Spatial and Non-Spatial Panel Data Estimators: Simulation Study
and Application to Personal Income in U.S. States
AHMED H. YOUSSEF1, MOHAMED R. ABONAZEL1, OHOOD A. SHALABY2
1Department of Applied Statistics and Econometrics, Faulty of Graduate Studies for Statistical
Research, Cairo University, Giza, EGYPT
2National Center for Social and Criminological Research, Giza, EGYPT
Abstract:- The spatial analysis aims to understand and explore the nature of entanglements and interactions
between spatial units’ locations. The analysis of models involving spatial dependence has received great
attention in recent decades. Because ignoring the presence of spatial dependence in the data is very likely to
lead to biased or inefficient estimates if we use traditional estimation methods. Therefore, this paper is an
attempt to assess the risks involved in ignoring the spatial dependence that characterizes the panel data by using
a Monte Carlo simulation (MCS) study for two of the most common spatial panel data (SPD) models; Spatial
lag model (SLM) and spatial error model (SEM), by comparing the performance of two estimators; i.e., spatial
maximum likelihood estimator (MLE) and non-spatial ordinary least squares (OLS) within-group estimator,
across two levels of analysis; Parameter-level in terms of bias and root mean square error (RMSE), and model-
level in terms of goodness of fit criteria under different scenarios of spatial units N, time-periods T, and spatial
dependence parameters, by using two different structures of spatial weights matrix; inverse distance, and
inverse exponential distance. The results show that the non-spatial bias and RMSE of are functions of the
degree of spatial dependence in the data for both models, i.e., SLM and SEM. If the spatial dependence is
small, then the choice of the non-spatial estimator may not lead to serious consequences in terms of bias and
RMSE of . On the contrary, the choice of the non-spatial estimator always leads to has disastrous
consequences if the spatial dependence is large. On the other hand, we provide a general framework that shows
how to define the appropriate model from among several candidate models through application to a dataset of
per capita personal income (PCPI) in U.S. states during the period from 2009 to 2019, concerning three main
aspects: educational attainment, economy size, and labour force type. The results confirm that PCPI is spatially
dependent lagged correlated.
Key-Words: - Per capita personal income, spatial autoregressive combined model, spatial lag model, spatial
error model, spatial weights matrix, simulation.
Received: August 29, 2021. Revised: May 21, 2022. Accepted: June 12, 2022. Published: July 1, 2022.
1 Introduction
Panel data refer to cross-section units (e.g.,
individuals, groups, countries, companies) observed
over several time periods. In a SPD setting, the
cross-section observations are associated with a
particular location in space. The data can be
observed either at point locations (e.g., housing
data) or aggregated over regular or irregular areas
(e.g., countries, regions, states, counties). The
structure of the interactions between each pair of
spatial units is represented by a spatial weights
matrix. On a somewhat more formal level, in spatial
econometrics, these interactions may relate to the
models’ dependent variable, to the exogenous
variables, to the disturbance term, or various
combinations of them.
The analysis of models involving spatial
dependence has received great attention in recent
decades. Because ignoring the presence of the
spatial dependence in the data is very likely to lead
to inefficient or biased estimates if we use
traditional estimation methods, such as OLS. For
instance, when the spatial dependence exists in the
data, then this may be an additional source of
variation. As we know, ignoring the source of
variation can lead to biased estimates, and also the
traditional estimators are no longer efficient due to
changes in asymptotic variance-covariance matrices
(VCMs). Therefore, alternative estimation methods
had to be developed that take into account spatial
dependence that characterizes the data to obtain
more accurate results.
Spatial econometrics was first studied in the 1950s.
It was named by Jean Paelinck in the early 70s.
Many works have been published since the launch
of Cliff and Ord’s seminal work, see [1]. More
recently, researchers have recognized the
importance of introducing this approach in the case
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of panel data models to take the advantages
provided by these models, where SPD models have
the same structure of panel data model which
capture spatial interactions across spatial units and
over time. With panel data available, we can not
only improve the efficiency of estimates but also
investigate some issues that cannot be addressed by
the cross-section data, such as heterogeneity and
dependence across time. Besides the additional
information regarding the use of the cross-sectional
dimension of the data enables accounting for the
presence of unobservable heterogeneity among
cross-section units. Also, access to information
included in both cross-sectional and temporal
dimensions enables us to model dynamic relations,
see [2, 3, 4, and 5].
Since many applications in spatial econometrics are
currently based on panel data. In addition, the
attention to the space of geographical units and the
interaction between them has become an important
feature of the empirical work, see [6, 7, and 8],
because this type of data possesses information
about the location of the observations that may
constitute an additional source of variation, and
ignoring this variation may lead to biased estimates,
see [9]. Therefore, many of researchers are trying to
propose estimation methods that allow for the
existence of spatial dependence in panel data
models, see [10, 11, and 12].
Although a reasonable amount of literature has been
devoted to reviewing the spatial econometrics
techniques in the last decades, this paper focus some
of the recent theoretical advances in this research
area. For this purpose, as follows:
(1) Studying the effect of ignoring spatial
dependence in the data by comparing the
performance of the spatial and non-spatial
estimators for two specifications of SPD
models, i.e., SLM and SEM, through a MCS
study under different scenarios of N, T, spatial
dependence degree, and spatial weights matrix.
(2) Investigating the influence of the structure of
spatial weights on the performance of spatial
estimators and the goodness of fit model
through a simulation study.
Applying the SPD modeling to analyze the
determinants of PCPI in U.S. states, and
providing a general framework of how to select
the appropriate SPD model among several
candidates.
This paper is divided into 9 sections as follows:
Section 2 introduces the specification of SPD
models and some other related terminology, sections
3 illustrates the assumptions of the SPD models,
section 4 provides a brief summary for spatial
weights matrix, section 5 explains the SLM and its
estimation methods, section 6 explains the SEM and
its estimation methods, section 7 provides a MCS
study, section 8 presents our application to personal
income in U.S. States, finally, section 9 includes the
concluding remarks.
2 The Specification of SPD Models
As we mentioned previously; spatial econometrics
focuses on interaction effects among geographical
units, such as counties, regions, etc. in modelling
terms; Elhorst [13] defined three types of interaction
effects to explain why an observation related to a
specific location may be dependent on observations
at other locations as stated in
(1) Endogenous interaction effects among the
dependent variable (y): measures the
dependency of unit (A) in dependent variable y
on other units in the same dependent variables.
This effect can be denoted by y.
(2) Exogenous interaction effects among the
independent variables (X): measures the
dependency of unit (A) in dependent variable y
on other units in the explanatory variables X.
This effect can be denoted by X.
(3) Interaction effects among the error terms
(u): refers that units may behave similarly
because they have the same unobserved
characteristics or face similar unobserved
environments. This effect can be denoted by
u.
A full static model with the above three types of
interaction effects can be expressed as:
(1)
where is a vector consisting of one
observation of the dependent variable for every
spatial unit in the sample at time
, is a matrix of exogenous
explanatory variables, reflects the error terms
specification of the model, which is assumed to be
spatially correlated, and is a (N1) vector of
disturbance terms, whose elements have zero
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mean and finite variance . is the spatial
autoregressive coefficient, is the spatial
autocorrelation coefficient, and are
vectors contain the response parameters of
the exogenous explanatory variables. Any vector or
a matrix pre-multiplied by denotes its spatially
lagged value. is a (NN) non-negative matrix of
known constants describing the spatial arrangement
of the units in the sample. Where the element in
the matrix represents the prior strength of the
interaction between spatial unit (row) and spatial
unit (column). In other words, the elements of,
are non-zero if and j are neighbors. By
convention, a self-neighbor relation is excluded, so
the diagonal elements of are zero.
To generalize the spatial weights matrix in panel
data settings, the weights are assumed to remain
constant over time, then the full (NT ×NT) weights
matrix becomes:
(2)
Model (1) can be rewritten in a reduced form as
follows:
(3)
where:
(4)
(5)
Table 1 summarizes the main spatial regression
models. These models can be treated as fixed effects
(FE) or random effects (RE).
In this paper, we will focus on studying two
specifications of SPD models that suffered from
some econometric problems; SLM, and SEM which
mentioned in Fig. 1. In the first case; it must be
dealt with the endogeneity of the spatial lag (SL),
and in the second case; the non-spherical nature of
the error VCM must be taken into account.
3 Models Assumptions
Even though there are different SPD model
specifications, there are some basic common
features for all of them. The following common
assumptions will be used throughout the text for
static SPD models. In addition to these, specific
assumptions for some models will be listed
when needed.
A1. Assumptions of Spatial Weights Matrices:
(1) The spatial weights matrix is non-
stochastic matrix with zero diagonals.
(2) The spatial transformation matrices (i.e.,
)) are invertible on the compact parameter
spaces of spatial parameters and.
(3) The admissible parameter space for the true
spatial parameters and is[-1, 1].
(4) Row sums of the matrices
before is
row-standardized, are uniformly bounded (UB)
in absolute values as N goes to infinity.
A2. Assumptions of the Error Components: The
relevant disturbances, i.e.
are across and with zero mean, and
finite variance, and their higher than fourth
moments exist, i.e., for some c>0.
A3. Assumptions on Covariates: The regressors
are non-stochastic and have full rank and their
elements are UB in absolute value.
A4. Assumption of N and T: Most studies assume
that N is large while T can be finite or large. The
case of finite N and large T is of less interest as the
incidental parameter problem doesn’t occur in this
situation.
These assumptions are frequently made in spatial
econometrics. For cross-sectional models; see [14],
or [15], among others. For panel data models; see
[11], or [16], among others.
4 Spatial Weights Matrix
A spatial weights matrix is a representation of the
spatial structure in a particular data. It is a key
element in spatial models, which represents the
spatial dependence structure between locations
exogenously, see [17] and [18]. In other words,
Anselin [19] mentioned that the weights matrix is
the formal expression of spatial dependence
between observations.
The problem of choosing the optimal weights matrix
is still in the developing phase. In this paper, we
focus only on two structures of the spatial weights
matrix as follows:
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Table 1. SPD Models with Different Combinations of Spatial Interactions
Type of Model
Spatial Interaction Effects
Term
Number
SLM
Spatial Lag Model
1
SEM
Spatial Error Model
1
SAC
Spatial Autoregressive Combined Model
&
2
SLX
Spatial Lag of X Model
K
SDM
Spatial Durbin Model
&
K+1
SDEM
Spatial Durbin Error Model
&
K+1
GNS
General Nesting Spatial Model
& &
K+2
Fig. 1: SPD Models under Consideration.
Note: Where is called the time-invariant individual (spatial) effects or individual-specific effects, and is
independent and identically distributed disturbances.
(1) The inverse distance weights: This method
relies on a simple transformation by taking the
inverse of the distance separating the spatial
units, and respects the Tobler’s law: the weights
are greater (smaller) as the units are spatially
closer (further apart) as in the following
equation:
(6)
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where is the distance between region and
region , and
denotes a threshold distance (or
bandwidth).
(2) The inverse exponential distance weights:
Another possible transformation of the distance
can be defined as:
(7)
This transformation gives more weights to spatially
close units and fewer weights to units that are
further apart, for more details, see [9].
5 Spatial Lag Panel Data Model
A SLM or spatial autoregressive (SAR) model
includes a spatially lagged dependent variable on
the right-hand side of the regression specification, as
follows:
(8)
The SLM can be treated as:
5.1 Fixed Effects
By adding the time-invariant individual FE,, to the
model (8), the SLM can be rewritten after stacking
the observations across individual and time as:
(9)
where is (N1) vector contains spatial specific
effects, , is a (T×1)
vector of ones and is a Kronecker product. This
model suffers from As well known, when,
there is no consistent estimator of the individual FE,
due to the incidental parameter problem, in another
words, the No. of parameters goes to when N
goes to ∞. [20] used the transformation in (10) to
eliminate the FE from the model (9) and using these
transformed variables to estimate the parameters by
using ML estimation.
(10)
The transformed model for (9) can be written as:
(11)
where:
,
(12)
Besides the incidental parameter problem, the
endogeneity of
violates the assumption
of the standard regression model
that
. Therefore, the focus
in this section will base on ML estimation because
the MLE account for the endogeneity of
,
also, the No. of researches considering IV/GMM
estimators of SPD models is still sparse. In this
context, Elhorst [10] suggested a concentrated
likelihood function that can be maximized from the
residuals of the OLS regression of on and
the residuals of the OLS regression of
on . Then the MLE of is obtained by
maximizing the following concentrated log-
likelihood function:
(13)
where C is a constant not depending on. This
maximization problem is only solved numerically,
since a closed-form solution for doesn’t exist.
Therefore, an iteration procedure must be used,
which require to be initially fixed to calculate
and. Finally, and are obtained from the
first-order conditions of the likelihood function by
replacing with its numerically estimated value, see
[17].
5.2 Random Effects
In contrast to the FE approach, the RE models do
not have a problem with a large N. In this context,
the SLM can be written in a stacked form across
individual and time as:
(14)
Assuming that, the unobserved individual effects,
are uncorrelated with the other explanatory
variables in the model, and
.
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Additionally, the idiosyncratic error term,
, an independent from each other. The log-likelihood of the model (14) is:
(15)
where the transformed variables are defined as:
(16)
and is defined as:
(17)
By using a similar procedure in SLM with FE, we
can estimate , but the subscript * must be
replaced by Given can be
estimated by maximizing the concentrated log-
likelihood function with respect to .
(18)
where the element of is defined as follows:
(19)
6 Spatial Error Panel Data Model
The SAR specification for error vector in time
can be defined as:
(20)
The SEM can be treated as:
6.1 Fixed Effects
By adding the time-invariant individual FE,, to the
model (20):
(21)
To eliminate the individual FE, the model (21) is
transformed according to the same -
transformation which used for SLM and which
defined in (10).
As mentioned in [21], the estimation procedures of
SEM with FE can be summarized as follows:
(1) Estimated OLS residuals of the transformed
variables can be used to obtain an initial
estimate of .
(2) The initial estimate of can be used to compute
a (spatial) feasible generalized least squares
(FGLS) estimator of and and a new set of
estimated GLS residuals.
(3) Then an iterative procedure can be used: the
concentrated likelihood and the GLS estimators
are alternately computed until convergence.
Lee and Yu [16] proved that the estimation of the
SLM or SEM with spatial FE, which is based on the
Q0-transformation, produces biased estimates for
if N is large and T is fixed, and they called this
procedure the direct approach. Starting with the
SAC model, and using asymptotic theory, Lee and
Yu [16] suggested two methods to obtain consistent
results, as follows:
(1) The first method: Instead of demeaning, they
proposed an alternative procedure to eliminate
the spatial FE, reducing the number of
observations available for estimation by one
observation, i.e., from NT to N(T-1)
observations. This procedure is called the
transformation approach.
(2) The second method: It is a bias correction
procedure for the parameter’s estimates
obtained by the direct approach based on ML
function that is obtained under the
transformation approach. The biases of the SAC
model [16] can be conducted on the SLM and
SEM models. Where the of obtained by
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the direct approach will be biased. This bias can
easily be corrected by:
(22)
This bias correction will have no any effect if T is
large, for more details, see [22].
6.2 Random Effects
In this section, we focus on the approach of Kapoor
et al. [11] for specifying the SEM with RE which
can be written after stacking across individual and
time as follows:
(23)
where is a (N×1) vector of cross-sectional random
components, , and the vectors
are independent of each other and the
regressor matrix X. The second line in (23) can be
written in a reduced form as follows:
(24)
The corresponding error VCM is:
(25)
where is VCM of. Since are
independent, it implies that:
(26)
where is defined in (10) and is defined as:
(27)
Since then:
(28)
(29)
Kapoor et al. [11] proposed a generalization of
generalized method of moments (GMM) estimator
provided in [23] for estimating the SAR parameter
and the two variance components of the disturbance
process
. Therefore, to estimate the model
(23), Kapoor et al. [11] defined three sets of GMM
estimators based on the following moment
conditions for T ≥ 2:
(30)
where:
(31)
To estimate
, follow the steps:
(1) The First Step:
o The first three moment conditions can be used
to obtain the first set of GMM estimators for
.
o The initial estimates obtained
are
then used to provide an estimate for
based on the fourth-moment condition.
(2) The Second Step: Under the normality
assumption of innovation Kapoor et al. [11]
derived the VCM of the sample moments at the
true parameter values Ȩ, whose inverse is to be
used as the optimal weighting matrix in a GMM
estimator.
The second set of GMM estimators is then
defined as the nonlinear least squares estimators
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based on all moment conditions weighted by the
optimal weighting scheme
.
(3) The Third Step: the third set of GMM
estimators is suggested because of
computational considerations and is based on a
simpler weighting matrix. The third set of
GMM estimators uses all moment conditions
but a simplified weighting scheme.
(4) The Fourth Step: The FGLS for can be
obtained based on consistent estimates for
that result from previous steps.
7 Monte Carlo Simulation Study
In this section, we focus on trying to achieve two
main objectives in two specifications of SPD
models, i.e., SLM and SEM with FE, as follows:
(1) Comparing between the finite sample properties
of the spatial MLEs (transformation approach),
which will be referred to in an abbreviated
manner as (the spatial estimator), and the non-
spatial OLS within-group estimator, or in a
short way (the non-spatial estimator), under
different values for temporal and cross-sectional
dimensions (N and T), spatial parameters, and
spatial weights matrix.
(2) Verifying the impact of the spatial weights
structure on the goodness of fit model and
performance of estimates in the used SPD
models. Where some researches indicate that the
choice of the spatial weights matrices are crucial
and can affect the findings of the research [24].
7.1 Design of the Simulation
We relied on the general methodology of MCS for
Mooney [25], which has been mentioned in most
empirical studies, see e.g., [26]. Fig. 2 provides a
summary of the used algorithm in our simulation
study.
7.2 Simulation Results
This section summarizes the results obtained from
our simulation described previously. Each model
focuses on the comparison between the non-spatial
and spatial estimators in terms of bias and RMSE
of, in addition to, provides the comparison among
the two structures of spatial weights matrix in terms
of bias and RMSE of . Besides, it furnishes
information on the goodness of fit criteria for each
model and the relationship between biases of
ignoring spatial and the degree of spatial
dependence in the data.
Table 2. Design of Our MCS
Design Factor
Levels
No. of Levels
Type of SPD Models
SLM and SEM
2
Value of Parameter
4
Type of Weights Matrix for Data
Generation
W1 or W2
2
No. of Spatial Units
N={5, 20, 35, 60}
4
No. of Time-periods
T={10, 30, 50}
3
Coordinates of Distance
1
Spatial FE
1
5
1
Explanatory Variable
1
Error Terms
1
No. of Unique Simulations
SLM: 4 (values of ) × 2 (types of W) × 4 (values of N) × 3 (values of T)
= 96 Simulations.
SEM: 4 (values of ) × 2 (types of W) × 4 (values of N) × 3 (values of T)
= 96 Simulations.
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No. of Replications
R=1000
Total No. of Simulations
SLM: 96×1000= 96000
SEM: 96 ×1000= 96000
Fig. 2: Algorithm of Our Simulation Study
*
Note: W1: Inverse distance weights and W2: inverse exponential distance weights.
7.2.1 Simulation Results of Spatial Lag Model
The findings show that the bias and RMSE of
resulting from ignoring the presence of spatial
dependence in SLM is a function of the degree or
magnitude of spatial dependence in the data. In
other words, If the spatial dependence is small, i.e.,
, then the consequences of choosing
the non-spatial estimator are not great, where the
non-spatial bias and RMSE of may be equal to or
less than the spatial bias and RMSE of in some
cases. Quite the contrary, the non-spatial estimator
choice definitely brings dire consequences in terms
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of bias and RMSE of when is large, see Table 3
- Table 6. On average, the spatial estimator of
performs mostly satisfactory, where it produces a
19.32% bias less than the non-spatial estimator
when is small. This percentage reaches 82.02%
when is large. While, it produces a 19.41% RMSE
less than those of the non-spatial estimator when is
small. This percentage reaches 80.58% when is
large.
On the other hand, i.e. model-level, the spatial
Akaike information criterion (AIC) and Bayesian
information criterion (BIC) are always less than
their non-spatial counterparts when is large.
Additionally, the spatial model always performs
better than the non-spatial model when N and T are
small regardless of the spatial dependence strength
and the structure of the spatial weights matrix.
Table 3. Simulation Results of SLM and N = 5
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.146
0.153
0.211
0.241
156.1
161.9
0.229
0.261
158.8
172.2
-0.8
0.163
0.171
0.310
0.339
158.3
164.0
1.295
1.306
215.6
229.0
0.2
0.110
0.115
0.150
0.182
154.3
160.0
0.175
0.207
159.2
172.6
0.8
0.031
0.032
0.111
0.139
151.6
157.3
0.473
0.510
342.3
355.7
30
-0.2
0.074
0.081
0.074
0.092
490.8
499.8
0.119
0.140
466.2
487.2
-0.8
0.080
0.089
0.099
0.122
493.7
502.7
0.869
0.874
642.3
663.4
0.2
0.057
0.062
0.065
0.081
487.6
496.6
0.067
0.084
480.5
501.6
0.8
0.016
0.018
0.079
0.099
484.7
493.7
2.329
2.335
1080.2
1101.2
50
-0.2
0.083
0.087
0.073
0.088
815.8
826.3
0.147
0.160
771.0
795.7
-0.8
0.084
0.089
0.113
0.128
821.7
832.2
1.010
1.012
1077.8
1102.5
0.2
0.065
0.068
0.052
0.064
810.3
820.9
0.078
0.093
795.8
820.4
0.8
0.019
0.020
0.054
0.068
803.9
814.4
1.497
1.501
1783.2
1807.8
W2
10
-0.2
0.117
0.123
0.209
0.240
157.4
163.1
0.259
0.289
162.6
176.0
-0.8
0.104
0.111
0.286
0.317
160.0
165.7
1.656
1.666
247.5
260.9
0.2
0.094
0.099
0.151
0.183
155.2
161.0
0.188
0.219
163.1
176.5
0.8
0.027
0.028
0.112
0.140
152.8
158.5
0.495
0.533
354.5
367.9
30
-0.2
0.049
0.057
0.071
0.089
492.0
501.0
0.131
0.152
476.6
497.6
-0.8
0.042
0.049
0.083
0.104
494.0
503.0
1.105
1.110
716.0
737.1
0.2
0.042
0.048
0.065
0.081
489.1
498.1
0.067
0.084
491.4
512.5
0.8
0.013
0.015
0.076
0.095
486.3
495.4
2.441
2.448
1109.9
1131.0
50
-0.2
0.060
0.063
0.068
0.082
817.5
828.0
0.171
0.182
792.1
816.8
-0.8
0.047
0.051
0.090
0.105
822.1
832.7
1.312
1.314
1222.4
1247.1
0.2
0.051
0.053
0.052
0.064
812.6
823.1
0.088
0.102
817.8
842.4
0.8
0.015
0.016
0.051
0.064
807.2
817.8
1.460
1.465
1842.5
1867.2
Table 4. Simulation Results of SLM and N = 20
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.180
0.185
0.068
0.085
644.9
654.8
0.072
0.089
608.7
681.3
-0.8
0.142
0.150
0.089
0.106
653.2
663.1
0.402
0.409
791.3
863.9
0.2
0.169
0.172
0.059
0.073
635.0
644.9
0.061
0.076
599.9
672.4
0.8
0.064
0.064
0.059
0.074
619.4
629.3
0.062
0.077
798.0
870.5
30
-0.2
0.195
0.197
0.074
0.082
1941.7
1954.9
0.065
0.074
1788.2
1884.9
-0.8
0.156
0.158
0.111
0.118
1973.8
1987.0
0.459
0.461
2355.9
2452.6
0.2
0.176
0.177
0.039
0.048
1912.8
1925.9
0.038
0.046
1791.6
1888.3
0.8
0.062
0.062
0.042
0.052
1868.7
1881.9
0.375
0.378
3358.8
3455.5
50
-0.2
0.142
0.143
0.024
0.031
3291.3
3306.0
0.048
0.055
2970.2
3078.2
-0.8
0.110
0.112
0.034
0.041
3320.8
3335.5
0.392
0.393
3915.9
4023.8
0.2
0.131
0.132
0.033
0.040
3262.2
3276.9
0.025
0.031
2997.9
3105.9
0.8
0.047
0.047
0.076
0.082
3219.1
3233.8
0.750
0.751
6010.7
6118.7
W2
10
-0.2
0.032
0.035
0.066
0.083
656.9
666.8
0.162
0.176
728.0
800.6
-0.8
0.010
0.012
0.084
0.102
659.9
669.8
4.222
4.226
1509.7
1582.3
0.2
0.033
0.035
0.058
0.073
652.3
662.1
0.067
0.083
727.8
800.4
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Volume 21, 2022
0.8
0.010
0.011
0.061
0.077
648.5
658.4
3.677
3.681
1568.7
1641.3
30
-0.2
0.034
0.035
0.062
0.071
1987.6
2000.8
0.185
0.190
2126.0
2222.7
-0.8
0.009
0.009
0.084
0.094
1998.6
2011.8
4.663
4.664
4490.6
4587.3
0.2
0.038
0.039
0.034
0.042
1972.2
1985.4
0.045
0.054
2132.2
2228.9
0.8
0.015
0.016
0.054
0.064
1948.0
1961.2
3.800
3.802
4631.2
4727.9
50
-0.2
0.026
0.027
0.024
0.030
3325.7
3340.4
0.162
0.165
3505.0
3613.0
-0.8
0.006
0.007
0.027
0.033
3337.9
3352.7
4.199
4.199
7378.2
7486.2
0.2
0.030
0.031
0.037
0.044
3306.1
3320.9
0.056
0.063
3530.2
3638.1
0.8
0.013
0.013
0.087
0.093
3273.6
3288.3
3.944
3.945
7728.5
7836.5
Table 5. Simulation Results of SLM and N = 35
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.281
0.284
0.094
0.107
1111.8
1123.4
0.072
0.084
1050.4
1193.1
-0.8
0.235
0.239
0.124
0.136
1133.1
1144.7
0.388
0.392
1271.7
1414.4
0.2
0.258
0.259
0.064
0.077
1089.1
1100.7
0.058
0.071
1040.7
1183.4
0.8
0.094
0.094
0.047
0.060
1055.8
1067.4
0.069
0.083
1337.2
1479.9
30
-0.2
0.223
0.224
0.028
0.035
3430.8
3445.6
0.046
0.054
3080.3
3263.7
-0.8
0.182
0.184
0.043
0.051
3471.0
3485.8
0.293
0.295
3731.5
3914.9
0.2
0.213
0.214
0.024
0.031
3389.1
3403.9
0.031
0.039
3072.6
3256.0
0.8
0.086
0.086
0.056
0.063
3312.7
3327.6
0.068
0.075
4666.9
4850.3
50
-0.2
0.176
0.177
0.032
0.038
5770.8
5787.2
0.034
0.040
5109.9
5312.2
-0.8
0.142
0.143
0.022
0.027
5815.5
5831.9
0.242
0.243
6186.3
6388.6
0.2
0.175
0.175
0.051
0.055
5727.4
5743.8
0.020
0.025
5110.3
5312.6
0.8
0.077
0.077
0.084
0.087
5624.9
5641.3
0.214
0.215
8192.9
8395.2
W2
10
-0.2
0.035
0.037
0.066
0.079
1146.9
1158.5
0.224
0.231
1280.1
1422.9
-0.8
0.008
0.009
0.077
0.092
1153.5
1165.1
4.998
5.001
2741.9
2884.7
0.2
0.044
0.045
0.048
0.059
1135.2
1146.7
0.048
0.060
1255.7
1398.4
0.8
0.019
0.019
0.060
0.073
1120.5
1132.1
2.646
2.650
2683.2
2826.0
30
-0.2
0.015
0.016
0.025
0.032
3490.6
3505.5
0.177
0.180
3768.2
3951.6
-0.8
0.004
0.004
0.027
0.034
3497.8
3512.6
4.776
4.777
8097.2
8280.5
0.2
0.017
0.018
0.030
0.037
3480.3
3495.2
0.037
0.044
3762.2
3945.6
0.8
0.005
0.006
0.046
0.054
3475.4
3490.3
3.423
3.424
8223.3
8406.7
50
-0.2
0.014
0.014
0.045
0.050
5826.4
5842.8
0.138
0.140
6200.7
6403.0
-0.8
0.003
0.004
0.039
0.046
5838.3
5854.7
4.355
4.355
13290.4
13492.7
0.2
0.016
0.016
0.060
0.064
5815.7
5832.1
0.076
0.079
6223.6
6425.9
0.8
0.006
0.006
0.082
0.086
5797.5
5813.9
3.967
3.968
13711.6
13913.9
Table 6. Simulation Results of SLM and N = 60
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.407
0.409
0.061
0.071
1875.2
1888.4
0.037
0.045
1785.3
2057.9
-0.8
0.387
0.389
0.091
0.100
1921.1
1934.3
0.149
0.155
2028.1
2300.7
0.2
0.360
0.361
0.038
0.048
1830.6
1843.8
0.032
0.041
1779.0
2051.6
0.8
0.122
0.122
0.034
0.043
1764.7
1777.9
0.042
0.052
2060.6
2333.2
30
-0.2
0.311
0.312
0.027
0.032
5859.6
5876.1
0.023
0.029
5236.3
5577.0
-0.8
0.268
0.270
0.018
0.023
5938.3
5954.8
0.141
0.143
5957.6
6298.3
0.2
0.302
0.303
0.042
0.047
5775.9
5792.4
0.018
0.023
5217.4
5558.1
0.8
0.120
0.120
0.041
0.046
5607.2
5623.7
0.068
0.071
6065.9
6406.7
50
-0.2
0.355
0.355
0.015
0.018
9641.3
9659.4
0.018
0.022
8695.1
9067.5
-0.8
0.328
0.329
0.037
0.041
9830.3
9848.3
0.125
0.126
9958.6
10331.0
0.2
0.319
0.319
0.024
0.028
9471.0
9489.0
0.014
0.018
8688.5
9060.9
0.8
0.113
0.113
0.050
0.053
9173.9
9191.9
0.128
0.129
12227.8
12600.2
W2
10
-0.2
0.030
0.031
0.054
0.064
1969.5
1982.7
0.159
0.164
2250.7
2523.3
-0.8
0.007
0.008
0.083
0.094
1982.3
1995.5
6.479
6.481
4784.6
5057.2
0.2
0.034
0.035
0.032
0.040
1952.3
1965.5
0.159
0.164
2256.7
2529.3
0.8
0.013
0.013
0.082
0.093
1928.3
1941.5
6.619
6.620
4855.7
5128.3
30
-0.2
0.018
0.018
0.037
0.043
5992.1
6008.6
0.109
0.111
6533.1
6873.9
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.56
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Volume 21, 2022
-0.8
0.004
0.005
0.022
0.028
6014.0
6030.4
5.810
5.811
14017.7
14358.4
0.2
0.020
0.021
0.063
0.067
5970.1
5986.5
0.194
0.195
6567.5
6908.2
0.8
0.008
0.008
0.109
0.112
5930.0
5946.5
6.770
6.770
14356.6
14697.3
50
-0.2
0.023
0.023
0.016
0.020
10005.4
10023.4
0.130
0.131
10856.8
11229.2
-0.8
0.006
0.007
0.044
0.049
10059.1
10077.2
6.040
6.040
23327.3
23699.7
0.2
0.023
0.024
0.026
0.031
9962.1
9980.1
0.177
0.178
10922.2
11294.6
0.8
0.008
0.008
0.073
0.076
9896.2
9914.2
6.662
6.663
23906.3
24278.7
As for the level of comparison between the spatial
weight structures, Fig. 3 displays the pattern of
spatial bias and RMSE of under the influence of N
and T for each structure of the spatial weights
ignoring the values of . The results indicate that
each structure of the weights matrix has a specific
pattern of the spatial bias and RMSE of . In all
structures, the pattern of spatial bias and RMSE of
are irregular with increasing N or T, however, it can
be argued that the spatial estimator produces a large
bias and small RMSE when T is small.
On the other hand; Fig. 4 displays the spatial bias
under the influence of N and λ for each structure of
the spatial weights ignoring the values of T. our
conclusion from this Figure can be pointed that each
structure of spatial weights has a specific pattern of
the spatial bias according to the used values of N
and. Interestingly, the result of the spatial bias is
in favor of W2.
Fig. 5 shows the behavior of the spatial RMSE
across different combinations of N, T, and the
structure of the weights matrix. The results confirm
that the results of RMSE are always in favor of
W2.
For a comparison between the two structures of
weights matrix in terms of the spatial AIC and BIC.
The results show that the two structures of weights
matrix seem to provide very similar levels of spatial
AIC and BIC, However, the SLM based on W2
returns a slightly higher value of spatial AIC and
BIC in all cases, see Table 3 - Table 6.
W1
W2
Bias
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RMSE
Fig. 3: The Spatial bias and RMSE of in SLM at Different Values of N, T, and W for All
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Volume 21, 2022
Fig. 4: Bias in SLM at Different Values of N,, and W for All T
Fig. 5: RMSE in SLM at Different Values of N, T, and W for All
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Volume 21, 2022
Table 7. Simulation Results of SEM and N = 5
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.202
0.259
0.101
0.127
144.4
150.1
0.107
0.134
152.3
165.7
-0.8
0.208
0.261
0.098
0.122
143.0
148.7
0.137
0.172
167.8
181.2
0.2
0.152
0.197
0.102
0.126
144.7
150.4
0.101
0.127
148.1
161.5
0.8
0.043
0.058
0.091
0.113
144.6
150.4
0.161
0.203
207.7
221.1
30
-0.2
0.104
0.130
0.068
0.087
428.1
437.2
0.071
0.089
438.6
459.7
-0.8
0.112
0.140
0.063
0.079
427.8
436.8
0.082
0.101
486.0
507.1
0.2
0.076
0.096
0.063
0.079
428.0
437.0
0.064
0.082
435.7
456.7
0.8
0.023
0.030
0.056
0.072
428.3
437.3
0.136
0.171
675.6
696.7
50
-0.2
0.081
0.104
0.050
0.062
712.6
723.2
0.051
0.063
726.2
750.8
-0.8
0.089
0.111
0.048
0.060
709.7
720.2
0.057
0.071
801.8
826.4
0.2
0.061
0.077
0.044
0.054
711.5
722.1
0.045
0.056
723.3
747.9
0.8
0.016
0.021
0.041
0.051
712.0
722.5
0.086
0.111
1141.9
1166.5
W2
10
-0.2
0.179
0.223
0.101
0.127
144.5
150.3
0.108
0.135
152.6
165.9
-0.8
0.160
0.200
0.095
0.118
143.5
149.3
0.148
0.186
173.2
186.6
0.2
0.140
0.180
0.102
0.126
144.7
150.4
0.101
0.126
148.3
161.7
0.8
0.042
0.057
0.089
0.110
144.6
150.3
0.167
0.211
211.2
224.6
30
-0.2
0.094
0.117
0.068
0.087
428.2
437.3
0.071
0.090
439.3
460.4
-0.8
0.087
0.108
0.062
0.077
428.2
437.3
0.088
0.108
503.1
524.1
0.2
0.072
0.090
0.062
0.079
428.0
437.0
0.064
0.082
436.3
457.4
0.8
0.022
0.029
0.055
0.070
428.2
437.3
0.143
0.180
687.0
708.1
50
-0.2
0.073
0.092
0.050
0.062
712.8
723.3
0.051
0.064
727.3
752.0
-0.8
0.067
0.084
0.047
0.058
710.2
720.8
0.061
0.076
830.4
855.0
0.2
0.057
0.072
0.044
0.054
711.6
722.1
0.045
0.056
724.3
748.9
0.8
0.016
0.020
0.040
0.050
711.9
722.5
0.089
0.114
1161.0
1185.7
Table 8. Simulation Results of SEM and N = 20
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.135
0.169
0.055
0.070
570.1
580.0
0.054
0.068
594.6
667.1
-0.8
0.131
0.162
0.053
0.066
569.5
579.4
0.058
0.073
629.3
701.9
0.2
0.116
0.146
0.054
0.067
568.7
578.6
0.052
0.065
584.8
657.4
0.8
0.042
0.056
0.051
0.064
570.7
580.6
0.054
0.069
601.9
674.5
30
-0.2
0.075
0.095
0.033
0.040
1705.8
1719.0
0.033
0.041
1737.6
1834.4
-0.8
0.074
0.093
0.030
0.036
1705.4
1718.5
0.033
0.040
1842.3
1939.0
0.2
0.063
0.079
0.031
0.039
1706.8
1720.0
0.031
0.039
1726.1
1822.9
0.8
0.021
0.026
0.030
0.038
1705.2
1718.4
0.041
0.051
2018.2
2115.0
50
-0.2
0.057
0.071
0.023
0.029
2839.8
2854.5
0.023
0.029
2877.0
2984.9
-0.8
0.060
0.074
0.024
0.030
2840.4
2855.1
0.027
0.035
3049.6
3157.6
0.2
0.046
0.059
0.024
0.030
2839.6
2854.3
0.024
0.030
2864.0
2972.0
0.8
0.016
0.020
0.023
0.029
2840.6
2855.3
0.035
0.044
3485.8
3593.8
W2
10
-0.2
0.051
0.063
0.055
0.070
570.4
580.3
0.056
0.070
606.8
679.4
-0.8
0.019
0.024
0.046
0.058
569.9
579.8
0.127
0.160
977.0
1049.6
0.2
0.051
0.064
0.053
0.066
568.5
578.4
0.053
0.067
599.0
671.5
0.8
0.017
0.022
0.045
0.056
570.1
579.9
0.129
0.161
983.5
1056.0
30
-0.2
0.028
0.035
0.032
0.040
1706.1
1719.3
0.034
0.042
1776.0
1872.7
-0.8
0.011
0.014
0.025
0.031
1705.6
1718.7
0.080
0.101
2880.2
2976.9
0.2
0.028
0.035
0.031
0.039
1706.7
1719.9
0.033
0.040
1765.3
1862.0
0.8
0.009
0.012
0.025
0.032
1704.9
1718.1
0.093
0.116
2957.5
3054.2
50
-0.2
0.022
0.027
0.023
0.028
2840.0
2854.7
0.024
0.030
2940.6
3048.6
-0.8
0.009
0.011
0.021
0.026
2840.9
2855.7
0.065
0.082
4776.2
4884.2
0.2
0.020
0.026
0.023
0.029
2839.4
2854.1
0.025
0.031
2928.9
3036.9
0.8
0.008
0.010
0.019
0.025
2840.5
2855.2
0.071
0.089
4947.0
5054.9
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Table 9. Simulation Results of SEM and N = 35
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.127
0.159
0.041
0.052
997.1
1008.7
0.042
0.052
1035.4
1178.1
-0.8
0.140
0.173
0.041
0.051
995.0
1006.6
0.042
0.053
1072.0
1214.7
0.2
0.114
0.146
0.042
0.053
994.1
1005.7
0.040
0.050
1025.5
1168.2
0.8
0.041
0.054
0.043
0.053
997.8
1009.4
0.045
0.057
1067.4
1210.2
30
-0.2
0.072
0.089
0.024
0.030
2981.8
2996.6
0.024
0.030
3030.6
3214.0
-0.8
0.076
0.094
0.023
0.029
2981.9
2996.8
0.025
0.031
3141.4
3324.8
0.2
0.060
0.074
0.024
0.030
2981.1
2996.0
0.024
0.030
3012.7
3196.1
0.8
0.020
0.026
0.024
0.030
2984.5
2999.4
0.026
0.033
3258.7
3442.1
50
-0.2
0.055
0.069
0.019
0.023
4968.4
4984.8
0.019
0.024
5024.1
5226.3
-0.8
0.058
0.072
0.018
0.023
4967.1
4983.5
0.020
0.025
5205.0
5407.3
0.2
0.044
0.055
0.019
0.024
4968.9
4985.3
0.019
0.024
5004.0
5206.3
0.8
0.016
0.020
0.019
0.023
4973.0
4989.4
0.022
0.027
5583.4
5785.7
W2
10
-0.2
0.039
0.048
0.040
0.050
997.3
1008.9
0.043
0.054
1060.1
1202.9
-0.8
0.015
0.018
0.033
0.041
996.5
1008.0
0.113
0.141
1731.0
1873.7
0.2
0.038
0.047
0.042
0.052
994.2
1005.8
0.041
0.052
1048.4
1191.2
0.8
0.013
0.016
0.036
0.045
996.0
1007.5
0.120
0.153
1765.1
1907.9
30
-0.2
0.021
0.026
0.024
0.030
2982.3
2997.1
0.024
0.031
3102.1
3285.5
-0.8
0.008
0.010
0.018
0.023
2982.9
2997.8
0.063
0.079
5115.0
5298.4
0.2
0.021
0.026
0.023
0.029
2981.0
2995.8
0.025
0.031
3084.3
3267.7
0.8
0.007
0.009
0.020
0.025
2983.9
2998.8
0.069
0.087
5257.8
5441.2
50
-0.2
0.017
0.021
0.019
0.023
4969.0
4985.4
0.020
0.025
5142.0
5344.3
-0.8
0.006
0.007
0.015
0.019
4968.8
4985.2
0.050
0.063
8484.0
8686.3
0.2
0.015
0.019
0.018
0.023
4969.0
4985.4
0.019
0.024
5123.5
5325.8
0.8
0.005
0.007
0.015
0.019
4972.3
4988.7
0.055
0.068
8776.9
8979.2
Table 10. Simulation Results of SEM and N = 60
W
T
Spatial Estimator
Non-spatial Estimator
AIC
BIC
AIC
BIC
W1
10
-0.2
0.118
0.147
0.033
0.041
1706.0
1719.2
0.033
0.041
1768.4
2041.0
-0.8
0.126
0.156
0.033
0.042
1705.3
1718.4
0.033
0.042
1811.5
2084.2
0.2
0.102
0.129
0.032
0.040
1706.4
1719.6
0.031
0.039
1760.7
2033.4
0.8
0.039
0.053
0.033
0.042
1705.5
1718.7
0.033
0.042
1781.7
2054.3
30
-0.2
0.068
0.085
0.018
0.023
5109.6
5126.1
0.019
0.023
5186.8
5527.5
-0.8
0.074
0.092
0.019
0.024
5107.5
5124.0
0.020
0.026
5316.6
5657.3
0.2
0.054
0.068
0.018
0.022
5111.7
5128.2
0.018
0.022
5164.0
5504.7
0.8
0.021
0.027
0.017
0.022
5114.8
5131.3
0.018
0.022
5239.5
5580.3
50
-0.2
0.053
0.066
0.014
0.018
8516.0
8534.0
0.014
0.018
8601.5
8973.8
-0.8
0.055
0.068
0.014
0.018
8515.7
8533.7
0.015
0.018
8812.2
9184.6
0.2
0.043
0.054
0.014
0.017
8521.8
8539.9
0.014
0.017
8578.4
8950.8
0.8
0.016
0.020
0.014
0.017
8517.8
8535.8
0.015
0.019
9000.2
9372.6
W2
10
-0.2
0.023
0.023
0.016
0.020
10005.4
10023.4
0.130
0.131
10856.8
11229.2
-0.8
0.006
0.007
0.044
0.049
10059.1
10077.2
6.040
6.040
23327.3
23699.7
0.2
0.023
0.024
0.026
0.031
9962.1
9980.1
0.177
0.178
10922.2
11294.6
0.8
0.008
0.008
0.073
0.076
9896.2
9914.2
6.662
6.663
23906.3
24278.7
30
-0.2
0.015
0.018
0.018
0.023
5109.9
5126.3
0.020
0.025
5355.9
5696.6
-0.8
0.005
0.006
0.016
0.019
5107.2
5123.7
0.063
0.079
9370.7
9711.4
0.2
0.014
0.017
0.017
0.022
5111.5
5128.0
0.019
0.023
5334.0
5674.7
0.8
0.005
0.006
0.014
0.017
5114.3
5130.8
0.057
0.071
9422.9
9763.6
50
-0.2
0.011
0.014
0.014
0.017
8516.4
8534.4
0.015
0.018
8882.5
9254.9
-0.8
0.004
0.005
0.011
0.014
8514.2
8532.2
0.043
0.054
15574.7
15947.1
0.2
0.011
0.014
0.014
0.017
8522.5
8540.5
0.014
0.018
8859.0
9231.4
0.8
0.004
0.005
0.012
0.014
8516.2
8534.2
0.046
0.058
15702.2
16074.6
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7.2.2 Simulation Results of Spatial Error Model
Our conclusion about the SEM is that the non-
spatial estimator choice of ignoring the presence of
spatial dependence in the data may not necessarily
bring tremendous drawbacks in terms of bias and
RMSE of when the value of the spatial
dependence is small, i.e., , where we
find the bias and RMSE of resulting from the non-
spatial estimator are less than or equal to their
counterparts of the spatial estimator in some cases.
In contrast, the bias and RMSE of resulting from
the non-spatial estimator tend to increase as the
spatial dependence increases, i.e.,
compared to their counterparts of the
spatial estimator. This pattern is happened
regardless of the spatial weights matrix used, see
Table 7 - Table 10. In general, we find that the
spatial estimator performs mostly acceptably, where
it produces biases that mostly are 5.71% lower than
those of the non-spatial estimator in case of low,
however, this percentage reaches 41.11% in case of
large on average. As for the model-level, we
compute AIC and BIC to assess the goodness of fit
for each regression. The results confirm that the
spatial model always performs better than the non-
spatial model regardless of the spatial dependence
strength and the structure of the spatial weights
matrix.
As for the level of comparison between the spatial
weights structures, Fig. 6 shows that the two spatial
weights matrices have the same pattern of the spatial
bias and RMSE of but with different values for
each of them. The spatial bias and RMSE of tends
to decrease when N or T increases.
On the other hand; Fig. 7 clears that W2 appears a
significant improvement in the results of bias of
compared to W1. There is no specific pattern to
show the relation between the degree of spatial
dependence in the data and the spatial bias of .
W1
W2
Bias
RMSE
Fig. 6: The Spatial bias and RMSE of in SEM at Different Values of N, T, and W for All
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Fig. 7: Bias in SEM at Different Values of N, , and W for All T
Fig. 8: RMSE in SEM at Different Values of N, T, and W for All
Fig. 8 shows that the results of RMSE of are
always in favor of W2 across all combinations of N
and T. In addition to, the RMSE of gradually
decreases when N or T increases in the two
structures of weights.
For a comparison between the two structures of
weights matrix in terms of the spatial AIC and BIC.
The results show that the two structures of weights
matrix seem to provide very similar levels of spatial
AIC and BIC except for N is large and T is small,
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where the SEM based on W2 returns much higher
value of spatial AIC and BIC in this case, see Table
7 - Table 10.
8 Application to Personal Income in
U.S. States
This paper is an attempt to use the SPD approach for
investigating the determinants of PCPI in U.S.
States. As we know, panel data models have played
an important role in the literature of analysing
determinants of PCPI. So our contribution in this
paper focuses on adding the spatial dimension to the
analysis to enable us to model the spatial
dependence.
Annual data are collected from U.S. census bureau
and U.S. bureau of economic analysis (BEA) for 48
U.S. states over 11 periods from 2009 to 2019. The
total sample size NT equals 528 without any
missing data. Four model specifications; non-SPD
models, SLM, SEM, and SAC, are utilized with
individual FE and RE setting under different
structures of spatial weights matrix. Additionally,
attention was paid to direct and indirect effects
estimates of the independent variables. The second
objective of this application is to show how to select
the appropriate model to fit the data. Today, the
researcher in the spatial econometrics has the
possibility to choose from many models. First, he
should ask himself whether there are spatial effects,
or not, and, if so, which type of spatial interaction
effects should be accounted for a (1) spatially
lagged dependent variable, (2) spatially
autocorrelated error term, or (3) combination of
them. Second, he asks himself whether they should
be treated as FE or RE.
Table 11. Definition of the Variables
Dimension
Variable Name
Variable Name on
the Site
Definition
Measuring
Unit
Source
Dependent
Variable
Per Capita Personal
Income
Personal income in
a specific region
divided by its
population
thousand
dollars
U.S.
BEA
Educational
Attainment
Some College, No
Degree
Percentage of
individuals without
a degree
%
U.S.
Census
Bureau
Bachelor's Degree
Percentage of
individuals with
bachelor's degrees
%
Graduate or
Professional Degree
Percentage of
individuals with
graduate or
professional degree
%
Economy's Size
Per Capita Real
GDP by State
Real GDP per capita
thousand
dollars
U.S.
BEA
Labor Force
Type
Population
Population
No. of population
100 thousand
persons
Non-farm
Non-farm
Employment
No. of non-farm
jobs
Un-employ
Unemployment
Rate
Percentage of
unemployed persons
in the total labor
force
%
U.S.
Census
Bureau
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Table 12. Descriptive Statistics of the Variables (NT=528)
Variable Name
Mean
Std. Dev.
Min.
Max.
PCPI
45.84
8.71
29.86
7.297
21.34
2.87
15.0
27.90
18.47
2.96
10.40
26.60
10.96
2.79
6.30
20.30
GDPPC
50.08
9.44
33.15
76.36
Population
65.29
71.20
5.60
395.12
Non-farm
37.90
40.83
3.72
243.63
Un-employ
7.06
2.64
2.60
15.10
Table 13. Summary Statistics of the Straight-Line Geographic Distances between Centroids of U.S. States, in
kilometers
Min. of all
Distances
Mean of all
Distances
Max. of all Distances
Std. Dev. of all
Distances
80.41
1676.80
4231.84
948.20
Links
W1: Inverse Distance
W2: Inverse Exponential
Total No. of Links
210
104
Min. No. of Links
1
1
Mean No. of Links
4.38
2.2
Max. No. of Links
11
5
Threshold Distance
519
Source of data: https://www.mapdevelopers.com/distance_from_to.php (Accessed Date: 29/8/2019).
8.1 Data Description
8.1.1 Economic Data
To model regional PCPI, we take into account 7
explanatory variables as in Table 11. The dataset is
limited by the amount of information available for
states involved.
8.1.2 Data of Spatial Weights Matrix
The data of the straight-line geographical distances
between centroids of U.S. states, which are
summarized in Table 13, are used to create the
spatial relations based on inverse distance and
inverse exponential distance weights. The threshold
distance is calculated by the max-min criterion. The
two used spatial weights matrices are row
standardized to facilitate interpretation, see [27].
8.2 Testing the Multicollinearity
The first step of data processing is to try to ensure
that there is no high linear correlation between
independent variables.
We used the most common methods to detect the
multicollinearity: (1) Pearson correlation matrix
between each pair of variables and (2) the variance
inflation factor (VIF), see [28], [17], and [29].
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Table 14. Pearson Correlation Matrix and VIF
Variable
ND
BD
GD
GDPPC
Population
Non-farm
Un-employ
ND
1
BD
-0.21a
1
GD
-0.54a
0.72a
1
GDPPC
-0.30a
0.60a
0.56a
1
Population
-0.19a
0.10c
0.15a
0.20a
1
Non-farm
-0.20a
0.14b
0.18a
0.24a
0.99a
1
Un-employ
0.04a
-0.45a
-0.21a
-0.28a
0.18a
0.14a
1
VIF1
1.60
3.10
3.34
1.75
239.93
240.23
1.66
VIF2
1.60
3.10
3.23
1.72
----
1.15
1.38
Notes: VIF1: is VIF for all variables, VIF2: is VIF after removing “Population”. The superscripts a, b, and c indicate
statistical significance at the 0.001, 0.01, and 0.05 levels, respectively.
Table 14 shows that there is a strong linear
correlation between (population and non-farm jobs).
Additionally, the results of VIF for the first time
with all independent variables (VIF1) confirmed
that there is a multicollinearity problem between
independent variables; where in most empirical
studies, the general rule of thumb is that VIF values
exceeding 5 need further investigation, while VIF
values exceeding 10 indicate to serious
multicollinearity requiring correction, see [17]. If
two independent variables are almost linearly
correlated, we can eliminate one of them to combat
multicollinearity, see [28]. Therefore, we drop
(population) from the model. All values of new VIF
(VIF2) less than 5 confirmed on there is no
multicollinearity.
8.3 Hausman Specification Test
Because ignoring spatial dependence may lead to
biased and inefficient estimates, therefore, panel
data models are applied with/ without spatial effects
to avoid these shortcomings, and then allow the data
to determine the most appropriate approach. Before
applying the framework in Fig. 9, the Hausman
specification test is conducted to compare between
RE and FE estimators. Hausman [30] developed this
test for non-SPD model. Mutl and Pfaffermayr [31]
showed how to apply this procedure to a spatial
framework.
The results of the all estimated spatial and non-
spatial panel data models are reported in Table 15
according to two methods of wights matrices in
context of FE and RE settings which tested by
Hausman specification test. As shown in Table 15,
the null hypothesis of the Hausman test is rejected at
the 0.001 level of significance for all models,
indicating that FE specifications are more suitable
than RE specifications.
8.4 Testing the Spatial Dependence
As a next step, we need to capture spatial
dependence in the data. Therefore, the following
framework in Fig. 9 is proposed. Lagrange
Multiplier tests, i.e., LM -lag, LM-error tests, and
their robust counterparts (RLM), can be applied to
specify whether the estimation of a spatial model is
warranted. If the null hypotheses of LM tests are
rejected for the absence of SL or spatial error (SE)
in the model, it proves that SPD is a suitable method
for the analysis. Burridge [32] and Anselin [19]
proposed LM tests for a spatially lagged dependent
variable and SE correlation term in the case of
cross-sectional data. The hypotheses for the LM
tests are:
For SLM:
, henceforth
For SEM:
henceforth
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Anselin et al. [33] also proposed robust LM (RLM)
statistics for a spatially lagged dependent variable in
the local presence of SE autocorrelation and another
one for SE autocorrelation in the local presence of a
spatially lagged dependent variable. In other words,
the hypotheses for RLM tests are:
For SLM:
henceforth
For SEM:
henceforth
Table 15. Results of Estimated Panel Data Models
Variable
Non-spatial
W1: Inverse Distance
W2: Inverse Exponential
FE
RE
SLM
SEM
SLM
SEM
FE
RE
FE
RE
FE
RE
FE
RE
ND
-0.32b
-0.37a
-0.18c
-0.17 c
-0.06
-0.30c
-0.20c
-0.20c
-0.25c
-0.31b
BD
1.39a
0.83a
0.40a
0.39a
0.22
0.56a
0.74a
0.68a
1.23a
0.68a
GD
2.47a
1.69a
1.11a
0.99a
0.68a
1.31a
1.42a
1.29a
2.25a
1.47a
GDPPC
0.27a
0.25a
0.25a
0.27a
0.30a
0.35a
0.27a
0.27a
0.30a
0.32a
Non-farm
0.09a
0.02c
0.14a
0.08a
0.14a
0.02
0.13a
0.08a
0.08a
0.003
Non-employ
-0.39a
-0.95a
-0.02
-0.14b
-0.24a
-0.78a
-0.08
-0.22a
-0.46a
-0.89a
Intercept
-14.3b
12.9a
-14.8 a
-10.9a
17.36a
14.52a
-16.6a
-10.9b
-10.8c
14.14a
----
----
0.59a
0.58a
----
----
0.42a
0.42a
----
----
----
----
----
----
0.89a
0.61
----
----
0.33a
0.40
Hausman
22.88a
68.40a
189.78a
116.89a
236.47a
Note: The superscripts a, b, and c indicate statistical significance at the 0.001, 0.01, and 0.05 level, respectively. Non-
spatial models are estimated by Within-group OLS for FE and GLS for RE. Spatial models are estimated by ML
(transformation approach) for all models except for SEM-RE is estimated by GMM.
Recently, Anselin et al. [6] also developed the
classical LM tests for SPD models, and Elhorst [10]
developed the robust counterparts of these LM tests
for SPD models.
Table 16 shows that the classical LM tests of SL and
SE terms are significant at the 0.001 level.
However, the RLM tests are significant for SL term
but not significant for SE term in all structures of
spatial weights matrices. Therefore, our model will
include SL term and exclude SE term. To be more
certain of which terms are included in our model,
we'll compare between the SLM and SEM models
with SAC, which includes the two types of spatial
dependence terms, in terms of goodness of fit
criteria to select the best model for the data as in
Table 17.
In some way, this is in line with some of the applied
literature that estimated the SAC if the researcher
doesn’t have a strong prior in favour of either SLM
or SEM. In other words, an empirical strategy could
be to start from the most general specification, SAC,
along with the appropriate type of individual model,
SLM or SEM, and let the data tell us which of the
two spatial processes – if any and if not both – is
more appropriate, by looking at the significance of
spatial coefficients and goodness of fit criteria, for
more details, see [34].
8.5 Model Selection
As in a lot of empirical researches, the models are
comparable in terms of AIC, and BIC. These criteria
are one of the best methods to select the most
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adequate weighting matrix, see [35] and [36]. Table
17 shows that the values of the goodness of fit
criteria for the non-spatial model are much bigger
than for all SPD models. As expected, the SLMs
record the smallest values of AIC and BIC
compared with their counterparts for SEM and SAC
models. Therefore, we can say that SLM is the most
adequate model among the candidate models. On
purely statistical grounds, the SLM based on W1
returns a slightly lower AIC and BIC compared to
W2.
Table 16. Results of LM Tests with Different Spatial Weights Matrices
Test
W1: Inverse Distance
W2: Inverse Exponential
Lag
Error
Lag
Error
LM
73.70a
20.41a
51.16a
12.86a
RLM
53.88a
0.32
38.42a
0.13
Note: The superscript a indicates statistical significance at the 0.001 level.
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Fig. 9: Our Strategy for Selecting the Most Appropriate SPD Model
Note: and: The Lagrange Multiplier Tests for a Spatially Lagged Dependent Variable and Spatial
Error Correlation Respectively - and : The Robust Counterparts of these Tests - SLM: Spatial Lag
Model - SEM: Spatial Error Model - SAC: Spatial Autoregressive Combined Model - AIC: Akaike Information
Criterion - BIC: Bayesian Information Criterion.
Table 17. Results of Estimated SPD Models with Spatial FE
Criterion
Non-
spatial
W1: Inverse Distance
W2: Inverse Exponential
SLM
SEM
SAC
SLM
SEM
SAC
AIC
1813.19
1390.42
1525.50
1392.42
1463.27
1621.76
1465.27
BIC
1843.08
1423.81
1558.89
1429.98
1496.66
1655.15
1502.83
8.6 Interpretation of the Results
The coefficients interpretation in the linear
regression model is not complicated. Since the
model is linear in parameters and assumes that the
observations are independent, the parameter can be
explained as the partial derivative of the response
variable with respect to the independent variable.
When we consider SPD models, the interpretation
needs more proper considerations to fully interpret
the effect of changes, direct and spillover effects,
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must be obtained and interpreted as the coefficients
of model, see [37].
In SLM, the direct effect is the average of main
diagonal elements of the matrix in (32), and the
spillover effect is the average of row off-diagonal
elements in the same matrix, see [38] and [39].
(32)
Table 18 provides the measures of direct, spillover,
and the total effect of each regressor to assess the
magnitudes of impacts arising from changes in the 6
independent variables under the study.
The first column of Table 18 express about the
direct effects, which measure how much the
dependent variable, PCPI, changes in a state when a
given independent variable changes in that same
state. The second column refers to the spillover
(indirect) effects of changes in our independent
variables; Finally, the last column shows the point
estimates of the total effects that are defined as the
sum of the direct and indirect effects. We note that
all effects of four explanatory variables are
statistically significant at the 0.001 level and one
variable is significant at 0.05.
In general, we can conclude the following points
from Table 18:
(1) The direct effect of increasing ND in a specific
state by 1% directly reduces PCPI by $214.5 in
the same state. Also, the indirect effect of
increasing ND in neighboring states is negative
on the PCPI by $229.0. The total effect of ND is
negative and consists mostly of indirect effect.
(2) The BD has a positive direct and indirect effect
on PCPI, indicating that we would expect an
increase in PCPI in states with a high level of
BD. The magnitude of the indirect effect
produced from BD increases by a very small
amount over the magnitude of the direct effect,
indicating that the direct and indirect effects of
this variable are almost equal.
(3) The direct and indirect effects of GD are
positive; this refers to that the increase in GD in
a specific state by 1% directly increases the
PCPI by $1302.8 in the same state and
indirectly increases it in other states by $1391.1.
(4) The direct and indirect effects of GDPPC are
positive; as the GDPPC increases by $1000 in a
particular state, the PCPI will increase by
$302.5 on average in the same state, and
increase by $323.0 on average in other states.
(5) The number of non-farm jobs has a positive
direct and indirect effect on the PCPI; when the
number of non-farm jobs increases by 100,000
jobs in a particular state, the PCPI increases by
$159.0 in the same state, and increase by $169.8
on average in other states.
(6) The impact of the unemployment rate on PCPI
is not significant.
Table 18. Direct and Indirect Effects of SLM with
Spatial FE and using W1
Variable
Direct
Effects
Spillover/
Indirect
Effects
Total
Effects
ND
-0.2145c
-0.2290c
-0.4435c
BD
0.4741a
0.5062b
0.9803a
GD
1.3028a
1.3911a
2.6939a
GDPPC
0.3025a
0.3230a
0.6255a
Non-farm
0.1590a
0.1698a
0.3289a
Un-employ
-0.0290
-0.0310
0.0601
Note: The superscripts a, b, and c indicate statistical
significance at the 0.001, 0.01, and 0.05 level
respectively.
9 Conclusions
This paper is an attempt to assess the risks involved
in ignoring the spatial dependence that characterizes
the data. Due to the importance of this topic, our
contribution is not limited by this empirical
application, but also we conduct a MCS study to
evaluate the performance of both the spatial MLE
(transformation approaches) and the non-spatial
OLS (within-group) estimator for two specifications
of the most common spatial data generating
processes (DGPs), i.e., SLM and SEM with spatial
FE under different scenarios of N, T, and spatial
dependence parameters. Besides, we employ two
structures of weights matrices; i.e., inverse distance
(W1) and inverse exponential distance (W2), to draw
and estimate each DGP aiming to compare the
impact of these structures on each model.
In a summary way, the following points can be
concluded from our simulation study:
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(1) On the Parameter-Level: The non-spatial bias
and RMSE of are functions of the degree of
spatial dependence in the data for both models,
i.e., SLM and SEM. The choice of the non-
spatial estimator may not lead to serious
consequences in terms of bias and RMSE of
when the spatial dependence is small. On the
contrary, the choice of the non-spatial estimator
always leads to has disastrous consequences if
the spatial dependence is large.
(2) On the Model-Level: The SLM always
performs better than its non-spatial counterpart
when is large in terms of AIC and BIC.
However, the spatial AIC and BIC in SEM are
always much less than their non-spatial
counterparts in all cases.
(3) For a comparison between the two structures of
weights matrix in terms of the spatial AIC and
BIC. The results of SLM show that the two
structures of weights matrix seem to provide
very similar levels of spatial AIC and BIC,
However, the SLM based on W2 returns a
slightly higher value of spatial AIC and BIC in
all cases. This result is also true for SEM except
for N is large and T is small, where the SEM
based on W2 returns much higher value of
spatial AIC and BIC in this case.
On the other hand; our empirical study confirms the
following points:
(1) PCPI is spatially dependent lagged correlated.
(2) There are no differences among the two used
structures of spatial weights matrix in terms of
the inference drawn from Hausman and LM
tests, the number of significant variables, and
their significance levels. However, the
differences among the two used structures can
be confined in the values resulting from each
procedure in our analysis not in the conclusion,
for example, the W1 yields a higher
improvement in terms of goodness of fit criteria.
In the future, this work can obviously be
extended along many dimensions, for example;
1. The set of maintained assumptions in our
simulation study can be made more general.
Here the first extension that can be addressed is
to allow for the RE specification. In other
words, a MCS study can be performed to
compare the performance of the estimators used
in non-spatial and SPD models with RE settings.
2. It could be interesting to consider studying the
estimation procedures used in dynamic SPD
models or SPD models with random
coefficients, see [40, 41, 42].
3. We here study only the time-invariant spatial
weights matrix; therefore, it is important to
study the situation of the spatio-temporal
weights matrix that allows decomposing the
spatial effects when the spatial relations are
being collected continuously over time.
4. The models under consideration in our study
can be extended to include other elements. In
particular, it would be helpful to consider a SL
in the explanatory variables, or there is a linear
relationship (multicollinearity) between the
explanatory variables, see [43, 44, 45, 46].
5. A final remark needs to be made concerning the
interpretation of parameter estimates in the SPD
models. Unlike in OLS regressions, parameter
estimates in SPD models that contain SL of the
dependent variable have not a direct
interpretation due to the embedded feedback
effects among spatial units. Therefore, LeSage
and Pace [38] developed summary measures
reflecting the impact of change in explanatory
variables on the dependent variable as we
cleared previously. To meaningfully interpret
parameter estimates in the context of the
proposed SPD models, the development of
summary measures appears to be a promising
area for future researches.
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