J. Nig. Soc. Phys. Sci. 5 (2023) 1514

Journal of the
Nigerian Society

of Physical
Sciences

The Efficiency of the K-L Estimator for the Seemingly Unrelated
Regression Model: Simulation and Application

Oluwayemisi Oyeronke Alabaa,b,∗, B. M. Golam Kibriab

aDepartment of Statistics, University of Ibadan, Nigeria
bDepartment of Mathematics and Statistics, Florida International University, Miami, FL, USA

Abstract

This paper considers the Ridge Feasible Generalized Least Squares Estimator (RFGLSE), Ridge Seemingly Unrelated Regression RS UR and
proposes the Kibria-Lukman K LS UR estimator for the parameters of the Seemingly Unrelated Regression (SUR) model when the regressors of the
models are collinear. A simulation study was conducted to compare the performance of the three different types of estimators for the SUR model.
Different correlation levels (0.0, 0.1, 0.2, · · · , 0.9) among the independent variables, sample sizes replicated 10000 times and contemporaneous
error correlation (0.0, 0.1, 0.2, · · · , 0.9) among the equations were assumed for the simulation study. The efficiency of the three (RFGLSE, RS UR,
and K LS UR estimators for SUR, when the predictors are correlated, was investigated using the Trace Mean Square Error (TMSE). The results
showed that the K LS UR estimator outperformed the other estimators except for a few cases when the sample size is small.

DOI:10.46481/jnsps.2023.1514

Keywords: K − LS UR; Multicollinearity; Ridge feasible generalized least squares estimator; Seemingly Unrelated Regression; Trace Mean
Square Error

Article History :
Received: 22 April 2023
Received in revised form: 21 May 2023
Accepted for publication: 24 May 2023
Published: 14 June 2023

c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: B. J. Falaye

1. Introduction

One of the most ingenious and groundbreaking research in
econometrics is combining different single equations into a sys-
tem of equations to improve the efficiency of the parameter es-
timation [1], [2]. The Seemingly unrelated regression (SUR)
model with M equations and T observations is given as,

Y = Xβ + ε (1)

∗Corresponding author tel. no: +2348052172452
Email address: oluwayemisioyeronke@yahoo.com (Oluwayemisi

Oyeronke Alaba )


y1
...

ym


mn × 1

=


X1 · · · 0
...
0 · · · Xm


mn ×

∑
ki


β1
...
βm

∑
ki × 1


ε1
...
εm


mn × 1

(2)

where yi is an nm × 1 vector of observations on the ith response
variable, Xi is a fixed mn×

∑
ki matrix of explanatory variables,

βi is a
∑

ki × 1 vector of unknown regression parameters, εi is
an nm×1 vector of disturbances such that cov(ε) = E [ε′ε]⊗In,
E(ε) = 0.

The Ordinary Least Squares (OLS) estimator is widely used
to estimate the unknown regression parameters one equation

1



Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 2

at a time since each equation is a classical regression. The
SUR estimator simultaneously captures different regression
equations. However, the efficiency gain in SUR is premised
on the high level of contemporaneous correlation between
or among each classical regression equation. Notable works
on the efficiency gained in SUR which takes cognizance of
the contemporaneous correlation of error terms in the joint
equations include [3], [4], [5], [6], [7], [8], [9], [10], [11]
among others. The Generalized Least Squares Estimator
(GLSE) is used to estimate the variance-covariance matrix of
the disturbances in SUR model.

It is a statistical fallacy to assume that the relationship between
or among explanatory variables plays an insignificant effect on
the error structure of the model. The severity of the correlation
levels among the predictors can affect the efficiency and
sensitivity of the estimators [12], [13]. Hence, the variance
of the estimator is inflated, unreliable inference and the
confidence interval due to multicollinearity is wider which may
increase the probability of a type-II error in hypothesis testing
of unknown parameters [14]. Numerous research on single
equation models when the problem of multicollinearity is
inherent are available in literature. Notable works include [15],
[16], [17], [18], [19], [20], [21], [22] among others. Studies on
multicollinearity on systems of equations of regression models
are still lacking or scarce in literature. However, a notable
exception is [23], [24].

Recently, shrinkage estimators such as ridge regression es-
timators have gained attraction among researchers such that
quite a number of exciting estimators emerged [17], [25],
[26]. [19] proposed the K-L estimator to tackle the correlated
regressors problem for the classical linear regression model,
which outperformed both the Generalized Least Squares Esti-
mator (GLSE) and Ridge Feasible Generalized Least Squares
Estimator
(RFGLSE). The objective of this paper is to develop an
estimator which is suitable for the joint modelling of the K-L
estimator, Kibria-Lukman Seemingly Unrelated Regression
K−LS UR estimator when the predictors are correlated as well as
compare the newly developed estimator with the existing Ridge
Seemingly Unrelated Regression estimator (RS UR) and Ridge
Feasible Generalized Least Squares Estimator (RFGLSE).

The organization of the paper is as follows: The estimators and
their Trace Mean Square Error (TMSE) expressions are given
in Section 2. A simulation study is presented in Section 3. To
illustrate the findings of the paper, real-life data are analysed
in Section 4. The paper ends with some concluding remarks in
Section 5.

2. Statistical Methodology

The GLSE is given as

β̂GLS E =
(
X′

(
Σ
−1
⊗ I

)
X
)−1

X′
(
Σ
−1
⊗ I

)
y (3)

The ridge parameter estimator is an important tool when ex-
planatory variables are correlated in classical linear regression
analysis. The ridge estimation technique was pioneered by [15]
and extended to the SUR model by [25], [26], [27]. The ridge
estimator for GLSE is given as:

β̂RGLS E =
[
X
′
(∑
−1
⊗I

)
X + G

]−1
X
′
(∑
−1
⊗I

)
y

=
[
X
′

ψ−1 X + G
]−1

X
′

ψ−1y where ψ =
∑
−1 IT , G = kI∑ ki

β̂RGLS E =
[
I∑ Ki +

(
X
′

ψ−1 X
)−1

G
]−1

β̂GLS E (4)

where G is a k × k matrix of non-negative elements character-
izing the estimator. To circumvent the problem that σ is un-
known, a Ridge Feasible Generalized Least Squares estimator
(RFGLSE) is given as:

β̂RFGLS E =
[
X
′
(
S −1 ⊗ I

)
X + G

]−1
X
′
(
S −1 ⊗ I

)
y

β̂RFGLS E =
[
I∑ ki +

(
X
′

ψ̃−1 X
)−1

G
]−1

β̂FGLS E (5)

where β̂FGLS E =
(
X
′
(
S −1 ⊗ I

)
X
)−1

X
′
(
S −1 ⊗ I

)
y and ψ̃−1 =

S −1 ⊗ I
From (1), given Λ as the diagonal matrix of the eigenvalues
and ψ a matrix whose columns are eigenvectors of X∗

′

X∗ of
the systems of equations, SUR. The canonical version of (1) is
defined as

Y∗ = Z∗α∗ + e∗ (6)

where Z∗ = X∗ψ,α∗ = ψ
′

β and Z∗
′

Z∗ =
(
ψ
′

X∗
′

X∗ψ
)

= Λ

The GLSE of SUR for α∗ is;

α̂∗GLS =
(
Z∗
′

Z∗
)−1

Z∗
′

Y∗ (7)

bias
(
α̂∗GLS

)
= E

(
α̂∗GLS −α

∗
)

(8)

V ar
(
α̂∗GLS

)
= E

(
α̂∗GLS −α

∗
)

E
(
α̂∗GLS −α

∗
)′

= σ2
(
ψ
′

X∗
′

X∗ψ
)−1

(9)

= σ2Λ−1

The ridge regression estimator is;

α̂RS UR =
(
Z∗
′

Z∗ + ψ∗Rψ∗
′
)−1

Z∗
′

Y∗ (10)

bias
(
α̂∗RS UR

)
= E

(
α̂∗RS UR −α

∗
)

(11)

using(4)and(7)

V ar
(
α̂∗RS UR

)
=

[
X
′
(∑
−1
⊗I

)
X + kIp

]−1
X
′
(∑
−1
⊗I

)
X

[
X
′
(∑
−1
⊗I

)
X + kIp

]−1
=

(
Λ + kIp

)−1
Λ

(
Λ + kIp

)−1 (12)
where Λ = X′(Σ−1 ⊗ I)X

T MS E
(
α̂∗RS UR

)
= σ2

(
Ip + kΛ−1

)−1
Λ−1

(
Ip + kΛ−1

)−1
+

(
−k

(
Λ + kIp

)−1)
α∗α∗

(
−k (Λ + kIk)

−1
)′ (13)

2



Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 3

Figure 1. Estimated TMSE when ρεM = 0.1

Following [19], we define the K-LSUR estimator as follows:

α̂∗K LS UR =
[(

Ip + kΛ
−1

)−1 (
Ip − kΛ

−1
)]
α̂∗ (14)

E
(
α̂∗K LS UR

)
=

(
Ip + kΛ−1

)−1 (
Ip − kΛ−1

)
E (α̂∗)

=
(
Ip + kΛ−1

)−1 (
Ip − kΛ−1

)
α̂∗

(15)

The K-L estimator is an unbiased estimator when k = 0

T MS E
(
α̂∗RS UR

)
= σ2

(
Ip + kΛ−1

)−1 (
Ip − kΛ−1

)
Λ−1

(
Ip − kΛ−1

)′
((

Ip + kΛ−1
)−1)′

+

[(
Ip + kΛ−1

)−1 (
Ip − kΛ−1

)
− Ip

]
α∗α∗

′
[(

Ip + kΛ−1
)−1 (

Ip − kΛ−1
)
− Ip

]′ (16)

3. Numerical Analysis

A simulation study is considered to compare the perfor-
mance of the estimators in this section. It consists of two parts
(i) Simulation study (ii) Discussion of Results.

3.1. Simulation Study

The Monte Carlo experiment was performed by generating
data according to the following algorithm.

1. Generate the explanatory variables from MV N3(0, Σx)

2. Set the true values of β to (1, 1, 1, 1)′

Figure 2. Estimated TMSE when ρεM = 0.4

Table 1. Description of Variables, Equations, Observations and Contemporane-
ous Correlations

Factors Symbol Design
No of equations M 3
No of observations T 20, 30, 50, 100
Correlation among the ex-
planatory variables

ρx 0.0, 0.1, 0.2,
0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9

Contemporaneous cor-
relation between corre-
sponding errors among the
equaions

ρΣ 0.0, 0.1, 0.2,
0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9

3. The variance-covariance is given as

∑
=


σ211 σ

2
12 σ

2
13

σ221 σ
2
22 σ

2
13

σ231 σ
2
32 σ

2
33

 =


1 0.6 0.8
1 0.7

1


such that

Y1 : N
(
X1β1,σ211

)
Y2 : N

(
X2β2,σ222

)
Y3 : N

(
X3β3,σ233

)
4. Simulate the vector random error from MV N3(0, Σe)

5. For a given X structure, transform the original model to
the canonical form.

6. Compute the trace mean squared error of
βK LS UR,βRFGLS E,
and βRS UR

7. Repeat the above step 10000 times.

3



Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 4

Table 2. Estimated TMSEs for the Different Methods when ρεM = 0.1, 0.2 and 0.3 at n = 20

n = 20
ρεm = 0.1 ρεm = 0.2 ρεm = 0.3

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 1.9671 31.4793 1078.181 2.9993 34.9221 34.9869 4.3360 33.2312 210.5745
0.1 2.2637 33.5409 40.7652 3.3615 34.4638 39.1302 4.7297 33.9507 179.0863
0.2 2.5411 33.5427 38.5592 3.7682 34.7496 47.0992 5.2687 36.4858 71.1576
0.3 2.8784 32.8007 49.5184 4.2931 35.8924 42.8646 7.7218 37.2722 53717.79
0.4 3.4011 33.7350 53.2065 5.2779 35.8924 75.3289 64.0115 37.3879 113.8371
0.5 4.0056 3.6415 2.5955 6.0753 37.7050 216.6385 8.4646 39.2840 208.3292
0.6 4.9863 3.5757 1.0886 214.7618 38.7659 295.7213 14.9161 40.7934 741.0087
0.7 7.5773 36.6885 1137.117 11.5006 40.7506 462.7717 345.8575 43.9871 1399.8543
0.8 20.4227 42.1282 1948.9526 45.0750 45.1791 16809.71 55.2401 51.3216 3303.2105
0.9 31.9145 51.3059 441.2463 979.4674 59.5731 27157.19 822.7517 70.6499 4580.6889

Table 3. Estimated TMSEs for the Different Methods when ρεM = 0.1, 0.2 and 0.3 at n = 30

n = 30
ρεm = 0.1 ρεm = 0.2 ρεm = 0.3

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 1.1817 32.6906 32.3140 1.4956 32.0123 31.5930 2.3545 33.3188 32.6339
0.2 1.1929 29.9211 29.4511 1.6637 32.7880 32.3673 2.1419 30.0010 32.0582
0.3 1.3553 22.4464 33.2031 1.8878 32.9291 32.5181 2.4342 34.7959 34.2291
0.4 1.5684 30.1094 29.5395 2.1817 31.1496 30.5211 2.8013 33.1929 32.2543
0.5 1.8712 20.1180 31.9485 2.6034 32.1032 31.3159 3.3439 33.6351 32.5898
0.6 2.3131 17.2247 16.8764 3.2240 32.7284 47.2943 4.1401 34.5030 36.8375
0.7 6.9799 31.2066 382.9524 4.3324 33.5406 47.9402 84.9530 35.5565 42.7001
0.8 4.6595 33.2122 1786.0465 6.7134 35.9082 166.2044 19.7855 36.5740 1303.1226
0.9 9.9985 37.0700 3186.9899 16.5210 42.0564 13172.89 528.8485 45.9672 3383.9887

Table 4. Estimated TMSEs for the Different Methods when ρεM = 0.1, 0.2 and 0.3 at n = 50

n = 50
ρεm = 0.1 ρεm = 0.2 ρεm = 0.3

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 0.6101 30.9197 30.7837 0.8323 17.4569 17.3667 1.0599 29.7809 29.5481
0.2 0.6797 32.7346 32.6091 0.9276 32.2903 32.1370 1.1776 33.7307 33.5492
0.3 0.7701 32.2188 32.0525 1.0484 31.8266 31.6524 1.3189 18.5846 18.3629
0.4 0.8906 30.1703 29.9848 1.2107 31.6182 31.4187 1.5282 31.8069 31.5339
0.5 1.0613 30.2766 30.0612 1.4362 19.7875 19.4246 1.8204 34.1597 33.8706
0.6 1.3208 33.5028 33.3809 1.7953 33.5506 33.2716 2.2618 34.4465 34.0654
0.7 1.7521 32.6755 32.2909 2.3778 34.2552 33.8549 3.0007 34.2756 33.6832
0.8 2.6126 34.5632 33.9872 3.5386 35.1982 34.4567 4.4997 36.0475 34.9669
0.9 5.1513 22.4060 20.8302 7.1461 38.2472 36.3301 9.1181 40.2009 38.1831

The simulation results were presented in Tables 2 to 13
for ρεM = (0.1, 0.2, 0.3),ρεM = (0.4, 0.5, 0.6) and ρεM =
(0.7, 0.8, 0.9) respectively. In addition, some results are illus-
trated graphically in Figures 1 to 4 for ρεM = 0.1, 0.4, 0.7 and
0.9 respectively. We considered R software to conduct the sim-
ulation study [27], [28].

4. Discussion of Results

From Tables 2 to 13 and Figures 1 to 4, we can see that
the proposed K LS UR estimator uniformly dominate the RFGLS
and RS UR estimator except when the sample size is small (n
= 20). Also, an increase in the ρxi xi increases the estimated
TMSE values of the estimators. Obviously, the TMSE values
were significantly large at sample size n = 20 when ρεM =
0.7, 0.8 and 0.9 than other corresponding TMSE values with
n = 30, 50, 100 for ρεM = {0.0, 0.1, 0.2, · · · , 0.9} and n = 20

4



Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 5

Table 5. Estimated TMSEs for the Different Methods when ρεM = 0.1, 0.2 and 0.3 at n = 100

n = 100
ρεm = 0.1 ρεm = 0.2 ρεm = 0.3

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 0.2958 31.1069 31.0652 0.4001 32.4019 32.3509 0.5034 32.0166 31.9597
0.2 0.3294 32.9002 32.8577 0.4439 31.5425 31.4946 0.5560 31.9488 31.8862
0.3 0.3730 31.0733 31.0145 0.5013 31.7451 31.6767 0.5259 30.9504 30.8669
0.4 0.4317 32.7167 32.6697 0.5793 32.3633 32.3027 0.68903 31.7897 31.7127
0.5 0.5141 30.5062 30.4347 0.6888 30.4318 30.3485 0.85712 30.8029 30.6942
0.6 0.6373 17.8189 17.7757 0.8544 32.3865 32.3051 1.06185 32.3417 32.2352
0.7 0.8467 32.4540 32.3723 1.1325 32.6137 32.4948 1.4077 33.2381 33.0755
0.8 1.2616 30.9976 30.7480 1.6856 31.4613 31.1143 2.0931 32.4275 31.9626
0.9 2.5207 32.4311 31.6849 3.3763 33.1247 32.1534 4.1961 33.8402 32.4072

Table 6. Estimated TMSEs for the Different Methods when ρεM = 0.4, 0.5 and 0.6 at n = 20

n = 20
ρεm = 0.4 ρεm = 0.5 ρεm = 0.6

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 9.7560 36.2392 256.4695 716.2386 35.4935 406.7923 12.0147 40.1432 308.325
0.1 6.4118 36.2199 459.8962 10.6846 37.8609 658.1879 12.6002 38.8413 253.003
0.2 7.7878 36.0814 193.1198 9.9320 39.0815 1772.816 739.9870 42.2980 171.349
0.3 14.6262 39.5822 284.7212 11.7001 40.6640 251.5625 27.9862 43.0574 692.950
0.4 16.2656 39.2694 248.8817 68.8660 42.2592 860.8801 21.2453 45.0407 477.386
0.5 12.6930 42.5478 2629.349 20.1253 45.4033 22608.84 25.9884 47.7564 4292.16
0.6 37.4753 43.3222 286.5887 78.1266 47.4747 252.3527 76.0018 51.3518 581.977
0.7 91.8338 48.2507 769.2762 186.599 53.0107 1751.206 12291.26 57.8010 7708.11
0.8 408.0553 57.7729 6626.982 482.442 65.6280 94973.63 7459.204 72.5054 1255.63
0.9 27135.45 82.4109 3004.294 8687.57 96.2975 18763.57 98272.26 96.1662 2682.17

Table 7. Estimated TMSEs for the Different Methods when ρεM = 0.4, 0.5 and 0.6 at n = 30

n = 30
ρεm = 0.4 ρεm = 0.5 ρεm = 0.6

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 2.1482 33.8490 33.4394 2.58122 33.6536 32.9963 3.0486 34.1725 33.4501
0.1 2.4351 29.8788 47.1569 2.83488 30.7532 29.9543 3.3416 31.2885 32.5218
0.2 2.2597 34.6346 33.5944 3.1033 31.2482 30.7619 3.5688 32.5834 31.1876
0.3 2.9817 35.3758 34.6432 3.5219 36.6019 35.7699 4.0414 36.7851 35.7306
0.4 3.4258 32.9304 32.1480 4.0364 33.0953 32.6963 4.6313 33.6982 48.0887
0.5 4.1175 37.4169 36.4355 4.7903 34.6699 35.0898 5.5344 34.1078 1134.971
0.6 5.0357 35.1816 266.3296 5.9218 35.9237 35.6257 7.0024 36.8212 38.8586
0.7 6.8262 37.1196 504.2674 8.0604 38.0785 683.5997 9.21488 39.7283 2209.564
0.8 12.1846 39.4336 26492.285 13.0759 41.5627 775.9910 16.8529 43.1572 330.2718
0.9 23.4235 52.9490 1302.4404 36.8351 56.4241 6609.2648 44.5576 57.7298 2350.237

for ρεM = {0.0, 0.1, 0.2, · · · , 0.7}. The TMSE values become
larger at n=20 as ρεM increases from 0.7 to 0.9. The tables and
figures consistently showed that the proposed K LS UR estimator
performs better than the RFGLS and RS UR estimators when
there exist moderate to high correlation among the regressors.
The plots of TMSE against various ρεM in Figures 1 to 4
showed that as the value of ρ increases, TMSE also increase,
while as the sample size increases, the TMSE value decreases.

The preferred TMSE values were at n =100 for ρεM
= 0.1 for K LS UR estimator. K LS UR produced the smallest
TMSE values when ρxi xi range from 0.1 to 0.9, such that the
corresponding TMSE values for K LS UR estimator increase as
the ρxi xi increase. However, it was noted that the ideal and
smallest TMSE value for K LS UR estimator occurred at ρεM =
0.1 and ρxi xi = 0.1. Concisely, K LS UR also produced small
TMSE values at ρεM = 0.1 and ρxi xi = 0.1, but not smaller in
comparison to the one produced by K LS UR estimator. Con-

5



Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 6

Table 8. Estimated TMSEs for the Different Methods when ρεM = 0.4, 0.5 and 0.6 at n = 50

n = 50
ρεm = 0.4 ρεm = 0.5 ρεm = 0.6

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 1.2231 31.8356 31.6180 1.4567 33.0326 32.7876 1.6850 30.8215 30.4527
0.1 1.2845 29.9504 29.6901 1.51162 30.2718 29.9303 1.7267 31.5870 31.1975
0.2 1.4195 32.2654 32.0162 1.6617 31.4414 31.1212 1.8969 31.7036 31.3102
0.3 1.5985 33.1219 32.8367 1.8689 33.8384 33.5117 2.1299 33.8781 33.4787
0.4 1.8389 32.0315 31.6715 2.1469 33.3783 32.9510 2.4424 32.6150 32.0562
0.5 2.1859 32.7642 32.3262 2.5474 32.0638 31.4425 2.9026 33.2641 32.5604
0.6 2.7172 34.1213 33.5881 3.1823 34.3878 33.7234 3.6293 35.1303 34.3261
0.7 3.6180 34.9418 34.1512 4.1749 35.4996 34.5134 4.7830 36.0520 34.80777
0.8 5.4192 36.9573 35.5262 6.3468 38.1684 36.3989 7.2398 38.6846 36.4323
0.9 11.0051 41.8592 41.3861 12.9492 43.3830 53.5260 15.1569 44.8944 47.0988

Table 9. Estimated TMSEs for the Different Methods when ρεM = 0.4, 0.5 and 0.6 at n = 100

n = 100
ρεm = 0.4 ρεm = 0.5 ρεm = 0.6

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 0.5570 31.1110 31.0429 0.65712 31.2067 31.1265 0.7562 32.5784 32.4877
0.1 0.6064 31.5343 31.4591 0.7069 31.4484 31.3653 0.80516 31.0438 30.9381
0.2 0.6666 32.7558 32.6821 0.7744 32.4927 32.4042 0.88786 32.4789 32.3799
0.3 0.7477 31.0310 30.9287 0.8670 30.8659 30.7475 0.9813 30.4286 30.3006
0.4 0.8616 32.4530 32.3617 0.9960 32.7987 32.6845 1.1274 32.8786 32.7485
0.5 1.0175 17.1745 17.1026 1.18127 31.1289 30.9470 1.3339 31.6014 31.3899
0.6 1.2645 32.7726 32.6332 1.4586 32.5574 32.3776 1.6516 32.0510 31.7438
0.7 1.6723 33.5980 33.3833 1.9303 33.4664 33.1906 2.1856 33.9971 33.6736
0.8 2.4884 32.3812 31.7566 2.8805 32.7172 31.9339 3.2131 19.5499 18.9774
0.9 4.9937 34.7593 32.9295 5.7754 35.4817 33.2136 6.5181 36.4971 33.78511

Table 10. Estimated TMSEs for the Different Methods when ρεM = 0.7, 0.8 and 0.9 at n = 20

n = 20
ρεm = 0.7 ρεm = 0.8 ρεm = 0.9

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.0 1.5970 4.0118 1.5448 27.0078 41.4233 161.6998 20.9443 44.5509 2102.610
0.1 13.3597 41.6366 7274.501 19.8636 43.4407 1096.485 45.4412 44.9649 366.5848
0.2 14.5438 42.4626 301.5057 21.2077 45.1824 402.2686 22.9364 46.0099 133.9393
0.3 15.8899 45.4626 125.4716 30.3843 47.2784 878.4280 109.9613 49.0023 644.4463
0.4 953.5130 47.2117 1997.0092 577.5404 49.5397 939.3170 1.1322 5.2701 2.0398
0.5 30.2281 51.3641 6113.3947 34.0842 53.2688 1525.4125 4787.2699 56.5041 2354.758
0.6 96.7198 53.3738 8236.2489 1440.019 57.9711 403.6016 481.5750 61.4689 10141.04
0.7 44014.384 62.7783 864.9972 9420.9595 67.5711 1731.6622 7184.6716 71.3247 34749.18
0.8 21669.976 79.6336 26152.861 47515.817 84.3908 36052.59 5962.1447 91.7812 717.1957
0.9 75800.340 127.1244 1039.1268 12340.321 137.575 1158.7834 50556.851 148.5709 9359.699

sequently, this implies that K LS UR is the preferred estimator
when considering SUR model when the regressors of the
model are collinear among the three considered estimators.
The strength of collinearity among the considered variables
for MV N3(0, Σe), m = 3 with varied for sample sizes 20 to
100 for K LS UR estimator relies on when n = 100 with ρεM =
0.1 and ρxi xi = 0.1. This implies that the strength and type
of dependency among the explanatory variables affect the

performance of each of the three estimators of K LS UR, RFGLS
and RS UR

As T increases, such that ρεM and ρxi xi decreases (or relatively
low), the TMSE values of K LS UR decreases. This implies that
there is a gain in the efficiency of the K LS UR estimator.

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Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 7

Table 11. Estimated TMSEs for the Different Methods when ρεM = 0.7, 0.8 and 0.9 at n = 30

n = 30
ρεm = 0.7 ρεm = 0.8 ρεm = 0.9

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 3.6831 32.4952 31.1870 4.0739 32.6618 30.9791 4.4489 32.9211 31.8359
0.2 4.0185 34.1073 32.5706 4.4305 33.9127 31.9412 4.8275 34.5264 32.4991
0.3 4.5332 37.4486 36.2957 5.0095 37.3701 35.8343 5.4564 38.6045 37.1221
0.4 5.1828 33.9950 40.1823 5.8834 34.7873 35.0340 6.2366 34.9937 36.4678
0.5 6.1682 36.0594 35.1923 6.8139 36.9741 98.5351 7.4014 37.2651 653.2791
0.6 7.6710 36.6365 59.8780 8.4420 38.3879 42.2726 9.1107 3.9449 1.0753
0.7 41.4184 40.5272 92.5242 11.4445 42.0848 4231.299 12.5689 42.5401 83.8831
0.8 28.8063 44.9286 137.4677 1023.891 46.0243 332.3192 23.1223 47.7619 11443.84
0.9 60.4966 62.9849 156.5118 701.8199 65.4741 234.6494 7.8273 6.8693 1.3251

Table 12. Estimated TMSEs for the Different Methods when ρεM = 0.7, 0.8 and 0.9 at n = 50

n = 50
ρεm = 0.7 ρεm = 0.8 ρεm = 0.9

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 1.9390 30.7288 30.2195 2.1375 32.1671 31.6753 2.3270 32.5277 31.9061
0.2 2.1207 31.4594 30.9896 2.7655 32.6984 31.8318 2.5422 34.9202 34.2994
0.3 2.3833 34.6569 34.1233 2.6170 34.4649 33.9328 2.8471 35.1956 34.5160
0.4 2.7287 33.9790 33.2775 2.9985 33.0041 32.2088 3.2564 33.0878 32.1194
0.5 3.2429 33.4776 32.5970 3.5620 33.9245 32.9419 3.8698 34.2694 33.1379
0.6 4.0373 34.8706 33.8102 4.4264 34.1091 32.6144 4.8349 36.7864 35.4188
0.7 5.3456 36.8501 35.2405 5.8171 37.0795 35.3813 6.3794 38.0211 35.8226
0.8 8.0694 39.0196 36.2068 8.8507 39.8438 36.5279 9.5831 39.7160 35.8735
0.9 16.8842 46.8999 215.3601 29.3585 48.1159 97.7010 51.5108 49.3468 56.2438

Table 13. Estimated TMSEs for the Different Methods when ρεM = 0.7, 0.8 and 0.9 at n = 100

n = 100
ρεm = 0.7 ρεm = 0.8 ρεm = 0.9

ρxi x j K LS UR RFGLS RS UR K LS UR RFGLS RS UR K LS UR RFGLS RS UR
0.1 0.8994 31.9839 31.8691 0.9882 31.6234 31.4959 1.0733 31.7436 31.5857
0.2 0.9785 33.1136 32.9996 1.0742 32.6911 32.5621 1.1653 32.2650 32.1150
0.3 1.0918 31.3878 31.2122 1.1969 31.2668 31.0811 1.2964 31.9378 31.7250
0.4 1.2536 32.6015 32.4556 1.3732 32.8539 32.6776 1.4871 32.2467 32.0410
0.5 1.4785 31.6940 31.4353 1.6218 31.1869 30.8941 1.7562 32.0679 31.7361
0.6 1.8292 31.9719 31.6237 2.0088 32.4433 32.0269 2.1661 32.3290 31.8782
0.7 2.4185 34.1733 33.7924 2.6589 34.4667 34.0165 2.8793 34.3942 33.8972
0.8 3.5981 33.4160 32.3067 3.9354 33.7854 32.4953 4.2864 34.0114 32.6416
0.9 7.2537 36.9970 33.7463 7.9236 37.7464 34.1527 8.6199 38.7249 34.4544

Table 14. Autocorrelation Test (Durbin-Watson Test)
Equation (model) Test Statistic p-Value
Brazil 2.3800 0.2846
India 1.9160 0.0658
Indonesia 2.4776 0.4547
South Africa 1.9669 0.1225
Turkey 1.5431 0.0032

5. Application

To further illustrate the results of the theoretical part of
this paper, we consider the dataset and structural model by

Table 15. Heteroscedasticity Test (Breusch-Pagan Test)
Equation (model) Test Statistic p-Value
Brazil 12.4250 0.1332
India 11.0220 0.2005
Indonesia 6.0900 0.6372
South Africa 5.6808 0.6829
Turkey 3.8404 0.8712

[12]. The study considered the foreign direct investment on

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Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 8

Table 16. Multicollinearity Test (Variance Inflation Factor)

Equation Deflator
Current
Account

Capital
Growth

Final
Consumption

Import of
Goods

Personal
Remittance

Total Reserves
Export of

Goods
Brazil 56.241 23.344 2.144 6.652 11.143 20.487 46.461 24.510
India 23.798 13.750 1.741 2.497 101.771 2.438 23.185 66.900
Indonesia 118.769 12.219 3.552 4.588 42.317 3.214 72.550 124.779
South Africa 29.423 39.671 4.278 10.284 109.568 10.157 49.739 45.688
Turkey 21.052 206.371 3.548 5.611 258.126 4.557 15.662 224.422

Table 17. Specification Test (Regression Equation Specification Error Test)
Equation (model) Test Statistic p-Value
Brazil 0.0591 0.943
India 1.0576 0.3913
Indonesia 0.98417 0.4148
South Africa 0.52381 0.6113
Turkey 1.0114 0.4059

Table 18. Shrinkage Parameter Estimators for the Life Study
K LS UR RFGLS RS UR

TMSE 35.00 40.52786 40.5278

Figure 3. Estimated TMSE when ρεM = 0.7

some economic and financial variables over the period of
twenty years between 2001 and 2019. The ”Fragile Five”
countries (cited in [12]) include Turkey (TUR), South Africa
(ZAF), Brazil (BRA), India (IND), and Indonesia (IDN).
Hence, M = 5 blocks, with measurements of T = 20 years per
equation. The raw data were extracted online from the World
Bank Indicators. The structural SUR model adopted is given as:

Figure 4. Estimated TMSE when ρεM = 0.9

F DIit = f (GRO, DF, CAB, FC, I PS, PRR, T R, EGS )

Fdii = β0i + β1iGro + β2i D f + β3iCab + β4i Fc + β5i I ps
+β6i Pr + β7iT r + β8i Egs + ei

(17)

i denotes countries (i= TUR, ZAF, BRA, IND, IDN).

Fdii is the Foreign Direct Investment of the ”fragile five”
countries, Gro is the GDP per Capital growth, Df is the GDP
deflator, Cab is the Current Account Balance, Fc is the Gen-
eral Government Final Consumption Expenditure, Ips is the
Imports of Goods and Services, Prr is the Personal Remit-
tances Received, Tr is the Total Reserves and Egs is the Ex-
ports of Goods and Services. The assumptions in each equa-
tion and joint model were put into consideration. Normality,
homoscedasticity, multicollinearity and serial correlation of the
error term were examined. The results are available in Tables
14 to 18.

The null hypothesis for the Durbin-Watson test is that er-
rors are random and independent. A significant p-value in this
test rejects the null hypothesis that the time series is not auto-
correlated. Table 5 suggests a rejection of the null hypothesis
for Turkey at the significance level, that is, p-value = 0.0032
≺0.05. This implies that each equation satisfied the assumption
of non-autocorrelation.

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Alaba & Kibria / J. Nig. Soc. Phys. Sci. 5 (2023) 1514 9

The null hypothesis for the Breusch-Pagan Test is that
there is no homoscedasticity (that is, there is presence of het-
eroscedasticity). Since, the p-value for each of the ”fragile five”
country is greater than 0.05. We did not reject the null hypoth-
esis in each equation, so the assertion of homoscedasticity in
each equation is satisfied.

The Variance Inflation Factor (VIF) makes it possible to
measure how many times the variance of the regression coef-
ficients will be for multicollinear data than for orthogonal/
canonical data. If VIF�10 this indicates multicollinearity. Con-
cisely, deflator, current account, total reserves; import and ex-
port of goods posed to be problematic (that is, multicollinearity
problem exist) as their VIF values were strictly greater than 10,
while others also call for little concern. This implies that the
problem of multicollinearity exists in the equations. Therefore,
the K LS UR estimator will be ideal to solve the problem.

The regression equation specification error test is meant
for testing the exogeneity of explanatory variables. The null
hypothesis for the test is that there is no correlation between
the error term and the explanatory variables or that linearity
exist in the functional form of the regression model, that is,
E [εi | Xi] = 0. Since, the p-value for each of the ”Fragile Five”
country is greater than 0.05 in table 14, this suggests that there
is no correlation between the error term and the explanatory
variables for all the countries (Turkey, Brazil, India, Indonesia,
and South Africa).

The shrinkage parameter estimator is designed for model
reduction for each of the M equations in order to absolve only
significant explanatory variables in line with their number of
observations. The shrinkage estimator makes it possible to stan-
dardize each reduced equation. From Table 18, the shrinkage
parameters of the K LS UR estimator via the TMSE produced the
smallest TMSE value of 35.00 compared to RFGLS and RS UR
that gave higher TMSE of 40.52786 and 40.5278 respectively.
This makes the K LS UR estimator to possess a higher efficiency
gain than the two other estimators. Summarily, K LS UR = 35.00
indicates that the estimator outperforms RFGLS and RS UR.

6. Conclusion

The seemingly unrelated regression model is an exciting
and celebrated model when the error structure of joint mod-
els are correlated. We considered the ridge feasible general-
ized least squares estimator, ridge seemingly unrelated regres-
sion and the Kibria-Lukman K LS UR estimator for estimating the
parameters of the seemingly unrelated regression model when
the regressors of the model are collinear. Findings from this
comparative study revealed that K LS UR estimator is a better al-
ternative when compared with ridge and ridge feasible general-
ized least squares, which are often used to tackle the problem
of multicollinearity in single and joint equations respectively.
Simulation and real-life data were used for assessment. Note
that we have considered one of many possible estimators of the
ridge parameter k. The conclusion of the paper may change if
we consider different values of k and such possibility is under
the current investigation.

Acknowledgements

This paper was partially completed while the first author
was visiting Prof. B. M. Golam Kibria, Florida international
University in the month of April-May, 2021.

Availability of Data and Codes: Available on request.

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