1. Introduction
Population growth is a
major problem for any country, and has a significant
impact on its economic and social development. This has huge implications for
government plans, economic growth, and social welfare. Population forecasting helps you understand
the future trend of economic growth and better allocate existing resources. Thus, the population
forecasting procedure can help control the increase and decrease in the population. It can provide
plentiful labor resources for all domains of life and prevent unwanted effects on the population, such as
aging and impairment [1].
Many researchers have
done a lot of papers works on the analysis and forecasting of population data.
In [2], the authors made an empirical study to forecast the population of
Pakistan from 1951 to 2007 by applying the Box-Jenkins methodology. Results
concluded that the ARIMA (1,2,0) is a suitable model and can be applied for
forecasting for the next 20 years. The author in [3], applied the Box-Jenkins
methodology to forecast the census data of Iraq. The study concluded the
continuous increasing trend that the ARIMA (2,2,0) is the best model to be used
for forecasting the population of Iraq during the period 2011 to 2020. Authors
in [4], used a mathematical model for dynamic population growth to obtain a
prediction model for the rural population in Shaanxi province. The study
results showed that, for the coming 15 years, the ratio of the rural elderly
population in Shaanxi Province will get larger and grow, the ratio of the
old-age dependency will increase too, and the ratio of the labor force will
decrease. In reference [5], the authors used structural population data for
regions, states, and provinces from the
Population Census of China. A model of population, development, and
environment was used to explain the population change criteria. The analysis
concluded that the population of China is expected to increase first and then
decrease from 2010 to 2050. In recent years, the authors in [6] proposed the
ARIMA model for forecasting the population change in China in the next few
years depending on the feature extraction and the analysis of the principal components.
The analysis showed a downward trend in the population. In [7], a study was
carried out using the logistic model and least square model to do a comparative
study for making predictions of population growth between Bangladesh and India
at the end of the 21st century. It was found that the projection data from 2000
to 2020 using the established models closely match the real data. In [8], the
author compared three methods for forecasting the population of Iraq consisting Markov chains, artificial neural networks, and
ARIMA methodology based on some statistical criteria and using real data of the
population of Iraq during the period 1977-2007. The ARIMA (1,1,1) model was
selected from the comparison as the best one for predictions. based on the
specified model, The population of Iraq was forecasted during the period 2008
to 2030. An
objective strategy for forecasting multi-regional population growth was put
forth by the researchers in [9]. The findings indicate that this work can
continue to serve as a neutral reference for urban and regional planning. The
ARIMA Model was used by the researchers in [10] to
predict the growth of the urban population in the Philippines. It has been
demonstrated that the ARIMA(20,1,10) model
is the most accurate for predicting the rise of the nation's urban population. In [11], the author
used the Malthusian model, Unary linear regression model, Logistic model, and Gray prediction model to
offer forecasts of the populations of 210 prefecture-level cities. According to
the findings, there is
a growing demographic disparity between cities, and the overall urban population
tends to increase in
middle-tier cities while decreasing in high-tier and low-tier cities.
The research aims
first, to review time-series forecasting methods through Box-Jenkins models.
Second, to test the possibility of applying the ARIMA method in predicting the
percentage of Iraqi citizens who are in the countryside. Third, to determine
the optimal model among the ARIMA models for forecasting the proportion of the
rural population. Finally, to forecast the proportion of the rural population
until 2030. The rest of the study is organized as follows. In section 2, the
methodology of the study will be discussed. In section 3, the application of
Box-Jenkins on the real data will be done. Finally, in section 4, some
conclusions and recommendations will be present.
2. Box-Jenkins Methodology
The Box-Jenkins analysis technique refers to an
analyzing method of identifying the model, estimating, diagnostic, and using it
to forecast by using integrated autoregressive, moving average (ARIMA) time
series models. The procedure is suitable for medium to long length time series
(i.e; not less than 50 observations). The general
equation model ARIMA (P, I, Q) model for y can be presented as [12]:
Here ∅ and θ terms indicates the parameters to be
find and a is a normal error and the
value of its mean is zero and its distribution is independent and identical, P
is a full behind value number of 
it refers to the rank of the dimensions of
autoregressive (AR), I represent the number of differences of w, and Q
represents the number of full behind values of the error expressions showing
the rank of the moving average (MA) dimension of the specified model. The term
integrated indicates that to get a forecast for y from the determined model it
is required to integrate the forecast .
It must be taken into consideration that the
initial identification model imposes us to temporary consideration of ARIMA
models that will be efficiently suited and checked [9]. The Box-Jenkins
supposes stationarity of the sequential data. A series is said to be strictly
stationary when satisfying the two conditions, a fixed mean and variance as a
first condition, and the existence of a constant auto-covariance structure as a
second condition. If the second condition is not satisfied, then the series has
weak stationarity (also called second-order stationarity) [13]. In time series analysis, if the values of
the autocorrelations start high and decline quietly, then the series is said to
be non-stationary, and the methodology of Box-Jenkins induce taking difference
for once or more if needed to gain stationarity. Also, the variance of the
inaccuracy of the specified model should be constant which indicates that the
variance of values is identical for each subgroup and it is not depending on
the time point or time level. When this condition is not met, then a suitable
transformation using the Box-Cox test is required to solve un-stabilizing the
variance [14].
A.
Identification
The identification step for building and
selecting the suitable values (P, Q) of the specified ARMA model
for understanding the stationarity of time series yt
is executed on the grounds of its properties, including the mean, the function
of autocorrelation ACF and the function of partial autocorrelation PACF. For
autoregressive process AR, the values pattern of ACF are infinite (i;e. exponential and/or sine-cosine wave decay). While PACF
is finite (i;e. cut off at lag P). For moving average
process MA, the values pattern of ACF and PACF are the opposite of the AR
process. For the ARMA process, the ACF and PACF pattern of their values are
infinite (i;e. exponential and/or sine-cosine wave
decay) [15].
B.
Estimation
In addition to the need for stability in the
Box-Jenkins models, it also has to be capable of invertibility. This means that
the latest data are more steadily metric than more remote data; the parameters
used in the model turn down from the most recent data down to the further past
data [16]. Many techniques of estimation have been used
in time series analysis such as backcasting,
cross-validation, and out-of-sample procedure. Other important estimation
methods for building and fitting models using the Box-Jenkins methodology are
ordinary least square, maximum likelihood, non-linear estimation, and moments
method [12],[17], [18]. To obtain the perfect and best model from a
combination of some adequate models, some criteria measure of accuracy and
goodness of fit can be used. In this paper, we depend on two important
statistics: the Root Mean Square Error RMSE which has the characteristic of
being easy handling mathematically, and the Akaike Information Criterion AIC
which represents a formula of the variance of the model residuals, disciplined
by the number of parameters estimated. The appropriate model is selected based
on the lowest value of these criteria [19],[20].
C.
Diagnosing
A selected model should be studied cautiously
to test the verification of model adequacy. If the adequacy of the model is
confirmed, then the residual series should behave as white noise. The residuals
of both ACF and PACF can be used to check the proximity of at to white noise.
The work is done by studying the autocorrelation scheme of the residuals to
consider if there are additional large correlation values that exist. If all
the ACF and PACF values are small, then the model is mentioned to be adequate
and forecasts are generated. Otherwise, the values of P and/or Q should be
adjusted and the re-estimation of the model is needed [21].
It is important to mention for a statistic test
to detect autocorrelation or test the combined hypothesis that all m of the rk coefficients of the correlation is jointly
equal to zero using the Box-Pierce test which was introduced and developed by [22]. The test is a way to check for the absence
of sequential autocorrelation, up to a specified lag k. The formula can be
showed as:
Where T represents sample size and m shows
maximum lag length. It is be able to say that the test is approximately
Chi-square distribution under the null hypothesis that all values of m
autocorrelation coefficients are zero [23].
D.
Forecasting
Once a suitable model has been chosen depending
on the scientific Box-Jenkins technique and its parameters have been estimated
successfully, the model has a right to be used and make forecasts. The
efficiency of the specified model can only be completely rated after the real
data for the forecast period have become obtainable [20]. To
achieve successful outcomes, we used Statgraphics 18
for data analysis forecasting.
3. Application
The data used in this study consist of 60
values of the rural population of Iraq over the period (1960-2019). Data are
taken from (macrotrends website [24]). To
analyze and predict the total population of the rural area of Iraq over the
next ten years (assuming that the decreasing of rural population data is
affected only by time, and it is free from outside intervention). Table 1 shows the rural population data during the
time interval 1960 to 2019 and the selected data from 1960 to 2015 is for
processing and analysis, while the remaining values of 2016 to 2019 were
allocated for verification. By using the Statgraphics
statistical software, a broken line diagram of the rural population of Iraq
from 1960 to 2015, is shown in Figure 1.
Table 1: The rural population of Iraq
from 1960 to 2019 in percentage.
|
Year |
Pop. |
Year |
Pop. |
Year |
Pop. |
|
1960 |
57.101 |
1975 |
38.621 |
1990 |
30.294 |
|
1961 |
55.568 |
1976 |
37.602 |
1991 |
30.478 |
|
1962 |
54.022 |
1977 |
36.596 |
1992 |
30.663 |
|
1963 |
52.468 |
1978 |
35.822 |
1993 |
30.848 |
|
1964 |
50.907 |
1979 |
35.148 |
1994 |
31.034 |
|
1965 |
49.349 |
1980 |
34.479 |
1995 |
31.22 |
|
1966 |
48.132 |
1981 |
33.817 |
1996 |
31.407 |
|
1967 |
47.055 |
1982 |
33.161 |
1997 |
31.595 |
|
1968 |
45.98 |
1983 |
32.512 |
1998 |
31.612 |
|
1969 |
44.911 |
1984 |
31.868 |
- |
- |
|
1970 |
43.846 |
1985 |
31.233 |
- |
- |
|
1971 |
42.786 |
1986 |
30.604 |
- |
- |
|
1972 |
41.731 |
1987 |
29.983 |
2017 |
29.722 |
|
1973 |
40.687 |
1988 |
29.928 |
2018 |
29.527 |
|
1974 |
39.649 |
1989 |
30.111 |
2019 |
29.322 |
Fig.1. Annual variation of rural population in
Iraq (% of total population)
From Fig.1., it can be seen that the series contains a long-term trend of
curve decrement from 1960 to 1988, and from 1989 to 2002 the slight increasing
events of the rural population ratio have appeared. From 2003 till 2015, the
curve tended to decrease gradually which is preliminarily conclude with the
non-stationary of the sequence. The two functions so-called the Autocorrelation
Function ACF and the Partial Autocorrelation Function PACF are two useful tools
to check whether the series is stationary or not. Figure 2 shows the lags and
the values of the two functions.
Fig.2. ACF and PACF of
the rural population in Iraq during 1960 - 2015
In fig.2., the ACF
values show steadily declining from the beginning to the end, and four values
are out of the confidence interval. The value of the Box-Pierce statistic with
eighteen lags is equal to 269.491 (p-value = 0.00), which is highly
significant. While in the case of PACF, only one spike can be seen at lag 1
which is out of the limits. On the other hand, the Box-Cox transformation value
was (-0.431) and its interval was (-2.941, 2.042) which includes the value
zero. This recommended that the log transformation is a suitable choice to make
the series stationary in variance before taking the difference of the series.
After taking the log transform on the real data and checking again the ACFs and
PACFs several times, we deduced that the series should be differenced twice to
get stationarity in the mean. Figures3 and 4 show the transformed series
respectively.
Fig.3. Log of rural
population in Iraq after 2nd differencing during 1960-2015.
Fig.4. ACF and PACF of log of the rural population in Iraq
after 2nd differencing during 1960-2015.
Fig.3. present the
trend of the rural population after differencing twice the natural logarithm of
the series during the specified period. Fig.3. is unusual as that of Figure 1,
and its pattern is approximate stationary and has no more trend. Figure 4 and
its upper part is the ACF of the 2nd difference of natural logarithm of the
rural population. Except for the first lag, all the spikes at different lags
are within the confidence limits. Figure 4 and its lower part is the sample
PACF of the same series shows that all its spikes at different lags are inside
the confidence interval except one spike at lag 1. On the other hand, the test
concerning Box-Pierce takes a value of 8.78021 (p-value = 0.964619), which is
greater than 0.05 meaning that the series is random. From figures 3 and 4 and
the box-Pierce test, we conclude that the series is stationary, and different
stationary models can be applied to this series.
After obtaining
stationarity, we proceed to fit a suitable ARMA model to the adjusted series.
The study uses the Root Mean Square Error RMSE as a measure of accuracy and
Akaike Information Criterion AIC as a measure of the goodness of fit of the
model which was mentioned in the section devoted to the theoretical side to
select the best model order. Table 2 shows some combinations and adequate ARIMA
with the estimated criteria.
Table 2. Some adequate
ARIMA models with their criteria values
|
Model |
RMSE |
AIC |
MAE |
|
ARIMA(0,2,1) |
0.0958 |
-4.6561 |
0.0309 |
|
ARIMA(1,2,0) |
0.0975 |
-4.62092 |
0.0356 |
|
ARIMA(2,1,0) |
0.0964 |
-4.60626 |
0.0455 |
|
ARIMA(0,2,2) |
0.0967 |
-4.60281 |
0.0316 |
|
ARIMA(1,2,1) |
0.0966 |
-4.60249 |
0.0313 |
It is clear from the
table II that out of five specified models, the ARIMA (0,2,1) model is more
appropriate to be adopted and explain the natural characteristic properties of
the rural population because it has the lowest values of the RMSE, AIC and MAE
comparing to the other models. Table 3. presents the estimated parameter of the
ARIMA (0,2,1) model.
Table 3. Parameter
estimation of ARIMA (0,2,1)
|
Parameter |
Estimate |
Standard Error |
t-test |
P-Value |
|
MA (1) |
-0.422137 |
0.125993 |
-3.35047 |
0.001493 |
The next step after estimating the parameter of
the ARIMA (0,2,1) model is to check for randomness using the residuals graph of
the ACF and PACF as shown in figure 5.
Fig.5. Residuals of the ACF and PACF for the adjusted
data
From fig.5., one can
see that all the autocorrelation coefficients values of ACF and PACF are
non-statistically significant, concluding that the current series may well be
white noise. Furthermore, the test statistics for the randomness of residuals
using Box-Pierce was equal to (1.4282) and the P-value was (1.000) which
exceeds 0.05. Thus, the hypothesis concerning the randomness of the series
cannot be rejected at 95% or higher of the confidence level.
After proceeding with
the main steps of building an ARIMA model including data preparation, model
selection, parameter estimation, and diagnostics, the last and the important
step remains, which is forecasting. As we mentioned before that the rural
population values of 2016 to 2019 will be allocated for verification. Table 4
presents the comparison between the forecasted values by the ARIMA (0,2,1)
model with the real values for the specified period.
Table 4. The forecasted values by the ARIMA
(0,2,1) model with real data during 2016-2019
|
Year |
Real value |
Forecast |
Lower 95% limit |
Upper 95% limit |
|
2010 |
30.897 |
30.967 |
30.774 |
31.162 |
|
2011 |
30.732 |
30.915 |
30.411 |
31.427 |
|
2012 |
30.568 |
30.862 |
29.976 |
31.775 |
|
2013 |
30.405 |
30.810 |
29.483 |
32.197 |
|
2014 |
30.242 |
30.758 |
28.944 |
32.685 |
|
2015 |
30.079 |
30.706 |
28.367 |
33.237 |
|
2016 |
29.906 |
29.917 |
29.740 |
30.094 |
|
2017 |
29.722 |
29.755 |
29.297 |
30.220 |
|
2018 |
29.527 |
29.594 |
28.790 |
30.421 |
|
2019 |
29.322 |
29.434 |
28.234 |
30.685 |
From table 4, we can
record some notes. First, from 2010 to 2019, all the true values and predicted
values are very close to each other. Second, all these observed values fall
inside the confidence interval. Third, from 2016 to 2019 the percentage of
decreasing of the rural population in Iraq concerning the real values is
(1.95%) while for the predicted values is (1.61%). Thus, we can say that ARIMA (0,2,1) model is
the best and appropriate to be used to forecast annually rural population data,
and during the period 2016 to 2030, there will be (7.3%) decrease in the rural
population, and the percentage of the rural population of Iraq in 2030 would be
(27.732) persons. Table V. shows the forecasted values (in percentages) of the
rural population in Iraq with their confidence intervals and the Figure6
presents the forecasts for the log of the annually rural population data from
the period 2016 to 2030 depending on the ARIMA (0,2,1) model.
Table 5. The forecasted
values by the ARIMA (0,2,1) model during the period 2016-2030
|
Year |
Forecast |
Lower 95% limit |
Upper 95% limit |
|
2020 |
29.275 |
27.641 |
31.007 |
|
2021 |
29.117 |
27.016 |
31.381 |
|
2022 |
28.960 |
26.367 |
31.807 |
|
2023 |
28.803 |
25.699 |
32.283 |
|
2024 |
28.648 |
25.016 |
32.807 |
|
2025 |
28.493 |
24.322 |
33.380 |
|
2026 |
28.339 |
23.620 |
34.002 |
|
2027 |
28.186 |
22.913 |
34.673 |
|
2028 |
28.034 |
22.204 |
35.394 |
|
2029 |
27.882 |
21.496 |
36.166 |
|
2030 |
27.732 |
20.790 |
36.991 |
Fig.6. Forecasts for
the annually rural population of Iraq from 2016 to 2030 using ARIMA (0,2,1).
4. Conclusion
In
this paper, with the help of the Statgraphics
program as statistical software, the ARIMA (0,2,1) model is determined as an
adequate model according to the total rural population data of Iraq country
from 1960 to 2015. To test whether the chosen parameters of the model are
sensible, the rural population data of Iraq from 2016 to 2019 are validated.
The analysis results show that the model has a good fitting effect because the
decrease ratio of the rural population of Iraq between the real values and
forecasted values during the specified time interval was close to each other
and all values were placed within the confidence limits. Finally, we predict
the rural population data of Iraq in the next years. the prediction results
show that the rural population of Iraq is keeping decrease generally year by
year. During the period 2020 to 2030, the percentage of the decrease of the
rural population will be (7.3%), and the size of the rural population of Iraq
in 2030 would be (27.732%).
It's important to note that the outcomes of
this investigation and the forecasting model developed and selected by ARIMA
(0,2,1) are remarkably comparable to those of the study carried out in accordance
with [3] and the basis of which the ARIMA (0,2,2) prediction model was
constructed. The previous study was used to forecast Iraq's population census,
which is progressively growing, whereas the present study focuses on the
country's rural population, which is steadily declining. This disparity in
results between the two studies is the result of this divergence.
5. Recommendations
1- The selected
model can be used for forecasting the rural population of Iraq in the future.
2- We recommend
doing some other studies for the same data using different methods such as
multiple regression analysis, wavelet, and neural network analysis to compare
the results of the current ARIMA for assessment and select the best one for
forecasting the rural data of Iraq.
3- It is hoped
that the current study can provide an important theoretical reference for the
planning issues and adjustment of relevant policies in the rural population of
Iraq.
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