American Journal of Applied Mathematics
Volume 4, Issue 3, June 2016, Pages: 124-131

 Methodology Article

Application of Statistical Methods of Time-Series for Estimating and Forecasting the Wheat Series in Yemen (Production and Import)

Douaik Ahmed1, Youssfi Elkettan2, Abdulbakee Kasem2

1The National Institute of Agronomic Research (INRA), Rabat, Morocco

2Department of Mathematics Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco

Email address:

(D. Ahmed)
(Y. Elkettan)
(A. Kasem)

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:

2.3. White Noise

White noise is a stationary process such that:

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

Figure 1. Graph of time series of wheat(product and import)from 1961 to 2010.

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

Figure 2. Autocorrelation of time series of wheat(product and import).

Figure 3. Autocorrelation partial of time series of wheat(product and import).

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.

Figure 4. Dickey-Fuller test of time series of wheat product.

Figure 5. Dickey-Fuller test of time series of wheat import.

When we execute the first differences, we note of figure 6 and 7 that a stationary series.

Figure 6. Dickey-Fuller test of first differences of wheat product.

Figure 7. Dickey-Fuller test of first differences of wheat import.

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.

Figure 8. Cooparison of model.

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:

Figure 9. Forecasting of series of the wheat (product and import).

Figure 10. Graph forecasting of series of the wheat (product and import).

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.


References

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  2. Philippe Marier, cours Prévision de la demande, Consortium de Recherche Université Laval.
  3. Lahoussaine Baamal, (2012), Cours d’analyse des Séries Chronologiques.Université Ibn Tofail,Kénitra.
  4. E. Ostertagova,O. Ostertag,The Simple Exponential Smoothing Model, http://www.researchgate.net/publication/256088917
  5. Rodolphe Palm,(2007), Etude des séries chronologiques par méthodes de lissage, Faculté Universitaire des Sciences agronomiques, Unité de Statistique, Informatique et Mathématique appliquées, Belgique.
  6. Bary Adnan Majed, (2002), Statistical forecasting methods-1, Université du Roi Saoud.
  7. Ruey S. Tsay, (2005), Analysis of financial time series, Wiley,Hoboken, New Jersey.
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