American Journal of Innovative Research and Applied Sciences.ISSN 2429-5396Iwww.american-jiras.com
ORIGINAL ARTICLE

| Mounir Boumhamdi *1 | and |Aziz Atmani 1 || Mounir Boumhamdi *1 | and |Aziz Atmani 1 |
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<sect1><title>1. UniversityHassan II | Laboratoire d'actuariat, criminalité financière et migration internationale1. UniversityHassan II | Laboratoire d'actuariat, criminalité financière et migration internationale| B.P 8110 Oasis | Casablanca | Maroc || Received | 10 October 2018 | | Accepted 19 November 2018 | | Published 25 November 2018 | | ID Article | Mounir-ManuscriptRef.14-ajira101118 |AbstractBackground: North Africa has witnessed a political and economic evolution and very important strategic location. Objective: The aim of this work is the modeling and the simulation of gross domestic product (GDP), PPP (current international dollar) growth in North African countries from 1990 to 2017.Methods: Using mixed effects model theory. In our case, we will use linear mixed effects models for longitudinal data covering the last twenty seven years. Results: This model provides a good fit to describe the law of evolution of GDP (PPP based). Conclusion: The Linear Mixed Effects Models can help to characterize and to understand many complex linear economical processes.Keywords: Linear mixed effects model, GDP (PPP based), North Africa, Longitudinal data.INTRODUCTIONNorth Africa has a very important strategic location for its proximity to Europe and its openness to sub-Saharan Africa. This part of the world has witnessed a political and economic evolution. The purpose of this work is the modeling of the Gross Domestic Product in current international dollar. An international dollar has the same purchasing power over GDP as United States which is the main economic standard for measuring economic output produced within a country. This modeling covers all North African Countries except Libya due to lack of data. The mathematical models allow the analysis and interpretation of the observed data because they describe the evolution law as a function of only a few parameters that can be statistically compared. For repeated measurements data, mixed-effects models offer a flexible and powerful tool in which population characteristics are modeled as mixed effects and unit-specific variation is modeled as random effects. Linear mixed-effects (LME) models [1, 2, 3, 4] and nonlinear mixed-effects (NLME) models [5, 6, 7] are widely used in longitudinal data analysis. The overall objectives of this paper is to make a linear mixed-effects model for describing GDP growth as a function of time (years), taking individualization (Inter-individual variation, Intra-individual variation) into account. Thus, the linear mixed-effects model introduced in this paper provides a good fit for data.Finally, the analysis of this research was accomplished with the lme4 package [8] for R statistical software.Generalized linear mixed-effect modelsSuppose that a study is based on
N
N
individuals and that we seek to build a global model for all thecollected observations for the
N
N
individuals. Inspired from the work [9], we denote
y
ij
{y} rsub {ij}
the
j
j
observation taken of individual
i
i
and
t
ij
(
1
)
,
…
,
t
ij
(
m
)
{t} rsub {ij} rsup {(1)} ,…, {t} rsub {ij} rsup {(m)}
the values of the
m
m
explanatory variables for individual
m
m
.If we assume that the parameters of the model can vary from an individual to another, then for any subject
i
,
1
≤
i
≤
N
,
i , 1≤i≤N,
the model is:
y
ij
=
b
i
0
+
b
i
1
t
ij
(
1
)
+
b
i
2
t
ij
(
2
)
+
…
+
b
ℑ
t
ij
(
m
)
+
ϵ
ij
,
1
≤
j
≤
n
i
(
2.1
)
{y} rsub {ij} = {b} rsub {i0} + {b} rsub {i1} {t} rsub {ij} rsup {(1)} + {b} rsub {i2} {t} rsub {ij} rsup {(2)} +…+ {b} rsub {im} {t} rsub {ij} rsup {(m)} + {ϵ} rsub {ij} , 1≤j≤ {n} rsub {i} (2.1)
To begin with, suppose that each individual parameter
b
ik
{b} rsub {ik}
can be broken down into a fixed component
β
k
{β} rsub {k}
and an individual component
r
ik
{r} rsub {ik}
additively:
b
ik
=
r
ik
+
β
k
(
2.2
)
{b} rsub {ik} = {r} rsub {ik} + {β} rsub {k} (2.2)
Where
b
i
k
{b} rsub {i k}
represents the deviation of
r
i
k
{r} rsub {i k}
from the value
β
k
{β} rsub {k}
in the population for individual
i
i
and
r
i
k
{r} rsub {i k}
is a random variable normally distributed with mean
0
0
. Using this parameterization, the model becomes:
y
ij
=
β
i
0
+
β
i
1
t
ij
(
1
)
+
β
i
2
t
ij
(
2
)
+
…
+
β
ℑ
t
ij
(
m
)
+
r
i
0
+
r
i
1
t
ij
(
1
)
+
…
+
r
ℑ
t
ij
(
m
)
+
ϵ
ij
,
1
≤
j
≤
n
i
(
2.3
)
{y} rsub {ij} = {β} rsub {i0} + {β} rsub {i1} {t} rsub {ij} rsup {(1)} + {β} rsub {i2} {t} rsub {ij} rsup {(2)} +…+ {β} rsub {im} {t} rsub {ij} rsup {(m)} + {r} rsub {i0} + {r} rsub {i1} {t} rsub {ij} rsup {(1)} +…+ {r} rsub {im} {t} rsub {ij} rsup {(m)} + {ϵ} rsub {ij} , 1≤j≤ {n} rsub {i} (2.3)
We can then rewrite the model in matrix form:
y
i
=
T
i
β
+
T
i
r
i
+
ϵ
i
,
(
2.4
)
{y} rsub {i} = {T} rsub {i} β+ {T} rsub {i} {r} rsub {i} + {ϵ} rsub {i} , (2.4)
Where:
y
i
=
(
y
i
1
y
i
2
⋮
y
i
n
i
)
{y} rsub {i} = left (matrix {matrix {{y} rsub {i1} ## {y} rsub {i2}} ## matrix {⋮ ## {y} rsub {i {n} rsub {i}}}} right )
,
T
i
(
1
1
t
i
1
(
1
)
t
i
2
(
1
)
…
⋯
t
i
1
(
m
)
t
i
2
(
m
)
⋮
1
⋮
t
i
n
i
(
1
)
⋱
…
⋮
t
i
n
i
(
m
)
)
{ T} rsub {i} left (matrix {matrix {matrix {1 ## 1} # matrix {{t} rsub {i1} rsup {(1)} ## {t} rsub {i2} rsup {(1)}}} # matrix {matrix {… ## ⋯} # matrix {{t} rsub {i1} rsup {(m)} ## {t} rsub {i2} rsup {(m)}}} ## matrix {matrix {⋮ ## 1} # matrix {⋮ ## {t} rsub {i {n} rsub {i}} rsup {(1)}}} # matrix {matrix {⋱ ## …} # matrix {⋮ ## {t} rsub {i {n} rsub {i}} rsup {(m)}}}} right )
,
β
=
(
β
i
1
β
i
2
⋮
β
ℑ
)
β= left (matrix {matrix {{β} rsub {i1} ## {β} rsub {i2}} ## matrix {⋮ ## {β} rsub {im}}} right )
,
r
i
=
(
r
i
1
r
i
2
⋮
r
ℑ
)
{r} rsub {i} = left (matrix {matrix {{r} rsub {i1} ## {r} rsub {i2}} ## matrix {⋮ ## {r} rsub {im}}} right )
, and
ϵ
i
=
(
ϵ
i
1
ϵ
i
2
⋮
ϵ
i
n
i
)
{ϵ} rsub {i} = left (matrix {matrix {{ϵ} rsub {i1} ## {ϵ} rsub {i2}} ## matrix {⋮ ## {ϵ} rsub {i {n} rsub {i}}}} right )
,where
y
i
{y} rsub {i}
is the
n
i
{n} rsub {i}
vector of observations for individual
i
i
,
T
i
{T} rsub {i }
is the
n
i
×
d
{n} rsub {i} ×d
design matrix (with
d
=
m
+
1
d=m+1
),
β
β
is a d-vector of fixed effects (i.e. common to all individuals of the population),
r
i
{r} rsub {i}
isa d-vector of random effects (i.e. specific to each individual) and
ϵ
i
{ϵ} rsub {i}
is a
n
i
{n} rsub {i}
-vector of residual errors.The model is called linear mixed effects model because it is a linear combination of fixed and randomeffects. The random effects are assumed to be normally distributed in a linear mixed effects model:
r
i
∼
N
(
0
d
,
Ω
)
,
(
2.5
)
{r} rsub {i} ∼N left ({0} rsub {d} ,Ω right ) , (2.5)
Where
Ω
Ω
is the
d
×
d
d×d
variance-covariance matrix of the random effects. This matrix is diagonal if the components of
r
i
{r} rsub {i}
are independents.The vector of residual errors
ϵ
i
{ϵ} rsub {i}
is also normally distributed:
ϵ
i
∼
N
(
0,
Σ
i
)
,
(
2.6
)
{ϵ} rsub {i} ∼N left (0, {Σ} rsub {i} right ) , (2.6)
</sect2><sect2><title>3. MODELING3. MODELING3.1 DataThe data used in this paper comes from the World Bank and the International Monetary Fund. They describe the evolution of the Gross Domestic Product (current international dollar) in the countries of North Africa during the last twenty-seven years.Our data has 60 rows and 3 columns, while the columns containing the grouping factor (indicating the subject), the predictor(s) and the response. In our case, the grouping factor is Country, while that the second contains the Year (the predictor) and the third its GDP in current international Dollar (the response). The plot of the data show in Figure 1.Figure 1: The figure presents Gross Domestic Product (GDP, current $) versus Year. 3.2 ModelsThe mixed effects model combines a model for the fixed effects and a model for the random effects.Let us see some possible combinations.3.2.1 Model 1For the first proposed linear mixed effects model, we assume that gross domestic product (GDP) and growth rate (i.e. intercept and slope) may depend on the individual, we have 5 individuals (5 countries
i
=
1,2
,
…
,
5
i=1,2,…,5
),
y
i
=
b
i
0
+
b
i
1
t
ij
+
ϵ
ij
,
1
≤
j
≤
27
(
3.1
)
{y} rsub {i} = {b} rsub {i0} + {b} rsub {i1} {t} rsub {ij} + {ϵ} rsub {ij} , 1≤j≤27 (3.1)
Where
t
ij
{t} rsub {ij}
is a regression variable representing "Year".Table 1: The table presents the fixed effects results of fitting model 1.Parameter Estimate Std. Error t value Correlation

β
0
{β} rsub {0 }
-26299290120371 3453948510583 -7.514
β
1
{β} rsub {1 }
13247968623 1721065234 7.613

Figure 2: The figure presents the predicted Gross Domestic Product (GDP) versus observed GDP.3.2.2 Model 2We assume that the intercept randomly varies between individuals and the GDP increases with the same rate for all the individuals. The mathematical representation of this model is:
y
ij
=
b
0
+
b
1
t
ij
+
r
i
0
{y} rsub {ij} = {b} rsub {0} + {b} rsub {1} {t} rsub {ij} + {r} rsub {i0}
+
ϵ
ij
,
1
≤
i
≤
5
∧
1
≤
j
≤
27
(
3.2
)
{ϵ} rsub {ij} , 1≤i≤5 and 1≤j≤27 (3.2)
Where
t
ij
{t} rsub {ij}
is a regression variable represents "Year".Table 2: The table presents the fixed effects results of fitting model 2.Parameter Estimate Std. Error t value Correlation

β
0
{β} rsub {0 }
-26299290120359 3500173713954 -7.514 -0.999
β
1
{β} rsub {1 }
13247968623 1740248929 7.613

3.2.3 Model 3In this model, we assume that the intercepts are the same for all individuals but different growth rates for individuals (we put a random effect on the slopes). The mathematical representation of this model is:
y
ij
=
b
0
+
b
1
t
ij
+
r
i
1
t
ij
{y} rsub {ij} = {b} rsub {0} + {b} rsub {1} {t} rsub {ij} + {r} rsub {i1} {t} rsub {ij}
+
ϵ
ij
,
1
≤
i
≤
5
∧
1
≤
j
≤
27
(
3.3
)
{ϵ} rsub {ij} , 1≤i≤5 and 1≤j≤27 (3.3)
Table 3: The table presents the fixed effects results of fitting model 3.Parameter Estimate Std. Error t value

β
0
{β} rsub {0 }
12898374004 16898205283 0.763

3.3 Comparing linear mixed effects modelsBayesian information criterion (BIC) [10] and Akaike's information criterion (AIC) [11] are used to compare several models.Table 4: The table presents the results of ANOVA.Model AIC BIC logLik deviance

Model 1 3240.6 3253.2 -1614.3 3228.6Model 2 3238.3 3246.6 -1615.1 3230.3Model 3 3134.5 3145.0 -1562.3 3124.5

The best model, according to (AIC) and (BIC), is the Model 3 that assumes same fixed intercept and a random slope. The plot of individual predicted GDP versus observed GDP (Figure 3):Figure 3: The figure presents the predicted Gross Domestic Product (GDP) versus observed GDP.CONCLUSIONThe modeling and simulation of the Gross Domestic Product (GDP) data using Linear Mixed-Effects Models proved to be successful. We can say that the Linear Mixed Effects Models can help to characterize and to understand many complex linear economical processes. This study has shown the effectiveness of the linear mixed effects models as a new approach to explain GDP data. The final proposed model (model 3) has proven good estimations for all parameters in the model.5. REFERENCESLaird N.M and Ware J.H: Random-effects models for longitudinal data. Biometrics. 1982; 38(4): 963-974.Ware J.H: Linear models for the analysis of longitudinal studies. The Annals of Statistics. 1985; 39(2): 95-101.Diggle P, Heagerty P and Liang K.Y and Zeger S.L. Analysis of longitudinal data, 2002.Bates D, Maechler M, Bolker B.M and Walker S.C. Fitting Linear Mixed-Effects Models Using lme4. 2015; 67(1): 1-48.Davidian M and Giltinan D.M. Non-linear models for repeated measurement data, 1995Ware J.H. Linear models for the analysis of longitudinal studies. The Annals of Statistics.1985; 39(2): 95-101.Davidian M and Giltinan D.M. Non-linear models for repeated measurement data, 1995.Bates D, Maechler M, Bolker B and Walker S. Linear Mixed-Effects Models Using Eigenand S4, R package version 1.1-10, 2015; 39(2): 95-01. Available on: http://CRAN.R-project.org/package=lme4.Pinheiro J.C and Bates D.M. Mixed-Effects Models in S and S-PLUS, 2000.Schwarz G: The Annals of Statistics.1978; 6(2): 461-464.Akaike H. A new look at the statistical model identification. IEEE Trans. Autom. Control. 1974; 19(6): 716-723.List of acronym:GDP: Gross Domestic Product is a monetary measure of the market value of all the final goods and services produced in a period of time, often annually or quarterly. Nominal GDP estimates are commonly used to determine the economic performance of a whole country or region, and to make international comparisons.AIC: The Akaike information criterion: is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.BIC: The Bayesian information criterion is a criterion for model selection among a finite set of models; the model with the lowest BIC is preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC).