Pigeon Pea Arima 3mb2h

International Review of Business and Finance ISSN 0976-5891 Volume 2 Number 1 (2010), pp. 97–102 © Research India Publications http://www.ripublication.com/irbf.htm

Use of the ARIMA Model for Forecasting Pigeon Pea Production in India Rachana Wankhade1, Suvarna Mahalle2, Sonal Gajbhiye3, and V.M. Bodade4 1,2,3,4

Senior Research Fellow (Agril. Econ.) Dr. PDKV, Akola, India

Abstract Pigeon pea is an important pulse crop in India and being the largest producer of Pigeon pea in the world, India contributes to around 85 percent of the world’s total production. It is imperative to assess scientifically the accurate future production potentials of this crop on the basis of past trends. Thus, the study has been made to forecast the production of Pigeon pea in India up to the year 2015.by using ARIMA (autoregressive integrated moving average) model. At present it is the most sophisticated model for forecast. The results showed that production of Pigeon pea in India would be 2.73452 M. Tons in 2015. In the year 2009 price of Tur dal touched the sky, price of any commodity ultimately related to production. So it is necessary to study the future movement of Pigeon pea production. Keywords: ARIMA; Forecasting; Production; Pigeon pea; India.

Introduction Pigeon pea is an important pulse crop in India and being the largest producer of Pigeon pea in the world, India contributes to around 85 percent of the world’s total production. It is a protein rich staple food and consumed in the form of split pulse as Dal. Pigeon pea is second largest pulse crop in India, ing about 20 percent of total pulse production, The crop ranks sixth in the world in dry land legume production. Auto Regressive Integrated Moving Average (ARIMA) is the most general class of models for forecasting a time series. Different series appearing in the forecasting equations are called “Auto-Regressive” process. Appearance of lags of the forecast errors in the model is called “moving average”. (ARIMA) model was introduced by Box and Jenkins in 1960 for forecasting variables. In the year 2009 price of Tur dal touched the sky, price of any commodity ultimately related to

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production. So it is necessary to study the future movement of Pigeon pea production by using past trends

Methodology Respective time series data for this study were collected from Government Publications, (Agricultural Statistics at a glance,2008).Box and Jenkins (1976) linear time series model was applied. Auto Regressive Integrated Moving Average (ARIMA) is the most general class of models for forecasting a time series. Different series appearing in the forecasting equations are called “Auto-Regressive” process. Appearance of lags of the forecast errors in the model is called “moving average” process. The ARIMA model is denoted by ARIMA (p,d,q), where “p” stands for the order of the auto regressive process, ‘d’ is the order of the data stationary and ‘q’ is the order of the moving average process. Δdyt =δ+θ1 Δdyt-1+θ2 Δdyt-2+----- + θpyt-p+e t-1 α et-1-α2 et-2 αq et-2 (1) Where, Δd denotes differencing of order d,i.e., Δyt = yt-yt-1, Δ2yt=Δyt-Δt-1 and so forth, Y t-1 ----- yt-p are past observations (lags), δ,θ1 -------- θp are parameters (constant and coefficient) to be estimated similar to regression coefficients of the Auto Regressive process (AR) of order “p” denoted by AR (p) and is written as Y = δ+θ1 y t-1+θ2 y t-2 + ---------- + θpy t-p +et (2) Where,et is forecast error, assumed to be independently distributed across time with mean θ and variance θ2e, et-1, et- 2 ------ et-q are past forecast errors, α1, --------- αq are moving average (MA) coefficient that needs to be estimated. While MA model of order q (i.e.) MA (q) can be written as Yt = et-α1 αt-1 - α2et-2 ------------- αqet-q (3) The major problem in ARIMA modeling technique is to choose the most appropriate values for the p, d, and q. This problem can be partially resolved by looking at the Auto correlation function (ACF) and partial Auto Correlation Functions (PACF) for the series (Pindyk & Rubinfeld, 1991). The degree of the homogeneity, (d) i.e. the number of time series to be differenced to yield a stationary series was determined on the basis where the ACF approached zero. After determining “d” a stationary series Δd yt its auto correlation function and partial autocorrelation were examined to determined values of p and q, next step was to “estimate” the model. Using the results of ARIMA (p,q,d), forecast the Tur production in India from 2008 up to 2015

Result and Discussion To fit an ARIMA model requires a sufficiently large data set. In this study, we used the Pigeon pea production data for the period 1950-51 to 2007-2008. The maximum production of Pigeon pea in India was obtained in 2007-08 year (3.09 million tones) and minimum in 1950-51 year (1.72 million tones). The ARIMA model was applied according to three steps namely Identification, Estimation and Verification. Last fifty seven years data of Pigeon pea production was used for modeling

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purpose. The model specification involved the plots of the auto correlation function (ACF), partial auto correlation function (PACF) and the plot of the differenced series. Auto correlation function indicated the order of the autoregressive components ‘q’ of the model, while the partial correlation function gave an indication for the parameter p. First step was to check the stationarity of the data. The time series plot for production showed an increasing trend. Auto correlation function of the series showed non-stationary as auto correlation function did not fall as quickly as the log (k) increased. To make the series stationary for production, difference series was used and first difference series of Pigeon pea production showed stationarity. After this the values of autoregressive (AR) parameter “p” and moving average (MA) parameter ‘q’ was determined from correlograms of partial autocorrelation function and the auto correlation function, respectively. Partial auto correlation function of the first differenced series of production was used to determine the parameter “p”. It was observed that partial auto correlation function fell after lag 2. Thus the value of “p” was decided 2 for production which gave good results consequently, the respective values of p,d,q were determined for ARIMA i.e. ARIMA (2,1,0)

Model estimation With the help of SPSS computer package ARIMA (1, 1, 1) models was found to be estimated for Pigeon pea Production in India. Diagnostic checking For diagnostic checking of the estimated models, different diagnostic checks were applied for whether these were properly fitted or not. Which is given in table 1.

Table 1: Values of AIC and BIC. ARIMA (p d q) 111 211 212 101

AIC 19.79 21.03 21.32 23.49

BIC 25.92 29.20 31.54 29.67

It is shown from the table that the most suitable model is ARIMA (1, 1, 1) this model has the lowest AIC and BIC values

Residual Analysis One of the indicators of the properly fitted model is that of scattered residuals in a rectangular shape around the zero at horizontal level. The time series plot of residuals of production data showed scatter trend, therefore, models were fitted properly by residual

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Non- Seasonal lag

AR1 MA1 Constant Number of Residual Number of parameters Residual Df Adjusted Residual sum of Squares Residual sum of Squares Residual variance Model std. Error Log – likelihood Akaike’s Information Criterion (AIC) Schwartz’s Bayesian Criterion (BIC)

Estimates -0.15554435 0.70368794 0.01569487

Std Error T 0.18045731 -0.8619454 0.13216211 5.3244301 0.00989606 1.5859713 57 2 54 4.2483160

Approx sig 0.39252919 0.00000202 0.11858501

0.07741866 027824208 -6.8987496 19.797499 25.926653

Table 3: Autocorrelations and partial autocorrelations of residuals. Lag Autocorrelation Std. Error Box-Ljung Statistic 1 -0.028 0.129 0.046 2 -0.077 0.128 0.413 3 0.053 0.127 0.589 4 0.041 0.126 0.695 5 0.146 0.124 2.077 6 -0.018 0.123 2.099 7 -0.052 0.122 2.279 8 -0.078 0.121 2.696 9 0.091 0.119 3.282 10 -0.100 0.118 4.005 11 -0.060 0.117 4.267 12 -0.026 0.116 4.318 13 -0.117 0.114 5.369 14 -0.006 0.113 5.371 15 0.050 0.112 5.571 16 -0.052 0.110 5.790

df 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 16.000

Sig. 0.830 0.813 0.899 0.952 0.838 0.910 0.943 0.952 0.952 0.947 0.961 0.977 0.966 0.980 0.986 0.990

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Lag Partial Autocorrelation Std. Error 1 -0.028 0.132 2 -0.078 0.132 3 0.049 0.132 4 0.038 0.132 5 0.158 0.132 6 -0.006 0.132 7 -0.034 0.132 8 -0.106 0.132 9 0.071 0.132 10 -0.132 0.132 11 -0.034 0.132 12 -0.041 0.132 13 -0.096 0.132 14 -0.034 0.132 15 0.077 0.132 16 -0.028 0.132

Figure 1: ACF of residual of fitted Figure 2: PACF of residual of fitted ARIMA model. ARIMA model.

Table 4: Forecasted values of Pigeon pea production in India up to year 2014-15. Year Estimated Production Lower CL Upper CL 2008-09 2.56797 2.00479 3.13115 2009-10 2.66730 2.09709 3.23752 2010-11 2.66999 2.07604 3.26394 2011-12 2.68771 2.07331 3.30210 2012-13 2.70309 2.06784 3.33833 2013-14 2.71883 2.06286 3.37480 2014-15 2.73452 2.05787 3.41116

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Forecasts of Pigeon pea production ARIMA (1,1,1) was taken for seven year ahead and forecasts for Pigeon pea production which are given in table 4 along with 95% confidence interval values. Forecasts of Pigeon pea production will increases to some extent in future i.e. 200809 is 2.49479 million tones up to the year 2014-2015 it will be accepted 2.73452 million tones. With lower and upper limits of 2.05787 million tones and 3.41116 million tones respectively.

Conclusions ARIMA model offers a good technique for predicting the magnitude of any variable. Its strength lies in the fact that the method is suitable for any time series with any pattern of change and it does not require the forecaster to choose a prior value of any parameter. Its limitations include requirement of a long time series. Often it is called a 13 ‘Black Box’ model. Like any other method, this technique also does not guarantee perfect forecasts. Nevertheless, it can be successfully used for forecasting long time series data. In our study ARIMA (1, 1, 1) model was best suited for estimation of Pigeon pea production data. From the forecast values obtained the developed model, it can be said that forecasted production will increases to some extent in future i.e. In 2008-09 production of Pigeon pea was 2.49479 million tones up to the year 2014-2015 it will be 2.74 million tones.

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