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# Sensitivity of estimators to three levels of correlation between error terms.

1.0 Introduction

Beginning with the method developed by [15] for solving the problem of single equation bias, econometricians have devoted considerable effort to developing additional methods for estimating the structural parameters of simultaneous equation models [28], [16], [25] and [27]. While it has been fairly easy to develop the asymptotic properties of these estimators, a distinguishing characteristic Noun 1. distinguishing characteristic - an odd or unusual characteristic
distinctive feature, peculiarity

characteristic, feature - a prominent attribute or aspect of something; "the map showed roads and other features"; "generosity is one of his best
of econometric e·con·o·met·rics
n. (used with a sing. verb)
Application of mathematical and statistical techniques to economics in the study of problems, the analysis of data, and the development and testing of theories and models.
models is that they are invariably in·var·i·a·ble
Not changing or subject to change; constant.

in·vari·a·bil
based upon small samples of data and thus, the asymptotic properties of the various estimators are not necessarily the best guide in selecting the appropriate estimating procedure. One approach to this problem has been the derivation derivation, in grammar: see inflection.  of the exact finite-sample properties of some estimators by [30], [31], [32]) and [19]. Relatively little is known about the finite sample distributions of the various estimators. The exact finite sample distributions of limited-information maximum likelihood estimates and two-stage least squares estimates have been derived by Basmann in certain special cases ([30], [32]). He found that these distributions do not always possess finite moments of low order; in certain cases even the mean does not exist. An alternative approach to uncovering the small sample properties of various structural equation estimators has been to conduct sampling experiments with the aid of more or less artificial models. The most notable among these have been studies by [11], [19] and [29]. Several small models are examined in these studies from various points of view; the general conclusions emerging from them are excellently summarized by [9].

Another approach, which is generally applicable to all estimators, has been to conduct sampling experiments with different simultaneous equation models using small samples of data which have been artificially generated [34], [30], [6], [33], [20], [23], [10] and [11]. More recent work has been done by ([1]; [2]; [4]), [5], [8], and [7]. The net result of all these studies has been to show that there exist no clear guidelines for the choice of an estimator for econometric models. The general consensus of opinion, however, is that, thus far, two-stage least squares is the cheapest, easiest, and most efficient estimator in most situations [24]. A different approach to the simultaneous equation bias problem is the full information maximum likelihood (FIML FIML Full Information Maximum Likelihood
FIML Football Is My Life (fantasy football league)
) estimation method [3].

It has been shown by [13] that the full-information maximum likelihood method of estimating the coefficients of structural equations is a generalization of the least squares principles. These estimates are consistent and efficient. Nevertheless, the properties of other types of estimator continue to be of interest because of the computational difficulty of obtaining full-information estimates ([12]; [14]). Noteworthy among alternative methods are limited-information maximum likelihood, indirect least squares, two-stage least squares, direct least squares (the last two being special cases of the general k-class of estimators), three-stage least squares, linearized and several others ([17]; [18]; [21]; [26]). With the exception of direct least squares these methods also possess the properties of consistency although they yield biased estimates in finite samples [22].

Compared to the instrumental variables methods (2SLS (Selective Laser Sintering) See laser sintering and 3D printing.  and 3SLS), the FIML method has these advantages and disadvantages:

(1) FIML does not require instrumental variables.

(2) FIML requires that the model include the full equation system, with as many equations as there are endogenous endogenous /en·dog·e·nous/ (en-doj´e-nus) produced within or caused by factors within the organism.

en·dog·e·nous
1. Originating or produced within an organism, tissue, or cell.
variables. With 2SLS or 3SLS you can estimate some of the equations without specifying the complete system.

(3) FIML assumes that the equations errors have a multivariate normal distribution. If the errors are not normally distributed, the FIML method may produce poor results. 2SLS and 3SLS do not assume a specific distribution for the errors.

(4) The FIML method is computationally expensive A computationally expensive algorithm is one that, for a given input size, requires a relatively large number of steps to complete; in other words, one with high computational complexity. .

The random deviates on which the selection of error terms in Monte Carlo Monte Carlo (môNtā` kärlō`), town (1982 pop. 13,150), principality of Monaco, on the Mediterranean Sea and the French Riviera.  studies is based are usually assumed to be pair wise uncorrelated. This is not always true although the correlation coefficients are usually small. Since random deviates will loose the quality of randomness if they are forced to be orthogonal, the objective of this paper is focused on investigating the sensitivity of estimators of a two-equation model in the presence of three levels of unintended correlation between pairs of normal deviates used in the Monte Carlo experiment.

2.0 The Model

Numerous methods have been developed for estimating the coefficients of a system of simultaneous linear structural equation of the form

By + [GAMMA]z = u (2.0)

It is assumed that z is a vector of exogenous Exogenous

Describes facts outside the control of the firm. Converse of endogenous.
variables (assumed to be identical in repeated samples and not to contain lagged values of endogenous variables), u is a vector of jointly normally distributed error terms with mean zero and covariance matrix In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.  [SIGMA], y is a vector of endogenous variables, and B (nonsingular) and r are matrices of coefficients.

Assume the following two-equation model

[Y.sub.1t] = [[beta].sub.12][Y.sub.2t] + [[gamma].sub.11] [X.sub.1t] + [U.sub.1t]

[Y.sub.2t] = [[beta].sub.2t][Y.sub.1t] + [[gamma].sub.22] [X.sub.2t] + [[gamma].sub.23] [X.sub.3t] + [U.sub.2t] (2.1)

where the Y's are the endogenous variables, X's are the predetermined variables Predetermined variables are variables that were determined prior to the current period. In econometric models this implies that the current period error term is uncorrelated with current and lagged values of the predetermined variable but may be correlated with future values.  and U's are the random disturbance terms, [beta]'s and [gamma]'s are the parameters. The first equation is over-identified while the second equation is a just identified equation. The error terms were not independent. ([2]; [4])

The reduced form In social science and statistics, particularlly econometrics, a reduced form equation is a method of dealing with endogeneity. A reduced form equation is defined by James Stock & Mark Watson (2007) in the following way:  equation of the above equation (2.1) is given as

By = [GAMMA]x + u y = [B.sup.-1][GAMMA]x + [B.sup.-1]u = [PI]x + v (2.2)

Where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]

But,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

The reduced form of equations (2.3) and (2.4) are

[y.sub.1t] = [[PI].sub.11] [X.sub.1t] + [[PI].sub.12] [X.sub.2t] + [[PI].sub.13] [X.sub.3t] + [V.sub.1t] (2.5)

[y.sub.2t] = [[PI].sub.21] [X.sub.1t] + [[PI].sub.22] [X.sub.2t] + [[PI].sub.23] [X.sub.3t] + [V.sub.2t] (2.6)

3.0 Design of Experiments

Three arbitrary levels of correlation between pairs of random deviates are assumed. These three scenarios of correlation are then used to generate pairs of normal deviates of sizes N = 15, 25 and 40 with 100 replications. Each set of normal deviates with the different sample sizes are then transformed using the upper ([P.sub.1]) triangular matrix In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis. . The procedure was repeated using the lower triangular matrix ([P'.sub.1]), such that in each case, [OMEGA] = [P.sub.1][P'.sub.1]

To generate the data, the structural equations (2.1) were transformed to the reduced form, error terms for sample sizes of fifteen, twenty-five and forty were produced by a random normal deviate Normal deviate

Related: Standardized value
generator and values for the endogenous variables were calculated. For each sample size, hundred sets of data were generated, with the vectors of exogenous variables remaining the same for each set of data. Five estimators are used in this experiment; they are Ordinary Least Squares (OLS OLS Ordinary Least Squares
OLS Online Library System
OLS Ottawa Linux Symposium
OLS Operation Lifeline Sudan
OLS Operational Linescan System
OLS Online Service
OLS On Line Support
OLS Online System
), Two Stage Least Squares (2SLS), Limited Information Maximum Likelihood (LIML LIML Limited Information Maximum Likelihood ), Three Stage Least Squares (3SLS) and Full Information Maximum Likelihood (FIML).

In assessing the performance for the various estimators, an examination of the means and standard deviations of the estimates of structural parameters was made and from this some summary statistics were prepared. These permitted evaluations on the basis of two criteria, smallest bias and smallest standard deviation. A combined or scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g.  measure of these two criteria could be Root Mean Square Error (MSE MSE Mouse (computer)
MSE Materials Science & Engineering
MSE Mean Squared Error
MSE Mean Square Error
MSE Master of Science in Engineering
MSE Manufacturing Systems Engineering
MSE Mechanically Stabilized Earth
) or Mean Absolute Error (MAE (1) (Metropolitan Area Exchange) Originally known as Metropolitan Area Ethernets, MAEs are junction points on the Internet where data is exchanged between carriers. See IXP and NAP. ). One investigator has stated that on a priori a priori

In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience.
grounds it is hard to choose between these measures [[10], p. 12]; therefore, a summary statistics using two measures; total absolute bias and sum of squared residuals are included for this study.

4.0 Simulation Result

Tables 1 and 2 contain summaries of the performance of estimators using total absolute bias (TAB) of estimates. To reduce the dimension of the results displayed in tables 1 and 2, the total absolute biases are summed across correlation levels for each estimator; this will facilitate a study of the asymptotic behavior of TAB for each estimator, computation of the average bias for each estimator and its dispersion over sample sizes, all of which will also help in ranking the estimators under [P.sub.1] and [P.sub.2] in increasing order of average of total absolute bias. Tables 1 and 2 are used to generate table 3.

The entries in the rows of table 3 for [P.sub.1] show that the sums of total absolute bias decrease as the sample size increases for OLS, 2SLS, LIML and 3SLS, the sums do not reveal any such asymptotic behavior for FIML where the sample size 25 appears to be a turning point (maximum bias for FIML). For [P.sub.2] the row entries reveal asymptotic behavior for 2SLS, LIML and FIML while 3SLS has sample size 25 as a convex Convex

Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds.
turning point and the sums increase as the sample size increases for OLS. This result shows that estimates of absolute bias are sensitive to changes in the sample sizes.

It is also of interest to rank the estimators on the basis of the magnitude of total absolute bias and to examine the dispersion of the estimates using the coefficient of variation Coefficient of Variation

A measure of investment risk that defines risk as the standard deviation per unit of expected return.
.

These averages and the coefficients of variation of the 3 estimates for each estimator are displayed in table 4 for [P.sub.1] and [P.sub.2].

Using the Average Total Absolute Bias (ATAB) and its Coefficient of Variation (CV) presented in table 4, the five estimators are ranked as shown in table 5 in increasing order of bias and coefficient of variation under [P.sub.1] and [P.sub.2].

It is noteworthy in respect of average absolute bias that the five estimators rank uniformly under [P.sub.1] and [P.sub.2]. This finding clearly shows that the ranking of the estimators in terms of the magnitude of the average total absolute bias is invariant (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the invariant.  to the choice of the upper ([P.sup.1]) or lower ([P.sub.2]) triangular matrix.

It is also remarkable that whereas the average absolute biases of the other four estimators range between 9 and 15, those of FIML maintain a very distant fifth position with 40 and about 50 for [P.sub.1] and [P.sub.2] respectively. The poor ranking of FIML in this situation of correlated disturbances and over-identified equation may be attributed to the fact that it uses more information as an estimator than any of the other four estimators. The only remarkable uniformity in the ranking of estimators on the dispersion of the total absolute bias is the fact that the 3SLS and FIML are in the fourth and fifth positions respectively under [P.sub.1] and [P.sub.2].

Finally, a decision on the best estimator for this model cannot be taken on the basis of our findings on total absolute bias alone. This is because the yardstick is the total absolute bias of two equations, which differ in their identifiability status. In estimating multi-equation models, the choice of estimator is equation specific. Hence, the findings here will have to be reconciled with findings elsewhere before a prescription of best estimator of each equation can be suggested.

To further study the asymptotic behavior as well as the sensitivity of each estimator to changes in TAB of estimates over replication, tables 1 and 2 are used to chart the behavior of estimators over correlation coefficients and sample sizes and these are presented in table 6 for both [P.sub.1] and [P.sub.2] respectively.

The entries show that under [P.sub.1], for OLS, the model absolute bias decreased consistently as correlation changes over the three ranges rose consistently for 3SLS and attained a minimum (V) as correlation changes from high negative value through low negative or positive values to high positive values for 2SLS and LIML. The behavior is inconclusive FIML. Under [P.sub.2], the findings are generally less conclusive, however, model absolute bias is downward sloping for OLS (similar to behavior in [P.sub.1]) and has a convex behavior with the turning point at the middle interval for 2SLS, LIML and 3SLS at N = 25 and N = 40 respectively.

Theoretically, one expects the "V" trend to be the most frequent since that would imply that total absolute bias is a minimum when correlation of the error term is smallest (negative or positive). This is reflected to a large extent by estimates of 2SLS and LIML based on [P.sub.1] and 2SLS, LIML, and 3SLS based on [P.sub.2].

The sum of squared residuals of each equation for all the five estimators are displayed in tables 7 and 9. These tables are arranged to facilitate the study of the asymptotic distribution In mathematics and statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables

Zi

for i
of the sum of squared residuals. They reveal changes in the estimates of RSS (Really Simple Syndication) A syndication format that was developed by Netscape in 1999 and became very popular for aggregating updates to blogs and the news sites. RSS has also stood for "Rich Site Summary" and "RDF Site Summary.  as N increases at different levels of error correlation.

For OLS, LIML and FIML, the RSS obtained in equation two, the just identified equation, are smaller at all levels of error correlation than those obtained in equation one, the over-identified equation. For 2SLS and 3SLS, the estimates obtained in equation one are smaller than those obtained in two.

An overview of these tables reveals that RSS for OLS follow a consistent pattern column-wise, i.e. for the two equations and at all levels of correlation coefficient Correlation Coefficient

A measure that determines the degree to which two variable's movements are associated.

The correlation coefficient is calculated as:
, RSS increase as sample size increases for both [P.sub.1] and [P.sub.2].

As expected the RSS displayed in these tables (7 and 9) for [P.sub.1] and [P.sub.2] are fairly uniform row-wise for all estimators except the FIML where estimated RSS vary sample sizes. Also the RSS for FIML are remarkably higher than for the other four estimators.

As before, to gain some insight into the behavior of the estimated RSS as correlation of the error term changes from r<-0.05, through -0.05<r<0.05 to r>0.05; the relevant charts are displayed in tables 8 and 10 (using the results of the sum of squared residuals of estimates displayed in tables 7 and 9) for the three sample sizes given 100 replications for [P.sub.1] and [P.sub.2]. For example, in table 7 for EQ1, N = 15 RSS fell from 8.469256 to 7.697955 and fell further to 7. 222594 across the three levels of correlation coefficient, this is represented by the trend "\". At N=25, for the same equation RSS maintained the downward trend "\". This is repeated for each parameter to obtain the different trends shown in tables 8 and 10.

In tables 8 and 10 for the two equations under [P.sub.1], the downward sloping trend is most frequent for OLS, which implies that, the RSS decrease consistently as correlation coefficient changes from highly negative, through feeble to highly positive range. For FIML, identical results are obtained for the two equations and triangular matrices.

It is also worth mentioning that, the trends under 2SLS, LIML and 3SLS are similar for the two equations when both [P.sub.1] and [P.sub.2] are considered.

The most popular chart in respect of the two equations under both [P.sub.1] and [P.sub.2] is 'V' followed by the downward trend "\". These tables also reveal that the results obtained for OLS under [P.sub.1] are similar to those obtained under [P.sub.2].

On the behavior of RSS as correlation coefficient changes through the three cardinal levels, OLS estimator shows the most stable pattern of declining RSS i.e. the downward sloping ("\") trend (6/6 for both equations [P.sub.1] and [P.sub.2]). The 2SLS, LIML and 3SLS estimators also have a concave Concave

Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex.
("V") trend predominantly for sample sizes 15 and 25 for equation 1, 25 and 40 for equation 2 under [P.sub.1]. This pattern is repeated for these estimators under [P.sub.2] except for sample size 40 of the first equation.

The frequencies of the four trends (\, /, [LAMBDA The Greek letter "L," which is used as a symbol for "wavelength." A lambda is a particular frequency of light, and the term is widely used in optical networking. Sending "multiple lambdas" down a fiber is the same as sending "multiple frequencies" or "multiple colors. ] and V) are relatively more uniform under [P.sub.1] than under [P.sub.2]. This suggests that the identifiability status of the two equations affects the behavior of RSS under [P.sub.1] than [P.sub.2] in some respects.

The marginal totals of three tables (6, 8 and 10) of frequencies of four correlation-based charts (\, /, [LAMBDA], V) of behavior of the two attributes are displayed in table 10. These percentages show the frequencies of these charts for both equations.

There is a remarkable uniformity in the column-wise comparison of the entries in table 10 for both criteria of [P.sub.1] where the frequencies are similar for the two charts (\, V). The upward sloping chart (representing increasing values of TAS TAS
abbr.

2. true airspeed
or RSS across the three correlation levels) and the convex chart (representing maximum values of TAS or RSS at the middle interval) are less frequent than the other two charts (\, V) which have relatively high frequencies for both equations in [P.sub.1] and [P.sub.2].

5.0 Conclusion

The sensitivity of the simultaneous equation techniques to violation of mutual independence of random deviates in a two-equation model has been investigated. Based on TAB, it can be concluded that since the 3SLS estimator has the minimum ATAB for both [P.sub.1] and [P.sub.2] is the best followed by LIML which is also closely followed by 2SLS. The OLS is however, on top of the group when comparing the performances of the estimators using coefficient of Variation followed by 2SLS and LIML. To further [micro]examine the sensitivity of each estimator to changes in TAB of estimates over 100 replications, a detailed table presentation of the behavior of estimators over correlation coefficients and sample sizes are charted and presented in table 6 for both [P.sub.1] and [P.sub.2]. The model absolute bias for 2SLS and LIML attained a minimum at the feebly correlated region while OLS performed poorly with an increasing TAB as the correlation changes over the three cardinal points cardinal points
Noun, pl

the four main points of the compass: north, south, east, and west
. The behavior of FIML revealed no reasonable pattern.

Using the RSS, the performances of 2SLS, LIML and 3SLS are similar for both equations and triangular matrices.

Best RSS estimates of 2SLS, LIML, and 3SLS are found in the feebly correlated region which is consistent with the theory. That is, the "V" trend is expected to be the most frequent since that would imply that residual sum of squares In statistics, the residual sum of squares (RSS) is the sum of squares of residuals,

In a standard regression model , where a and b
is a minimum when correlation of the error term is smallest (negative or positive).

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Application of scientific methods to management and administration of military, government, commercial, and industrial systems. It began during World War II in Britain when teams of scientists worked with the Royal Air Force to improve radar detection of
, 11, 747-758.

A.A Adepoju (1) and J. O Iyaniwura (2)

(1) Department of Mathematics, University of Mines and Technology The University of Mines and Technology is located at Tarkwa in the Western Region of Ghana. It is one of the new public universities in the country. The university came into being on 1 October 2001 when the Kwame Nkrumah University of Science and Technology School of Mines was , Tarkwa, Ghana Tarkwa is a town in the southwest of Ghana, located about 120 miles west of Accra. As of 2005, it was estimated to have a population of 40,397. Transport
Tarkwa is a junction on the Ghanaian Railways.

(2) Department of Mathematics and Statistics, Ogun State Ogun State is a state in South-western Nigeria. It borders Lagos State to the south, Oyo and Osun states to the North, Ondo State to the east and the republic of Benin to the west. Abeokuta is the capital and largest city in the state.  University, Ago-Iwoye, Nigeria
```Table 1: Summary of Total Absolute Bias R=100, [P.sub.1]

Level of                     OLS
correlation

N=15       N=25       N=40

r<-0.05        4.967447   4.948403   4.874522
-0.05<r<0.05   4.884578   4.88579    4.733118
r>0.05         4.84921    4.828668   4.423479

Level of                  2SLS
correlation

N=15       N=25       N=40

r<-0.05        4.902149   3.897816   3.881116
-0.05<r<0.05   4.635532   3.492337   3.616084
r>0.05         5.10576    4.186388   3.698753

Level of                    LIML
correlation

N=15       N=25       N=40

r<-0.05        4.384517   4.600761   3.574813
-0.05<r<0.05   3.393374   3.043991   2.933429
r>0.05         4.947764   3.40555    3.146334

Level of                     3SLS
correlation

N=15       N=25       N=40

r<-0.05        3.996025   2.280115   2.760661
-0.05<r<0.05   4.027558   2.899392   2.803212
r>0.05         4.996303   4.00182    3.257218

Level of                     FIML
correlation

N=15        N=25        N=40

r<-0.05        11.514893   23.234947   9.408441
-0.05<r<0.05   12.582233   16.593795   12.561232
r>0.05         14.484919   11.052439   9.298833

Table 2: Summary of Total Absolute Bias R=100, [P.sub.2]

Level of                    OLS
correlation

N=15       N=25       N=40

r<-0.05        4.888746   4.890336   5.044038
-0.05<r<0.05   4.85784    4.865581   5.015919
r>0.05         4.851891   4.877528   4.933268

Level of                     2SLS
correlation

N=15       N=25       N=40

r<-0.05        4.096107   4.339463   4.412196
-0.05<r<0.05   4.715642   3.671604   3.555401
r>0.05         3.947009   4.117722   3.6645

Level of                    LIML
correlation

N=15       N=25       N=40

r<-0.05        3.785076   4.293223   4.867604
-0.05<r<0.05   5.078825   3.403852   2.982053
r>0.05         4.066545   3.736673   3.103178

Level of                    3SLS
correlation

N=15       N=25       N=40

r<-0.05        2.761579   3.095991   4.088159
-0.05<r<0.05   3.142008   1.725094   2.910107
r>0.05         4.554659   3.070647   3.73892

Level of                     FIML
correlation

N=15        N=25        N=40

r<-0.05        15.690135   18.081122   11.662543
-0.05<r<0.05   21.060479   27.288149   11.39745
r>0.05         23.081417   11.108718   9.666032

Table 3: Sums of Total Absolute Bias over Correlation Levels,
Replication Numbers or Sample Sizes.

Triangular   REPLICATIONS        OLS               2SLS
Matrix

SAMPLE SIZES

15    25    40    15    25    40

[P.sub.1]        100        14.   14.   14.   14.   11.   11.
70    66    03    64    58    19

[P.sub.2]        100        14.   14.   14.   12.   12.   11.
60    63    99    76    13    63

Triangular   REPLICATIONS        LIML              3SLS
Matrix

SAMPLE SIZES

15    25    40    15    25    40

[P.sub.1]        100        12.   11.   9.6   13.   9.1   8.8
72    05     5    02     8     2

[P.sub.2]        100        12.   11.   10.   10.   7.8   10.
93    43    95    46     9    74

Triangular   REPLICATIONS        FIML
Matrix

SAMPLE SIZES

15    25    40

[P.sub.1]        100        38.   50.   31.
58    88    27

[P.sub.2]        100        59.   56.   32.
83    48    73

Table 4: Average Total Absolute Bias and their Coefficient
of Variation ([P.sub.1] and [P.sub.2]).

Triangular           OLS      2SLS     LIML     3SLS     FIML
Matrix

[P.sub.1]    Mean   14.46    12.47    11.14    10.34    40.24
C.V    0.0260   0.1515   0.1380   0.2251   0.2463

[P.sub.2]    Mean   14.74    12.17    11.77     9.70    49.68
C.V    0.0147   0.0465   0.0878   0.1620   0.2974

Table 5: Ranking of Estimators under [P.sub.1] and [P.sub.2] on
ATAB and CV.

ATAB                     CV

[P.sub.1]   [P.sub.2]   [P.sub.1]   [P.sub.2]

3SLS          3SLS         OLS         OLS
LIML          LIML        2SLS        2SLS
2SLS          2SLS        LIML        LIML
OLS            OLS        3SLS        3SLS
FIML          FIML        FIML        FIML

Table 6: Trends of Total Absolute Bias as Error Correlation changes
from 'High Negative through Small (negative and positive) to High
Positive Values, R = 100.

Estimator        [P.sub.1]                [P.sub.2]

Sample size (N)          Sample size (N)

15      25      40      15         25      40

OLS         \       \       \       \          \       2SLS        V       V       V    [LAMBDA]      V       V
LIML        V       V       V    [LAMBDA]      V       V
3SLS        /       /       /       /          V       V
FIML        /    [LAMBDA]   \       /       [LAMBDA]
Table 7: Summary of Sum of Squared Residuals for Three Correlation
Levels R=100, [P.sub.1].

Estimator        Level of                    EQ1
correlation

N = 15     N = 25     N = 40

OLS             r < -0.05       8.469256   14.53042   23.15642
-0.05 < r < 0.05   7.697955   13.86687   21.90421
r > 0.05       7.222594   11.99807   18.88565

2SLS            r < -0.05       52.04317   90.07682   109.1199
-0.05 < r < 0.05   26.93059   81.35832   140.1711
r > 0.05       37.66998   85.28595   99.51579

LIML            r < -0.05       184.874    1105.567   458.6458
-0.05 < r < 0.05   134.3037   483.5175   758.5876
r > 0.05       1121.227   1104.505   542.7323

3SLS            r < -0.05       52.04317   90.07682   109.1199
-0.05 < r < 0.05   26.93059   81.35832   140.1711
r > 0.05       37.66998   85.28595   99.51579
FIML            r < -0.05       1399.557   17482.48   1659.51
-0.05 < r < 0.05   3899.056   11258.74   8061.006
r > 0.05       5747.848   3371.886   2351.284

Estimator        Level of                    EQ2
correlation

N = 15     N = 25     N = 40

OLS             r < -0.05       5.633654   9.655884   16.66429
-0.05 < r < 0.05   5.196804   9.082502   15.82505
r > 0.05       4.765228   7.69349    14.32214

2SLS            r < -0.05       45.97432   115.9413   201.2111
-0.05 < r < 0.05   51.94936   76.6478    115.6186
r > 0.05       73.40602   192.3063   151.4921

LIML            r < -0.05       45.97432   115.9413   201.2111
-0.05 < r < 0.05   51.94936   76.6478    115.6186
r > 0.05       73.40602   192.3063   151.4921

3SLS            r < -0.05       83.76712   909.1305   600.1468
-0.05 < r < 0.05   276.3245   106.6174   248.7424
r > 0.05       485.2463   458.2371   2334.198

FIML            r < -0.05       882.1201   11472.43   841.4681
-0.05 < r < 0.05   3494.945   7905.957   5532.736
r > 0.05       4886.049   2625.551   1729.908

Table 8: Charts of the Behavior of RSS of Estimators over Correlation
Coefficients for each Sample Size R = 100; [P.sub.1].

Estimator         EQ1                  EQ2

Sample size          Sample size

15   25      40      15   25      40

OLS         \    \       \       \    \       2SLS        V    V    [LAMBDA]   /    V       V
LIML        V    V    [LAMBDA]   /    V       V
3SLS        V    V    [LAMBDA]   /    V       V
FIML        /    \    [LAMBDA]   /    \    [LAMBDA]

Table 9: Summary of Sum of Squared Residuals for Three
Correlation Levels R = 100, [P.sub.2].

Estimator   Level of correlation                 EQ1

N = 15     N = 25     N = 40

OLS              r < -0.05         8.025363   14.39986   22.30994
-0.05 < r < 0.05     7.672212   13.98346   22.12155
r > 0.05         7.518787   11.88107   19.20711

2SLS             r < -0.05         61.16581   69.22513   268.809
-0.05 < r < 0.05     36.15682   84.48975   108.098
r > 0.05         94.27346   108.824    112.8809

LIML             r < -0.05         247.6202   349.9958   10355.51
-0.05 < r < 0.05     1165.518   417.5886   361.349
r > 0.05         2263.259   604.6652   436.645

3SLS             r < -0.05         61.16581   69.22513   268.809
-0.05 < r < 0.05     36.15682   84.48975   108.098
r > 0.05         94.27346   108.824    112.8809

FIML             r < -0.05         3810.179   13611.22   5834.852
-0.05 < r < 0.05     16084.06   37128.56   4677.572
r > 0.05         20454.36   2092.687   3030.157

Estimator   Level of correlation                 EQ2

N = 15     N = 25     N = 40

OLS              r < -0.05         5.93094    10.02278   17.42814
-0.05 < r < 0.05     5.257079   9.217827   16.65403
r > 0.05         4.756709   7.553808   13.31428

2SLS             r < -0.05         88.53055   231.2958   125.553
-0.05 < r < 0.05     195.3029   105.2122   107.1845
r > 0.05         226.2144   334.4002   221.0916

LIML             r < -0.05         88.53055   231.2958   125.553
-0.05 < r < 0.05     195.3029   105.2122   107.1845
r > 0.05         226.2144   334.4002   221.0916

3SLS             r < -0.05         632.2091   1106.055   182.4555
-0.05 < r < 0.05     274.0143   259.8756   127.3117
r > 0.05         3144.892   1141.369   402.9815

FIML             r < -0.05         2999.988   9414.314   3631.625
-0.05 < r < 0.05     12926.5    28985.17   2890.633
r > 0.05         15550.94   1522.433   2260.823

Table 10: Charts of the Behavior of RSS of Estimators over
Correlation Coefficients for each Sample Size R = 100; [P.sub.2].

Estimator            EQ1                  EQ2

Sample size         Sample size

15      25      40   15      25      40

OLS         \       \        \    \       \       2SLS        V       V        \    V       V       V
LIML        V       V        \    V       V       V
3SLS        V       V        \    V       V       V
FIML        /    [LAMBDA]    \    /   [LAMBDA]
Table 11: Summary of Frequencies of Correlation-based Charts of

Attribute   Table              \                       /

[P.sub.1]   [P.sub.2]   [P.sub.1]   [P.sub.2]

TAB           6         27          27          27          13
RSS         8 & 10      27          37          17           7

Attribute   Table         [LAMBDA]                     V

[P.sub.1]   [P.sub.2]   [P.sub.1]   [P.sub.2]

TAB           6          7          20          40          40
RSS         8 & 10      17           7          40          50
```
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