Forecasting of Post-Covid-19 Import Value Index in Nigeria using Box-
Jenkins Methodology
OGUNLADE TEMITOPE OLU
Department of Mathematics, Ekiti State University,
Ado-Ekiti, NIGERIA
&
Landmark University SDG 3 Research Group (Good Health and Well-being),
Department of Physical Sciences (Mathematics Programme),
Landmark University, Omu-Aran, Kwara State,
NIGERIA
AKINDUTIRE OPEYEMI ROSELYN
Department of Accounting (Actuarial Science Unit), Ekiti State University,
Ado-Ekiti, NIGERIA
FAWEYA OLANREWAJU
Department of Statistics, Ekiti State University,
Ado-Ekiti, NIGERIA
BALOGUN KAYODE OGUNTUASE
Federal School of Statistics, Oyo State,
NIGERIA
OKORO JOSHUA OTONRITSE
Landmark University SDG 3 Research Group (Good Health and Well-being),
Department of Physical Sciences (Mathematics Programme),
Landmark University, Omu-Aran, Kwara State,
NIGERIA
Abstract: - This study employed the Box-Jenkins Methodology otherwise known as the Autoregressive
Integrated Moving Average (ARIMA) modelling to model and forecast the series for the period of 2018 to
2030. The results indicated an upward trend with fluctuations while the series was stationary at first difference,
i.e the series was I(1) . Based on the Akaike information criterion (AIC) and Bayesian Information Criteria
(BIC) choice criteria, it was found that ARIMA (2, 1, 2) model was better suited to the import value index (IVI)
series. Diagnosed check of the model reveals that the error was random, normally distributed and there was no
serial correlation, in the same vein, thirteen years forecast was made which shows fluctuation pattern in import
value index (IVI) series.
Key-Words: -Import value index, COVID-19, Forecast, ARIMA, Box-Jenkins, Autocorrelation.
Received: June 10, 2021. Revised: February 7, 2022. Accepted: February 21, 2022. Published: March 24, 2022.
1 Introduction
The ratio which explains the percentage change in
nominal value to base value is said to be import
value index (IVI) [1]. The index tells the percentage
of import at a given point in time relative to its base
point. However, when the number of exported
goods in a country is far more than the imported
goods, such country is said to be experiencing
economic development. Presently, the advent of
coronavirus disease 2019 (COVID-19), a global
pandemic has created uncertainties in international
trade, especially among African countries (for
instance, Nigeria, South Africa and Angola) whose
markets are driven by the exportation of
commodities whose prices have crashed in the
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international market [2]. In the same vein, it is very
essential for a country to involve in foreign trade
which thereafter bring about economic growth
development [3, 4]. It is generally suggested that
most developing countries recorded an indefinite
reduction which contributed substantially to the
breakdown of oil prices on the market in their
foreign exchange revenue from the early 1980s: A
forecast on import value index (IVI) using
Autoregressive Integrated Moving Average
(ARIMA) model.
The transportation of produce, human and resources
(financial and nonfinancial) through national
disturbances, has been disposed of, particularly in
recent time. Academic, trade and technical studies
from various economies have emphasized foreign
trade on a global scale [5] and external financial
flowsas determinants of buoyant growth in countries
that are keen on identifying and taking advantage of
opportunities. Besides, they argue that agro-based
market is a stimulus for expansion, particularly in
countries whose main source of incomes (national
and foreign) and employment generations are from
agricultural produces. They further note that
agricultural trade creates varieties of choices for
consumers. The country having bountiful products
for farming; nevertheless, the country’s most
agricultural export commodity have not been
completely handled for industrial and agricultural
business yet. Before the advent of oil discovery and
extraction in the country in 1960s, agriculture was
the giant wellspring of exportation, but it has been
hijacked by the crude oil. Thus, agricultural
activities have gradually declined, particularly
during the 1970s’ oil boom. Ever since, Nigeria is a
net importer of grain and agricultural products.
Therefore, due to over-dependence on oil and the
decrease in agricultural production, Nigeria has
started importing goods which can be produced
locally, however, loosed its potentials of agriculture
[6-10]. Active players in the agricultural sector have
claimed that, if assisted by strong, effective and
long-lasting government agricultural policies, only
the middle belt of the country could supply the rice
demand for all of West Africa. Low cereal yield in
Nigeria are due to higher production costs, lack of
fertilizers, failure to maintain irrigation facilities,
and lack of labour. Management methods such as
weeding, transplanting and harvesting rely on
minimal family labour. Presently, a number of
related formal models have been formulated to
forecast some selected cereals such as maize,
sorghum etc. In this study, we are applying the
univariate time series model to justify truly whether
past values of Nigeria Cereals Production (CP)
series can predict its current and future values using
methodology technically known as ARIMA
modeling.
2 Materials and Methods
An annual time series data on import value index
(IVI) in Nigeria was used for this research work. In
this study, an annual time series data on food
production index (FPI) in Nigeria ranges from 1980
to 2017 was originated from the record of World
Bank through [11]. In this study, Box-Jenkins
methodology which is also known as ARIMA
modeling propounded by Box and Jenkins [12] was
used in analysing the import value index in Nigeria.
According to Pankratz [13], the autoregressive
integrated moving average (ARIMA) model reveals
the relationship between the time series data and its
former valence. There are four major steps in
creating a good model which are; identification,
estimation, diagnostic checking and forecasting
[12]. The methodology is however focusing on
making non-stationary time series data stationary by
differencing. The general equation of the ARIMA
(p, d, q) model is as follows;
(1)
Also, studies [14,15] showed that the mathematical
equation above can be expressed using lag
polynomial as shown in (2).
(2)
More so, (2) express polynomial factorization
property with p=p'-d which is given written as;
(3)
Where L is the lag operator, are the parameters of
the autoregressive part of the model, the are the
parameters of the moving average part, are the
error terms which are generally assumed to be
independent, identically distributed (i.i.d). Also, p
means the number of preceding (“lagged”), X values
to be added or subtracted from X in the model, in
order to make better predictions based on local
growth time or decrease in data capturing of the
ARIMA’s auto-regressive existence. More so, d is
the number of times that variations need to be made
between the data to generate a stationary signal that
i.e a signal that has constant mean over time
covering the integrated (I) essence of ARIMA. And
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q represents the number of lagged values for the
error word, which captures the moving average
(MA) portion of ARIMA.
2.1 Model Modification
Box-Jenkins methodology is one of the popular
method used in fitting an appropriate models to a
given time series. Thus, AR and MA of orders p and
q must be critically examine the Sample
Autocorrelation Function (SACF) as well as the
Sample Partial Autocorrelation Function (SPACF)
for prior to the estimation of ARMA (p, q) for a
given time series .The SAFC is used for detecting
the order q of the MA term while the SPACF is used
for detecting the order p of the AR term from the
sample correlogram. By sample correlogram, we
mean a plot of SACF against the lags k [16].
Mathematically, the sample Autocorrelation
Function (SAFC) at lag k is defined as:
=
=
, k = 0, 1, 2, …
(4)
Where:
is the sample covariance at lag k,
is
the sample variance and
is the sample mean.
The most popular model selection criteria with
specific utility are Akaike Information Criterion
(AIC) and the Bayesian Information Criterion (BIC)
which is otherwise known as Schwarz Information
Criterion (SIC) [17]. The mathematical expression
for AIC and BIC are as follow:
AIC = 2k – 2ln(
) (5)
BIC = kln(n) – 2ln(
) (6)
Where
is the maximized value of the likelihood
function of the model M, i.e
= p(x /
, M), also
are the parameter values that maximize the
likelihood function, x is the observed data, n is the
number of effective data point in x and k is the
number of estimated parameters in the model.
Konishi and Kitagawa [18] derived the BIC to
approximate the distribution of the data, integrating
out the parameters using Laplace’s method which is
given below;
P(x / M) = , M)π(θ / M)dθ (7)
Where π(θ / M) is the prior for θ under model M.
Another model selection criterion which is also
common but less used is the Hannan–Quinn
information criterion (HQC) [19]. The
mathematically expression is as follows;
(8)
Where: is the log-likelihood, k is the number
of parameters and n is the number of observations.
After a tentative model has been identified, the next
step is to estimate the parameters in the model.The
last steps in creating a good model is forecast.
Therefore, forecast is then made for the series
3 Data Analysis
This section focuses on the data analysis and the
interpretation of results carried out on Import Value
Index (IVI) series using GRETL (Gnu Regression,
Econometrics and Time – series Library) version
1.9.8.
Fig. 1: Time series plot of Import Value Index (IVI)
Fig. 1 shows that the import value index (IVI) is increasing with time (i.e upward trend) although
0
100
200
300
400
500
600
700
1980 1985 1990 1995 2000 2005 2010 2015
IVI
TIME
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with fluctuations. This means that the series will
need to be differenced for it to become stationary.
To determine the order of integration(d), this study
conducts unit root tests such as Augmented Dickey-
Fuller (ADF) and Phillips-Perron (PP) tests. Tables
1a and 1b indicate the results of unit root tests
conducted on the IVI series.
Table 1a. Augmented Dickey-Fuller (ADF) Test for IVI Series
ADF
Test Variable
ADF Statistics
Critical Values
p-value
Integration
Order
IVI
-6.439140
-2.945842
<0.0001***
I(1)
Table 1b. Phillips-Perron (PP) Test for IVI Series
PP
Test Variable
PP Statistics
Critical Values
p-value
Integration
Order
IVI
-6.424313
-2.945842
0.0001***
I(1)
The results from tables 1a and 1b as shown above
reveals that the IVI series is differenced stationary
series of order one {I(1)}. It can be deduced from
their p-values (0.0001) since both less than the
chosen level of significance (=0.05). The ACF and
PACF are employed to determine the type and order
of the model since the series has become stationary
[20,21].
Fig. 2: ACF and PACF for the IVI stationary series after first difference
Figure 2 reveals that the spikes of the ACF and
PACF are decaying exponentially while spikes of
ACF are significant at lag 1, lag 2, lag 3 and lag 4
respectively and the PACF has only two significant
spikes at lag 1 and lag 4. Consequently, ARIMA (p,
d, q) model has any possible order from p=1, 2, 3, 4
of AR term and q=1, 2, 3 and 4 of MA term is
therefore suggested to be fitted for the data.
According to the result revealed by the ACF and
PACF, selection criteria such as Akaike Information
Criteria (AIC), Bayesian Information Criteria (BIC)
and Hannan–Quinn information criterion (HQC)
were also employed for further detection of order p
and q. This is important so as to select the best model
from all the possible fitted models as it is shown in
the table 2 below.
-1
-0.5
0
0.5
1
0 1 2 3 4 5 6 7 8
lag
ACF for IVI
+- 1.96/T^0.5
-1
-0.5
0
0.5
1
0 1 2 3 4 5 6 7 8
lag
PACF for IVI
+- 1.96/T^0.5
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Table 2. Possible Fitted ARIMA (p, d, q) Models
It can be seen from table 2 that ARIMA (2, 1, 2)
model has the smallest values of the AIC, BIC and
HQC selection criteria. Thus, it can be confirmed as
the best model for the import value index (IVI)
series. Since the best model has been known.
Therefore, model estimation of ARIMA (2, 1, 2)
model can be done which is presented on Table 3.
Table 3. The estimates of AR and MA terms of ARIMA (2, 1, 2) model
(9)
Table 3 shows the estimates of AR and MA terms of
ARIMA (2, 1, 2) model. It was shown that prior two
years values t-2 (i.e2015 and 2016) of IVI series for
the AR term and the random shocks in the MA term
are related to the IVI in current time t (i.e 2017). The
p-values from their test statistic also proved that it
was statistically significant since the chosen
significant level ( = 0.05) is greater than their p-
values.
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Fig. 3: Correlogram residuals of the fitted model
Figure 3 shows that the spikes for both ACF and
PACF of the residual correlogram are not
statistically significant. It indicates that ARIMA
(2, 1, 2) model is a best fit to the Import Value
Index (IVI) series.
Table 4. Forecast of the Import Value Index (IVI) series based on Thirteen Years
Year
Prediction
Std.Error
[95% Conf. Interval]
2018
483.868299
59.573133
367.107104 -600.629494
2019
441.446518
77.458684
289.630288 -593.262749
2020
519.444463
91.937944
339.249403 -699.639522
2021
527.610735
109.35016
313.288352 -741.933117
2022
466.123909
121.41926
228.146533 -704.101285
2023
517.983243
130.45924
262.287830 -773.678657
2024
561.811925
142.46949
282.576865 -841.046986
2025
501.863493
153.00903
201.971298 -801.755688
2026
518.378736
160.23799
204.318054 -832.439418
2027
584.135975
169.2281
252.454995 -915.816956
2028
543.68251
178.82857
193.184959 -894.180061
2029
526.179453
185.41616
162.770459 -889.588447
2030
595.703891
192.45468
218.499641 -972.908141
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
lag
Residual ACF
+- 1.96/T^0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 1 2 3 4 5 6 7 8
lag
Residual PACF
+- 1.96/T^0.5
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Fig. 4: Forecast model values superimposed on the original values’ plot
4 Discussions
This study investigated the import value index
(IVI) in Nigeria using ARIMA modelling
technique. The stationary condition of import
value index (IVI) series was fore determined by
using time series plot as shown in Figure 1. The
figure indicated enough facts that the series was
not stationary at level. The augmented Dickey-
Fuller (ADF) and Philips-Perron (PP) tests were
used to ascertain the suitability of the order of
integration (d) of the variables or the unit roots.
The ADF and PP test statistics as shown in table
1a and 1b respectively revealed that all the series
were stationary at first difference, that is I(1).
However, Box-Jenkins methodology was
adopted in this study and all steps laid down for
modelling cycle were meticulously applied. It
was established that ARIMA (2, 1, 2) model was
the best model for the IVI data used as indicated
in Figure 2 and Table 2 respectively.
Furthermore, Table 3 revealed the estimation of
the model where two years past values t-2
(i.e2015 and 2016) of IVI series for the AR term
and the random shocks in the MA term are
related to the IVI series in the current time t
(2017) which was statistically significant as their
p-values less than the level of significance. More
so, Figure 3 indicates the residual correlogram
obtained from the estimated model which
showed that spikes of ACF and PACF are not
statistically significant thus connote that
ARIMA (2, 1,2) model is a perfect fit for the IVI
series.
Lastly, forecast of thirteen years was made for
IVI series from 2018-2030 as shown in Table 4,
the forecast series were within the95%
confidence bounds. Thus, it reveals that
theforecast values were good. More so, there
was fluctuation in the forecast series which
shows that the series will continue fluctuating
for these forecasted time period.This is an
indication of present COVID-19 condition.
However, this holds true as there is a significant
tendency toward import of goods at the moment.
In the same vein, Fig. 4 indicates that there will
a continuous fluctuation for the forecast time
period of the import value index.
5 Conclusion
Overall, this study models and forecasts import
value index (IVI) in Nigeria for the periods 2018
to 2030 employing ARIMA modeling
techniques which is also known as Box-Jenkins
methodology. The findings show that rate of IVI
in recent years has currently been affected by the
present COVID-19 pandemic. Model
identification in one of the stages of the
0
100
200
300
400
500
600
700
800
900
1000
2000 2005 2010 2015 2020 2025 2030
TIME
IVI
forecast
95 percent interval
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modeling techniques shows that IVI series was
not stationary at level due to the result of ADF
test as well as the time plot as it was shown in
Table 1a and Figure 1 respectively. The IVI
series was found stationary at its first difference
and it was the ARIMA (2, 1, 2) model found to
be the best fit for the data based on the selection
criteria employed i.e AIC and BIC. Furthermore,
model estimation was the next stage where the
IVI of current is shown to be related to one
period lag of its own value and one period lag of
error term. More so, it was shown in the
correlogram of the residuals (Figure 3) that error
emanated from ARIMA (2, 1, 2) model was
random, normally distributed and absence of
serial correlation. In the same vein, forecast of
thirteen years was made which reveals
fluctuation in import value index (IVI) series
and therefore, indicates that the series will
continue fluctuating for the forecast time period.
It is therefore recommend that Nigeria should
limit the level of importation of goods due to the
availability of some commodities in the country
as this renders the home-made goods valueless.
Furthermore, level of risk exposure will be
deteriorating through limiting the level of goods
importation to the country.
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[18] Konishi S and Kitagawa G. (2008):
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
All Authors carried out the conceptualization of
the research and data collection. Akindutire
Opeyemi carried out the Statistical analysis and
prepared the original Draft. Ogunlade Temitope,
carried out the paper reviewing. Faweya
Olanrewaju, Balogun Kayode and Okoro Joshua
O had the general project administration
Sources of Funding for Research Presented in
a Scientific Article or Scientific Article Itself
No funding to declare
Creative Commons Attribution License 4.0
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This article is published under the terms of the
Creative Commons Attribution License 4.0
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