# Forecasting of Paddy Production in Sri Lanka: A Time Series Analysis using ARIMA Model

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3 Forecasting of Paddy Production in Sri Lanka METHODOLOGY The econometric model of Autoregressive Integrated Moving Average (ARIMA) was used to fit the data set which is complementary to the trend regression approach and forecasting of the concerned variable to the near future (Box et al., 994). ARIMA (Box-Jenkins model) method is an extrapolation method for forecasting and, like any other such method, it requires only the historical time series data for the variable under forecasting. Among the extrapolation methods, this is one of the most sophisticated methods, as it incorporates the features of all such methods, does not require the investigator to choose the initial values of any variable and values of various parameters a priori and it is robust to handle any data pattern. As one would expect, this is quite a difficult model to develop and apply as it involves consecutive four stages such as Identification process, Estimation, Diagnostic testing and Forecasting (Janacek and Swift, 993). Theoretical basis of Time-series analysis A time series is a set of values of a continuous variable Y (Y, Y2,..., Yn), ordered according to a discrete index variable t (, 2,..., n). The term time-series comes from econometric studies in which the index variable refers to intervals of time measured in a suitable scale. In general, in a given time series the following can be recognized and separated into the following components. ) A regular, long-term component of variability, termed trend, that represents the whole evolution pattern of the series; 2) A regular, short-term component whose shape occurs periodically at intervals of lags of the index variable, currently known as seasonality, because this term is also derived by applications in economics; 3) An AR(p) autoregressive component of p order, which relates each value Zt =Yt (trend and seasonality) to the p previous Z values, according to the following linear relationship. () where Ф i (i=,..., p) is the parameter to be estimated and t is a residual term; and 4) A MA(q) moving average component of q order, which relates each Zt value to the q residuals of the q previous Z estimates where θi (i=,..., q) parameter to be estimated... (2) In mathematical terms, therefore, a seasonal ARIMA model is an ARIMA (p, d, q) model whose residuals εt can be further modeled by an ARIMA (P, D, Q)s structure with linear operators (P, D, Q) being functions of the Bs operator. Since annual value of the concerned variable was used, univariate, non-seasonal ARIMA (p, d, q) was used to fit the data sets. Data and data analysis Time series forecasting analysis utilized the secondary data of the Department of Census and Statistics of Sri Lanka for the time period of 952 to 2. Time series forecasting analysis 23

4 Sivapathasundaram and Bogahawatte was carried out using statistical software of SPSS. The time series analysis was done to fulfill the objective of detecting the trend and predicting future changes of paddy production for the three leading years in Sri Lanka by developing a model according to the behavior of the data. The econometric model of ARIMA was used to fit the data set. Since annual value of the concerned variable was used, a univariate non-seasonal ARIMA (p, d, q) was estimated. Modeling was done by four stages such as Identification process, Estimation, Diagnostic testing and Forecasting. Identification process Annual values were plotted to identify the variation of the data (i.e. trend, seasonality, cyclical and extreme events). Autocorrelation function was used to identify the stationarity of the series (µ = and σ 2 constant throughout the observation), non-stationarity series was converted to stationary by taking the differences (i.e. st difference of Y t+ Y t, 2 nd difference of Y t+2 Y t or first difference of the st differenced series). All data were checked for the pure randomness (White noise), hypothesis of randomness. The null hypothesis, H: No autocorrelation up to a given lag significantly differs from zero. That type of series did not fit ARIMA. Tentative AR/MA/ARMA/ARIMA was selected using Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF). Parameter estimation Both AR and MA parameters were estimated. The Null Hypothesis for selection of the parameter was: H: All considered parameters should be elaborating the condition with the value is between - to +. Various non-seasonal ARIMA models were tried and each of the parameters were estimated. Diagnostic checking Adequacy of the model was checked by using Box Pierce (Ljung Box) Chi Square statistic residuals with null hypothesis of H: Residuals follow white noise series and H was rejected at P<.5. Sum of Squares residual of the best model was of the lower value. AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) also used to select the best model. Normal probability value was taken to check the residuals. Forecasting Selected best fitting model was used to forecast the future data for the three leading years. Simple linear regression model was fitted to auto correlated annual series to determine coefficients and trend. RESULTS AND DISCUSSION ARIMA model was estimated only after transforming the variable under forecasting into a stationary series whose values vary over time only around a constant mean and constant variance. The most common method is to check stationarity through examining the graph or time plot of the data. Fig. revealed that the data was non-stationary. Non-stationarity in mean was corrected through appropriate differencing of the data. In this case difference of order was sufficient to achieve stationarity in mean. 24

5 Forecasting of Paddy Production in Sri Lanka The newly constructed variable X was examined for stationarity. The graph of X was stationary in mean. The next step was to identify the values of p and q. For this, the autocorrelation and partial autocorrelation coefficients of various orders of X were computed (Table 2). The ACF and PACF (Fig. 2 and 3) showed that the order of p and q can at most be. Then entertained three tentative ARIMA models were used to choose the model which has the minimum AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). The models were ARIMA (2,, ), ARIMA (2,, ) and ARIMA (2,, 2). Among the above models, ARIMA (2,, ) was the most suitable model as this model has the lowest AIC and BIC values. 4 Production ( Mt) Year Fig.. Time plot of paddy production data ACF of st Difference Data (With 5% significance for the autocorrelation)..5 Coefficient Upper Confidence Limit Lower Confidence Limit ACF Lag Number Fig. 2. ACF of differenced data 25

6 Sivapathasundaram and Bogahawatte PACF of st Differenced Data (With 5% significance for the autocorrelation) Partial ACF Coefficient Upper Confidence Limit Lower Confidence Limit Lag Number Fig. 3. PACF of differenced data Table 2. Autocorrelations and partial autocorrelations Lag Autocorrelation Std. Error Lag Partial Autocorrelation Std. Error Model estimation and verification Model parameters were estimated using SPSS package. Results of estimation are reported in Table 3. The model verification is concerned with checking the residuals of the model to see if they contain any systematic pattern which still can be removed to improve on the chosen ARIMA. This is done through examining the autocorrelations and partial autocorrelations of the residuals of various orders. For this purpose, the various correlations up to 4 lags were computed and the same along with their significance was tested by using Box-Ljung test and the results are shown in Table 3. Results indicated that none of these correlations were significantly different from zero at a reasonable level. This proved that the selected ARIMA model is an appropriate model. The ACF and PACF of the residuals (Fig. 4 and 5) also 26

7 Forecasting of Paddy Production in Sri Lanka indicated goodness of fit of the model. So the fitted ARIMA model for the production data (Equation 3), Z t = Z t-.669 Z t Z t-3 +ε t..(3) Table 3. Estimates of the ARIMA model Estimates Std Error t Approx Significance Nonseasonal Lags AR AR Constant Number of Residuals 58 Number of parameters 2 Residual Df 55 Adjusted Residual SS Residual SS Residual Variance Model Std. Error Log-Likelihood -4.3 Akaike s Information 88.6 Criterion (AIC) Schwarz s Bayesian Criterion (BIC) Coefficient Upper Confidence Limit Lower Confidence Limit.5 ACF Lag Number Fig. 4. ACF of residuals 27

8 Sivapathasundaram and Bogahawatte Partial ACF Lag Coefficient Upper Confidence Limit Lower Confidence Limit Fig. 5. PACF of residuals Forecasting with ARIMA model ARIMA models are developed basically to forecast the corresponding variable. There are two kinds of forecasts: sample period forecasts and post-sample period forecasts. The former are used to develop confidence in the model and the later to generate genuine forecasts for use in planning and other purposes. The ARIMA model can be used to yield both these kinds of forecasts. Sample period forecasts: The sample period forecasts were obtained simply by plugging the actual values of the explanatory variables in the estimated Equation 3.The explanatory variables here were the lagged values of Zt and the estimated lagged errors. To judge the forecasting ability of the fitted ARIMA model, important measures of the sample period forecasts accuracy were computed. The Mean Absolute Percentage Error (MAPE) for paddy production in Sri Lanka was.5. This measure indicated that the forecasting accuracy was high. Post sample forecasts: The principal objective of developing an ARIMA model for a variable is to generate post sample period forecasts for that variable. This was done through using Equation 3. The forecasts for paddy production during 2 to 23 are given in Table 4 where it has the increasing trend but it has decreased for the year 2 and 22 and increased for the year 23 compared to 2. Fig. 6 and 7 showed the time series plot and trend analysis plot of the forecasted paddy production respectively. Table 4. Estimated values of paddy production and 95% confidence limits (CL) Year Estimated Production ( Mt) Lower CL Upper CL

9 Forecasting of Paddy Production in Sri Lanka Time Series Plot for Production (with forecasts and their 95% confidence limits) Production ( Mt) Time (Years) Fig. 6. Time series plot of the paddy production Trend Analysis Plot Linear Trend Model Yt = *t Production ( Mt) o c ti o r Variable Actual Fits Forecasts Accuracy Measures MAPE.5 MAD 78.3 MSD Year Fig. 7. Trend analysis plot of the paddy production CONCLUSIONS The developed model for paddy production was found to be ARIMA (2,, ). From the forecast available by using the developed model, it can be seen that forecasted production for the years 2 and 22 was lower than in 2 but in later year in 23 the production was higher. Overall it has an increasing trend. The validity of the forecasted values can be checked when the data for the lead periods become available. The model can be used by researchers for forecasting of paddy production in Sri Lanka. However, it should be updated continuously with the incorporation of recent data. 29

10 Sivapathasundaram and Bogahawatte REFERENCES Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (994). Time Series Analysis: Forecasting and Control. (3 rd ed.). Prentice Hall Press, Englewood cliffs, New Jersey. Central Bank of Sri Lanka. (2). Annual Report, Colombo, Central Bank of Sri Lanka. Department of Census and Statistics, Agriculture Census, 2 [on line]. [Accessed on..2] Available at De Silva, D.A.M. and Yamao, M. (29). Rice pinch to war thrown nation: an overview of the rice supply chain of Sri Lanka and the consumer attitudes on government rice risk management. J. of Agric. Sci. 4(2), Dayaratna Banda, O.G., Jayawickrama, J.M.A. and Ranathilaka, M.B. (28). Sense and nonsense of rice price controls in Sri Lanka. The Peradeniya J. of Economics. 2(&2), Janacek, G.J. and Swift, L. (993). Time Series: Forecasting, Simulation, Applications. Ellis Horwood, New York, London, Partheepan, K. and Jeyakumar, P. (25). Development of a time series model to forecast climate data in Batticaloa district, Sri Lanka. Proceedings of the Water Professional Day, Postgraduate Institute of Agriculture, Peradeniya. October, 25. pp Raghavender, M. (2). Forecasting paddy production in Andhra Pradesh with ARIMA model. Int. J. of Agric. and Stat. Sci. 6(), Thattil, R.O. and Walisinghe, W.M.P.K. (23). Forecasting paddy yields. In: Abeysiriwardena, D.S. de Z., Dissanayaka, D.M.N. and Nugaliyadde, L. (Ed.) Rice Congress 2. Department of Agriculture, Sri Lanka. Weerahewa, J., Kodithuwakku, K.A.S.S. and Ariyawardana, A. (2). The fertilizer subsidy program in Sri Lanka. 2 pp. In: Andersen, P.P. (Ed.) Food Policy for Developing Countries: The Role of Government in the Global Food System. Ithaca, New York. 3

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