Forecasting for the Imported Weight of Equipment to Cargo of Sulaimani International Airport

Handreen Tahir Abdulla1,Renas Abubaker Ahmed2, Aras Jalal Mhamad3,4* 1Department of IT and Informatics – College of Commerce – University of Sulaimani, Sulaymaniyah City, Iraq, 2Department of Statistic and Informatics – College of Administration & Economics-University of Sulaimani, Sulaymaniyah City, Iraq, 3Department of Statistic and Informatics – College of Administration & Economics-University of Sulaimani, Sulaymaniyah City, Iraq, 4Accounting Department – College of Administration & Economics-University of Human Development, Sulaymaniyah City, Iraq. *Email: HYPERLINK “mailto:[email protected]” [email protected]

Abstract

The problem of a weight of imported equipment in the airport with their effects on economic situation is one of the most important problems that challenges faced in the airports of the region especially international Sulaymaniyah airport due to the increasing air travels as well as the increasing corruption in the most governmental section. This study aims to analyze the time series of weight of imported equipment in international Sulaymaniyah airport for the period between (Jan; 2010 to Nov; 2017) using the modern style in analyzing the time series which is (Box-Jenkins) method for the accuracy and flexibility it has in addition to its high efficiency in analyzing the time series. In this study, we are interested in forecasting the weight of imported equipment of international Sulaymaniyah airport using Box- Jenkins method. The results of this study showed that the suitable and efficient model to represent the data of the time series according to AIC, RMSE, MAPE and MAE criteria with the smallest values is the seasonal model of lag 12 (SARIMA (1,1,0)×(1,0,1)12).

According to the results of SARIMA (1,1,0)×(1,0,1)12, the amounts of the weight of imported equipment of international Sulaymaniyah airport have been forecasted for the period from Nov; 2017 to Oct; 2018 (the forecasting was done for 12 months). Those values showed harmony with their counterparts in the original time series. It provided us with a future image of the reality of the monthly weight of imported equipment of international Sulaymaniyah airport.

Finally, we recommend decision makers and the interested people to adapt to formulate strategic plan depend mainly on the scientific method in forecasting of the monthly weight of imported equipment since there is a real problem facing international Sulaymaniyah airport throughout the upcoming years.

Key words: Time Series Analysis, SARIMA model, Forecasting

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1. Introduction

Economic problems have an effect on our region (Kurdistan) especially Sulaymaniyah province such as another country in the world, in our region the airports are one of the most important centers which improve some of these problems according to their incomes, in addition, the statistical tools could be analyzed these problems especially when time is a significant factor in them. Time series analysis is one of the powerful statistical tools that is used to forecasting the weight of imported equipment which are causes of changing the economic situation in our region, therefore the main objective of this study is to forecasting of weight of imported equipment and determining their economic effects according to their weights by using seasonal autoregressive moving average model, thus these weight of imported equipment are reasons for improving economic situation at Sulaymaniyah province.

2. Materials and Methods

2.1 Time series Analysis

A time series is a sequence of data variables, which is consisting of successive observations on a quantifiable variable(s), that is making an over a time interval 6. Usually, the observations are chronological and taken at regular intervals (days, months, years). Time series data are also often seen naturally in many field areas including; (Economics, Finance, Environmental, and Medicine) Time series can be represented as a set of observations X_t, each one being recorded at a specific time t 6, and written as:

{X1, X2, …, Xt } or {XT}, where T = 1, 2,…t

and Xt is the value of X at time t, then the goal is to create a model of the form:

Xt=f(Xt-1, Xt-2, …, Xt-n)+et ………………. (2.1)

Where Xt-1 is Xt variable for values of lag 1 that is the previous observations value, Xt-2 is the Xt variable for values of lag 2 means two observations value ago, etc., and it represents noise value which doesn’t follow a pattern of predictable. The Xt value is usually highly correlated with Xt-cycle if a time series is following a pattern repeating, where the cycle was an observations number in a regular cycle 8.

2.2 Component of Time Series

A basic step in choosing an appropriate model and forecasting procedure to a time series is to consider the type of data patterns exhibited from the time series plots. Traditional methods of time series analysis are mainly concerned with decomposing the variation in series 11.The sources of variation in terms of patterns in the time series data are mostly classified into four main components. The components are:

2.2.1 Trend

A time series data may show upward or downward trend for a period of years and this may be due to factors such as an increase in population, change in technological progress, large-scale shift in consumer demands, etc. The increase or decrease in the movements of a time series is called Secular trend 4.

2.2.2 Seasonal Variation

In a time series the variations of seasonal are a short-term fluctuation that occurs in a year periodically, then it repeats year after year. The customs of people or conditions are responsible for a pattern of repetitive of seasonal variations as a major factor. In general, seasonality is defined as a pattern that repeats itself over fixed intervals of time 4.

2.2.3 Cyclical Variations

The changes in the economic cycles from the rise and fall beyond the year and statement of the function of the pocket or sinus with a difference in length and capacity and includes several five stages in the full cycle is the initial rise – retreat – recession – recovery – (8-10 years) due to many factors such as government policy, international relations, etc. The length of the trade cycle is measured over the period of time between successive stages of successive boom or recession. The following figure shows a model. 4.

2.2.4 Irregular Variation

These changes are irregular to the time series movements of the top and bottom after excluding the other changes and the general trend. These changes are caused by uncontrollable factors such as earthquakes, volcanoes, floods, wars, bank bankruptcy and the like. It is obviously unpredictable for irregularity and for the small time period it is easy to influence when studying the other elements of the time series and is often referred to as Residual Variations because they include the remaining factors that are not referred to in the three elements of the above series. Or chance, and the following figure shows a model of random change. 4.

2.3 Models for Stationary Time Series

2.3.1 Stationary Series:

Stationary series is known as having a constant mean and variance during passing time, which means the series is in a balanced statistically situation. There are two types of stationary series:-

Perfect stationary: it is a series that describes in having possibility attribute that change in time does not effect on its pattern, also maybe there is a perfect stationary series if there is no trend increasing or decreasing in the series 10.

Weakly stationary: this type of stationary means that the series has a constant mean with a variable variance of series gaps.

2.3.2 Augmented Dickey–Fuller test

Augmented Dickey–Fuller tests the hypothesis which is stated that the series is not stationary, can be tested in regression equation 9.

?Xt=?0+at+?1Xt-1+i=1p?i?Xt-i+?t ………………. (2.2)Where a random walk, at=at-1+a?t is allowed 9.

2.3.3 Autocorrelation Function (ACF)

The autocorrelation function measures the degree of correlation between neighboring observations in a time series. The autocorrelation coefficient is estimated from sample observation using the formula 10:

?k=r-2n(Xt-?x)(Xt+k-?x)r=1n(Xt-?x)2 ………………. (2.3)Thus, the autocorrelation function at lag k is defined as:

?k=?k?0 , k=0,±1, ±2, …2.3.4 Partial Autocorrelation Function (PACF)

The partial autocorrelation function at lag k is the correlation between Xt and Xt-k after removing the effect of the intervening variables Xt-1, Xt-2, ….., Xt-k+1 which locate within (t, t-k) period, partial autocorrelation function will be donated by ???, PACF is calculated by iteration 17.

?00=1 ?11=?1?kk=?kk-j=1k-1?k-1,j?k-j1-j=1k-1?k-1,j?j , k=2,3,… ………………. (2.4)Therefore ?kj=?k-1,j-?kk?k-1, k-1 , j=1,2, …., k-12.4 Box-Jenkins Models:

This is a methodology that George-Box and Gwilyn Jenkins at 1970 applied to time series data. Box and Jenkins popularized an approach that combines the moving average and the autoregressive approaches 3. A Box-Jenkins model explains that the time series is stationary or not. Box and Jenkins is recommended that the non-stationary differencing one or more times series to obtain stationarity, with the “I” standing for “Integrated” of an ARIMA model. A Box-Jenkins methodology is a powerful approach to the solution of many time series analysis problems 18. This methodology depends on parts of procedure which is autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) that can be explained as follow:

Autoregressive (AR) model

The order of this type model depends on the number of the significant partial autocorrelation function (PACF), its order is denoted by (P), the AR model can be written as follow 1, 10:

Xt=?1Xt-1+?2Xt-2+ . . . + ?pXt-p+ at ……………. (2.5)

By using back shift operator equation (2.6) can be rewrite as follow:

?p(?p)Xt = at …………………. (2.6)

Where: ?p?p=( 1 – ?1?1 – . . . – ?p?p) Xt : is the origin series.

at : is white noise, at ~ N (0,?a2)

?p : is the estimated PACF.

To find Variance-Covariance the equation (2.6) should be multiplied by (Xt-k) and taking expectation so we get:

EXtXt-k=E(?1Xt-1Xt-k+ ?2Xt-2Xt-k + . . . + ?pXt-pXt-k+atXt-k ) ..…(2.7)

Note:

E(XtXt-k ) = ?k .

E(atXt-k) = 0 .

Then:

?k = ?1?k-1+?2?k-2 + . . . +?p?k-p ; K ; 0 …..…………. (2.8)

To get the ACF the equation (2.8) should be divided by the variance of the series (?o).

Pk = ?1Pk-1+ ?2Pk-2 + . . . + ?pPk-p ……………. (2.9)

Note:

?k?o = Pk, ?o = ?X2

Then the PACF for the AR(P) model can be estimated by using Yule-Walker equations

Pj = ?kPj-1+ ?k(k-1)Pj-2 + . . . + ?kkPj-p ……………. (2.10)

Moving Average (MA) Model:

The order of moving average model depends on the number of significant ACF and –?1 is the coefficient of dependency of observations (Xt) on the error term et and the previous error term et-1, the MA(q) model can be write as follow 10:

Xt=at-?1at-1-?2at-2- . . . – ?qat-q ……………. (2.11)

Equation (2.11) can be rewrite with back shift operator as follow:

Xt=?(B)at …………………. (2.12)

Where:

?B= 1 – ?1? – . . . – ?q?qThe Var-Cov of MA(q) model is:

?k = ?a 2(-?k+ ?1 ?k+1+ . . . + ?q ?q-k );K=1, 2, …,q0;K;q ……. (2.13)

?o = ?a 2 i=aq?i2 …………………… (2.14)

Note:

?0 = 1.

And the ACF is:

Pk = -?k+ ?1 ?k+1+ . . . + ?q ?q-k1+?12+?22+…+?q2;k=1,2,…,q0;k;q ………… (2.15)

Autoregressive Moving Average Model (ARMA)

There is large family of models which is named “Autoregressive-Moving Average Models” and abbreviated by ARMA. Many of researches in different application fields prove that ARMA models fits more than other traditional methods for forecasting 10. The ARMA model is a more general model as a mixture of the AR(p) and MA(q) models and it is called an autoregressive moving average model (ARMA) of order (p,q). The ARMA(p,q) is given by 19:

?p(B)Xt=?q(B)at …..………….. 2.16

Where:

?pB= 1 – ?1? – . . . – ?p?p?qB= 1 – ?1? – . . . – ?q?qWe write equation (2.16) as:

Xt=?1Xt-1+. . . + ?pXt-p+ at-?1at-1 – . . . – ?qat-q …(2.17)

2.4.4 The Autoregressive Integrated Moving Average Models (ARIMA)

If a non-stationary time series which has variation in the mean is differenced to remove the variation the resulting time series is called an integrated time series. It is called an integrated model because the stationary model which is fitted to the differenced data has to be summed or integrated to provide a model for the non-stationary data. Notationally, all AR(p) and MA(q) models can be represented as ARIMA(1,0,0) that is no differencing and no MA part. The general model is ARIMA (p,d,q) where p is the order of the AR part, d is the degree of differencing and q is the order of the MA part 11.

Wt= ?dXt= (1-B)dXtThe general ARIMA process is of the form

Xt=t=1p?iXt-1+t=1q?iat-1+?+at ………………. (2.18)2.5 Estimating the Parameters of an ARMA Model

The procedure for estimating the parameters of the ARMA model is an iterative method. The residual sum of squares is calculated at every point on a suitable grid of the parameter values, and the values that give the minimum sum of squares are the estimates. For an ARMA(1,1) the model is given by

Xt-?= ?1(Xt-1-?)at+?1at-1 ………………. (2.19)

Given N observation X1,X2,…,XN, we guess values for ?, ?1, ?1, set a0=0 and Y0=0 and then calculate the residuals recursively by

a1=X1-?a2=X2-?-?1X1-?-?1 aN=XN-?-?1X1-?-?1aN-1The residual sum of squares t=1Nat2 is calculated. Then other values of ?, ?1, ?1, are tried until the minimum residual sum of squares is found 7, 12.

Note: It has been found that most of the stationary time series occurring in practices can be fitted by AR(1), AR(2), MA(1), MA(2), ARMA(1,1) or white noise models that are customarily needed in practice 10.

2.6 Seasonal Time Series Models

Seasonal time series refers to similar or almost similar patterns which a time series appears to follow during corresponding months of successive years. The movement is usually due to the recurring events which takes place annually or quarterly as the case may be. The plot of the series against time checks for non-seasonal or seasonal changes that reveal non-stationarity. Peaks in a series for every 12 months would indicate annual seasonality whereas peaks in a series of every 3 months would indicate quarterly seasonality. Seasonal models have pronounced regular ACF and PACF patterns with a periodicity equal to the order of seasonality. The number of times per year that seasonal variations occur also matters. If the seasonality is annual, the seasonal variation ACF spikes are heightened patterns at seasonal lags over and above the regular non-seasonal variation once per year. If the seasonality is quarterly, there will be prominent ACF spikes four times per year 5.

2.7 Seasonal Autoregressive Integrated Moving Average (Seasonal ARIMA) Models

An important tool in modeling non-stationary seasonal processes is the seasonal difference. The seasonal difference of period s for the series {Xt} is denoted by ?sXt and is defined as:

?sXt=Xt-Xt-s ………………. (2.20)

For a series of length n, the seasonal difference series will be of length n-s that is s data values are lost due to seasonal differencing. In a non-stationary seasonal model, a process {Xt} is said to be a multiplicative seasonal ARIMA model with non-seasonal orders p, d and q, seasonal orders P, D, and Q, and seasonal period s if the differenced series:

Wt=?d?SDXt ………………. (2.21)

Satisfies an ARMA (p×q)(P×Q)s model with seasonal period s. We say that {xt} is an ARIMA (p,d,q)(P,D,Q)s model with seasonal period s. The seasonal part of a seasonal ARIMA model has the same structure as the non-seasonal part: It may have an AR factor, an MA factor, and/or an order of differencing. Box and Jenkins have generalized the ARIMA model to deal with seasonality and defined a general multiplicative seasonal ARIMA model in the form:

?B?B1-B12Xt=?(B)(B12)at ………………. (2.22)

Where B denotes the backward shift operator, ?, ?, ?, and ? are polynomials of order p, P, q, and Q respectively and at is the purely random process with mean zero and constant variance ?a2. Consider a Seasonal ARIMA (0, 1, 1) × (0, 1, 1)12 model for instance. The model specification is:

1-?121-?Xt=1-?1?1-?1?12at………………. (2.23)

By expansion, we have:

Xt=Xt-1+Xt-12-Xt-13+at-?1at-1-?1at-12+?1?1at-12The estimates of the parameters of the model in (2.23) may be obtained using the method of maximum likelihood 5, 15.

2.8 Model Selection Criteria

2.8.1 Akaike Information Criterion AIC

Akaike Information Criterion AIC 2 is defined as

AIC = -2 log L + 2p ………………. (2.24)

where L is the maximized likelihood function and p is the number of effective parameters. The best model is the one with the smallest AIC. The likelihood function part reflects the goodness of fit of the model to the data, while 2p is described as a penalty. Since L generally increases with p, AIC reaches the minimum at a certain p. AIC is based on the information theory.

2.8.2 Mean Absolute Percentage Error (MAPE)

The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of accuracy of a method for constructing fitted time series values in statistics, specifically in trend estimation. It usually accuracy is defined by the formula:

M=1nt=1nAt-FtAt ………………. (2.25) Where At is the actual value and Ft is the forecast value 14.

2.8.3 Mean Absolute Error (MAE)

The Mean Absolute Error (MAE) is defined mathematically by equation below:

MAE=1nt=1net ………………. (2.26)Where et is the error term and n is the number of forecasting 18.

2.8.4 Root Mean Square Error (RMSE)

The formula for computing RMSE is

RMSE=1Ni=1NXi-Xi2 ………………. (2.27)Where: Xi = actual value

Xi= predicted value

N = number of forecasted time period 16.

2.9 Models Forecasts

The main goal of constructing a model for a time series is to make future predictions for a given series. It also plays a significant role in assessing the forecasts accuracy. The ultimate test of an ARIMA model is power or ability to forecast. In order to obtain a forecast with a minimal errors, there are seven features of a good ARIMA models taken into account 13, 15. First, a good model is parsimonious. That is, it has the smallest number of coefficients which explain the data set. Secondly, a good autoregressive (AR) model must be stationary. Thirdly, the moving average (MA) of the model should be invertible. Fourth, a good model should have high quality estimates of its coefficients (AR and MA). Fifth the residuals of a good model should be independent. Sixth, a good model should be normally distributed. Lastly, good fitted model has sufficiently optimal forecast errors 13. From the existing theory of the series up to time t, namely, X1,X2, X3, …, Xt-1,Xt, we can forecast the value of Xt+h?, that will happen h time units ahead. In this case, time t is the forecast origin and h is the lead time forecast. This forecast is denoted and estimated as

XtL=EXt+hX1,X2, ……,Xt………………. (2.28)

Once an adequate and satisfactory model is fitted to the series of interest, forecasts can be generated using the model 13.

xt=?1xt-1+…+?pxt-p+a1-?1at-1…-?qat-q…………. (2.29)

The one-step ahead forecast for time t ??1 is given by:

xt+1=?1xt+…+?pxt-p+1+at+1-?1at…-?qat-q+1…. (2.30)

3.1 introduction

Descision making is so sensitive when it is based on the forecasting methods, therefore the researcher should be careful during use the forecasting methods, these methods are depending on the number of observation and its run term.

Also the method of estimation is important to estimate the parameters of the model in use, in this study maximum likelihood estimation method have been used to estimate the parameter of the best model that is adequate the data under consideration.

3.2 Data Discription

The data set used for the analysis in this study came from the international Sulaymaniyah airport of Sulaymaniyah city which is contained one variable and deals with monthly weight of imported equipment since January 2010 up to November 2017. These data are measurements of the wiehgt of imported equipment in Sulaymaniyah city from international Sulaymaniyah airport.

3.3 Applications

The time series plots are display observations on the y-axis against equally spaced time intervals on the x-axis. They are used to evaluate patterns, knowledge of the general trend and behaviors in data over time. The time series plot of monthly wiehgt of imported equipment in Sulaymaniyah city is displayed in Figure 3.1 below:

Figure 3.1: Time series plot of monthly wiehgt of imported equipment in Sulaymaniyah city

Figure 3.1 indicates that the time series is not stationary series. The plot shows consistent pattern of short-term changes for data which indicates the existence of seasonal fluctuations. This series varies randomly over time and there is seasonal fluctuations. For further testing of the stationary of the time series, we applied Augmented Dickey- Fuller test for monthly wiehgt of imported equipment in Sulaymaniyah international airport. The augmented Dickey Fuller tests the hypotheses which is stated that the series is non–stationary series. Table 3.1 shows the results of ADF of the data of the time series of wiehgt of imported equipment.

Table 3.1: Dickey-Fuller test for monthly wiehgt of imported equipment in Sulaymaniyah international airport

Test t-Statistic P-value

ADF 0.647249 0.9903

Table (3.1) shows that the p-value of the Dickey-Fuller test equals 0.9903 and it is greater than the value of ????0.05. This result indicates that the time series of monthly wiehgt of imported equipment is not stationary, and demonstrates this results by the examination of the autocorrelation and partial autocorrelation functions as shown below.

Figure 3.2: Autocorrelation Function for the monthly wiehgt of imported equipment in Sulaymaniyah international airport

Figure 3.3: Partial Autocorrelation Function for the monthly wiehgt of imported equipment in Sulaymaniyah international airport

All the above results and plots confirm that the time series data is not stationary at the level, and need some treatments to be transformed to a stationary series. Therefore, we used many transformations and we found that the most suitable transformation is by differencing the series. We note that the time series for the first differenced series in Figure 3.4 indicates that the series is stationary.

Figure 3.4: Time series plot of the first difference of monthly wiehgt of imported equipment in Sulaymaniyah city

Table 3.2: Dickey-Fuller test of the first differences for monthly wiehgt of imported equipment in Sulaymaniyah international airport

Test t-Statistic P-value

ADF -6.592854 0.000

Table (3.2) shows that the p-value of the Dickey-Fuller test equals 0.000 and it is less than the value of ????0.05. This result indicates that the non-stationarity hypotheses of the differenced monthly wiehgt of imported equipment in Sulaymaniyah international airport is rejected and this demonstrates by estimating the autocorrelation partial autocorrelation function (ACF and PACF) for the first differenced series in Figure 3.5 and 3.6

Figure 3.5: Autocorrelation Function for the first differenced series of the monthly wiehgt of imported equipment in Sulaymaniyah international airport

Figure 3.6: Partial Autocorrelation Function for the first differenced series of the monthly wiehgt of imported equipment in Sulaymaniyah international airport

The results above demonstrates the success of differencing of the time series data of the monthly wiehgt of imported equipment in Sulaymaniyah international airport. Thus, the series became stationarity.

3.4 Model Identification

This section shows how we determine the order of the seasonal ARIMA model. We computed all relevant criteria to select the best seasonal ARIMA model for the data of weight of imported equipment. Those are the ACF and PACF in addition to RMSE, MAE, MAPE, and AIC criteria. To take a decision must be looking at all the plots of autocorrelation function and the partial autocorrelation coefficients of the series as shown in the figure (3.5 and 3.6) respectively, It can be seen from the autocorrelation coefficients and the partial autocorrelation coefficients of the series that it is necessary to consider the seasonal changes when identifying and estimating the model. Weight of imported equipment data is monthly and according to the identification criteria, the following models have been examined and estimated as shown in table (3.3) below. The best seasonal model is chosen through the RMSE, MAE, MAPE and AIC criteria if it shows the lowest values of these criteria as it is shown in table (3.3).

Table 3.3: SARIMA Models Criteria for the monthly Weight of imported equipment in Sulaymaniyah international airport

Model RMSE MAE MAPE MPE AIC

ARIMA(1,1,0)x(1,0,1)12 763875 494826 90.1219 -21.3587 27.1568

ARIMA(0,1,1)x(1,0,1)12 797370 494338 86.9252 -20.3465 27.2427

ARIMA(1,1,1)x(1,0,1)12 807813 484720 83.7792 -21.056 27.2902

ARIMA(0,1,0)x(0,0,0)12 995642 451982 66.5704 -29.1702 27.6223

It is shown in table (3.3) that the SARIMA (1,1,0)x(1,0,1)12 model produced the value of each RMSE, MAE, MAPE and AIC criteria with the smallest values. This means that the SARIMA (1,1,0)x(1,0,1)12 model is the best among all the other models, which is the most suitable model that can be obtained for the monthly weight of imported equipment in Sulaymaniyah international airport.

3.5 Parameters Estimation:

Since we concluded in the previous section that the SARIMA (1,1,0)x(1,0,1)12 model is the best model with the smallest value of RMSE, MAE, MAPE and AIC criteria, the parameters had been estimated using the method of maximum likelihood estimation as it is the best and most appropriate method of estimation. The results of the parameters estimation of the model are shown in table (3.4) below.

Table 3.4: Parameter Estimates of SARIMA (1,1,0)x(1,0,1)12 Model Estimate model coefficients

Parameter Estimate Stnd. Error t P-value

AR(1) 0.09964 0.11864 0.839849 0.403244

SAR(1) -1.36688 0.0294053 -46.4843 0.000000

SMA(1) -1.3409 0.0258532 -51.8661 0.000000

It is shown in table (3.4) that the p-value for the parameters SAR(1), SMA(1) coefficients are less than ????0.05. This indicates that these coefficients are significantly different from zero. As it is shown for this model, the RMSE, MAPE, MAE and AIC criteria are the smallest values among the other models. Thus, the final model is SARIMA (1,1,0)x(1,0,1)12.

3.6 Forecasting and Prediction of future water production

After getting the final model SARIMA (1,1,0)x(1,0,1)12 of the data of the monthly weight of imported equipment in Sulaymaniyah international airport that has been expressed above, the researcher used it for forecasting future quantities weight of imported equipment. We forecasted quantities of monthly weight of imported equipment in Sulaymaniyah international airport in 2017-2018 for 12 months. The forecasting of time series for monthly weight of imported equipment in Sulaymaniyah international airport have been plotted as in figure (3.7).

Figure 3.7: Plot of the data and the forecasts with 95% confidence interval are represented

Figure 3.7 shows the result that the series of the forecasted values follow the same behavior of the original series of weight of imported equipment in Sulaymaniyah international airport. The results of the forecasted values in table (3.5) for the year 2017-2018 are all between the upper and lower boundaries of the 95% confidence intervals. This confirms that the forecasting is very efficient.

Table 3.5: Forecast future value with the lower and upper 95% confidence interval

Period Forecast Lower Limit 95.0% Upper Limit 95.0%

Nov – 2017 9587980 8039000 11137000

Dec – 2017 10662100 8359780 12964400

Jan – 2018 11341400 8468350 14214500

Feb – 2018 11547900 8199210 14896600

Mar – 2018 11290800 7526080 15055600

Apr – 2018 11069300 6930070 15208500

May – 2018 9192720 4710200 13675200

Jun – 2018 10638500 5837160 15439800

Jul – 2018 10832100 5731890 15932400

Aug – 2018 9852610 4470010 15235200

Sep – 2018 9155000 3504150 14805800

Oct – 2018 3854720 -2052210 9761660

Table 3.5 shows that the quantities of monthly weight of imported equipment in Sulaymaniyah international airport in 2017 – 2018 for 12 months have been forecasted. It is also shown from these results that the forecasted values are all between the upper and lower boundaries of the 95% confidence intervals. This supports that the forecasting is efficient.

4. Conclusion and Recommendations

4.1 Conclusion

From the previous results, the following conclusions can be summarized:

The statistical tests show that the time series of the monthly weight of imported equipment in Sulaymaniyah international airport is stable. In addition to have the seasonal changes. It repeats itself every 12 months.

The best and most efficient model is SARIMA (1,1,0)×(1,0,1)12 among the possible models which was chosen using the balancing standards (the smallest value of each : AIC, RMSE, MAPE and MAE criteria).

Parameter Estimates of SARIMA (1,1,0)×(1,0,1)12 Model are significant except AR(1) thus, the SARIMA (1,1,0)×(1,0,1)12 is efficient.

According to SARIMA (1,1,0)×(1,0,1)12, the monthly weight of imported equipment in Sulaymaniyah international airport for the year 2017-2018 for 12 months have been forecasted. The forecasted values are showed harmony with its counterparts in the original series values. Moreover, the forecasted values are all between the upper and lower boundaries of the 95% confidence intervals. Thus, it provided a future image of the reality of monthly weight of imported equipment.

4.2 Recommendations:

Through the results that have been reached, we recommend the following:

Adopting the results of this research and the adopted formula of forecasting by the related agencies because it uses the suitable scientific style in forecasting as well as taking in account that there is a real problem facing Sulaymaniyah international airport according to weight of imported equipment through the upcoming years which is the decrease in the monthly weight of imported equipment in Sulaymaniyah international airport. Furthermore, it helps the officials and the decision makers in finding solutions and quick alternatives to face this problem and putting the future plans of the monthly weight of imported equipment in Sulaymaniyah international airport to stop aggravating the problem.

Using this method in deducting the standard method and improving it to forecast the monthly weight of imported equipment that is predicted to be arranged every year. This is done according to the development of the actual series of the monthly weight of imported equipment in Sulaymaniyah international airport.

Generalizing this study to similar studies on the cities and districts level and the other cities level and comparing between them.

Weight of imported equipment at each airport is a vital element of the economic development and progress, thus it should be available plan to use this fortune from monthly weight of imported equipment.

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