Fitting ARIMA Model and Forecasting for the Tea Production, and Internal Consumption of Tea (Per year) and Export of Tea

Every year, in Bangladesh the production has increased by 1.84 % and contributes 1.37 in export in the word tea trade and earns nearing 1775 million Taka. The internal consumers of the country are presently consuming about 98% of its produce. The consumption is increasing day by day mainly due to the rapid increase in population. This study considered the published secondary data of yearly tea production in Bangladesh over the period 1990 to 2015 by the Bangladesh Tea Board. We are forecasting the production of tea and internal consumption of tea by using Auto-Regressive Integrated Moving Average (ARIMA) for the next 5 years. Run test, Jarque and Bera test criteria are used for the adequacy of the fitted model which followed by the residual analysis. The comparison between the original series and forecasted series are showing the same trends which indicate the fitted model is statistically well and appropriate for the forecast the productions of tea in Bangladesh.


INTRODUCTION
Tea is one of the best export items of Bangladesh. But exporting tea of the country has been declined day by day for the calm growth of production rising consumption and strong competition from other tea exporting courtiers in the world. Bangladesh attester substantial changes for exporting tea to several countries over time . It is necessary to search new customers and explore an incursive policy for cope with similar changes in the world tea market. This paper gives several suggestions and examines present situation for raising tea export. Domestic tea consumption has increased steadily. Likely to remain this trend and even may increase day by day. In the face of rising domestic consumption, a sustained increase in tea production can be stable -level of tea export maintained . Various aspects of tea production and export have been dealt with, among others, by Khalid (1975), Majid (1991) and Sabur (2002). 9 UniversePG l www.universepg.com

Dickey-Fuller unit root test -Dickey and Fuller
suggested that under the null hypothesis the estimated coefficient of 1  t y in the model above follows the tau ( ) statistic is known as Dickey-Fuller test. Allowing the various possibilities DF test is estimated in three different forms, under different null hypothesis t y is a random walk:  is a pure white noise error and where In that procedure we compute t Ẑ by successive substitution for each value of the MA parameters and choose the set of values of the parameters that minimizes the error sum of squares  2 t Z . For ARMA models, both the AR and MA parts are to UniversePG l www.universepg.com be estimated with the procedure discussed above. are constants. The series is said as AR (p) process ARMA (p,q) process -The time series   t y is an ARMA (p,q) process if it stationary and satisfies for every t,  1 1 have no common factors. ARIMA (p,d,q) process -If the ARMA (p,q) process is also integrated process than it is known as ARMA process of order (p,d,q). Here d represents the number of difference the needed to be stationary. ARIMA process is used when data is non-stationary.

Diagnostic Checking
Analysis and Findings -A time series is told to be stationary if it's mean, variance and covariance function does not depend on time, and otherwise it is non-stationary. The time series plot of the data is given below :   Fig 1: Time series Plot of the data. UniversePG l www.universepg.com Comments: From the above time plot we see that the mean and variance of the tea production, internal consumption, export and export price of Bangladesh from 1990-2014 is not same over time. That is the mean and variance has changed over time. So from the properties of non-stationary time series we can tell that the above time series is nonstationary process. We know that in the time series analysis it is possible to make the non-stationary time series into stationary time series. In order to make the above non stationary time series into stationary process we have to make some differences of the tea production. Comments: From the above figure we see that variance are almost stable here and the production of tea changes equally over time i.e, mean and variance are about to same. So, we can ask that the series is Stationary.
The Auto-Correlation Function of ARMA -We get the following acf of i st difference of tea production, We get the following acf of i st difference of Export of tea,

Fig 4:
ACF of export of Tea after taking one difference.

The Partial Auto-Correlation Function of an ARMA
Comments: From the above figure of ACF curve of first difference on tea production is statistically significant at the lag 1. So we can mention that the production of tea follows AR (1) Process.
We get the following pacf of i st difference of tea production,

Fig 5:
PACF of tea production after taking one difference (thousand metric tons).
Comments: From the above figure of Partial autocorrelation function of tea production we can mention that the series is a Moving Average Process of order 0.
We get the following pacf of export of tea, UniversePG l www.universepg.com Comments: From the above figures it is evident that of ACF and PACF curve of first difference on tea production is Statistically significant at the lag 1.Therefore the time series of production of tea follows ARIMA (1,1.0). Again, From the above figures it is evident that of ACF and PACF curve of first difference on export tea from Bangladesh is Statistically significant at the lag 1.Therefore the time series of production of tea follows ARIMA (1,1.0). Comments: Since the null hypothesis is not accepted so we can say that the series of tea production after taking first differences is stationary.

Data = Export tea
Consider the hypothesis: At 5% level of significance we will conduct the test.
We get the following P-Value of the Dickey-Fuller test.
P-Value: 0.0106 This is less than level of significance (0.05) So, we are not accepting the null hypothesis.

Comments:
Since the null hypothesis is not accepted so we can ask that the series of export tea after taking first differences is stationary.
Augmented Dickey-Fuller Test -Consider the hypothesis, 0 H : The tea production is non-stationary.
1 H : The tea production is stationary.

Data: Tea production
Dickey-Fuller = -0.9184, Lag order = 1, p-value = 0.9326 At 5% level of significance the p-value of Augmented Dickey-Fuller Test is 0.9326, which is greater than 0.05. So the null hypothesis is not rejected.
That means, the tea production is non-stationary.
Augmented Dickey-Fuller Test -Consider the hypothesis, 0 H : The export of tea is non-stationary. 1 H : The export of tea is stationary.

Data: Export of Tea
Dickey-Fuller = -3.529, Lag order = 2, p-value = 0.04172 At 5% level of significance the p-value of Augmented Dickey-Fuller Test is 0.04172, which is smaller than 0.05. So the null hypothesis is rejected.
That means, the export of tea is stationary.
Fitted ARIMA Model for tea data -The fitted ARIMA (1,1,0) model for the production of tea which we find from R-Console is given below: The fitted ARIMA (1,1,0) model for export of tea which we find from R-Console is given below: X (t) = 14.486+1.975*X(t-1)

Diagnostic checking for tea production in
Bangladesh -At this time we have to perform diagnostic checking for the fitted model. Now from R3.0.2 we have found the diagnostic checking plot, which is given below: Comments: From this plot we can observe that the plot of the standardized residual and the ACF plots are correctly specified. From the plot of P-value we can view that all p-values are above the level of significance. So, we can interpret that the residual is white noise, hence the model is also correctly estimated.
Diagnostic checking for the internal consumption of tea -At this point we have to perform diagnostic checking for the fitted model. Now from R3.0.2 we have found the diagnostic checking plot, which is given below: Comments: From this plot we can observe that the plot of the standardized residual and the ACF plots are correctly specified. From the plot of P-value we can see that all p-values are above the level of significance. So we, can interpret that the residual is white noise, hence the model is also correctly estimated.
Diagnostic checking for the exported tea from Bangladesh -At this point we have to perform diagnostic checking for the fitted model. We have found the diagnostic checking plot, which is given below: Comments: From this plot we can notice that the plot of the standardized residual and the ACF plots are correctly specified. From the plot of P-value we can observe that all p-values are above the level of significance. So we, can interpret that the residual is white noise, hence the model is also correctly estimated. From the above analysis we may say that the production of tea and the amount of exported tea are non-stationary process.
After taking successive differences we have made all the non-stationary time series into stationary process. In these analysis, production of tea follows ARIMA (0,1,1) model, export of tea follows ARIMA (0,1,1) model. After diagnostic checking for all types of data of tea we have found that all the models are appropriate. UniversePG l www.universepg.com

CONCLUSION
The main concept of this work is forecasting a group of associated variables thus making use of all possible information such as correlation. The correlations among the export variables may be high or comparatively low . In this project, we focus better forecast which can give strong correlation among these associated export variables. In multivariate time series models, the vector autoregressive model is used for forecasting. For this we find out the forecasted values. We measure the forecasting accuracy by using mean error (ME), mean square error (MSE), Root Mean Squared Error (RMSE), mean absolute error (MAE), mean percent error (MPE), mean absolute percent error (MAPE), mean absolute percent error (MSPE) and so on. We also select a proper model for forecasting on the basis of model selection criterions such as AIC and BIC, HQ and SC of the model. The others highly correlated variables forecast in multivariate forecast are good. The forecasting accuracy ME, MSE, RMSE, MAE, MPE, MAPE and so on all are not so very distance in multivariate forecast. Finally we conclude that before forecasting we should check correlations among the variables, if existing highly correlation then multivariate time series is better than univariate time series. So all over we conclude that if we want to forecast, at first checking correlations among the variables. If there existing highly correlated variables then we should use multivariate forecast for these variables. If there do not exist highly correlated variables then we should use univariate forecast for these variables. Since our case study export variable are highly correlated to production and internal consumption for the selected export variable than univariate time series analysis using ARIMA model.