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The objective of present study was to analyse the behaviour of onion prices in Kurnool market and forecasting the prices for the future. Based on secondary data from January 2003 to December 2017, the future prices were predicted for the months of January to June, 2018 by employing the Auto Regressive Integrated Moving Average (ARIMA) technique. The annual increase in prices of onion in Kurnool market was observed to be ₹ 6.22 per quintal per annum. The highest seasonal index was observed in the month of August and lowest seasonal index was recorded in May. Price cycles were not identified in onion prices. Maximum R-Square (62.34), minimum Mean Absolute Percentage Error (MAPE) (34.96), Root Mean Square Error (RMSE) (454.71) and Mean Absolute Error (MAE) (263.19) was used as a criteria to select the best model for price forecasting. Based on the above criteria the model (1,1,1) (1,1,1) was found to fit the time series to predict future prices. The forecasted price of onion would be ranging from ₹ 2956 to ₹ 1651 per quintal for the months from January to June 2018 respectively.

Onion has become an almost indispensable part of the Indian diet and also its prices are highly volatile. Onion price fluctuations are occurring all over Indian markets and they are causing damage to both onion producers and consumers. The ARIMA model is commonly used in price time series prediction, especially for series that has a cyclic or seasonal pattern. At the same time, Box-Jenkins ARIMA model give the good representation of short time forecasting. The principle of the model contains filtering out the high-frequency noise in the data, detecting local trends based on liner dependence and forecasting the trends. Despite its high predictive performance, the model has some limitations which decrease its scope of application. The model assumes a linear relationship between the dependent and independent variables while the actual data often present non-linear relationships. Besides, the model assumes that the mean and variance of response series are independent of time, which means stationary. Thus, more than one model should be tested to choose a better one. Forecasting of prices of perishable agricultural commodities is very difficult because they are not only governed by demand and supply but also by so many other factors which are beyond control like weather vagaries, storage capacity, transportation

The main objective of present research was to analyse the price behaviour and forecasting of onion prices in Kurnool market of Andhra Pradesh state.

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

Monthly data Y_{t}_{t}_{t} × C_{t} × I_{t}

where,

Y_{t} = Time series data on prices at time period

T_{t} = Trend component at time period

S_{t}= Seasonal variations at time period

C_{t}= Cyclical movements at time period 't'

Over a long period, time series is likely to show tendency to either increase or decrease over time. Price trend explains the general direction of the movement of prices over long period of time. Ordinary least square method was employed to ascertain the trend in prices by estimating the intercept (a) and slope coefficient (b) in the following linear functional form:

Trends in prices of onion in Kurnool market

Seasonal indices of onion prices in Kurnool market

Cyclical indices of onion prices in Kurnool market

Irregular indices of onion prices in Kurnool market

Autocorrelation and Partial Autocorrelation coefficients of onion prices in Kurnool market

Autocorrelation and Partial Autocorrelation coefficients of residual of ARIMA (1,1,1) (1,1,1) model for the onion prices

Ex-ante and Ex-post forecast of monthly prices of onion in Kurnool market

Average of percentage centered 12 months moving average and computation of seasonal index for observation

Seasonal indices (%) in prices of onion in Kurnool market

Residual analysis of monthly prices of onion

Conditional least square estimates of onion prices

Ex-ante and Ex-post forecast of monthly prices of onion

where,

Y_{t} = Trend value at time

X_{t} = period (Serial number assigned to the t^{th}month)

e_{t} = Random disturbance term (assumption of zero mean and constant variance)

The goodness of fit of trend line to the data was tested by computing the multiple coefficient of determination (R^{2}).

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 _{t}_{t} and the irregular variation I_{t} _{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 _{t}) is divided by corresponding (S_{t}_{t} corresponding to time point

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 _{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}

Symbolically,

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}

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

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)

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

The parameters of the tentatively identified model were estimated and are presented in

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

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.