There are a lot of study have been done by the researcher to fit an ARIMA model in the agriculture sector in all over the world for different types of agricultural crops. ARIMA model is used in different agriculture sector to forecast agricultural productions. The relevant work for forecasting by using Box-Jenkins (1970) ARMA model, from which we get the idea about forecasting techniques for different types of agricultural productions forecasting such as Goodwin and Ker (1998) added new dimensions to the evolution of this literature. They introduced a univariate filtering model, an ARIMA (0, 1,2) to best represent crop yield series. Mohammed Amir Hamjah (2014) has used Box-Jenkins ARIMA model to forecast different types of Seasonal rice productions in Bangladesh. From his study, it was found that the best selected ARIMA model for Aus productions is ARIMA (2,1,2), for Aman productions is ARIMA (2,1,2) and, for Boro productions is ARIMA (1,1,3). Rachana et al. (2010), used ARIMA models to forecast pigeon pea production in India. Badmus and Ariyo ARIMA (1,1,1) and ARIMA (2,1,2) for cultivation area and production resrespectively. Falak and Eatzaz (2008), analyzed future prospects of wheat production in Pakistan. Applying ARIMA model. Hossian et. al. (2006) forecasted three different varieties of pulse prices namely motor, mash and mung in Bangladesh with monthly data from Jan 1998 to Dec 2000; Wankhade et al. (2010) forecasted pigeon pea production in India with annual data from 1950- 1951 to 2007-2008; Mandal (2005) forecasted sugarcane production. Rahman (2010) fitted an ARIMA model for forecasting Boro rice production in Bangladesh. M. A. Awal and M.A.B. Siddique’s study was carried out to estimate growth pattern and also examine the best ARIMA model to efficiently forecasting Aus, Aman and Boro rice productions in Bangladesh. Nasiru Suleman and Solomon Sarpong (2011) made a paper with the title “Forecasting Milled Rice Production in Ghana Using Box-Jenkins Approach”. The analysis is revealed that ARIMA (2, 1, 0) is the best model for forecasting milled rice production.
4. Data Source and Used Software The crop data-sets are available from Bangladesh Agricultural Ministry’s website named as www.moa.gov.bd. This analysis has completely done by statistical programming based open source Software named as R with the version 2.15.1. The additional library packages used for analysis are forecast and tseries.
5. Methodology A time series is a set of numbers that measures the status of some activity over time. It is the historical record of some activity, with measurements taken at equally spaced intervals with a consistency in the activity and the method of measurement.
Box and Jenkins procedure’s steps i. Preliminary analysis: create conditions such that the data at hand can be considered as the realization of a stationary stochastic process. ii. Identification: specify the orders p, d, q of the ARIMA model so that it is clear the number of parameters to estimate. Recognizing the behavior of empirical autocorrelation functions plays an extremely important role. iii. Estimate: efficient, consistent, sufficient estimate of the parameters of the ARIMA model (maximum likelihood estimator). iv. Diagnostics: check if the model is a good one using tests on the parameters and residuals of the model. Note that also when the model is rejected, still this is a very useful step to obtain information to improve the model. v. Usage of the model: if the model passes the diagnostics step, then it can be used to interpret a phenomenon, forecast.
5.4. Procedure of Maximum Likelihood Estimation (MLE) Method The advantage of the method of maximum likelihood is that all of the information in the data is used rather than just the first and second moments, as is the case with least squares. Another advantage is that many large-sample results are known under very general conditions. For any set of observations, Y1, Y2, …,Yn time series or not, the likelihood function L is defined to be the joint probability density of obtaining the data actually observed. However, it is considered as a function of the unknown parameters in the model with the observed data held fixed. For ARIMA models, L will be a function of the ’s, θ’s, µ, and given the observations Y1, Y2, …,Yn. The maximum likelihood estimators are then defined as those values of the parameters for which the data actually observed are most likely, that is, the values that maximize the likelihood function.