In each equation of simultaneous Equation model, the exogenous variables need to satisfy all the basic assumptions of linear regression model and be non-negative especially in econometric studies. This study examines the performances of the Ordinary Least Square (OLS), Two Stage Least Square (2SLS), Three Stage Least Square (3SLS) and Full Information Maximum Likelihood (FIML) Estimators of simultaneous equation model with both normally and uniformly distributed exogenous variables under different identification status of simultaneous equation model when there is no correlation of any form in the model. Four structural equation models were formed such that the first and third are exact identified while the second and fourth are over identified equations. Monte Carlo experiments conducted 5000 times at different levels of sample size (
n = 10, 20, 30, 50, 100, 250 and 500) were used as criteria to compare the estimators. Result shows that OLS estimator is best in the exact identified equation except with normally distributed exogenous variables when
. At these instances, 2SLS estimator is best. In over identified equations, the 2SLS estimator is best except with normally distributed exogenous variables when the sample size is small and large,
and
; and with uniformly distributed exogenous variables when
n is very large,
, the best estimator is either OLS or FIML or 3SLS.
Normally Distributed Exogenous Variables Uniformly Distributed Exogenous Variables Identification Status Estimators Exact Identified Equation Over Identified Equation1. Introduction
A simultaneous equation system is a regression equation system where two types of variables (the endogenous, the predetermined or exogenous variable) appear with disturbance terms. [1] defined simultaneous equation as the process of modeling more than one equation at a time; a multi-equation modeling. It is proposed to solve the problem of correlation.
In a multi-equation model, the dependent variable Y appears as endogenous variable in one equation and as explanatory variable in another equation of the model. The X variable appears as the explanatory variable in the equations. This creates problems of equation identification, multicollinearity and choice of estimation techniques among others.
Identification problem which creates estimation problems has the same features with multicollinearity [2] . Correlation between the pairs of exogenous or independent variables is an important problem in econometrics especially in single equation estimation. The impact of multicollinearity is less serious when attention is focused on predicting or forecasting values of the endogenous variables than when the analyst is interested in estimating the parameters [3] .
Econometric variables are often non-negative and can exhibit violation against the normality assumption of classical models which inevitably influences the performances of the estimation techniques. This study therefore examines the performances of four (4) common estimation techniques namely; Ordinary Least squares (OLS), Two-stage Least squares (2SLS), Three-stage Least squares (3SLS) and Full Information Maximum Likelihood Estimators under both normally and uniformly distributed exogenous variables for different equation identification status. Here it is assumed that there is no form of correlation in the simultaneous equation model.
Most economic data are often positive [4] and correlated [5] . However, various works done in the recent time especially on correlation studies on simultaneous equation model have being with normally distributed exogenous variables, exhibiting both positive and negative values [6] [7] .
2. Methodology
The methodology followed in this research work is as follows:
The Model and Its Description
Consider the simultaneous Equation model of the for
where is an endogenous variable,;and are the exogenous variables
where and are the structural parameters of the model. Equations (i) and (iii) are exactly identified while Equations (ii) and (iv) were over identified by both order and rank condition.
For the simulation study, Equation (1) was expressed as follows:
This can be written in matrix form as:
where
, ,
, and
Now, from (5),
Equation (6) was used to generate the endogenous variables by taking the true value of the parameters as:
Monte Carlo experiments were performed 5000 times for seven sample sizes (n = 10, 20, 30, 50, 100, 250 and 500) when there is no correlation of any form among exogenous variables and the error terms. The two forms of exogenous variables are generated to follow normal and uniform distribution.
3. Data Generation of Exogenous Variables
The generation was done as follows:
1) Normally Distributed Exogenous Variables
The exogenous were generated to be normal with mean zero and variance unity i.e. and is the value of correlation between the two variables i and j, [8] . In this study, , was adopted as a situation with no forms of any correlation between the exogenous variables.
2) Correlated Uniformly Distributed Variables
Using the generated normally distributed exogenous variables above, , I = 1, 2, 3, 4; we further utilize the properties of random variables that cumulative distribution function of normal distribution produces U(0, 1) without affecting the correlation among the variables to generate correlated uniformly distributed exogenous variables, , I = 1, 2, 3, 4; [9] .
3) Generation of Correlated Error Terms
Equation provided by [8] was modified when the mean of the error terms are zero and variance is unit (1). Also, is the value of correlation between the two error terms i and j. In this study, , was adopted as no form of correlation between the error terms.
4) Method of Generating the Data of Endogenous Variables
Equation (6) was used to generate the endogenous variables by taking the true value of the parameters as:
5) Criteria Used for Assessment of the Estimators
To assess the performance of the estimators, the finite properties were used (Bias, Absolute bias, Variance and Mean Square Error):
Mathematically, MSE is the addition of the Bias squared to the variance. For any estimator of in model (1),
,
where R = 5000
as a measure of dispersion.
In this paper, is used to represent any of the parameters in (1).
4. Results and Discussion4.1. Performance of the Estimator Based on the Bias Criterion
The performances of the estimators was examined, ranked and summed over all the parameters in each equation, having examined and summed the rank of the bias of each parameter over all the parameters in each equation the outcome is given in Table 1.
The preferred estimators for different identification status differ. The preferred estimators are also slightly affected by the two exogenous variables.
In the exact identified equation with normally distributed exogenous variables, OLS or 2SLS or both are generally preferred except for very large samples sizes. At this instance, 3SLS is preferred. Also, with uniformly distributed exogenous variables, OLS or 2SLS estimators are generally preferred.
In the over identified equation with normally distributed exogenous variable, 2SLS estimator is generally preferred except when and when. At these instances, FIML estimator is preferred.
Also, with uniformly distributed exogenous variable 2SLS estimator is generally preferred except at very large sample sizes when and when. At these instances, the FIML estimator is preferred.
4.2. Performances of the Estimators Based on Absolute Bias Criterion
The performances of the estimators was examined, ranked and summed over all the parameters in each equation, having examined and summed the rank of absolute bias of each parameter over all the parameters in each equation the outcome is given in Table 3.
From Table 3, at each level of sample size the sum rank were further added over the equations and preferred estimator under both exact and over identification model with the two exogenous variables were bolded. Table 4 gives the summary of the findings as follows:
The preferred estimators differ in term of identification status. The preferred estimators are slightly affected
Summary of the total ranks of the parameter based on bias criterion when there is no correlation
N
Estimator
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
10
OLS
6
6
12
9
12
21
5
4
9
9
10
19
2SLS
6
6
12
6
9
15
7
8
15
5
4
9
3SLS
13
13
26
12
6
18
12
15
27
4
9
13
FIML
15
15
30
3
3
6
16
13
29
12
7
19
20
OLS
8
8
16
12
12
24
4
5
9
9
9
18
2SLS
5
4
9
6
9
15
8
7
15
6
3
9
3SLS
11
13
24
9
3
12
12
14
26
3
6
9
FIML
16
15
31
3
6
9
16
14
30
12
12
24
30
OLS
5
4
9
12
12
24
6
4
10
12
11
23
2SLS
7
8
15
6
3
9
6
8
14
3
3
6
3SLS
16
13
29
9
6
15
13
12
25
7
6
13
FIML
12
15
27
3
9
12
15
16
31
8
10
18
50
OLS
8
7
15
12
12
24
6
9
15
12
12
24
2SLS
4
5
9
6
5
11
6
4
10
9
3
12
3SLS
12
12
24
9
4
13
12
13
25
6
9
15
FIML
16
16
32
3
9
12
16
14
30
3
6
9
100
OLS
16
16
32
12
12
24
7
7
14
12
12
24
2SLS
7
4
11
6
3
9
5
5
10
5
4
9
3SLS
5
10
15
3
6
9
13
13
26
8
9
17
FIML
12
10
22
9
9
18
15
15
30
5
5
10
250
OLS
16
16
32
12
12
24
4
6
10
12
12
24
2SLS
5
12
17
6
3
9
8
6
14
9
7
16
3SLS
7
4
11
3
6
9
12
15
27
6
8
14
FIML
12
8
20
9
9
18
16
13
29
3
3
6
500
OLS
16
16
32
12
12
24
16
8
24
12
12
24
2SLS
5
12
17
6
5
11
12
4
16
9
5
14
3SLS
7
5
12
9
9
18
7
15
22
5
6
11
FIML
12
7
19
3
4
7
5
13
18
4
7
11
Source: Computed from simulated results of bias criterion.
by the two exogenous variables. In the exact identified equation with normally distributed exogenous variables under the absolute bias criterion the OLS estimators is preferred when and the 2SLS estimator when.
With uniformly distributed exogenous variables the OLS estimator is generally preferred.
In the over identified equation with normally distributed exogenous variables under the absolute bias criterion, the OLS estimator is preferred when the sample size is small; 2SLS estimator when; and FIML when.
With uniformly distributed exogenous variables the 2 SLS estimator is preferred over all the sample sizes. Thus, the preferred estimators are more stable in performance with uniformly distributed exogenous variables than the normally distributed exogenous variables.
4.3. Performances of the Estimators Based on Variance Criterion
The performances of the estimators was examined, ranked and summed over all the parameters in each equation, having examined and summed the rank of the variance of each parameter over all the parameters in each equation the outcome is given in Table 5.
From Table 5, at each level of sample size the sum rank were further added over the equations, and preferred estimator under both exact and over identification model with the two exogenous variables were bolded. Table 6 gives the summary of the findings as follows:
From Table 6, the following are observed about the preferred estimators under the variance criterion.
The preferred estimators differ in term of identification status. The preferred estimators are slightly affected by the two exogenous variables.
In the exact identified equation, the OLS estimator is generally preferred in all the sample sizes for both exogenous variables. In the over identified equation with normally distributed exogenous variables, OLS or 2SLS estimator are preferred when; but for, the OLS or FIML estimator are preferred. With uniformly distributed exogenous variables the OLS or 2SLS estimators are generally preferred. Moreover, the 3SLS estimator replaces the 2SLS when the sample sizes is very large,.
4.4. Performances of the Estimators Based on Mean Squared Error Criterion
The performances of the estimators was examined, ranked and summed over all the parameters in each equation, having examined and summed the rank of the mean squared error of each parameter over all the parameters in each equation the outcome is given in Table 8.
From Table 8 the following are observed about the preferred estimators under the mean squared error criterion.
The preferred estimators differ in term of identification status. The preferred estimators are slightly affected by the two exogenous variables. In the exact identified equation with normally distributed exogenous variables, the OLS estimator is generally preferred except when the sample size is large. At these instances 2SLS is generally preferred. With uniformly distributed exogenous variables, the OLS estimator is generally preferred. In the over identified equation with normally distributed exogenous variables, the OLS estimators is preferred when; 2SLS when, and FIML when. With uniformly distributed exogenous variables the OLS estimators is preferred except when the sample size is large,. At these instances, the 2SLS estimator is generally preferred. The performance of the preferred estimators is more stable with uniformly distributed exogenous variables than the normally distributed exogenous variables.
4.5. Performances of the Estimators Based on All the Criteria
From Table 10, the following are observed about the preferred estimators under the overall criteria.
The preferred estimators differ in term of identification status. The preferred estimators are slightly affected by the two exogenous variables.
In the exact identified equation with normally distributed exogenous variables the OLS estimator is generally preferred except when the sample size is large when. At these instances, 2SLS is preferred. With uniformly distributed exogenous variable, the OLS estimator is preferred for all sample sizes.
In the over identified equation with normally exogenous variables the OLS estimators is preferred when the samples sizes is small; 2SLS when, and; 3SLS when and FIML estimator with.
With uniformly distributed exogenous variable, the 2SLS estimator is preferred except when the sample sizes is large. At this instance the 3SLS estimator is preferred.
From Table 1, at each level of sample size the sum rank were further added over the equations, and preferred estimator under both exact and over identification model with the two exogenous variables were bolded. Table 2 gives the summary of the preferred estimators.
From Table 3, at each level of sample size, the sum rank were further added over the equations, and preferred estimator under both exact and over identification model with the two exogenous variables were bolded. Table 4 gives the summary of the preferred estimators.
From Table 5, at each level of sample size the sum rank were further added over the equations. The preferred
Summary of the preferred estimators based on bias criterion
Sample sizes
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
10
OLS\2SLS
FIML
OLS
2SLS
20
2SLS
FIML
OLS
2SLS\3SLS
30
OLS
2SLS
OLS
2SLS
50
2SLS
2SLS
2SLS
FIML
100
2SLS
2SLS\3SLS
2SLS
2SLS
250
3SLS
2SLS\3SLS
OLS
FIML
500
3SLS
FIML
2SLS
2SLS\FIML
Source: Table 1.
Performances of the estimator based on absolute bias criterion
N
Estimator
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
10
OLS
4
4
8
3
3
6
4
4
8
4
3
7
2SLS
8
8
16
6
6
12
8
8
16
5
6
11
3SLS
12
13
25
9
9
18
12
14
26
9
9
18
FIML
16
15
31
12
12
24
16
14
30
12
12
24
20
OLS
4
4
8
10
12
22
4
4
8
5
6
11
2SLS
8
8
16
3
3
6
8
8
16
4
4
8
3SLS
12
12
24
6
6
12
12
12
24
9
8
17
FIML
16
16
32
11
9
20
16
16
32
12
12
24
30
OLS
4
4
8
12
12
24
4
4
8
9
5
14
2SLS
8
8
16
3
3
6
8
8
16
3
4
7
3SLS
12
13
25
9
6
15
12
12
24
6
9
15
FIML
16
15
31
6
9
15
16
16
32
12
12
24
50
OLS
5
4
9
12
12
24
4
4
8
6
9
15
2SLS
7
8
15
6
3
9
8
8
16
3
4
7
3SLS
12
12
24
3
6
9
12
13
25
9
7
16
FIML
16
16
32
9
9
18
16
15
31
12
10
22
100
OLS
16
10
26
12
12
24
4
4
8
10
11
21
2SLS
4
4
8
6
3
12
8
8
16
5
3
8
3SLS
8
12
20
6
9
15
12
13
25
4
7
11
FIML
12
14
26
3
6
9
16
15
31
11
9
20
250
OLS
16
16
32
12
12
24
4
4
8
12
12
24
2SLS
4
6
10
9
5
14
8
8
16
3
3
6
3SLS
8
8
16
6
9
15
12
13
25
6
8
14
FIML
12
10
22
3
4
7
16
15
31
9
7
16
500
OLS
16
16
32
12
12
24
10
5
15
12
12
24
2SLS
4
6
10
9
5.5
14.5
6
7
13
9
3
12
3SLS
8
8.5
16.5
3
8
14
10
14
24
5
8
13
FIML
12
9.5
21.5
3
4.5
7.5
14
14
28
4
7
11
Source:Computed from simulated results of Absolute bias criterion.
Summary of the preferred estimators based on absolute bias criterion
Sample sizes
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
10
OLS
OLS
OLS
2SLS
20
OLS
2SLS
OLS
2SLS
30
OLS
2SLS
OLS
2SLS
50
OLS
2SLS\3SLS
OLS
2SLS
100
2SLS
FIML
OLS
2SLS
250
2SLS
FIML
OLS
2SLS
500
2SLS
FIML
2SLS\OLS
FIML\2SLS\3SLS
Source: Table 3.
Performances of the estimators based on variance criterion
n
Estimator
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
10
OLS
4
4
8
3
9
12
4
4
8
3
8
11
2SLS
8
8
16
6
8
14
8
8
16
6
5
11
3SLS
13
15
28
9
7
16
12
15
27
9
10
19
FIML
15
13
28
12
6
18
16
13
29
12
7
19
20
OLS
4
4
8
3
9
12
4
4
8
3
7
10
2SLS
8
8
16
6
8
14
8
8
16
6
4
10
3SLS
12
12
24
9
5
14
12
12
24
9
7
16
FIML
16
16
32
12
8
20
16
16
32
12
12
24
30
OLS
4
4
8
3
9
12
4
4
8
3
8
11
2SLS
8
8
16
6
4
10
8
8
16
6
4
10
3SLS
12
13
25
9
7
16
12
12
24
9
7
16
FIML
16
15
31
12
10
22
16
16
32
12
11
23
50
OLS
4
4
8
3
9
12
4
4
8
3
9
12
2SLS
8
8
16
9
5
14
8
8
16
6
4
10
3SLS
13
12
25
6
6
12
12
13
25
9
9
18
FIML
15
16
31
12
10
22
16
15
31
12
8
20
100
OLS
4
4
8
3
9
12
4
4
8
3
9
12
2SLS
14
8
22
12
4
16
8
8
16
8
4
12
3SLS
8
13
21
9
8
17
12
13
25
7
9
16
FIML
14
15
29
6
9
15
16
15
31
12
8
20
250
OLS
44
4
8
3
9
12
4
4
8
3
9
12
2SLS
8
12
20
12
5
17
8
8
16
6
7
13
3SLS
12
10.5
22.5
9
8
17
12
14
26
9
8
17
FIML
16
13.5
29.5
6
8
14
16
14
30
12
6
18
500
OLS
4
4
8
3
9
12
4
4
8
3
9
12
2SLS
8
10
18
12
5
17
8
8
16
12
5
17
3SLS
12
12
24
9
10
19
13
13.5
26.5
6
6
12
FIML
16
14
30
6
6
12
15
14.5
29.5
9
10
19
Source:Computed from simulated results of variance criterion.
estimators under both exact and over identification model with the two exogenous variables were made bold. Table 6 gives the summary of the preferred estimators.
From Table 7, at each level of sample size the sum rank were further added over the equations and preferred
Summary of the preferred estimators under variance criterion
Sample sizes
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
10
OLS
OLS\2SLS
OLS
OLS\2SLS
20
OLS
OLS\2SLS\3SLS
OLS
OLS\2SLS
30
OLS
2SLS\OLS
OLS
2SLS\OLS
50
OLS
OLS\2SLS\3SLS
OLS
2SLS\OLS
100
OLS
OLS\FIML
OLS
OLS\2SLS
250
OLS
OLS\FIML
OLS
OLS\2SLS
500
OLS
OLS\FIML
OLS
OLS\2SLS
Source:Table 5
Performances of the estimators based on mean squared error criterion
n
Estimator
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
Equation (1)
Equation (3)
Tot
Equation (2)
Equation (4)
Tot
10
OLS
4
4
8
3
3
6
4
4
8
3
3
6
2SLS
8
8
16
6
6
12
8
8
16
6
6
12
3SLS
13
15
28
9
9
18
12
15
27
9
9
18
FIML
15
13
28
12
12
24
16
13
29
12
12
24
20
OLS
4
4
8
3
3
6
4
4
8
3
3
6
2SLS
8
8
16
6
6
12
8
8
16
6
6
12
3SLS
12
12
24
9
9
18
12
12
24
9
9
18
FIML
16
16
32
12
12
24
16
16
32
12
12
24
30
OLS
4
4
8
3
12
15
4
4
8
3
2
6
2SLS
8
8
16
3
3
9
8
8
16
6
6
12
3SLS
12
13
25
9
6
15
12
12
24
9
9
18
FIML
16
15
31
12
9
21
16
16
32
12
12
24
50
OLS
4
4
8
6
12
18
4
4
8
3
3
6
2SLS
8
8
16
9
3
12
8
8
16
6
6
12
3SLS
13
12
25
3
6
9
12
13
25
9
10
19
FIML
15
16
31
12
9
21
16
15
31
12
11
23
100
OLS
13
4
17
12
12
24
4
4
8
3
4
7
2SLS
11
8
19
9
3
12
8
8
16
8
5
13
3SLS
5
13
18
6
9
15
12
13
25
7
9
16
FIML
11
15
26
3
6
9
16
15
31
12
12
24
250
OLS
16
16
32
12
12
24
4
4
8
3
12
15
2SLS
4
8
12
9
4.5
13.5
8
8
16
6
3
9
3SLS
8
6.5
14.5
6
9
15
12
14
26
9
7
16
FIML
12
9.5
21.5
3
4.5
7.5
16
14
30
12
8
20
500
OLS
16
16
32
12
12
24
4
4
8
12
12
24
2SLS
4
6
10
9
3
12
8
8
16
9
3
12
3SLS
8
7.5
15.5
6
9
15
13
14.5
27.5
3
7
10
FIML
12
10.5
22.5
3
6
9
15
13.5
28.5
6
8
14
Source: Computed from simulated results of Mean squared error criterion
estimator under both exact and over identification model with the two exogenous variables were bolded. Table 8 gives the summary of the preferred estimators
4.6. Overall Examination of the Model Parameters
The overall summary of the performances of the estimators having examined and obtained by adding the total ranks over all the criteria is given in Table 9
From Table 9, at each level of sample size the sum of the rank were further added over all the criteria and preferred estimator under both exact and over identification model with the two exogenous variables were bolded. Table 10 gives the summary of the preferred estimators
Summary of the preferred estimators when there is no correlation under mean squared error criterion
Sample sizes
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
10
OLS
OLS
OLS
OLS
20
OLS
OLS
OLS
OLS
30
OLS
2SLS
OLS
OLS
50
OLS
3SLS\2SLS
OLS
OLS
100
OLS\2SLS\3SLS
2SLS\FIML
OLS
OLS
250
2SLS\3SLS
FIML
OLS
2SLS
500
2SLS
FIML\2SLS
OLS
3SLS\2SLS
Source: Table 7
Overall performances of the estimators based on all criteria
N
Estimator
Normal
Uniform
Exact
Over
Exact
Over
10
OLS
36
45
33
43
2SLS
60
53
63
43
3SLS
107
70
107
68
FIML
117
72
117
86
20
OLS
40
64
33
45
2SLS
57
47
63
39
3SLS
96
56
98
60
FIML
127
73
126
96
30
OLS
33
75
34
54
2SLS
63
34
62
35
3SLS
104
61
97
62
FIML
120
70
127
89
50
OLS
40
78
39
57
2SLS
56
46
58
41
3SLS
98
43
100
68
FIML
126
73
123
74
100
OLS
83
84
38
64
2SLS
60
49
58
42
3SLS
74
56
101
60
FIML
103
51
123
74
250
OLS
104
84
34
75
2SLS
59
53.5
62
44
3SLS
64
56
104
61
FIML
93
46.5
120
60
500
OLS
104
84
55
84
2SLS
55
54.5
61
55
3SLS
68
66
100
46
FIML
93
35.5
104
55
Source: Computed from Table 1, Table 3, Table 5 and Table 7
Overall summary of the best estimators on the basis of all criteria
Sample sizes
Normal Exogenous Variables
Uniform Exogenous Variables
Exact
Over
Exact
Over
10
OLS
OLS
OLS
2SLS\OLS
20
OLS
2SLS
OLS
2SLS
30
OLS
2SLS
OLS
2SLS
50
OLS
3SLS
OLS
2SLS
100
2SLS
2SLS
OLS
2SLS
250
2SLS
FIML
OLS
2SLS
500
2SLS
FIML
OLS
3SLS
Source: Table 4 and Table 95. Conclusions
The performances of the estimators are affected by the distribution of the exogenous variables. The best estimators are more stable over the levels of sample size with uniformly distributed exogenous variables than the normally distributed exogenous variables
In exact identified equation with normally distributed exogenous variables the following were observed
For low sample sizes, OLS is the best performed estimator. In medium sample sizes, OLS or 2SLS is the best performed estimator and 2SLS estimator is the best performed in the large sample sizes. Whereas with uniformly distributed exogenous variables, OLS is the best performed estimator in the entire sample sizes category
In over identified equation with normally distributed exogenous variables, the following are observed
For low sample sizes, OLS or 2SLS is the best performed estimator, in medium sample sizes 2SLS\3SLS is the best performed estimator and FIML estimator performed best in the large sample sizes. Whereas with uniformly distributed exogenous variables, 2SLS is the best in low and medium sample sizes category, but 2SLS/ 3SLS estimator is the best in large sample sizes category
Hence, when there is no correlation of any form in the model the performances of the estimators are affected by the distribution of the exogenous variables in simultaneous equation models
ReferencesSchmidt, J.S. (2005) Econometrics. McGraw-Hill International Edition.Koutsoyainnis, A. (1977) Theory of Econometrics. 2nd Edition, Palgrave Publishers Ltd., New York.Odunta, E.A. (2004) A Monte Carlo Study of the Problem of Multicollinearity in a Simultaneous Equation Model. Unpulished Ph.D. Thesis, Department of Statistics University of Ibadan, Ibadan.Kmenta, J. and Gilberet, R.F. (1967) Small Sample Properties of Alternative Estimators of Seemingly Unrelated Regression. Journal of the American Statistical Association, 63, 1180-1200.
http://dx.doi.org/10.1080/01621459.1968.10480919Chatterjee, S. and Ali, S.H. (2012) Regression Analysis by Example. 5th Edition, John Wiley and Sons, Hoboken.Johnson, T.L., Ayinde, K. and Oyejola, B.A. (2010) Effect of Correlations and Equation Identification Status on Estimators of a System of Simultaneous Equation Model. Electronic Journal of Applied Statistical Analysis EJASA, 3, 115-125. http://siba-ese.unile.it/index.php/ejasa/indexAyinde, K., Johnson, T.L. and Oyejola, B.A. (2011) Effect of Equation Identification Status and Correlation between Error Terms on Estimators of System of a Simultaneous Equation Model. American Journal of Scientific and Industrial Research, 2, 184-190. http://dx.doi.org/10.5251/ajsir.2011.2.2.184.190Ayinde K. ,et al. (2007)Equation to Generate Normal Variates with Desired Intercorrelations Matrix International Journal of Statistics and System 2, 99-111.Schumann, E. (2009) Generating Correlated Uniform Variates. http://comisef.wikidot.com/tutorial:correlated
. At these instances, 2SLS estimator is best. In over identified equations, the 2SLS estimator is best except with normally distributed exogenous variables when the sample size is small and large,
and
; and with uniformly distributed exogenous variables when
n is very large,
, the best estimator is either OLS or FIML or 3SLS.