computing the seasonal index, the original series is divided by the centered moving average. This gives the first estimate of seasonal component S_t..

It is always expressed in terms of percentages. In this process, we do not have moving average for the first six and last six months. These seasonal components are next arranged month-wise for each year (Table 1). The last row in the Table 1 give estimates of seasonal index for the 12 months adjusted for their total to 1200 or averaged to 100. The last row in the Table. 1 gives the first estimates of seasonal variations. In order to obtain a better estimate i.e., stabilized seasonal indices we need to employ an interactive process as under. The original observation (Y_t) is divided by corresponding (S_t)) value and then obtain the residual (TCI)_t corresponding to time point t.

The residual series (TCI)_t thus obtained is subjected to the same process of determining 12 month centered averages as done earlier to obtain better estimates for trend cycle effect viz., (TC)_t. These revised estimates are next employed as above to generate a revised set of seasonal indices by dividing each observation (Y_t) by the corresponding (TC)_t value. This will lead to revise estimates of seasonal indices (S_t) as second interactive ones.

This interactive process is separately employed until stabilized seasonal indices are obtained i.e., two successive seasonal indices do not differ by more than five per cent.

Seasonal indices = Seasonal indices × correction factor

and

Correction factor = 1200 / Sum of seasonal indices

Cyclical variations are long term oscillatory movements with duration of greater than one year. The most commonly used method for estimating cyclical movement of time series is the residual method by eliminating the seasonal variation and trend. This is accomplished by dividing (Y_t)) by corresponding (S) for time 'if

Symbolically,

Irregular variations: It was estimated as residual component by using the estimates of model prices and cyclical components.

The details of ARIMA forecasting model are as follows:

Introduced by Box and Jenkins (1976), the ARIMA model has been one of the most popular approaches for forecasting. The ARIMA model is basically a data oriented approach that is adopted from the structure of the data itself. In an ARIMA model, the estimated value of a variable is supposed to be a linear combination of the past values and the past errors. Generally a time series can be modelled as a combination of past values and errors, which can be denoted as ARIMA (p,d,q) which is expressed in the following form

where Y_t and e_t are the actual values and random error at time t, respectively, Φi (i = 1,2,....., p) and θj (j = 1,2,...., q) are model parameters, p and q are integers and often referred to as orders of autoregressive and moving average polynomials respectively. Random errors are assumed to be independently and identically distributed with mean zero and constant variance. Similarly, a seasonal model is represented by ARIMA (p, d, q) × (P, D, Q), where P is the number of seasonal autoregressive (SAR) terms, D is the number of seasonal differences and Q is the number of seasonal moving average (SMA) terms. Basically this method has four steps identification of the model, estimating the parameters, diagnostic checking and forecasting

The results revealed that there was an increasing trend in onion prices in Kurnool market for the study period. The trend equation estimated for the present study was 264.39 + 6.22*t and graphically shown in Fig. 1. The annual increase in prices of onion in Kurnool market was observed to be ₹ 6.22 per quintal. Areef et al. (2019) revealed that the annual increase in prices of onion in Bangalore market was ₹ 6.92 per quintal for the period from Jan-2003 to Dec-2017 and it was found to be statistically significant. In order to analyse the seasonal variation in onion prices in the Kurnool market, seasonal indices were computed by adopting 12 months centered moving average method. The results (Table 2 and Fig. 2) revealed that the highest seasonal index was observed in August, followed by November and July as the indices stood at 140.80, 121.31 and 117.07 respectively. Lowest seasonal index was recorded in May with 62.72. The cyclical and irregular fluctuations in onion prices were graphically shown in Fig. 3 and Fig. 4 respectively. A definite periodic fluctuations were not identified in Kurnool market which was revealed by the absence of price cycles.

For forecasting onion prices in Kurnool market, ARIMA model was used after transforming the variable under forecasting into a stationary series. The stationary series is the one whose values vary over time only around a constant mean and constant variance. Identification of the model was concerned with deciding the appropriate values of (p,d,q) (P,D,Q). It was done by observing Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) values (Fig. 5). The Auto Correlation Function helps in choosing the appropriate values for ordering of moving average terms (MA) and Partial Auto-Correlation Function for those autoregressive terms (AR).

ARIMA model was estimated after transforming the variables under study into stationary series through computation of either seasonal or non-seasonal or both, order of differencing. Based on the maximum R-Square, minimum MAPE (Mean Absolute Percentage Error), RMSE (Root Mean Square Error) and MAE (Mean Absolute Error) the model (1,1,1) (1,1,1) was found to be fit the data and suitable to forecast future prices in Kurnool market (Table 3).

The parameters of the tentatively identified model were estimated and are presented in Table 4. The autocorrelation and partial autocorrelations of various orders of the residuals of ARIMA (1,1,1) (1.1,1) up to 36 lags were computed and are shown in Fig. 6. The figures showed that, the autocorrelation at lag 15 and 36 and partial autocorrelation functions at lag 15 and 20 were significantly different from zero and fell slightly outside the 95 per cent confidence interval, which indicated the presence of white noise error in the residuals.

Hence, except for lag 15, 20 and 36 autocorrelation was absent in the residuals. This showed that the selected ARIMA model was appropriate for forecasting the price of onion during the period under study. Both ex-ante and ex-post forecasting were done and it was compared with actual observations. The prices were forecasted up to June, 2018. The results of ex-ante and ex-post forecasted prices are presented in Table 5 and illustrated in Fig. 7. The forecast are also depicted that there is narrow variations in between the actual and forecasted values of prices of onion in the Kurnool market. According to the forecasts the price of onion would be ranging from ' 2956 to ' 1651 per quintal for the months from January to June 2018.

Reliable price forecast model enable the government to make appropriate decisions in advance like procurement, regulating export & imports and possibility of check on trader hoardings. The price seasonal indices and forecasted price information was more important for the farmer to selection of crop varieties, allocation of scarce inputs under different crops and adjusting the sowing & harvesting dates to get remunerative prices in a more rational way.