On New Two-Step GMM Estimation of the Panel Vector Autoregressive
Models with Missing observations
MOHAMED R. ABONAZEL*, MOHAMED ABDALLAH, EL-HOUSAINY A. RADY
Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical Research,
Cairo University, EGYPT
Abstract: Few estimation methods were discussed to handle the missing data problem in the panel data models.
However, in the panel vector autoregressive (PVAR) model, there is no estimator to handle this problem. The
traditional treatment in the case of incomplete data is to use the generalized method of moment (GMM)
estimation based on only available data without imputation of the missing data. Therefore, this paper introduces
a new GMM estimation for the PVAR model in case of incomplete data based on the mean imputation.
Moreover, we make a Monte Carlo simulation study to study the efficiency of the proposed estimator. We
compare between two GMM estimators based on the mean squared error (MSE) and relative bias (RB) criteria.
The first is the GMM estimation based on the list-wise (LW) and the second is the GMM estimation using the
mean imputation (MI) at multi-missing levels. The results showed that the MI estimator provides more
efficiency than the LW estimator.
Key-Words: Generalized method of moments; Mean Imputation; Missing Data; Two-Step Estimation; missing
observations.
Received: August 16, 2021. Revised: July 17, 2022. Accepted: August 17, 2022. Published: September 20, 2022.
1 Introduction
The regression of panel data differs from a regular
cross section or time series in that it has a double
subscript on its variables i and t, where the i
subscript represents the cross-section dimension
while t denotes the time series dimension, see [1, 2,
3, 4, 5]. Time series vector autoregressive (VAR)
models were discovered in the macro-econometrics
literature as an alternative to multivariate
simultaneous equation models, since all the
variables of VAR systems are treated as endogenous
variables [6].
The panel vector autoregressive PVAR models
include a lagged endogenous variable, and the first
difference of the error term will be correlated with
all of the explanatory variables. In this situation, the
estimators of PVAR models will be biased.
Yamamoto and Kunitomo [7] first derived the
asymptotic bias for the ordinary least squares
estimator of a multivariate autoregressive model
with a constant term. They reduced a model without
a constant term as a special case, and multivariate
autoregressive time series models could be treated
as similar to the idea of PVAR models. Missing data
patterns have effects on most applied studies in
economic fields. As a result, there is a wide
literature about how to treat the problem of missing
data. However, an efficient method to deal with
estimation in an arbitrary generalised method of
moment (GMM) setting with a general missing data
pattern is not available. Therefore, there are many
inefficient methods, such as complete-case analyses,
that dominate the empirical literature. In a survey
constructed by Abrevaya and Donald [8], they found
that few of the empirical research deals with missing
data, and in most of these cases, a complete-case
estimator is used, i.e., all incomplete observations
were discarded. Therefore, the main objective of
this paper is to present a Monte Carlo simulation
study to study the efficiency of the proposed
estimator suggested by Rady et al. [1].
The rest of the paper is organized as follows: section
2 introduces the PVAR model and its assumptions.
Section 3 provides the results of the Monte Carlo
simulation study. Finally, section 4 offers the
conclusions.
2 The PVAR model
In dynamic panel data (DPD), we assume the
observations are on many individuals, with many
observations on each individual, and the model of
interest is a regression model in which the lagged
value of the dependent variable is treated as one of
the explanatory variables. The error term in the
model is assumed to contain a time-invariant
individual effect as well as random noise [9, 10, 11,
12, 13, 14]. The basic problem faced in the
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estimation of DPD models is that a fixed effects
treatment tends to the within estimator (least-
squares after transformation to deviations from
means), which has inconsistent estimators because
the within transformation induces a correlation
between the lagged dependent variable and the
error, see [3, 12, 13, 14]. Holtz-Eakin et al. [15],
expanding on the Anderson-Hsiao [16] approach,
show how it is implemented to estimate a vector
autoregression with time-varying parameters.
Arellano and Bond [17] used Monte Carlo studies to
evaluate a GMM estimator that is like [15]
recommendation, and Kiviet [18] used the
simulations to compare these and many other
techniques, including the corrected least squares
dummy variable estimator, see [3, 12, 13]. Vector
autoregressions are now a standard part of the
applied econometrician's toolkit. Although their
interpretation in terms of causal relationships is
controversial,
Holtz-Eakin et al. [15] introduced an estimation and
testing for the PVAR model. They used an
estimation method as similar to that Anderson and
Hisiao [16]. Consider PVAR data with N units
observed for T + P consecutive time periods. Each
unit i, we observed M outcome variables yit1, ..., yitM,
where .
The behaviour of Yit = (Yit1,Yit2, ..., YitM)' and it is
described by the Pth order vector autoregression:
(1)
Where
; 1,..., , 1,...,T,
it i it
u i N t
ℎ lag order of , is ,
M-dimensional error term. ′,
and is a fixed effects in the
model.
2.1 Proposed Estimator Assumption
In general, the assumptions of the PVAR model are:
Assumption 1 (Condition of stationarity): The
roots of the given determinant:
(2)
This assumption refers that the process of vector
autoregressive is stable.
Assumption 2 (Regularity conditions): The υit has
finite eight-order moments and, as N →∞
Where . The conditions about the
moment in this assumption make sure that the
regular asymptotic behaviour of the least squares
estimator is standard [1].
Assumption 3 (The errors): The υit are independent
and identically distributed across i and t:
E[υit] = 0, E[υitυ'it] = Ω (3)
Where Ω is positive definite matrix and the
independence across time can allow for dependence
between υit and υit-P through their higher-order
moments. For simplicity, we can say this
assumption as, the error vectors are independent and
identically distributed (iid).
Assumption 4 The time dimension of panel is finite
with and the available observations are (Yi0,
Yi1, ..., YiT).
2.2 The Proposed Estimator
This section explains the proposed estimator was
introduced by Rady et al. [1]. Consider the standard
linear PVAR (1) model as discussed in (1)
(4)
Where Yi is a (possibly missing) scalar, Yi is a K-
vector of all lagged dependent variable. The first
element of Yi is 1; that is, the model is assumed to
contain an intercept. We assume the residuals only
satisfies the conditions in (3) to be a linear
projection, specifically
(5)
The variable mi indicates whether or not Yi is
missing for observational unit i:
The proposed weighted GMM estimator can be
introduced as
′
′′
′ (6)
Where the GMM estimator can be implemented
using instrumental variables methods. We can
define the instrumental-variable matrix Zi as
′
′
′
Which corresponds to (′ 0) using as instruments in
each time for which yit is observed. For the
statistical properties of this estimator, see [1].
3 Monte Carlo Simulation
A Monte Carlo simulation is a type of simulation
that relies on repeated random sampling and
statistical analysis to compute the results. This
method of simulation is very closely related to
random experiments, in which the specific result is
unknown in advance.
3.1 Design of the Simulation
The simulated dataset represents the generation of
different cross sections as a
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complete case, and it is assumed that the number of
endogenous variables is three () so, the
matrix is of order 3×3 and it consists of nine
coefficients to be estimated. The values of the
endogenous variable Y were generated as
independent normally distributed random variables.
The disturbances were generated as independent
normally distributed random variants, with mean
zero and standard deviation equal one. The
disturbances were allowed to differ for each cross-
sectional unit on a given Monte Carlo trial and were
allowed to differ between trials. The value of time T
was chosen to be fixed is equal nine. Moreover, it
was used the R codes employed by Muris [19] in
PVAR models and the paper considered by
Abonazel [20] that shows a new algorithm that
provides researchers with basics and advanced skills
about how to create their R-codes and then achieve
the simulation study for estimating the missing
observations and imputation in all percentages of
missingness and then estimate make simulation
using sample sizes (cross sections are 5, 10, 20, 30
and 40). The first step of this simulation is
examining the optimal lag of the PVAR model for
this data and the optimal lag length found to be
PVAR (1). After determining the optimal lag of the
model, now it said to be sure that the data generated
in a complete case is of order 1 so the matrix is of
order 3×3. Then it made new four datasets from the
complete dataset corresponding to four percentages
of missingness (10%, 20%, 30% and 40%). Then it
is estimated the PVAR model using (6). This
estimation using LW deletion and the proposed two-
step GMM estimator based on MI method to make a
comparison and these estimations are based on
many samples of size: 5, 10, 20, 30 and 40 then it is
estimated the MSE at each percentage of
missingness for each estimator, also, the MSE and
RB were calculated as:
Φ
Φ (7)
(8)
This RB is used to evaluate the percentage of the
bias for the missingness level, and MSE and RB are
used as criteria to determine the perfect usage of the
MI method, with which one of missingness is more
efficient at different sample sizes where chosen.
For each of the experimental settings, 500 Monte
Carlo trials (r = 500) were used in this simulation
because each trial takes the used package in the R
program and results were recorded and all
simulation results were conducted (see the
appendix) and the settings of the model and results
of the simulation study are discussed below.
3.2 Simulation Results for Different Sample Size
In this section, the basic objective is to study the
relationship between sample size and each of MSE
and RB under several percentages of missing
observations. These are made to evaluate the LW
imputation and MI method.
The MSE for missingness at multi percentage
levels
The following section presents the MSE at multi-
levels as discussed before. Figure 1 introduces the
MSE for the estimator
with GMM estimation of
PVAR with 10%, 20%, 30%, and 40% missing
observations, and the estimation provides a
comparison between LW imputation and MI among
the different sample sizes of 5, 10, 20, 30, and 40. It
was found that when the sample size increases, the
MSE decreases in both imputation methods, LW
and MI. Moreover, overall, the graph shows that MI
is more efficient than LW starting from sample size
5. It means when the sample size increases, the
MSE for the MI method becomes smaller than LW.
In the case of 20% of the missing observations It
was found that when the sample size increases, the
MSE decreases in both imputation methods, LW
and MI. Moreover, overall, the graph shows that MI
is more efficient than LW, which means when the
sample size increases, the MSE for the MI method
becomes
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Fig. 1: MSE of
missingness at multi percentage levels
smaller than the LW method and more efficient.
With 30% missing, starting from a sample of size
10, when the sample size increases, the MSE
decreases in both imputation methods, LW and MI.
Furthermore, the graph shows that MI is more
efficient than LW at the largest of all sample sizes,
which means if the sample size increases, the MSE
for the MI method becomes smaller than the LW
method, but LW is not efficient at very small
samples (5) when the missingness is at level
30%. When the missing observations are 40%, It
was seen when the sample size increased, then the
MSE decreased in both imputation methods LW and
MI except sample size of 30, and the graph shows
that MI is more efficient than LW at the most
sample sizes which means if the sample size
increase, the MSE for MI method becomes more
efficient than LW method when the missing
observations are 40%.
The RB for missingness at multi percentage
levels
The following section it is present the RB at multi-
levels of missing observations was discussed as
MSE before in the previous section. Figure 2
represents the RB for the estimator
with GMM
estimation of PVAR with missing observations at
multiple levels, and the estimation provides a
comparison between LW imputation and MI among
sample sizes of 5, 10, 20, 30, and 40 shown in the
graph. It was found that when the sample size
increases, the RB decreases in both imputation
methods, LW and MI. Moreover, the graph shows
that the MI method provides RB less than LW, so it
could be said that MI is more efficient than LW at
all sample sizes, which means when the sample size
increases, the RB for the MI method becomes
smaller than the LW method when 10% of
observations are missing.
With respect to 20% of the missing observations It
was found in the graph that MI is more efficient
than LW at sample sizes (5, 10, 20, 30, and 40),
which means the RB for the MI method becomes
smaller than the LW method at all samples with
20% missing observations, and it means that MI is
still more efficient than LW. With the view of
missing observations at 30%, it is observed that
when the sample size increases, the RB decreases in
both imputation methods, LW and MI. The graph
shows that MI is more efficient than LW at all
samples, which means if the sample size increases,
the RB for the MI method becomes smaller than the
LW method by a percentage of 30%.
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Fig. 2: RB of missingness at multi percentage levels
With respect to 40% missing observations, it was
found in the graph that MI is more efficient than
LW, which means the RB for the MI method
becomes smaller than the LW method at all samples
for the estimator with 40% missing observations.
This means that MI is more efficient than LW.
3.3 Simulation results for different percentage of
missingness fixed sample size
In the next section, the main purpose is to study the
relation between the missing observations
percentage through calculating MSE and RB in each
estimation and compare the LW imputation and MI
method for each sample size of sizes (5, 10, 20, 30
and 40) individually.
The MSE for each sample size
The following section is present the MSE at each
sample as discussed later.
Regarding Figure 3, it represents MSE for the
estimator
with GMM estimation of PVAR with a
sample size of 5, 10, 20, 30, and 40. The estimation
provides a comparison between LW imputation and
MI through the missing observation levels of 10%,
20%, 30%, and 40%. For a sample size of 5, it was
found that the MSE of the MI method is more
efficient than the LW imputation method at missing
observation levels of 10%, 20%, and 40%. Overall,
the graph shows that MI is more efficient than LW
at the most of all missing observation levels except
level 30%. It is shown that the difference in MSE
between LW and MI is very small, so, at this level,
the MSE is roughly similar, which means the MSE
for the MI method becomes smaller than the LW
method and more efficient. With respect to sample
size 10, it was seen that the MSE of the MI method
is more efficient than the LW imputation method at
all missing observation levels. Overall, the graph
shows that the MSE for the MI method becomes
smaller than the LW method and more efficient at
the start of the missing level. When the missing
level increases, the MSE in both LW and MI is
decreased, which means they are still better. By
viewing a sample of size 20 and the estimation
providing comparison between LW imputation and
MI through the missing observations of 10%, 20%,
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30%, and 40%, it was found that the MSE of the MI
method is more efficient than the LW imputation
method at all missing observation levels.
The graph shows that the MSE for the MI method is
still smaller than the LW method and more efficient,
and starting at a missing level of 20%, the MSE of
MI is decreased, but inversely, the MSE of LW is
increased, which means MI is efficient by increasing
the percentage of missing observations at sample
size of 20. Regarding a sample of size 30, and the
estimation provides a comparison between LW
imputation and MI through the missing observations
of 10%, 20%, 30%, and 40%, it was found that the
MSE of the MI method is more efficient than the
LW imputation method at missing observations of
10%, 30%, and 40%. Overall, the graph shows that
MI is more efficient than LW at the most of all
missing observation levels except level 20%. It is
shown that the difference in MSE between LW and
MI is small, so the MSE for the MI method is still
smaller than the LW method and more efficient at
the most of all percentage levels of missing
observations.
Moreover, when the missing percentage increases,
the MSE of LW increases but the MSE of MI
decreases, and the MI is still efficient when a large
missing observation level exists (30% and 40%).
For sample of size 40, the MI method is more
efficient than the LW imputation method at majority
of all missing observations levels except level 20%.
It is shown that the difference in MSE between LW
and MI is small, so the MSE for the MI method is
still smaller than the LW method and more efficient
at the most of all percentage levels of missing
observations. Also, as the percentage of missing
observations goes up, so does the MSE of LW. But
starting at level 20%, the MSE of MI goes down,
and MI still works well when a lot of observations
are missing, like 30% or 40%.
The RB for each sample size
The following section is showing the RB at each
sample size as previously discussed. With respect to
figure 4, it is to introduce the RB for the
estimator
with GMM. The estimation compares
LW imputation and MI using missing observation
levels of 10%, 20%, 30%, and 40%. It was found
that the RB of the MI method is more efficient than
the LW imputation method for most levels of
missing observations, except for level 20%, where
the RB in both methods is about the same. This
means that the RB for the MI method gets smaller
and more efficient at level 20%. At a sample size of
10, it was found that the RB of the MI method is
more efficient than the LW imputation method at
missing observations of 10%, 30%, and 40%.
Overall, the graph shows that MI is more efficient
than LW at the most of all missing observations
levels except level 20%. It is shown that the
difference in RB between LW and MI is so small
that at this level, the RB in LW and MI is roughly
approximately equal, which means the RB for the
MI method is still smaller than the LW method and
more efficient.
The estimation provides a sample size of 20, and the
estimation provides a comparison between LW
imputation and MI through the missing observations
of 10%, 20%, 30%, and 40%. The RB of the MI
method was found to be more efficient than the LW
imputation method at all missing observation levels.
The graph shows that the RB for the MI method is
still smaller than the LW method and more
efficient.
Regarding a sample of size 30, it was found that the
RB of the MI method is more efficient than the LW
imputation method at all missing observation
percentages, over all the graph shows that the RB
for MI method still smaller than LW method so the
MI is more efficient at all conditions about
missingness proportions at sample of size 30.
For a sample of size 40, the RB of MI method is
more efficient than LW imputation method at all
missing observation percentages, over all the graph
shows that the RB for MI method smaller than LW
method so, it could say that MI is more efficient at
all cases of missingness proportions at large sample
size as 40.
4 Conclusions
This paper presented a Monte Carlo simulation
study to study the two GMM estimators for the
PVAR models with missing observations. In our
simulation study, we used various sample sizes
(small, medium, and large). Furthermore, we ran
GMM estimation in the full model using full data
and then we eliminated some observations at multi-
missing levels. We used LW imputation based on
elimination, and we again ran GMM estimation
based on the MI at multi-missing levels. We
compare the proposed estimator with the LW
estimator based on the MSE and RB criteria. We
can summarize the final remarks as follows:
1- It was found that the small samples have a large
MSE while the large samples has small MSE in
each of LW and MI methods so, they are lead to the
negative relation between MSE and sample size
moreover, over all the results shows that MI method
provides more efficiency than LW at all sample
sizes which means when the sample size increase,
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the MSE for MI method becomes smaller than LW
method when the observations are missing so our
conclusion is the MI method in our estimator is
more efficient than the LW.
Fig. 3: MSE of
at various sample size
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Fig. 4: RB of
at various sample size
2- When we evaluate the RB among the various
sample sizes, It was found that when the sample size
increase, the value of RB decrease in both
imputation methods LW and MI moreover, the
results are show that MI method progress RB less
than LW method so, we can says that the MI is more
optimal than LW at all sample sizes which means
when the sample size increase, the RB for MI
method becomes smaller than LW method even
though the missing observations at very small
samples.
3- From studying the results of MSE for the
estimator
with GMM estimation of PVAR among
the various missing percentages as a comparison
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between LW imputation and MI through the missing
observations 10%, 20%, 30%, and 40% with
different sample sizes, it was found that the MSE of
MI method is more efficient than LW imputation
method at all missing observation levels except very
few cases whose studied , over all the results
supports that the MSE for MI method
4- Becomes still smaller than LW method and so
we can say that the MI more efficient than LW
particularly starting of missing level 20%, the MSE
in MI is more efficient.
5- By evaluating the results of the RB in our study
using GMM estimation of PVAR model with
several sample of sizes and the estimation provides
comparison between LW imputation and MI
through the missing observations 10%, 20%, 30%,
and 40%, it was found that the RB of MI method is
less than LW imputation method at most of all
missing observation percentages, over all those
results are introduce the MI as optimal method for
handling the missing observations in PVAR models.
6- Finally, we can conclude that the MI method is
more efficient than LW method in PVAR models if
the data set contains missing pattern and these
results are compatible with the estimator was
presented in of Rady et al. [1].
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Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Appendix
Table A.1 The MSE for the estimator with GMM estimation of PVAR with levels of missing observations
and comparison between LW and MI at n=5
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
8.4189
6.3545
8.5974
7.3949
7.0723
7.0002
6.8345
6.1951
14.6600
15.1800
13.3700
10.8200
9.7000
9.0800
6.1100
5.2800
12.1800
12.0200
9.2700
7.1800
6.0630
5.7200
5.0340
3.9100
20.4200
19.6000
12.9300
12.8400
10.5800
11.0400
10.1000
9.8100
36.0100
32.0500
26.5200
26.8200
27.9500
28.5600
37.3700
27.5300
32.0800
27.3500
34.4800
30.3800
32.7900
30.4900
35.6900
25.8100
39.7700
37.4300
28.4900
26.3900
30.3700
28.7600
25.1500
25.0200
27.8300
22.6400
19.7500
15.1400
15.9100
14.5800
29.3900
28.1200
19.2300
11.3800
31.1500
6.4500
26.6600
35.2000
17.8800
13.4400
Table A.2 The RB for the estimator with GMM estimation of PVAR with levels of missing observations and
comparison between LW and MI at n=5
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
214.28
189.06
204.80
179.43
204.65
201.42
181.43
176.26
77.61
81.14
88.62
82.75
93.44
91.80
96.23
85.31
122.29
107.16
129.58
114.34
124.10
111.90
119.67
116.71
166.19
180.66
179.81
196.75
175.50
173.73
137.11
187.54
84.43
83.12
88.53
87.03
98.61
97.33
96.51
90.27
123.47
103.73
134.51
123.84
144.14
143.53
127.26
100.44
114.02
95.87
109.99
109.06
107.76
102.21
125.82
126.07
257.79
165.39
251.49
189.14
211.17
133.66
195.40
139.55
238.52
122.17
222.95
199.13
223.42
206.63
418.23
331.00
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.79
Mohamed R. Abonazel,
Mohamed Abdallah, El-Housainy A. Rady
E-ISSN: 2224-2880
680
Volume 21, 2022
Table A.3 The MSE for the estimator with GMM estimation of PVAR with levels of missing observations
and comparison between LW and MI at n=10
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
2.5870
2.3250
8.2870
8.0700
7.2450
6.5421
5.9550
2.0210
11.1900
9.7400
8.9790
7.0000
6.0621
3.9600
3.4800
2.9800
10.5300
10.1700
8.9700
8.6500
5.2400
5.0710
4.8010
3.2300
15.7300
13.4600
12.0100
11.9000
9.9230
8.9000
8.0460
7.6600
30.0000
21.1200
25.8400
23.1000
22.2100
15.5400
25.0600
19.7400
17.8000
8.7400
28.5400
20.2700
29.8000
22.2600
21.3800
16.2100
27.5000
27.8500
22.4700
21.7900
22.8800
22.2200
24.2200
19.7100
26.0400
21.4700
15.8100
11.0700
15.7700
10.0400
26.6400
11.3100
12.1000
10.6800
30.6600
21.1500
22.4800
21.0700
15.2600
12.0100
Table A.4 The RB for the estimator with GMM estimation of PVAR with levels of missing observations and
comparison between LW and MI at n=10
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
137.11
96.37
124.62
118.08
192.70
145.66
166.44
108.49
66.06
60.04
77.89
69.56
80.85
77.48
94.38
81.25
130.56
98.54
113.73
102.53
114.09
104.35
120.73
98.45
127.51
124.30
177.02
98.78
153.39
110.82
134.51
82.56
73.48
60.20
78.38
74.40
93.84
83.99
77.62
63.37
99.93
94.41
132.95
90.24
125.25
103.07
122.39
92.52
87.45
86.51
98.86
79.01
97.34
84.30
105.96
102.68
105.20
96.86
118.36
109.68
150.49
99.05
127.40
124.48
203.18
114.38
179.09
165.22
219.91
138.13
298.33
216.06
Table A.5 The MSE for the estimator with GMM estimation of PVAR with levels of missing observations
and comparison between LW and MI at n=20
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
0.2374
0.1421
0.3274
0.1183
0.4327
0.0809
0.8377
0.0752
0.1354
0.1170
0.1833
0.1304
0.2552
0.1150
0.6729
0.1125
0.0362
0.0307
0.1503
0.0316
0.9296
0.0326
4.2738
0.0331
0.2474
0.1532
0.3429
0.1433
0.4521
0.1044
0.8321
0.1092
0.1749
0.1669
0.1967
0.1608
0.2804
0.1549
0.9990
0.1550
0.0398
0.0367
0.1623
0.0384
1.0208
0.0396
4.5601
0.0402
8.1621
5.1458
9.5889
5.3978
11.2147
4.3870
15.0301
5.1165
3.4784
2.9226
4.3058
2.8593
6.2234
2.0904
13.5185
2.6569
0.9914
0.4992
5.2104
0.4346
12.8616
0.4309
14.7841
0.3520
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.79
Mohamed R. Abonazel,
Mohamed Abdallah, El-Housainy A. Rady
E-ISSN: 2224-2880
681
Volume 21, 2022
Table A.6 The RB for the estimator with GMM estimation of PVAR with levels of missing observations and
comparison between LW and MI at n=20
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
127.47
87.65
111.39
102.17
135.52
112.80
96.56
89.07
43.37
38.38
68.56
61.24
66.70
66.17
70.65
62.66
75.95
87.24
108.78
90.89
109.42
92.13
95.35
93.64
101.59
77.20
115.33
75.52
126.99
66.33
116.82
67.26
67.10
54.63
78.55
74.72
88.63
77.76
76.29
59.54
77.90
90.24
116.58
84.71
101.99
96.63
98.06
97.42
86.44
85.06
79.86
76.46
82.49
79.31
104.37
100.67
79.58
62.79
94.03
80.97
94.20
90.35
107.34
104.43
139.83
108.75
122.44
114.42
177.34
111.59
259.59
102.81
Table A.7 The MSE for the estimator with GMM estimation of PVAR with levels of missing observations
and comparison between LW and MI at n=30
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
0.0367
0.0181
0.0672
0.2124
0.3860
0.2480
0.1045
0.2550
0.0483
0.0312
0.0868
0.0925
0.1231
0.0956
0.1494
0.1005
0.0007
0.0004
0.0028
0.0005
0.0084
0.0006
0.0185
0.0006
0.0356
0.1701
0.0639
0.0583
0.0908
0.0803
0.1101
0.2046
0.0395
0.0661
0.0701
0.0626
0.1031
0.0566
0.1263
0.0550
0.0008
0.0008
0.0028
0.0010
0.0080
0.0012
0.0207
0.0012
0.9028
1.0659
1.5729
2.1551
2.2108
2.0048
2.9942
2.0549
1.6281
1.0806
1.8222
1.5683
1.9443
1.7593
2.0253
0.2317
0.0262
0.0096
0.0722
0.0131
0.1518
0.0188
0.3511
0.0225
Table A.8 The RB for the estimator with GMM estimation of PVAR with levels of missing observations and
comparison between LW and MI at n=30
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
25.44
22.31
34.51
23.05
37.69
26.66
41.01
38.60
35.98
34.95
48.35
42.42
57.08
53.79
60.93
55.33
68.02
81.90
102.39
83.99
92.21
61.94
66.58
65.93
53.85
50.36
73.11
74.80
84.86
48.83
92.58
50.67
67.01
54.30
77.30
72.37
64.43
63.48
66.15
58.22
66.11
66.08
105.73
74.48
98.12
82.31
94.39
81.23
53.80
44.49
44.88
42.53
53.11
57.62
60.70
62.41
30.67
28.65
39.03
40.49
45.64
43.34
65.22
60.97
123.89
98.61
106.69
95.64
161.66
96.22
184.57
87.48
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.79
Mohamed R. Abonazel,
Mohamed Abdallah, El-Housainy A. Rady
E-ISSN: 2224-2880
682
Volume 21, 2022
Table A.9 The MSE for the estimator with GMM estimation of PVAR with levels of missing observations
and comparison between LW and MI at n=40
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
0.0920
0.0900
0.0300
0.1300
0.1500
0.1200
0.1700
0.1300
0.0300
0.0211
0.0410
0.0400
0.0700
0.0333
0.1000
0.0400
0.0058
0.0001
0.0025
0.0018
0.0020
0.0011
0.0012
0.0008
0.0200
0.0126
0.1000
0.0900
0.0813
0.0800
0.0900
0.0813
0.0200
0.0300
0.0300
0.0200
0.0400
0.0211
0.0800
0.0200
0.0008
0.0007
0.0008
0.0007
0.0006
0.0005
0.0006
0.0002
0.3800
0.1900
0.7700
0.7090
1.0900
0.8500
1.7100
1.0000
0.1600
0.0330
0.3500
0.5100
0.9540
0.7400
0.8600
0.0382
0.0100
0.0091
0.0200
0.0100
0.0300
0.0200
0.0700
0.0300
Table A.10 The RB for the estimator with GMM estimation of PVAR with levels of missing observations and
comparison between LW and MI at n=400
Missing%
10%
20%
30%
40%
Estimate
LW
MI
LW
MI
LW
MI
LW
MI
25.41
19.33
28.92
22.12
37.01
22.81
47.20
35.24
34.97
22.98
43.02
41.08
57.57
45.96
68.16
45.74
59.63
54.69
51.40
24.62
81.08
57.09
59.15
58.51
52.47
48.87
72.72
60.98
71.32
45.18
90.67
47.01
48.96
45.48
77.32
67.07
56.65
46.66
59.26
40.93
65.79
62.68
99.34
71.31
96.76
73.71
89.17
73.92
24.94
22.94
37.44
36.30
44.69
37.34
56.14
40.81
36.01
22.83
34.63
29.73
39.71
34.99
39.61
37.64
40.51
33.67
58.67
50.07
76.52
60.12
109.84
72.81
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.79
Mohamed R. Abonazel,
Mohamed Abdallah, El-Housainy A. Rady
E-ISSN: 2224-2880
683
Volume 21, 2022
