Methodology Article
Application of Statistical Methods of Time-Series for Estimating and Forecasting the Wheat Series in Yemen (Production and Import)
Douaik Ahmed^{1}, Youssfi Elkettan^{2}, Abdulbakee Kasem^{2}
^{1}The National Institute of Agronomic Research (INRA), Rabat, Morocco
^{2}Department of Mathematics Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco
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To cite this article:
Douaik Ahmed, Youssfi Elkettan, Abdulbakee Kasem. Application of Statistical Methods of Time-Series for Estimating and Forecasting the Wheat Series in Yemen (Production and Import). American Journal of Applied Mathematics. Vol. 4, No. 3, 2016, pp. 124-131. doi: 10.11648/j.ajam.20160403.12
Received: April 4, 2016; Accepted: April 15, 2016; Published: May 4, 2016
Abstract: Due to the importance of the wheat crop which represents 90% of the grain consumed, In this papers, we compared between the following statistical methods : Box and Jenkins model, exponential smoothing models (with trend and without seasonal) and Simple regression for estimating and forecasting to two time series of wheat(production and import). We reached to the following results: 1. Brown exponential smoothing model for modeling the imported wheat series. 2. ARIMA (1, 1, 1) model for modeling the product wheat series. For the wheat crop, the ratio of production to consumption is expected to reach 6.3% in 2015 and continues to decline even up to 5.4% in 2020. This means that the problem of food security well be worse in Yemen.
Keywords: Time Series, Wheat Crop, Forecasting, Box and Jenkins, Exponential Smoothing
1. Introduction
The human needs to know the past in order to predict the future to find optimal solutions of many problems which face humanity in this century. Yemen is one of the Arab countries where local demand for food is growing exponentially. Therefore, it suffers from a huge lack to cover all the population needs of foodstuffs especially wheat which represents staple food of most the population. Although in recent years the amount production of wheat compared with imported wheat reach in 2010 to 92%. According to what has mention above we compered these statistical methods of time series: Box and Jenkins methodology ,exponential smoothing model nd Simple regression to estimate and forecast the two wheat time series (import,product) from 1961 to 2010 of the Organization’s site of Food and Agriculture (FAO) and the Central Bureau of Statistics in Yemen. We used these programmes , and .
2. Theoretical Formulation
2.1. Holt and Brown’s Exponential Smoothing Method
In the case where the series has a trend, we can adopt the following prediction formula:
The values and are constantly updated by the following equations:
and
This forecast model is known as the model name of HOLT. A special case of model HOLT, called model BROWN or dual exponential smoothing is obtained when the smoothing constants and are related to the same parameter , by the relations: et . For these two models, we need to give initial values and to produce forecasts. Thus we take which equals the coefficient simple linear regression calculated on the basis of the first five values of the series. Thereafter, is deduced by the relation:, as the smoothing constants they are set by the user. In practice often gives a value to between 0.01 and 0.30. [3]
2.2. Stationary Process
A second process is stationary if:
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2.3. White Noise
White noise is a stationary process such that:
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This notion of white noise corresponds to the usual assumptions on residues in multiple regression. Random variables are also called random shocks. we implicitly assumes that random shocks follow a normal distribution
2.4. Autocorrelation
The autocorrelation function is the application of in defined by:
Measuring the correlation between and because:
2.5. Autocorrelation Partial
The partial autocorrelation function with delay is defined as the partial correlation coefficient between et the influence of other variables shifted by periods have been withdrawn.
2.6. Autoregressive Process AR(p)
Let a process , is said autoregressive process of order if
Where , white noise and are constants.
2.7. Moving Average Process (MA(Q))
Let a process , is said autoregressive process of order if
Where , white noise and are constants.
2.8. Moving Average Processes Autoregressive
A stationary process has an representation Minimum if it satisfies:
where
the polynomials and have their upper modules strictly roots to 1.
and have not common roots.
is a white noise of variance
2.9. Process ARIMA
A process is a process [Autoregressive integrated moving average] if it satisfies an equation of type :
where constant and is white noise.
2.10. Augmented Dickey–Fuller Test
The Augmented Dickey–Fuller test is a unit root test of the null hypothesis of unit root (or non stationarity). The test estimated three models:
The null hypothesis of test is the unit root hypothesis of the variable is the hypothesis . The test consists of comparing the estimated value Student associated with the parameter to the tabulated values of this statistic. The values tabulated for different test however tabulated values of Student test. The critical values of this statistic, denoted in the following, are given by MacKinnon (1996). The null hypothesis of non-stationary of the time series is rejected at the 5% level when the observed value of the Student’s t-test is less than the critical value tabulated by MacKinnon (1996) or
2.11. Box Jenkins Methodology
This is the technique for select the most appropriate or model for a given variable. It comprises four steps:
1. Identification of the model, this involves selecting the most appropriate lags for the AR and MA parts, as well as selecting if the variable requires first-differencing to become stationarity. The and are used to identify the best model. (Information criteria can also be used)
2. Estimation, this usually involves the use of a least squares estimation process.
3. Diagnostic testing, which usually is the test for autocorrelation. If this part is failed then the process returns to the identification section and begins again, usually by the addition of extra variables.
4. Forecasting, the models are particularly useful for forecasting due to the use of lagged variables.
3. Application
3.1. Graph Series
Through the graph figure, we observed a general upward trend over the period, this means that the series is not stationary.
3.2. Autocorrelation and Autocorrelation Partial
We examine the autocorrelation and partial autocorrelation function in figures 2 and 3 we observed that the estimated autocorrelation parameter decreases exponentially towards zero while that only the first partial autocorrelation parameter is not significant. To confirm the previous results we execute the Dickey-Fuller test and observed in Figure 4 and 5 that the series is not stationary.
When we execute the first differences, we note of figure 6 and 7 that a stationary series.
we note that the series is stationary. We deduce that d = 1 in the ARIMA model (p, d, q).
3.3. Identification and Selction of Model for Wheat Production Series
Although it appears that each partial autocorrelation parameter after the second parameter is not significantly different from zero at a = 0.05 but the autocorrelation function is gradually decreasing towards zero, this may be sufficient evidence that the random process is AR (1). For ensure we test the following statistical hypothesis:; , , , we deduce that the first partial autocorrelation parameter is not significantly different from zero at . We examining the autocorrelation partial parameters, we find that , that supports the possibility of using the AR (1) and therefore .
For import wheat series we get the same results. And then we compared between ARIMA models with the exponential smoothing(Holt,Brown)and simple regression. We get the following results.
We take 40 observation of the original series and forecast for the next ten years, then compare between models by and choose the best model. The results were as follows:
1. Brown’s exponential smoothing model for predict the series of wheat production.
2. The model for predict the series of wheat exports.
3.4. Tests of Residues
We test the best model:
• Graphic residues confidence limits,
• Graphic dispersion of points in parallel form residuals around zero
• Ljung-Box value is significant
• If the model realizes the previous tests, we use it to forecast.
3.5. Forecasting
Then we use the previous models to calculate the forecast from 2011 to 2020 and the results were as follows:
4. Conclusion
Wheat imports will increase from 2.9 million tonnes in 2011 to 4 million tonnes in 2020, where the proportion of imports was 92% in 2010 and it is expect that the wheat import proportion will increase to 94% in 2020. whereas, wheat production will drop by 6% during this decade.
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