Comparing MINQUE and IAUE Estimates of Variance …Comparing MINQUE and IAUE estimates of variance components 2729 variance is used, and also when other methods are used for the same

Applied Mathematical Sciences, Vol. 11, 2017, no. 55, 2727 - 2766

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2017.79290

Comparing MINQUE and IAUE

Estimates of Variance Components

of a Random Coefficient Model

Souha K. Badr

Department of Statistics

Faculty of Science, AL-Fisaliah

King Abdulaziz University, Jeddah, Saudi Arabia

Ahmed H. Youssef


Faculty of Science


Hanaa H. Abu-zinadah


Faculty of Science, AL-Fisaliah


Copyright © 2017 Souha K. Badr et al. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

An alternative way to estimate the coefficient variance in the Swamy’s RCR model

has been derived using Minimum Norm Quadratic Estimation (MINQUE), and the

Iteration Almost Unbiased Estimator (IAUE) methods. The estimators’

performance in the RCR model are examined in Monte Carlo study. The Monte

Carlo study provides some insight into how well the RCR model performs in small,

medium, and large samples in the case of random, mixed, and fixed coefficient

regression. We found that using MINQUE method to estimate the coefficient

variance has reduce the probability of having negative variance comparing with the

Swamy method. IAUE method was superior, since it gives zero percent of negative

2728 Souha K. Badr et al.

variance and has a low variation and a low bias in estimation coefficient parameters,

even in case of fixed coefficients.

Keywords: Mixed Model, Panel Data, Pooling Cross-section, Random Coefficient

Regression Model, Time Series Data, Variance Component Estimation, MINIQUE,

IAUE

1 Introduction

Statistical models can be characterized according to the type of data to which they

are applied. The field of survey statistics usually deals with cross-sectional data

describing each of many different individuals or units at a single point in time.

Econometrics commonly uses time series data describing a single entity. Most

researchers use methods of analysis devolved for either cross-sectional or time

series data.

There are varieties of statistical models that can be used to analyze above the cross-

sectional time-series data. Dielman (1989) gives a comprehensive review of the

statistical literature dealing with these models. One of these models is the Random

Coefficient Regression model RCR.

The random coefficient regression model (RCR) was originally proposed by C. R.

Rao (1965) and later extended by Swamy (1970, 1971, 1974). Swamy (1971) allows

for random variation in population regression coefficients over cross-sectional

units, and coefficients are viewed as fixed over time. Swamy treats both intercept

and slope as random variables that are distributed across units with the same mean

and the same variance-covariance matrix.

Negative values for estimated variances can arise in a random coefficient regression

model context, because of some parameter estimating methods. Wu (1992)

maintained that the reason for negative variance components lies in the use of the

estimation method itself, and once negative variance occurs, some other methods

should be used instead of the unsuccessful one.

Minimum norm quadratic unbiased estimator (MINQUE) is proposed as a method

of variance estimation in a series of papers starting with Rao (1970). The basic idea

for this method is to find unbiased quadratic estimators that are invariant, and to

minimize some matrix norms.

One of the major disadvantages associated with the MINQUE estimator is that

MINQUE sometimes produces negative variance components. Horn et al. (1975)

proposed an estimator that avoids the deficiencies of the MINQUE method called

the Almost Unbiased Estimator (AUE). Schaffrin (1983), and Lucas (1985)

completed the work of Horn and introduced the Iterated Almost Unbiased Estimator

(IAUE). This estimator, as Horn stated, overcomes the problem of the negative

variance and guarantees a positive value for the estimation.

In this article, we will apply these two methods of variance component estimation

on Swamy’s RCR model. A Monte Carlo simulation study is conducted to compare

the efficiency of Swamy’s RCR model when the original method of estimating

Comparing MINQUE and IAUE estimates of variance components 2729

variance is used, and also when other methods are used for the same model. The

results of the simulation are analyzed, discussed, and presented in the last part in

this article.

2 Random Coefficient Regression Model

Random Coefficient Regression Model (RCR), have a long history. Pioneering

work of Rubin, Klein, Wald and Theil in the late 1940s and early 1950s had little

practical impact and was ignored for some time. More comprehensive papers,

oriented toward practical applications, were written in the late 1960s by Rao, Fisk,

Hildreth and Houk, and Swamy. The pre-1970 literature is reviewed almost

completely by Swamy, a substantial body of theory was developed, and a number

of useful review papers appeared. The (RCR) model proposed by Swamy (1970)

has appeared in different literature and be used in a number of applications

including pooled cross-sectional and time series data. For example, Bones &

Frankfurter (1977), Mehta et al. (1978), Delicado and Romo (1999), Bhaum and

Gibbons (2001),Hobza and Morales (2011), and Cartwright and Riabko (2015).

Swamy Type Estimator

The RCR model applies to a set of N cross-sectional as a model

𝑦𝑖 = 𝑋𝑖𝛽𝑖 +∈𝑖 (1)

Where each 𝑦𝑖 represents the Tx1 vector of observations from the 𝑖𝑡ℎ cross-section,

for i=1,…N, each 𝑋𝑖 represents the TxK matrix of independent variables, 𝛽𝑖 is a

vector of unknown random parameters, and ∈i the Tx1 vector of random error

terms.

Swamy rewrites the coefficient 𝛽𝑖 as

𝛽𝑖 = �� + 𝑣𝑖 (2)

Where �� is a Kx1 vector of a fixed component, and the 𝑣𝑖 is a Kx1 vector of random

variables satisfy:

I. 𝐸(𝑣𝑖) = 0 II. 𝐸(𝑣𝑖𝑥𝑖𝑡

′ ) = 0

III. 𝐸(𝑣𝑖𝑣𝑗′) = {

∆, 𝑖𝑓 𝑖 = 𝑗0, 𝑖𝑓 𝑖 ≠ 𝑗

}

Some other assumptions on Swamy’s model are as following:

1- The number of cross-sections and the sample size of each cross- section

must be greater than the number of parameters needed to be estimated i.e

𝑁 > 𝐾 & 𝑇 > 𝐾.

2- The independent variables are non-stochastic in the sense that 𝑋𝑖 is fixed in

repeated samples on 𝑌𝑖. The rank of 𝑋𝑖 is K.

3- The ∈𝑖 is independently and identically distributed with

𝐸(∈𝑖)0 𝑎𝑛𝑑 𝐸(∈𝑖∈𝑖′) = 𝜎𝑖

2𝐼𝑇

4- The coefficient vectors 𝛽𝑖 are independently and identically distributed with

𝐸(𝛽𝑖) = �� 𝑎𝑛𝑑 𝐸(𝛽𝑖 − ��)(𝛽𝑖 − ��)′= ∆

5- The ∈𝑖and 𝛽𝑗 are independent for every i and j.


Using the equations in (1) and (2), the model can be written as

𝑦𝑖 = 𝑋𝑖(�� + 𝑣𝑖) +∈𝑖= 𝑋𝑖�� + 𝑒𝑖 (3)

where 𝑒𝑖 = 𝑋𝑖𝑣𝑖 +∈𝑖. The (N) equations can be re-written as;

𝑌 = 𝑋�� + 𝑒 (4)

Where 𝑌(𝑁𝑇×1) = [

𝑌1

𝑌2

⋮𝑌𝑁

] , 𝑌𝑖(𝑇×1)= [

𝑦𝑖1

𝑦𝑖2

⋮𝑦𝑖𝑇

], 𝑋𝑁𝑇×𝑁𝑘 = [

𝑋1 0 … … 00 𝑋2 … … 0⋮0

⋮0

⋮0

⋮…

⋮𝑋𝑁

],

��(𝑘×1) =

[ ��0

��1

⋮⋮

��𝑘]

, 𝑒(𝑁𝑇×1) = [

𝑒1

𝑒2

⋮𝑒𝑁

] , 𝑒𝑖(𝑇×1)= [

𝑒𝑖1

𝑒𝑖2

⋮𝑒𝑖𝑇

]

The error vector, e, is normally distributed with zero mean and variance-covariance

matrix Ω, given by:

𝐸(𝑒𝑒′) = 𝛺𝑁𝑇×𝑁𝐾 =

[ 𝑋1∆𝑋1

′ + 𝜎12𝐼𝑇 0 … 0

0 𝑋2∆𝑋2′ + 𝜎2

2𝐼𝑇 … 0⋮0

⋮0

⋮…

⋮𝑋𝑁∆𝑋𝑁

′ + 𝜎𝑁2𝐼𝑇]

Where the zeros are all TxT null matrices and ∆ is the variance-covariance matrix

of 𝛽𝑖. If Δ and 𝜎2 are known, the best linear unbiased estimator of �� is the

Generalized Least Squares (GLS) estimator

�� = (𝑋′𝛺−1𝑋)−1𝑋′𝛺−1𝑌 (5)

If Δ and 𝜎2 are unknown, which is typically the case, we should first estimate Δ

and 𝜎2, then we estimate �� by substituting the estimated Δ and 𝜎2 into (5) to get

the Feasible Generalized Least Squares (FGLS) estimator.

�� = (𝑋′��−1𝑋)−1

𝑋′��−1𝑌 (6)

Swamy (1970) suggests the following unbiased and consistent estimators:

��𝑖2 =

∈𝑖′∈𝑖

𝑇−𝐾 (7)

And

∆=𝑆��𝑖

𝑁−1−

1

𝑁∑ ��𝑖

2(𝑋𝑖′𝑋𝑖)−1𝑁

𝑖=1 (8)

Where

𝑆��𝑖= ∑ ��𝑖��𝑖′

𝑁𝑖=1 −

1

𝑁∑ ��𝑖

𝑁𝑖=1 ∑ ��𝑖′

𝑁𝑖=1 (9)


Also, Swamy shows that the FGLS estimator given in (6) is equivalent to

�� = {∑[∆ + ��𝑖2(𝑋𝑖′𝑋𝑖)

−1]−1

𝑁

𝑖=1

}

−1

∑[∆ + ��𝑖2(𝑋𝑖′𝑋𝑖)

−1]−1

𝑁

𝑖=1

��𝑖 (10)

Where ��𝑖 = (𝑋𝑖′𝑋𝑖)−1𝑋𝑖′𝑌𝑖 is the Ordinary Least Square (OLS) estimator of 𝛽𝑖.

Swamy showed that his estimator β is consistent as N → ∞ and T → ∞, and

asymptotically efficient as T → ∞ under certain conditions. The performance of β

in small samples was studied in Dielman (1992).The distribution of the coefficient

vector is invariant to translations along the time axis, and ∆ & ��2 are unbiased

estimators for the variance-covariance matrix ∆ and 𝜎2.

Issues with Swamy’s estimators

One particular problem with Swamy’s estimators is that the estimators of ∆,

equation (8), that he suggests may, at times, yield negative estimates of variances.

This is so because Swamy suggests an estimator for ∆ that is the difference of two

matrices. Griffiths in (1971) said that the greatest disadvantage with Swamy’s

estimator is the frequency with which it gives negative estimates. Not only because

negative values are meaningless, but because if retained they can lead to GLS

estimators of the 𝛽𝑖’s which perform extremely poorly in terms of MSE.

The gains that we earn in terms of quality of the estimate of �� justify additional

complexity in estimating ∆. There are some other alternative estimators that suggest

overcoming this problem. One of them is:

∆1= {∆ 𝑖𝑓 𝑎𝑙𝑙 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒

∆𝐴 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑑 𝑏𝑦 𝑧𝑒𝑟𝑜 𝑖𝑓 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠

(11)

Griffiths (1979) mentioned that changing negative estimates to zero is not

completely satisfactory, since this implies that the corresponding coefficients are

no longer random.

3 The Minimum Norm Quadratic Estimation

The method of minimum norm quadratic estimation (MINQUE) was introduced by

Rao (1971) for regression models with heteroscedastic variances. Later, a series of

papers was published (1971, 1972, 1973) to generalize the method for variance and

covariance component models and to compare MINQUE and modified MINQUE

estimators of heteroscedastic variances with the usual sample variances in the case

of replicated data. The basic idea for this method is to find unbiased quadratic

estimators that are invariant and to minimize some matrix norms.

The principle of MINQUE estimation state:

The quadratic form 𝑌′𝐴𝑌 is said to be the MINQUE (minimum norm quadratic

unbiased estimator) of ∑ 𝜌𝑖𝜎𝑖2

𝑖 if the matrix A is determined such that

‖(𝑈′𝐴𝑈 − ∆)‖ is a minimum subject to the conditions


𝐴𝑋 = 0

𝑡𝑟𝐴𝑉𝑖 = 𝜌𝑖 , 𝑖 = 1,2, … . 𝑘 (12)

MINQUE for RCR Model

Recall the random regression model equation (4),

𝑌 = 𝑋�� + 𝑒 (13)

Where Y represents NTx1 vector of observations, X represents NTxNK matrix of

independent variables, �� is a vector of unknown fixed parameters, and 𝑒 the NTx1

vector of random error terms.

From equation (13) we have that the i-th cross-section unit

𝑦𝑖 = 𝑋𝑖�� + 𝑋𝑖𝑣𝑖 + 𝜖𝑖 𝑖 = 1,2, … ,𝑁 (14)

where 𝑒𝑖 = 𝑋𝑖𝑣𝑖 + 𝜖𝑖 and the variables are defined as the same definitions and assumptions as in the

previous section.

from (13), and (14) we can rewrite the model as

𝑌 = 𝑋𝛽 + 𝑈1𝜉1 + 𝑈2𝜉2 + ⋯+ 𝑈𝑘𝜉𝑘 + 𝑈𝑘+1𝜉𝑘+1+∈ (15)

where

𝑈𝑖is 𝑁𝑇 × 𝑁𝑇 diagonal matrix whose corresponding diagonal elements of 𝑈𝑖 = 𝑋𝑖,𝑓𝑜𝑟 𝑖 = 1,2, …𝑘.

𝜉𝑖 is 𝐾 × 1 random vector has a multivariate normal distribution with mean �� and

covariance matrix ∆. Further, it is assumed that 𝜉𝑖 and 𝜉𝑗 𝑓𝑜𝑟 𝑖 ≠ 𝑗 are

uncorrelated.

Equation (15) can be expressed in a compact form as:

𝑌 = 𝑋𝛽 + 𝑈𝜉 (16)

Thus, generally we have 𝐸(𝑌) = 𝑋𝛽 and 𝐷(𝑌) = 𝛺 = ∑ ∆𝑖𝑉𝑖 + 𝜎𝑗2𝐼𝑇

𝑘𝑖=1 , 𝑓𝑜𝑟 𝑗 =

1, … . 𝑁,where 𝑉𝑖 = 𝑈𝑖𝑈𝑖′, D is called the dispersion matrix and the parameters

∆0, ∆1, … . , ∆𝑘are the unknown variance components whose values should be

estimated.

The quadratic form 𝑌′𝐴𝑌, in the observations, Y is said to be a MINOUE of ∑ 𝜌𝑖∆𝑖𝑖

if the symmetric matrix A is selected such that ‖(𝑈′𝐴𝑈 − ∆∗)‖ is a minimum

subject to the conditions

𝐴𝑋 = 0 and 𝑡𝑟𝐴𝑉𝑖 = 𝜌𝑖

For making the optimization easier, the squared Euclidean norm will be utilized.

Then we get

‖(𝑈′𝐴𝑈 − ∆∗)‖2 = 𝑡𝑟((𝑈′𝐴𝑈 − ∆∗)′(𝑈′𝐴𝑈 − ∆∗)) = 𝑡𝑟(𝐴𝑉𝐴𝑉) + ∆∗∗ (17)

where 𝑉 = ∑ 𝑉𝑖𝑘𝑖=1 = 𝑈𝑈′ and ∆∗∗ refer to a constant quantity and do not involve

A, so the second term is dropped.

Let A be a symmetric matrix and V be a symmetric and invertible matrix, then the

minimum 𝑡𝑟(𝐴𝑉𝐴𝑉), subject to invariant and unbiasedness criteria, is attained at:

𝐴 = ∑ 𝑎𝑖𝑅𝑉𝑖𝑅𝑘𝑖=1 (18)


where

𝑎 = 𝑆−1𝜌

𝑆𝑖,𝑗 = 𝑡𝑟(𝑄′𝑉−1𝑉𝑖𝑉−1𝑄𝑉𝑗), 𝑓𝑜𝑟 𝑖 𝑎𝑛𝑑 𝑗 = 1,… , 𝑘

𝑄 = 𝐼𝑁𝑇 − 𝑋(𝑋′𝑉−1𝑋)−1𝑋′𝑉−1

𝑅 = 𝑄′𝑉−1.

Consequently, the MINQUE of 𝜌′∆ is

𝑌′𝐴𝑌 = ∑ 𝑎𝑖𝑌′𝑅𝑉𝑖𝑅

𝑘𝑖=1 𝑌 = ∑ 𝑎𝑖𝑏𝑖

∗𝑘𝑖=1 = 𝑎′𝑏∗ = 𝜌′𝑆−1𝑏∗ (19)

where

𝑏∗ = 𝑌′𝑅𝑉𝑖𝑅𝑌.

By equating (4.26) with𝜌′∆ , we get:

∆𝑀𝐼𝑁𝑄𝑈𝐸1= 𝑆−1𝑏∗ (20)

Hsiao (1972, 1974) has proposed another way to find The MINQUE estimator for

RCR model by finding The MINQUE estimator for each cross-section as follows:

Let

𝜃𝑖 = ∑ 𝑋𝑖𝑗∆𝑖𝑋𝑖𝑗′ + 𝜎2𝐼𝑇

𝐾𝑗=1 = ��𝑖𝛤 𝑓𝑜𝑟 𝑖 = 1,… ,𝑁. (21)

where

��𝑖 is 𝑇 × 𝐾 matrix whose elements are squares of the corresponding elements of

𝑍𝑖 = 𝑋𝑖, and 𝛤 is the 𝐾 × 1 column vector with elements ∆1, … , ∆𝑘, so, the

MINQUE of ∑𝜌′𝜃𝑖is the quadratic form 𝑌𝑖′𝐴𝑌𝑖, such that the 𝑡𝑟 𝐴2 is minimized

subject to the conditions:

𝐴𝑋𝑖 = 0 (22)

∑ 𝜌𝑖𝜃𝑖𝑁𝑖=1 = 𝜌′��𝑖𝛤 (23)

where

𝜌′ = (𝜌1, … , 𝜌𝑁)′ following the computational procedure suggested by Rao, we have that:

𝑌𝑖′𝐴𝑌𝑖 = 𝜌′��𝑖(��𝑖

′��𝑖��𝑖)−1��𝑖

′��𝑖 (24)

where

𝑀𝑖 = 𝐼𝑇 − 𝑍𝑖(𝑍𝑖′𝑍𝑖)

−1𝑍𝑖′

𝑟𝑖 = 𝑌𝑖 − 𝑍𝑖��𝑖

��𝑖 is the OLS estimator for each cross-section

and ��𝑖, ��𝑖 denotes the corresponding matrix and vector obtained by squaring each

element in 𝑀𝑖 , 𝑟𝑖 respectively.

Equating (24) with 𝜌′��𝑖𝛤, the unbiased estimator of 𝛤𝑖 is

𝛤𝑖∗ = (��𝑖

′��𝑖��𝑖)−1

��𝑖′��𝑖. (25)

After applying equation (25) for each cross-section unit, we can obtain the

consistent estimator 𝛤 by taking the simple average as mentioned by Hsiao (1975).


4 Iteration Almost Unbiased Estimation

Three major disadvantages are associated with the MINQUE estimator:

1. MINQUE sometimes produces a negative variance component.

2. The S matrix, equation (20), is not always nonsingular.

3. MINQUE estimator needs a heavy calculation, which is time consuming.

Horn et al. (1975) proposed an estimator that avoids the deficiencies of the

MINQUE method. As Horn et al. stated, it overcomes the problem of the negative

variance and guarantees a positive value for the estimation, and exists under more

general conditions. It also needs less cost in computer time and storage than the

MINQUE estimator. They call the estimator, Almost Unbiased Estimator (AUE).

Since this estimator is proportional between a prior estimate and a true variance,

the bias introduced with the estimation is because of the failure to this proportional

factor to reach a unity, but this failure can be expected to be small. Hence, the

appellation “almost unbiased”.

Schaffrin (1983), and Lucas (1985) completed the work of Horn and introduced the

Iterated Almost Unbiased Estimator (IAUE). Like AUE, the estimation of IAUE

focuses on obtaining estimations for variance factors, which will approach unity as

prior estimates approach the true variance.

Hsu (2001) showed that the Helmert method is identical with the IAUE method. It

can easily be shown that a variance component obtained from the IAUE is always

positive provided that the resulting covariance matrix is positive definite. Hsu

(1999) indicates that it is possible for IAUE to produce some variance factors that

deviate significantly from unity due to inappropriate grouping. Egeltoft (1992),

Fotopoulos et al. (2005), Bahr et al. (2007), Amiri-Simkooei (2007), and Kall et al.

(2014) compare the former methods, which produce unbiased and invariant

estimates, but are very time-consuming, require a lot of computational resources,

and sometimes produce negative estimates, and the IAUE method, which although

is not always unbiased, does not produce negative variance component estimates,

demands fewer computational resources, and converges much more quickly than

other approaches.

IAUE for RCR Model

The working equation of IAUE as mentioned by Lucas (1985), and Hsu (1998) is

𝑓𝑖 =𝑌′𝑅∗𝑉𝑖𝑅

∗𝑌

𝑡𝑟(𝑉𝑖𝑅∗)

(26)

where

Y, and 𝑉𝑖 are as defined above.

𝑅∗ = 𝐷∗−1 − 𝐷∗−1𝑋(𝑋′𝐷∗−1𝑋)−1𝑋′𝐷∗−1

𝐷∗ = ∑ 𝜏𝑖𝑉𝑖𝐾𝑖=1 , 𝜏𝑖 is a prior estimate of ∆.

Computed 𝑓𝑖 based on 𝜏𝑖 will be updated based on the new value of 𝜏𝑖 until 𝑓𝑖′𝑠

approach one. Then ∆𝐼𝐴𝑈𝐸 can be expressed as:

∆𝐼𝐴𝑈𝐸= ∏ 𝑓𝑖𝑚−1𝑖=1 (27)

where m is the number of iterations.


5 Simulation Study for Swamy’s RCR Estimators

A Monte Carlo simulation trailer was conducted 10,000 times to investigate the

efficiency of the estimation of the RCR model with two parameters 𝛽0 𝑎𝑛𝑑 𝛽1. The

model under study is given by:

𝑦𝑖𝑡 = 𝛽0𝑖 + 𝛽1𝑖𝑥𝑖𝑡 +∈𝑖𝑡 (28)

for 𝑖 = 1,2…… ,𝑁 ; 𝑡 = 1,2, ……𝑇.

To perform the simulation, the model in equation (28) was generated as follows:

1- The value of the independent variable, 𝑥𝑖𝑡, was generated as independent

normally distributed random variables with mean 𝜇𝑥, set equal to five, and

variance𝜎𝑥2, set equal to three.

2- The error term, ∈𝑖𝑡, were generated as independent normally distributed

random variables, independent of the 𝑥𝑖𝑡 values, with mean set equal to zero,

and standard deviation 𝜎𝜖, set equal to either 1, 5, or 10.

3- Different values of N and T were chosen to be 5, 10, 15, 20, and 50 to represent

small, medium, and large samples for the number of individuals in each cross

section and number of cross sections. Where 5, and 10 were chosen to represent

the small sample, 15, and 20 represented a medium sample size, and finally 50

represented the large sample size.

4- The parameters, 𝛽0𝑖 and 𝛽1𝑖, were set at several different values to allow study

of the estimator where there was a small or a large variety between the cross

sections unit, and under the conditions where the model was both properly and

improperly specified.

To estimate the unknown regression parameters 𝛽° and 𝛽1,different methods of

estimating variance components were used. First, the Swamy method given in

equation (8). RWZ, which replaces the negative diagonal elements of ∆ by zero if

the Swamy method for estimating the variance coefficient has failed to produce

non-negative values, given in equation (11). The MINIQUE method given in

equation (20) was used to evaluate MINIQUE1, and equation (25) to evaluate

MINIQUE2. Finally, equation (26) is used to evaluate both IAUE1 and IAUE2,

each with a different initial value. IAUE1 uses the result of solving equation (20)

as its initial value, and a vector of unity is the initial value of IAUE2 as

recommended by Lucas.

The coefficient estimator ��0 and ��1are computed using different values of the

estimated coefficient’s variance, ∆. According to the method used to estimate ∆, we

will have different values for both ��0 and ��1.

The results are recorded in table (A.1) to table (A.7). Tables consist of five panels

for the different sample sizes (5, 10, 15, 20, and 50). In addition, each panel from

these panels will have three settings for the error standard deviation (1, 5, and 10).

Each setting for each panel will provide the results for six different methods

(Swamy, RWZ, MINQUE1, MINQUE2, IAUE1, and IAUE2). Each of the tables

provides the results for a particular scheme of the regression coefficients to show

the following information:


1- The coefficient mean estimators, ��0 and ��1, that are computed as in equation

(10). The values shown in the first row of each panel of each table are the

average over all 10,000 Monte Carlo trails at a particular setting.

2- The estimated variance of each coefficient, 𝑉𝑎𝑟(��𝑘) = ∆ 𝑓𝑜𝑟 𝑘 = 0,1

averaged over 10,000 trails is shown in the second row. The estimates ∆ are

computed using different methods.

3- The bias value of the coefficient mean estimators, ��0 and ��1, are computed

as

𝑏𝑖𝑎𝑠 (��) = �� − �� (29)

where �� is a vector of coefficient mean estimators and �� is the true vector of the

coefficients mean, the bias values are shown in row three of each panel.

4- The variation in estimating ��, is computed as the variance between the

estimated values of �� during the 10,000 trails, and recorded in the fourth

row of each panel.

5- The bias value of the estimate of the coefficient variances, ∆0 and ∆1, are

computed as

𝑏𝑖𝑎𝑠(∆) = ∆ − ∆ (30)

where ∆ is a vector of coefficient variances and ∆ is the true vector of the coefficient

variance, the bias values are shown in row five of each panel.

6- The variation in estimating ∆, is computed as the variance between the

estimated values of ∆ during the 10,000 trails, and recorded in the sixth row

of each panel.

7- The percentage of a negative variance estimate produced by the different

methods during the 10,000 trails were recorded in row seven in each panel.

6 Analytical Result for Simple Random Coefficient Models

In this section, we will use a Monte Carlo simulation of results to compare the

efficiency of Swamy’s RCR model when using different methods to estimate the

variance component. The behaviour of this model will be tested in small, medium,

and large samples for random, mixed, and fixed parameters. The results for these

models will be recorded in tables (A.1) to (A.7).

Table (A.1) displays the results of a simulation study when the mean and the

variance of the intercept parameter, 𝛽0, equals 10, and also the mean and variance

of the slope parameter, 𝛽1, is equal to the same value, i.e ��1 = 10 𝑎𝑛𝑑 ∆𝛽1= 10.

We will use the Swamy RCR technique to estimate the mean of the coefficient in

the model. To estimate the variance of the coefficient in the model the original

Swamy method for estimating the variance of the parameters will be used. In case

of appearing a variance with negative values during the 10,000-simulation trailer,

the alternative methods will be used for this trail.

As a guide to interpreting table (A.1), the results of using the Swamy method to

estimate the unknown parameters and their properties are recorded in the first

vertical panel of table (A.1). When 𝜎𝜀 = 1 and N=T=5, the average mean for 𝛽0and


𝛽1over all 10,000 Monte Carlo trails is 9.779 and 10.025 respectively. Note that

the true coefficient values for the mean for both parameters are 10, the average

variances are 9.981 and 10.002 for 𝛽0and 𝛽1 respectively. To estimate these values

from the generating samples, Swamy has failed around 10 % to have a non-negative

variance value. Note that this percentage should be zero. The variation in estimating

𝛽0 is equal to 721.616 and for 𝛽1 is 21.826, while the variation in estimating ∆0

and ∆1 are 97.173 and 50.787 respectively.

The results of estimating the unknown parameters and their properties when

replacing negative variance produced by using the Swamy method to estimate the

coefficient variance by zero value during 10,000-simulation trailer, are recorded in

the second vertical panel labeled “RWZ”. When 𝜎𝜀 = 1 and N=T=5, the average

mean for 𝛽0and 𝛽1over all 10,000 Monte Carlo trails are 9.712 and 10.037

respectively. The average variances are 10.165 and 10.002 for 𝛽0and 𝛽1

respectively. By default, this method does not produce negative variances value,

since it trades with the coefficient parameters as a fixed parameter with zero

variances. The variation in estimating 𝛽0 is increased to 1091.377 and for 𝛽1 to

33.092, while the variation in estimating ∆0 is decreased to 92.922 and is still the

same as Swamy for ∆1.

The Minimum Norm Quadratic Unbiased Estimator presented in equation (20), is

used to estimate the variance component in cases where Swamy produces negative

variance values. The results are recorded in the third panel in table (A.1) and labeled

“MINQUE1”. When 𝜎𝜀 = 1 and N=T=5, the average mean for 𝛽0 over all 10,000

Monte Carlo trails is increased to 11.236 and for 𝛽1 is decreased to 9.752. The

average variances are 10.276 and 8.95 for 𝛽0and 𝛽1 respectively. The variation in

estimating the coefficient parameters are higher compared to all other methods,

equal to 8985.615, and for 𝛽1 is 356.922, while the variation in estimating ∆0 and

∆1 is 91.369 and 55.368 respectively, with 8% of negative variances, which is the

second highest percentage of negative variance after Swamy.

Another way to calculate the Minimum Norm Quadratic Unbiased Estimator

presented in equation (25) is used in the case of negative Swamy’s variance, and

recorded in the fourth panel labeled “MINQUE2”. When 𝜎𝜀 = 1 and N=T=5, the

average mean for 𝛽0 and 𝛽1 over all 10,000 Monte Carlo trails are 10.036 and

9.983 respectively, with a decrease in the estimate of the variation compared with

Swamy. The average variances are 10.221 and 8.952 for 𝛽0and 𝛽1 respectively,

with the same variation as MINQUE1, and 3% of negative variance.

The result of the variance component produced from MINQUE1 is used as an initial

value to calculate the Iteration Almost Unbiased Estimator given in equation (26),

in the case of Swamy’s negative variance, and recorded in the fifth panel labeled

“IAUE1”. When 𝜎𝜀 = 1 and N=T=5, the average mean for 𝛽0and 𝛽1over all

10,000 Monte Carlo trails is 10.069 and 9.978 respectively, with the lowest

variation in estimating those parameters compared to all other methods. The

average variances are 10.27 and 9.055 for 𝛽0and 𝛽1 respectively, with a variation

in estimating ∆0 and ∆1 equal to 90.862 and 53.591 respectively. To estimate these

values from the generating samples, IAUE1 has 0% of negative variance.


Using a unity vector as an initial value to calculate the Iteration Almost Unbiased

Estimator given in equation (26), in the case of a negative Swamy variance, is

recorded in the sixth panel labeled “IAUE2”. As 𝜎𝜀 = 1 and N=T=5, the average

mean for 𝛽0 and 𝛽1 over all 10,000 Monte Carlo trails is 9.565 and 9.442

respectively. The average variances are 10.167 and 8.952 for 𝛽0and 𝛽1

respectively. The variation in estimating the coefficient parameters is equal to

57.468 and 5.793, while the variation in estimating ∆0 and ∆1 are 92.88 and 55.339

respectively, with 0% of negative variances.

As the variation in error term increases, IAUE2 has the worst estimators compared

to all other methods. The efficiency of Swamy estimators also becomes worse,

especially for the variation in estimating 𝛽 and ∆ and the percentage of negative

variance. MINQUE1 and MINQUE2 also increase the variation in estimating

parameters with the existence of negative variance less than Swamy. IAUE1 has a

zero percent of negative variance even with an increase in the error variation. In

addition, it is the least affected by an increase in the error variation in terms of

variation of estimation. RWZ is the second best method after IAUE1, but the big

issue with this method is dealing with coefficient parameters as fixed parameters

rather than random parameters.

Increasing both sample size and the number of cross sections, have improve the

estimators in all methods, except for IAUE2. For example, comparing a small

sample size, N=T=5, with a medium sample size, N=T=20, at the same standard

deviation of the error, 𝜎𝜀 = 5, reduce the negative variances produced by using the

Swamy method from 70% to 7%. The variation in estimating 𝛽1 is decreased from

7484.049 to 0.528 when using the RWZ method, and the variation in estimating ∆1

for MINQUE1 and MINQUE2 was 116.935 and 110.66 reduced to 17.255 and

17.119 respectively, and the absolute value for the average bias for 𝛽0 decreased

from 0.151 to 0.003 for IAUE1. Unfortunately, using the IAUE2 method does not

give any good results under any conditions.

Table (A.2) records the results of a simulation study when reducing the coefficient

variation of both parameters in order to study the effect of using different variance

component estimation methods on the behaviour of the RCR estimators when the

variances of both the intercept and the slope parameters are decreased from 10 to 1.

The mean of 𝛽0 and 𝛽1 are equal to 1.

Reducing coefficient variances led to producing a higher percentage of negative

variance when using the Swamy method, accompanied by a large variation in

estimating both 𝛽 and ∆. MINQUE1 and MINQUE2 have also increased in the

variation of estimating 𝛽 but with less negative variance compared to Swamy.

IAUE1 has the best results compared to other methods, for example as N=10 and

𝜎𝜀 = 1 , the absolute value of average bias for 𝛽0 and 𝛽1, when using IAUE1

method, is 0.023 and 0.005, with a variation in estimation of those parameters equal

to 7.068 and 0.358 respectively. The absolute value for the average bias of ∆0 and

∆1 is 0.026 and 0.006. IAUE1 never produces negative variances.

Table (A.3) displays the results of a simulation study when the mean of 𝛽0 and 𝛽1

is equal to 5, but the variance value for them is not equal; the variance of 𝛽0 equals

10 and the variance of 𝛽1 is 1. This model will allow us to study the effect of using


different variance component estimation methods on the behavior of the RCR

estimators when the variances of the slope parameter are decreased.

From table (A.3) when N=T=15 and 𝜎𝜀 = 5, the absolute values for the average

bias of 𝛽0 and 𝛽1 using the Swamy method, are 0.227 and 0.055, with a variation

in estimation equal to 3046.032 and 116.873 respectively. Using the same method,

the absolute values for average bias of ∆0 and ∆1 are 0.111 and 0.002, with a

variation in estimation equal to 122.583 and 0.466 respectively. Swamy has failed

around 23% to produce positive or zero variance. MINQUE1 has the second highest

percentage of negative variance equal to 8%. The absolute value for an average bias

of 𝛽0 and 𝛽1 using the same method, reduces to 0.064 and 0.017, with a big

difference in variation of estimation compared to Swamy, and is equal to 27.226

and 1.324 respectively. IAUE1 has even smaller results, with zero percent of

negative variance, and the absolute value for average bias of 𝛽0 and 𝛽1 is 0.019

and 0.008, with a variation in estimation equal to 7.244 and 0.393 respectively.

Using the same method, the absolute value for average bias of ∆0 and ∆1 is 0.608

and 0.075, with a variation in estimation equal to 98.985 and 0.345 respectively

The results of a simulation study when the variance of the intercept parameter is

decreased are recorded and displayed in table (A.4).The mean of 𝛽0 and 𝛽1 are

equal to 1, but the variances value for them are not equal; the variance of 𝛽0 equals

1 and the variance of 𝛽1 is 5. This model will allow us to study the effect of using

different variance component estimation methods on the behavior of the RCR

estimators when the variances of 𝛽0 is decreased.

From table (A.4) for a large sample size, N=T=50, we can see that Swamy has a

zero percent of negative variances at 𝜎𝜀 = 1 , but when the error variation increases

to 𝜎𝜀 = 5 Swamy has 21% percentage of negative variances during 10,000-

simulation trails. Again, MINQUE1 has the second highest percentage of negative

variance after the Swamy method. The bias and the variation in estimating 𝛽 and ∆

is higher than Swamy for both MINQUE1 and MINQUE2. IAUE1 still does well

compared to all other methods in the sense of the bias and the variation in estimating

𝛽, the variation in estimating ∆, and the percentage of negative variances. IAUE2

is the worst.

Mixed Model

In this section, we study the efficiency of the RCR model when using different

methods of variance component estimation. The behavior of this model will be

tested in small, medium, and large samples for the model that contains both random

and fixed parameters. The Monte Carlo simulation results for these models were

present in table (A.5) and (A.6). We will study two cases within these models, the

first, when the intercept parameter is fixed while the slope parameter is random.

The second case is when the intercept parameter is random while the slope

parameter is fixed.

Table (A.5) displays the results of a simulation study when the mean for both

coefficient parameters, 𝛽0 and 𝛽1, equal 10 and the variance of 𝛽0 equals 10, while

the variance of 𝛽1 is zero. This means that the fixed parameter in this model is the

slope parameter, 𝛽1. As above, the RCR model will be estimated using the Swamy


method to estimate the variance component, and in the case of a negative variance

appearing, the alternative method will be used. This model enables us to study the

efficiency of RCR estimators for Mixed RCR models, where the intercept

parameter 𝛽0 is random and the slope parameter 𝛽1 is fixed.

From table (A.5), when 𝜎𝜀 = 1 and N=T=15, the average mean for 𝛽0and 𝛽1over

all 10,000 Monte Carlo trails when we use the Swamy method to estimate the

variance component for the parameters, is 10.071 and 9.982 respectively. Note that

the true coefficient values for the mean for both parameters are 10, the average

variances are 9.944 and zero for 𝛽0and 𝛽1 respectively. To estimate these values


variance value for ∆𝛽1. Note that we deal with 𝛽1 as a fixed parameter. The variation

in estimating 𝛽0 is equal to 137.701 and for 𝛽1 is 5.675, while the variation in

estimating ∆0 and ∆1 is 16.576 and zero respectively. MINQUE1 produced 22.6%

of negative variances value for ∆𝛽1, and a smaller bias and variation of 𝛽 compared

to Swamy. MINIQUE2 has almost similar results as MINQUE1 with a small

percentage of negative variance equal to 15%. RWZ, IAUE1, and IAUE2 have 0%

of negative variances, almost the same results in the bias and the variation in

estimating 𝛽 between RWZ and IAUE1 methods. IAUE1 has a high bias and

variation of ∆0 and ∆1 equal to 4.942 and 0.551 for the bias, and the variation was

equal to 28.319 and 0.231 respectively. IAUE2 has the highest values in bias and

variation for estimating both 𝛽 and ∆.

Increasing the error variation even with a large sample size, makes the Swamy

method the worst method after the IAUE2 method, with a high percentage of

negative variance for both ∆0 and ∆1, a large bias and variation in estimating 𝛽, and

a large variation in estimating ∆.

Table (A.6) displays the results of a simulation study for a mixed model where both

coefficient parameters, 𝛽0 and 𝛽1 have a mean equal to 5. The variance of 𝛽1 is 1

and since we are dealing with a mixed model, in our case where the fixed parameter,

is the intercept parameter the variance of 𝛽0 will be equal to zero. This model will

be estimated using the RCR model using the Swamy method to estimate the

variance component, and in the case of negative variance appearing, the alternative

method will be used. This model enables us to study the efficiency of RCR

estimators for Mixed RCR models where the intercept parameter, 𝛽0, is fixed and

the slope parameter, 𝛽1, is random.

Table (A.6) shows a big variation in estimating, with a large bias in estimating the

coefficient parameters, and a high percentage for negative estimated variances

when using the Swamy method, especially for small and medium sample sizes.

IAUE1 and RWZ methods have the most suitable results among all other methods,

with 0% of negative estimated variances and small bias and variation in estimating

𝛽.

Fixed Model

In this section, we study the efficiency of the RCR model when using different

methods of variance component estimation. The behavior of this model will be tested in small, medium, and large samples for the model that contains fixed parame-


ters. The Monte Carlo simulation results for these models were present in table

(A.7).

Table (A.7) displays the results of a simulation study when the mean for

both coefficient parameters, 𝛽0 and 𝛽1, equal zero. Moreover, since we will deal

with a fixed model, then both parameters have a variance equal to zero. This model

will be estimated using the RCR model. The Swamy method will be used to

estimate the variance component, and in the case of appearing negative variance,

the alternative method will be used. This model enables us to study the efficiency

of RCR estimators for fixed models where the intercept parameter, 𝛽0, and the

slope parameter, 𝛽1, are fixed.

From table (A.7), when 𝜎𝜀 = 1 and N=T=10, the average mean for 𝛽0and 𝛽1over

all 10,000 Monte Carlo trails when we use the Swamy method to estimate the

variance component for the parameters is -2.346 and 0.446 respectively. Note that

the true coefficient values for the mean for both parameters are zero, the average

variances are 0.001 and zero for 𝛽0and 𝛽1 respectively. To estimate these values


variance value for ∆𝛽0 and 55% for ∆𝛽1

. Note that we deal with 𝛽0and 𝛽1 as fixed

parameters. The variation in estimating 𝛽0 is equal to 56899.9 and for 𝛽1 is 2099.2,

while the variation in estimating ∆0 and ∆1 are 0.659 and 0.001 respectively.

MINIQUE1 and MINIQUE2 have low percentages of negative variances compared

to Swamy. The absolute bias in estimating 𝛽0and 𝛽1 for MINQUE1 is equal to

0.049 and 0.014 and for MINQUE2 is equal to 0.045 and 0.005 respectively. The

variation in estimating those parameters for MINQUE1 is 199.4 and 6.347, and for

MINQUE2 is 91.822 and 2.002. For the RWZ method the absolute bias in

estimating 𝛽0and 𝛽1 is 0.293 and 0.055, with a variation in estimating parameters

equal to 911.401 and 32.095, the absolute bias in estimating∆0 and ∆1 is 0.316 and

0.012, with a variation in estimating parameters equal to 0.306 and zero. IAUE1 as

usual has zero percent of negative variances. The absolute bias in estimating 𝛽0and

𝛽1 is 0.069 and 0.01 with a variation in estimating parameters equal to 1.426 and

78.207, the absolute bias in estimating∆0 and ∆1 are 0.614 and 0.313, with a

variation in estimating parameters equal to 0.218 and 0.306.

Graphical Analysis

For further explanation we use 2D graphical figures to show the resulting variations

in RCR estimations when using different methods to estimate the variance

component. The figures are plotted regardless of whether the RCR model is the

right model to represent this data, to compare the efficiency of the different methods

even when we use it in an improper way. The IAUE2 method was excluded from

the graph, since it gives inappropriate results and that affects the shape of the graph.

Figure (B.1) shows the absolute bias of ��°, ��1, ∆𝛽°, and ∆𝛽1

and their variation

against different sample sizes respectively. While figure (B.2) shows the absolute

bias of ��°, ��1, ∆𝛽°, and ∆𝛽1

and their variation against different standard deviation

for disturbance respectively.


From figure (B.1), we can see how the absolute bias of the coefficient regression

parameters estimated by the Swamy method to estimate the variance component is

affected by sample size. As the sample size increases from 5 (small sample) to 50

(large sample), the absolute bias for both parameters rapidly decreases. For

example, when the sample size was 10 the absolute bias for ��° was close to 3, and

less than 0.5 when the sample size was 20, and almost zero when the sample size

increased to 50. Using the MINQUE1 and RWZ methods to estimate the variance

component also results in high bias values for the coefficient regression parameters,

especially when the sample size is 5, while MINQUE2 and IAUE1 have very low

bias values for the coefficient regression parameters compared with all other

methods. Using MINQUE1 as an initial value to compute IAUE1, produces

estimators with smaller bias values. Decreasing the sample size does not affect the

IAUE1 method as it does for other methods, as we can see that IAUE1 has bias for

both ��° and ��1 close to zero even with a small sample size.

The Swamy, RWZ, and MINQUE1 methods produce high variation values in

estimating the coefficient regression parameters as the sample size decrease, while

MINQU2 and IAUE1 have a very low variation, even with a small sample size

compared to other methods, with IAUE1 having the lowest variation between all

methods.

In estimating coefficient variances ∆𝛽° and ∆𝛽1

, Swamy has the lowest absolute bias

for both parameters compared to all other methods for all sample sizes. This bias is

low even with a small sample size. MINQUE1, MINQUE2, and IAUE1 produce

close values for coefficient variances estimates. Moreover, the variation in

estimating ∆𝛽° and ∆𝛽1

are very close for all methods for all sample sizes, except for

the sample size equal to five, while Swamy has a very high value compared to other

methods.

Figures (B.1.i) and (B.1.j) show that Swamy, MINQUE1, and MINQUE2 have all

produced negative variances, the percentage of which have reduced as the sample

size has increased. IAUE1 has never produced a negative variance, as it should do

over all of the different sample sizes. Note that we exclude the RWZ method from

these figures, since for this method we replace a negative variance with a non-

negative variance, i.e. zero.

From figure (B.2), we can see that as the disturbance standard deviation increased,

the absolute bias of the coefficient regression parameters and their variation

estimation increased for most methods, especially for Swamy, which has the highest

value for bias and variation in estimating β comparing it to all other methods.

IAUE1 is the method least affected by increasing the disturbance standard

deviation, with bias and variation in estimating β close to zero at different values

for disturbance standard deviation.

Again, Swamy has the lowest absolute bias for both parameters compared with all

other methods for all different error variances, while it has a high variation in

estimating ∆ which increased as error variances increased. MINQUE1, MINQUE2,

and IAUE1 have close bias and variation values. The variation for all methods are

almost the same, except for Swamy, which at 𝜎𝜀 = 10, has a very high value.


Swamy, MINQUE1, and MINQUE2 have produced negative variances. Swamy has

produced more than 50% of negative variances in estimating ∆𝛽° when 𝜎𝜀 = 10,

while IAUE1 has 0% for all different error variances. The RWZ method was

excluded for the same reason mentioned above.

7 Conclusion

Using the Swamy method to estimate the variance component in the RCR model

produces good estimators when the RCR model is the right model to represent the

data. Of course, reducing the sample size, or increasing the disturbance standard

deviation leads to a reduction in the efficiency of the Swamy model, since more

variances that have negative values were produced. When the RCR model is not the

right model to represent the data, Swamy has a very weak attitude regarding the

percentage of negative variances and the variation in estimating both β and ∆.

When the RCR model is the right model to represent the data, MINQUE1 and

MINQUE2 produce a lower percentage of negative variance in estimating ∆𝛽° than

Swamy did, but with higher absolute bias in estimating both β and ∆. When the

RCR model is the inappropriate model to represent the data, both methods are better

than Swamy regarding absolute bias and variation in estimating β and the

percentage of negative variances. In general, we can say MINQUE2 is doing better

than MINQUE1.

IAUE1 has a zero percentage of negative variance, with the bias in estimating β and

the variation in estimating both β and ∆ being the smallest, even if the RCR model

is not the right model to represent data. The weak points in using the IAUE1 method

to estimate the variance component in the RCR model are that firstly, it has a greater

bias in estimating ∆ compared to Swamy, and secondly, it needs a heavy

computation as we use the MINQUE method as its initial value.

References

[1] A. Amiri-Simkooei, K. Moghtased-Azar, R. Tehranchi, An Alternative Method

for Non-negative Estimation of Variance Components, Journal of Geodesy, 88

(2014), 427–439. https://doi.org/10.1007/s00190-014-0693-0

[2] C. Hsiao, Statistical Inference for a Model with Both Random Cross-Sectional

and Time Effects, International Economic Review, 15 (1974), 12-30.

https://doi.org/10.2307/2526085

[3] C. Hsiao, Some Estimation Methods for a Random Coefficient Model,

Econometrica, 43 (1975), 305-325. https://doi.org/10.2307/1913588

[4] C.R. Rao, Estimation of Variance and Covariance Components in Linear

Models, Journal of the American Statistical Association, 67 (1972), 112-115.

https://doi.org/10.1007/s00190-014-0693-0

https://doi.org/10.2307/2526085

https://doi.org/10.2307/1913588


https://doi.org/10.1080/01621459.1972.10481212

[5] D.B. Duncan, S.D. Horn, R.A. Horn, Estimating Heteroscedastic Variances in

Linear Models. Journal of the American Statistical Association, 70 (1975), 380–

385. https://doi.org/10.1080/01621459.1975.10479877

[6] J.R. Lucas, A Variance Component Estimation Method for Sparse Matrix

Applications, NOAA Technical Report NOS 111 NGS 33, National Geodetic

Survey, Rockville. 1985.

[7] L. W. Johnson, Regression with Random Coefficients, Omega, 6 (1978), 71-81.

https://doi.org/10.1016/0305-0483(78)90041-5

[8] M. Aitkin, D. Anderson, J. Hinde, Statistical Modelling of Data on Teaching

Styles, Journal of the Royal Statistical Society A (General), 144 (1981), 419 -461.

https://doi.org/10.2307/2981826

[9] M. H. Pesaran, T. Yamagata, Testing Slope Homogeneity in Large Panels,

Journal of Econometrics, 142 (2008), 50-93.

https://doi.org/10.1016/j.jeconom.2007.05.010

[10] M. R. Abonazel, A. Mousa, A. H. Youssef, A Monte Carlo Study for Swamy’s

Estimate of Random Coefficient Panel Data Model, InterStat Journal, 2011 (2011),

1-12.

[11] M.S. Abdallah, Z.A. Abdel_wahed, H.A. El_leithy, On Non-negative

Estimation of Variance Components in Mixed Linear Models, Journal of Advanced

Research, 7 (2016), 59-68. https://doi.org/10.1016/j.jare.2015.02.001

[12] P. Delicado, J. Romo, Goodness of Fit Tests in Random Coefficient

Regression Models, Annals of the Institute of Statistical Mathematics, 51 (1999),

125-148. https://doi.org/10.1023/a:1003887303233

[13] P. A. Cartwright, N. Riabk, Measuring the effect of oil prices on wheat futures

price, Research in International Business and Finance, 33 (2015), 355–369.

https://doi.org/10.1016/j.ribaf.2014.04.002

[14] P. A.V. B. Swamy, Efficient inferences in a random coefficient regression

model, Econometrica, 38 (1970), 311-323. https://doi.org/10.2307/1913012

[15] P. C. Austin, V. Goel, C. V. Walraven, An Introduction to Multilevel

Regression Models, Canadian Journal of Public Health, 92 (2001), 150-154.

https://doi.org/10.1080/01621459.1972.10481212

https://doi.org/10.1080/01621459.1975.10479877

https://doi.org/10.1016/0305-0483(78)90041-5

https://doi.org/10.2307/2981826

https://doi.org/10.1016/j.jeconom.2007.05.010

https://doi.org/10.1016/j.jare.2015.02.001

https://doi.org/10.1023/a:1003887303233

https://doi.org/10.1016/j.ribaf.2014.04.002

https://doi.org/10.2307/1913012


[16] R. D. Gibbons, D. K. Bhaumik, Weighted Random-Effects Regression Models

with Application to Interlaboratory Calibration, Technometrics, 43 (2001), 192-

198. https://doi.org/10.1198/004017001750386305

[17] R. G. Drynan, W. E. Griffiths, S. Prakash, Bayesian Estimation of a Random

Coefficient Model, Journal of Econometrics, 10 (1979), 201-220.

https://doi.org/10.1016/0304-4076(79)90005-8

[18] R. Hsu, An Alternative Expression for the Variance Factors in Using Iterated

Almost Unbiased Estimation, Journal of Geodesy, 73 (1999), 173–179.

https://doi.org/10.1007/s001900050234

[19] R. Hsu, Helmert Method as Equivalent of Iterated Almost Unbiased

Estimation, Journal of Surveying Engineering, 127 (2001), 79–89.

https://doi.org/10.1061/(asce)0733-9453(2001)127:3(79)

[20] R. L. Carter, M. C. K. Yang, Large sample inference in random coefficient

regression models, Communications in Statistics - Theory and Methods, 15 (1986),

2507-2525. https://doi.org/10.1080/03610928608829265

[21] R. P. Leone, H. D. Oberhelman, F. J. Mulhern, Estimating Individual Cross-

Section Coefficients from the Random Coefficient Regression Model, Journal of

the Academy of Marketing Science, 21 (1993), 45-51.

https://doi.org/10.1177/0092070393211006

[22] S.R. Searle, An Overview of Variance Component Estimation, Metrika, 42

(1995), 215–230. https://doi.org/10.1007/bf01894301

[23] T. E. Dielman, Pooled Cross-Sectional and Time Series Data Analysis, Marcel

Dekker, New York, 1989.

[24] T. Hobza, D. Morales, Small Area Estimation of Poverty Proportions under

Random Regression Coefficient Models, Chapter in Modern Mathematical Tools

and Techniques in Capturing Complexity Understanding Complex System,

Springer, 2011, 315-328. https://doi.org/10.1007/978-3-642-20853-9_22

[25] T. Kall, T. Oja, K. Tanavsuu, Postglacial land uplift in Estonia based on four

precise leveling, Tectonophysics, 610 (2014), 25-38.

https://doi.org/10.1016/j.tecto.2013.10.002

[26] V. V. Anh, T. Chelliah, Estimated Generalized Least Squares for Random

Coefficient Regression Models, Scandinavian Journal of Statistics, 26 (1999), 31-

46. https://doi.org/10.1111/1467-9469.00135

https://doi.org/10.1198/004017001750386305

https://doi.org/10.1016/0304-4076(79)90005-8

https://doi.org/10.1007/s001900050234

https://doi.org/10.1061/(asce)0733-9453(2001)127:3(79)

https://doi.org/10.1080/03610928608829265

https://doi.org/10.1177/0092070393211006

https://doi.org/10.1007/bf01894301

https://doi.org/10.1007/978-3-642-20853-9_22

https://doi.org/10.1016/j.tecto.2013.10.002

https://doi.org/10.1111/1467-9469.00135


[27] W. E. Griffiths, R. Drynan and S. K.B. Rao, Bayesian estimation of a random

coefficient model, Journal of Econometrics, 10 (1979), 201-220

https://doi.org/10.1016/0304-4076(79)90005-8

https://doi.org/10.1016/0304-4076(79)90005-8


Table (A.1) Results of RCR Estimation When 𝛽0~𝑁(10,10) and 𝛽1~𝑁(10,10)

SWAMY R W Z MINQUE1 MINQUE2 IAUE1 IAUE2

𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N

=

T

=

5

𝝈𝝐

=

1

�� 9.779 10.025 9.712 10.037 11.236 9.752 10.036 9.983 10.069 9.978 9.565 9.442

∆ 9.981 10.002 10.165 10.002 10.276 8.950 10.221 8.952 10.270 9.055 10.167 8.952

Bias of �� -0.221 0.025 -0.288 0.037 1.236 -0.248 0.036 -0.017 0.069 -0.023 -0.435 -0.558

Variation

of �� 721.616 21.826 1091.377 33.092 8985.615 356.922 72.289 4.185 55.537 3.566 57.468 5.793

Bias of∆ -0.019 0.002 0.165 0.002 0.276 -1.050 0.221 -1.048 0.270 -0.945 0.167 -1.048

Variation

of ∆ 97.173 50.787 92.922 50.787 91.367 55.368 91.849 55.339 90.862 53.591 92.880 55.339

%negative

variance 10.73% 0.01% 0% 0% 3% 5.03% 1.91% 1.2% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 5.191 11.057 19.158 8.265 9.770 9.928 10.270 9.924 10.151 9.965 7.353 6.923

∆ 4.641 9.655 63.897 10.232 75.358 6.217 66.116 6.547 60.750 6.746 67.114 7.195

Bias of �� -4.809 1.057 9.158 -1.735 -0.203 -0.072 0.270 -0.076 0.151 -0.035 -2.647 -3.077

Variation �� 207135.23 7989.74 200000.9 7484.049 110613.323 4288.823 2013.824 89.111 740.288 26.614 731.565 30.840

Bias of∆ -5.359 -0.345 53.897 0.232 65.358 -3.783 56.116 -3.453 50.750 -3.254 57.114 -2.805

Variation ∆ 27771.385 133.2108 14054.02 118.865 13830.206 116.935 13311.241 110.666 13803.536 108.157 21034.601 120.739

%negative

variance 55.98% 18.85% 0% 0% 17.13% 29.69% 14.07% 5.73% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 11.184 10.011 10.371 9.925 -1.116 12.078 11.373 9.650 10.015 9.978 6.842 6.856

∆ 6.235 9.825 160.731 12.997 203.175 10.593 188.908 10.816 151.854 10.187 5928.646 1001.589

Bias of �� 1.184 0.011 0.371 -0.075 -11.116 2.078 1.373 -0.350 0.015 -0.022 -3.158 -3.144

Variation

of �� 258480.478 5273.384 29222.901 987.966 8096812 236998 8271.280 508.970 890.850 29.918 810.659 31.594

Bias of∆ -3.765 -0.175 150.731 2.997 193.175 0.593 178.909 0.816 141.854 0.187 5918.646 991.589

Variation ∆ 136972.085 320.020 93834.944 308.746 136786.610 346.469 86597.467 287.221 93261.149 292.852 1472097991 38045969

%negative

variance 56.21% 35.2% 0% 0% 23.03% 26.77% 13.32% 13.95% 0.01% 0% 0% 0%

N

=

T

=

10

𝝈𝝐

=

1

�� 10.001 9.979 10.001 9.979 10.034 9.973 10.001 9.979 10.001 9.979 9.994 9.970

∆ 10.004 10.024 10.005 10.024 10.005 10.015 10.005 10.015 10.006 10.016 10.005 10.015

Bias of �� 0.001 -0.021 0.001 -0.021 0.034 -0.027 0.001 -0.021 0.001 -0.021 -0.006 -0.030

Variation �� 1.135 1.024 1.135 1.024 10.902 1.388 1.137 1.024 1.134 1.024 1.181 1.120

Bias of∆ 0.004 0.024 0.005 0.024 0.005 0.015 0.005 0.015 0.001 0.016 0.005 0.015

Variation

of ∆ 27.821 22.060 27.818 22.060 27.802 22.145 27.808 22.145 27.800 22.129 27.818 22.145

%negative

variance 0.09% 0% 0% 0% 0.01% 0.04% 0% 0.01% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 12.857 9.414 9.603 10.069 9.121 10.159 9.969 10.029 9.963 9.999 31.187 15.510

∆ 9.893 9.934 14.297 9.934 23.144 6.363 21.539 6.428 14.668 6.691 24.070 7.218

Bias of �� 2.857 -0.586 -0.397 0.069 -0.879 0.159 -0.031 0.029 -0.037 -0.001 21.187 5.510

Variation

of �� 124512.354 4974.336 1018.536 35.579 2813.485 110.055 1350.637 53.270 157.880 6.066 3614.858 264.154

Bias of∆ -0.107 -0.066 4.297 -0.066 13.144 -3.637 11.539 -3.572 4.668 -3.309 14.069 -2.782

Variation

of ∆ 540.618 27.625 357.868 27.614 260.937 41.855 260.690 41.048 347.242 37.896 815.761 34.209

%negative

variance 38.12% 0.12% 0% 0% 0.19% 13.65% 0.06% 7.01% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 13.421 9.355 10.148 9.952 7.556 10.487 10.306 9.920 10.128 9.960 44619.72 11911.06

∆ 9.606 9.975 32.067 10.014 79.712 6.022 71.835 6.364 32.293 6.327 27692.95 1426.831

Bias of �� 3.421 -0.645 0.148 -0.048 -2.443 0.487 0.306 -0.080 0.128 -0.040 44609.72 11901.06

Variation

of �� 52770.449 1961.913 648.642 25.480 38850.847 1602.002 3826.864 157.345 108.201 5.048 12356186592 412542364

Bias of∆ -0.394 -0.025 22.066 0.014 69.712 -3.978 61.835 -3.636 22.293 -3.673 27682.984 1416.831

Variation ∆ 5058.257 48.141 2675.037 47.292 2498.772 59.413 2268.614 55.756 2642.312 55.284 5053501756 6214327

%negative

variance 50.17% 3.39% 0% 0% 0.22% 19.46% 0.04% 11.3% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

15

𝝈𝝐 = 1

�� 9.996 9.998 9.996 9.998 9.996 0.998 9.996 9.998 9.996 9.998 9.996 9.998

∆ 10.057 10.052 10.057 10.052 10.057 10.052 10.057 10.052 10.057 10.052 10.057 10.052

Bias of �� -0.004 -0.002 -0.004 -0.002 -0.004 -0.002 -0.004 -0.002 -0.004 -0.002 -0.004 -0.002

Variation

of �� 0.724 0.658 0.724 0.658 0.724 0.658 0.724 0.658 0.724 0.658 0.724 0.658

Bias of∆ 0.057 0.052 0.057 0.052 0.057 0.052 0.057 0.052 0.057 0.052 0.057 0.052

Variation

of ∆ 16.242 14.411 16.242 14.411 16.242 14.411 16.242 14.411 16.242 14.411 16.242 14.411

%negative variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 10.097 9.987 10.009 10.005 9.311 10.139 10.368 9.914 10.051 9.996 205.535 14.914

∆ 9.979 10.007 10.856 10.007 15.588 8.073 14.785 8.112 11.053 8.249 90.339 8.699

Bias of �� 0.097 -0.013 0.009 0.005 -0.689 0.139 0.368 -0.086 0.051 -0.004 195.535 4.914

Variation �� 51.848 3.002 36.465 2.191 2659.141 93.851 1210.548 76.131 18.267 1.322 452470.041 176.275

Bias of∆ -0.021 0.007 0.856 0.007 5.588 -1.927 4.785 -1.888 1.053 -1.751 80.339 -1.300

Variation ∆ 133.243 16.403 108.928 16.403 102.602 28.795 95.684 28.196 104.798 26.099 77298.261 20.997

%negative

variance 19.66% 0% 0% 0% 0% 6.97% 0% 4.13% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 9.979 9.994 9.868 10.029 9.957 10.015 10.514 9.895 9.972 10.008 353429.44 12020.45

∆ 10.250 10.076 17.968 10.076 57.615 6.361 53.530 6.548 18.136 6.669 196363.3 1294.342

Bias of �� -0.021 -0.006 -0.132 0.029 -0.042 0.015 0.514 -0.105 -0.028 0.008 353419.44 12010.45

Variation �� 2405.235 93.249 155.341 6.109 344.494 16.355 925.999 39.267 9.175 0.958 550355156962 321444969

Bias of∆ 0.250 0.076 7.968 0.076 47.615 -3.639 43.530 -3.452 8.363 -3.331 196353.261 1284.342

Variation ∆ 1038.674 23.508 605.729 23.494 1602.997 41.290 1389.807 39.231 591.012 37.441 176733180989 3739488

%negative

variance 41.16% 0.12% 0% 0% 0% 15.39% 0% 11.03% 0% 0% 0% 0%

N = T =

20

𝝈𝝐 =

1

�� 10.005 10.008 10.005 10.008 10.005 10.008 10.005 10.008 10.005 10.008 10.005 10.008

∆ 9.981 9.993 9.981 9.993 9.981 9.993 9.981 9.993 9.981 9.993 9.981 9.993

Bias of �� 0.005 0.008 0.005 0.008 0.005 0.008 0.005 0.008 0.005 0.008 0.005 0.008

Variation

of �� 0.509 0.502 0.509 0.502 0.509 0.502 0.509 0.502 0.509 0.502 0.509 0.502

Bias of∆ -0.018 -0.006 -0.502 -0.006 -0.018 -0.006 -0.018 -0.006 -0.018 -0.006 -0.018 -0.006

Variation

of ∆ 11.491 10.459 11.491 10.459 11.491 10.459 11.491 10.459 11.491 10.459 11.491 10.459

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 10.040 9.997 10.023 10.0003 10.046 9.996 10.045 9.996 10.025 10.00004 374.861 11.647

∆ 9.806 9.986 9.951 9.986 11.722 9.275 11.571 9.282 10.024 9.342 148.571 9.491

Bias of �� 0.040 -0.002 0.023 0.0003 0.046 -0.003 0.045 -0.003 0.026 3.507𝑒−5 364.861 1.647

Variation

of �� 3.063 0.593 1.178 0.528 2.913 0.591 5.555 0.683 1.159 0.527 3.163𝑒6 50.843

Bias of∆ -0.193 -0.013 -0.048 -0.013 1.722 -0.724 1.571 -0.717 0.024 -0.657 138.571 -0.508

Variation

of ∆ 54.972 11.441 51.577 11.441 57.621 17.255 55.321 17.119 50.202 16.075 468530.181 13.980

%negative

variance 7.242% 0% 0% 0% 0% 2.628% 0% 2.057% 0% 0% 0% 0%

𝝈𝝐 =

10 �� 10.543 9.902 10.006 10.0008 10.031 9.998 10.134 9.977 10.015 9.999 69159.116 7238.752

∆ 9.683 10.038 14.650 10.038 51.628 6.510 47.778 6.674 15.026 6.833 36404.810 733.570

Bias of �� 0.543 -0.097 0.006 0.0008 0.031 -0.001 0.134 -0.022 0.015 -0.0008 69149.116 7228.752

Variation

of �� 1125.0004 36.129 5.366 0.657 68.548 2.926 27.748 1.650 3.685 0.613 15858467574 110798072

Bias of∆ -0.316 0.038 4.650 0.038 41.628 -3.489 37.778 -3.325 5.026 -3.166 36394.810 723.570

Variation

of ∆ 571.773 15.428 351.915 15.428 1630.149 34.565 1361.342 32.649 341.128 30.523 4604695739 1113180

%negative

variance 37.62 0% 0% 0% 0% 15.42% 0% 10.04% 0% 0% 0% 0%





𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

50

𝝈𝝐 = 1

�� 10.016 9.983 10.016 9.983 10.016 9.983 10.016 9.983 10.016 9.983 10.016 9.983

∆ 9.971 10.110 9.971 10.110 9.971 10.110 9.971 10.110 9.971 10.110 9.971 10.110

Bias of �� 0.016 -0.016 0.016 -0.016 0.016 -0.016 0.016 -0.016 0.016 -0.016 0.016 -0.016

Variation

of �� 0.201 0.209 0.201 0.209 0.201 0.209 0.201 0.209 0.201 0.209 0.201 0.209

Bias of∆ 0.028 0.110 -0.028 0.110 -0.028 0.110 -0.028 0.110 -0.028 0.110 -0.028 0.110

Variation ∆ 5.611 5.740 5.611 5.740 5.611 5.740 5.611 5.740 5.611 5.740 5.611 5.740


𝝈𝝐 =

5

�� 10.0007 9.977 10.0007 9.977 9.977 10.0007 10.0007 9.977 10.0007 9.977 10.0007 9.977

∆ 10.035 10.082 10.035 10.082 10.035 10.082 10.035 10.082 10.035 10.082 10.035 10.082

Bias of �� 0.0007 0.022 0.0007 -0.022 0.0007 -0.022 0.0007 -0.022 0.0007 -0.022 0.0007 -0.022

Variation

of �� 0.306 0.215 0.306 0.215 0.306 0.215 0.306 0.215 0.306 0.215 0.306 0.215

Bias of∆ 0.035 0.082 0.035 0.082 0.035 0.082 0.035 0.082 0.035 0.082 0.035 0.082

Variation ∆ 6.315 6.124 6.315 6.124 6.315 6.124 6.315 6.124 6.315 6.124 6.315 6.124

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 10.014 9.966 10.014 9.966 10.016 9.966 10.168 9.936 9.966 10.014 204253.519 1662.048

∆ 9.478 10.103 9.627 10.103 15.850 9.500 15.728 9.506 9.689 9.564 89434.122 164.859

Bias of �� 0.014 0.014 0.014 -0.033 0.016 -0.033 0.168 -0.063 0.014 -0.033 204243.519 1652.048

Variation

of �� 0.556 0.235 0.555 0.235 0.589 0.241 3.979 0.358 0.555 0.235 735805679721 43549521

Bias of∆ -0.521 0.103 -0.372 0.103 5.850 -0.499 5.728 -0.493 -0.310 -0.435 89424.122 154.859

Variation ∆ 8.159 6.535 8.106 6.535 48.687 6.187 47.370 6.188 8.105 6.186 9.870𝑒9 2.635𝑒4

%negative

variance 6.140% 0% 0% 0% 0% 3.728% 0% 3.070% 0% 0% 0% 0%


𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T = 5

𝝈𝝐 = 1

�� 0.929 0.999 0.704 1.056 -0.163 1.178 0.811 1.033 0.919 1.016 0.631 0.581

∆ 0.924 0.999 1.455 1.001 1.803 0.617 1.674 0.620 1.859 1.023 1.446 0.617

Bias of �� -0.071 -0.000 -0.296 0.056 -1.163 0.178 -0.189 0.033 -0.081 0.016 -0.369 -0.419

Variation �� 481.235 12.628 516.487 16.585 6575.22 391.109 113.669 4.442 78.304 2.853 78.131 3.033

Bias of∆ -7.643e-02

-2.887e-

05 0.455 0.001 0.803 -0.383 0.674 -0.379 0.859 0.023 0.446 -0.383

Variation ∆ 6.842 0.595 4.519 0.594 4.797 0.623 4.018 0.617 3.538 0.365 4.479 0.621

%neg var 41.2% 1.71% 0% 0% 11.76% 17.37% 3.55% 5.1% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 1.101 0.946 22.339 -2.377 19.937 -2.019 0.542 1.040 23.374 0.932 1.575 0.936

∆ 0.389 0.969 24.527 1.518 34.381 1.426 31.062 1.491 23.578 1.832 24.772 1.872

Bias of �� 0.101 -0.054 21.339 -3.377 18.937 -3.019 -0.458 0.040 0.374 -0.068 0.575 -0.064

Variation �� 13927.761 415.395 3333078.

7 76446.3 2501743.58 59000.95 7572.619 272.480 3633.998

118.7

85 3642.418 121.024

Bias of∆ -0.611 -0.031 23.527 0.518 33.381 0.426 30.062 0.491 22.578 0.832 23.772 0.872

Variation ∆ 4232.460 7.579 2080.306 5.069 2835.699 5.683 1945.490 4.644 2052.794 3.816 2179.569 6.734

%neg var 56.5% 41.96% 0% 0% 20.62% 25.08% 14.69% 11.77% 0% 0% 0% 0%

𝝈𝝐

=

1

0

�� -0.388 1.444 -0.387 1.271 -0.541 1.322 1.319 0.805 -0.573 1.334 57.408 78.552

∆ -7.124 0.873 496.091 23.788 529.115 24.847 523.316 25.036 492.571

24.20

8 1064.881 169.581

Bias of �� -1.388 0.444 -1.387 0.271 -1.541 0.322 0.319 -0.195 -1.573 0.334 56.408 77.552

Variation �� 380125.77 10780.8

6

138294.2

53

5587.72

2 35959.875 1478.565

51304.04

9 3259.662

18478.28

0

739.3

13 2022708 1727922

Bias of∆ -8.124 -0.127 495.091 22.788 528.115 23.847 522.316 24.036 491.571

23.20

8 1063.881 168.581

Variation ∆ 2011408.0

67

4558.04

3

1074521.

619

2525.35

6

1066316.95

8 2530.221

1050651.

576 2471.044

1076851.

668

2503.

343 97585076 5347288

%neg var 59.52% 58.88% 0% 0% 19.71% 28.29% 15.83% 12.47% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

10

𝝈𝝐 = 1

�� 1.099 0.977 0.959 1.009 1.015 0.997 0.563 1.094 0.977 1.005 0.832 0.788

∆ 0.971 0.993 1.052 0.993 1.248 0.794 1.188 0.797 1.262 0.994 1.053 0.794

Bias of �� 0.099 -0.023 -0.041 0.009 0.015 -0.003 -0.437 0.094 -0.023 0.005 -0.168 -0.212

Variation

of �� 937.431 36.979 15.299 0.647 3121.383 90.432 1573.326 73.376 7.068 0.358 7.111 0.517

Bias of∆ -0.029 -0.007 0.052 -0.007 0.248 -0.206 0.188 -0.203 0.026 -0.006 0.053 -0.206

Variation ∆ 1.366 0.244 1.152 0.244 0.919 0.358 0.962 0.354 0.879 0.197 1.150 0.359

%negative variance 20.65% 0.02% 0% 0% 0.18% 7.96% 0.03% 2.99% 0% 0% 0% 0%

𝝈𝝐

=

5

�� -1.311 1.344 0.436 1.082 0.156 1.127 0.713 1.007 0.165 1.129 32.629 3.051

∆ 0.727 0.991 6.264 1.021 19.464 0.679 17.018 0.791 6.703 1.182 150.979 3.044

Bias of �� -2.311 0.344 -0.564 0.082 -0.844 0.127 -0.287 0.007 -0.835 0.129 31.629 2.051

Variation

of �� 69662.826 1905.495 7247.355 192.481 8095.771 236.789 10389.923 357.215 6748.363 175.765 15194.225 190.176

Bias of∆ -0.273 -0.009 5.264 0.021 18.464 -0.321 16.018 -0.209 5.703 0.182 149.979 2.044

Variation ∆ 250.031 0.889 132.386 0.817 138.971 0.884 124.052 0.787 125.194 0.453 171328.453 12.655

%negative

variance 55.63% 12.41% 0.13% 20.7% 0% 11.48% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 0.327 1.125 0.799 1.019 0.729 1.056 1.452 0.891 1.029 0.986 3431.953 989.384

∆ 0.966 1.039 23.278 1.596 76.744 1.612 67.027 1.987 22.683 1.904 21612.99 1183.096

Bias of �� -0.673 0.125 -0.200 0.019 -0.270 0.056 0.452 -0.109 0.029 -0.014 3430.953 988.384

Variation

of �� 42361.539 1546.976 1169.258 35.151 379.898 17.508 1639.011 82.949 26.289 0.802 67067069 2583673

Bias of∆ -0.034 0.039 22.278 0.596 75.744 0.612 66.027 0.987 21.683 0.904 21611.989 1182.096

Variation ∆ 3449.968 7.454 1613.779 4.824 1894.981 5.106 1671.923 4.453 1591.793 3.657 2431396941 3441040

%negative

variance 54.88% 39.66% 0% 0% 0.25% 21.01% 0.16% 11.93% 0% 0% 0% 0%

N = T =

15

𝝈𝝐 =

1

�� 0.996 1.003 0.995 1.003 0.994 1.004 0.842 1.033 0.995 1.003 0.962 0.956

∆ 0.998 1.005 1.004 1.005 1.049 0.959 1.042 0.959 1.050 1.003 1.004 0.959

Bias of �� -0.004 0.003 -0.005 0.003 -0.006 0.004 -0.158 0.033 -0.005 0.003 0.038 -0.044

Variation

of �� 0.254 0.073 0.149 0.069 5.084 0.246 94.914 3.455 0.149 0.069 0.168 0.113

Bias of∆ -0.002 0.005 0.004 0.005 0.049 -0.041 0.042 -0.040 0.050 0.003 0.004 -0.041

Variation

of ∆ 0.489 0.153 0.474 0.153 0.428 0.189 0.431 0.189 0.426 0.146 0.474 0.189

%negative

variance 4.543% 0% 0% 0% 0% 1.757% 0% 1.043% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 0.458 1.096 1.524 0.906 0.973 1.003 1.389 0.918 0.991 1.001 30.003 1.188

∆ 1.210 1.019 3.707 1.020 15.066 0.646 13.239 0.721 4.181 1.081 130.310 1.261

Bias of �� -0.542 0.096 0.524 -0.094 -0.027 0.003 0.389 -0.019 -0.009 0.001 29.003 0.188

Variation

of �� 609.309 19.162 2144.166 66.882 83.493 2.777 2198.858 93.732 2.083 0.136 4499.030 0.551

Bias of∆ 0.210 0.019 2.707 0.020 14.066 -0.354 12239 -0.279 3.181 0.081 129.310 0.261

Variation

of ∆ 63.454 0.369 32.476 0.365 96.834 0.524 81.829 0.463 29.039 0.214 8.03e+04 0.497

%negative

variance 48.7% 1.7% 0% 0% 0% 15.371% 0% 10.286% 0% 0% 0% 0%

𝝈𝝐 =

10 �� -0.137 1.199 3.433 0.569 1.353 0.926 16.923 -1.910 1.071 0.987 11655.933 836.337

∆ 0.245 0.968 13.286 1.119 69.051 0.912 62.033 1.191 13.539 1.373 66679.069 875.238

Bias of �� -1.137 0.199 2.433 -0.430 0.353 -0.074 15.923 -2.910 0.071 -0.013 11654.933 835.337

Variation

of �� 2953.459 88.138 14568.094 458.842 384.358 15.457 1700533.60 56276.33 11.439 0.398 445311044.5 962632.1

Bias of∆ -0.755 -0.032 12.286 0.119 68.051 -0.088 61.033 0.191 12.539 0.373 66678.069 874.238

Variation

of ∆ 1152.025 1.933 538.839 1.475 1604.081 1.645 1385.088 1.555 525.064 0.907 1.41e+10 9.728e+05

%negative

variance 55.657% 25.057% 0% 0% 0% 22.4% 0% 15.2% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

20

𝝈𝝐 = 1

�� 1.0003 1.002 1.0004 1.002 1.0004 1.002 1.001 1.002 1.0004 1.002 0.997 0.998

∆ 1.0005 0.995 1.0008 0.995 1.004 0.991 1.004 0.991 1.004 0.995 1.0008 0.991

Bias of �� 0.0003 0.002 0.0004 0.002 0.0004 0.002 0.001 0.002 0.0004 0.002 -0.002 -0.001

Variation

of �� 0.076 0.051 0.075 0.051 0.078 0.051 0.086 0.051 0.075 0.051 0.077 0.055

Bias of∆ 0.0005 -0.004 0.0008 -0.004 0.004 -0.008 0.004 -0.008 0.004 -0.004 0.0008 -0.0008

Variation ∆ 0.257 0.108 0.256 0.108 0.252 0.112 0.253 0.112 0.252 0.108 0.256 0.112

%negative variance 0.414% 0% 0% 0% 0% 0.171% 0% 0.085 0% 0% 0% 0%

𝝈𝝐

=

5

�� 0.919 1.018 1.015 1.001 0.970 1.009 0.905 1.024 1.004 1.003 42.096 1.573

∆ 0.967 0.999 2.370 1.00006 13.466 0.615 12.304 0.668 2.823 1.038 162.916 1.551

Bias of �� -0.080 0.018 0.015 0.001 -0.029 0.009 -0.094 0.024 0.004 0.003 41.096 0.573

Variation

of �� 48.444 1.582 2.301 2.301 24.562 1.101 30.772 1.301 0.744 0.075 5505.083 1.028

Bias of∆ -3.220 -7.714 1.370 0.00006 12.466 -0.384 11.304 -0.331 1.823 0.038 161.916 0.551

Variation ∆ 22.650 0.230 11.639 0.230 112.0007 0.418 95.518 0.373 9.732 0.139 7.960 7.393

%negative

variance 45.642% 0.228% 0% 0% 0% 16.814% 0% 11.471% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 1.871 0.855 0.998 1.001 -0.925 1.382 0.839 1.026 1.0100 0.999 129928.716 1230.514

∆ 1.059 1.009 6.726 1.030 58.002 0.749 54.142 0.930 7.191 1.179 688799.498 1235.629

Bias of �� 0.871 -0.144 -0.001 0.001 -1.925 0.382 -0.160 0.026 0.010 -0.0004 129927.716 1229.514

Variation

of �� 1722.087 43.184 12.176 0.460 11608.102 452.030 256.889 10.842 2.368 0.127 40307097265 1962165

Bias of∆ 0.059 0.009 5.726 0.030 57.002 -0.250 53.142 -0.069 6.191 0.179 688798.498 1234.629

Variation ∆ 252.876 0.659 118.668 0.607 1861.528 0.771 1687.344 0.803 112.001 0.336 1.079 1.861

%negative

variance 52.1% 8.885% 0% 0% 0% 19.028% 0% 14.857% 0% 0% 0% 0%

N = T =

50

𝝈𝝐 =

1

�� 1.002 0.997 1.002 0.997 1.002 0.997 1.002 0.997 1.002 0.997 1.002 0.997

∆ 1.002 1.0005 1.002 1.0005 1.002 1.0005 1.002 1.0005 1.002 1.0005 1.002 1.0005

Bias of �� 0.002 -0.002 0.002 -0.002 0.002 -0.002 0.002 -0.002 0.002 -0.002 0.002 -0.002

Variation

of �� 0.025 0.020 0.025 0.020 0.025 0.020 0.025 0.020 0.025 0.020 0.025 0.020

Bias of∆ 0.002 0.0005 0.002 0.0005 0.002 0.0005 0.002 0.0005 0.002 0.0005 0.002 0.0005

Variation ∆ 0.190 0.186 0.190 0.186 0.190 0.186 0.190 0.186 0.190 0.186 0.190 0.186

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 1.010 0.995 1.010 0.995 1.071 0.983 1.002 0.999 1.010 0.995 1.010 0.995

∆ 1.065 0.994 1.167 0.994 6.244 0.801 6.101 0.808 1.369 1.002 1.369 1.002

Bias of �� 0.010 -0.004 0.010 -0.004 0.071 -0.016 0.002 -0.0002 0.010 -0.004 0.010 -0.004

Variation

of �� 0.123 0.024 0.122 0.024 4.882 0.217 41.276 1.597 0.121 0.024 0.121 0.024

Bias of∆ 0.065 -0.005 0.167 -0.005 5.244 -0.198 5.101 -0.191 0.369 0.002 0.369 0.002

Variation ∆ 0.461 0.164 0.435 0.164 23.734 0.139 22.631 0.139 0.455 0.164 0.455 0.164

%negative

variance 20.104% 0% 0% 0% 0% 10.833% 0% 7.395% 0% 0% 0% 0%

𝝈𝝐 =

10 �� 1.017 0.994 1.008 0.995 1.008 0.994 0.880 1.020 1.007 0.995 1.007 0.995

∆ 1.137 1.003 2.267 1.003 43.434 0.638 42.136 0.703 2.680 1.045 2.680 1.045

Bias of �� 0.017 -0.005 0.008 -0.004 0.008 -0.005 -0.119 0.020 0.007 -0.004 0.007 -0.004

Variation

of �� 0.444 0.034 0.394 0.032 7.763 0.289 18.478 0.806 0.392 0.032 0.392 0.032

Bias of∆ 0.137 0.003 1.267 0.003 42.434 -0.361 41.136 -0.296 1.680 0.045 1.680 0.045

Variation ∆ 4.572 0.220 3.117 0.220 922.670 0.169 874.185 0.189 3.092 0.226 3.092 0.226

%negative

variance 4.572% 0% 0% 0% 0% 21.048% 0% 16.044% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T = 5

𝝈𝝐 = 1

�� 4.846 5.028 4.747 5.046 5.133 4.989 5.285 4.908 4.884 5.020 3.776 2.947

∆ 9.999 0.982 12.442 1.046 11.599 0.840 11.589 0.840 11.766 1.231 11.382 0.849

Bias of �� -0.154 0.028 -0.253 0.046 0.133 -0.010 0.285 -0.092 -0.116 0.020 -1.224 -2.053

Variation

of �� 686.401 27.656 167.997 6.907 7065.117 334.422 1411.358 103.495 89.136 3.793 89.729 10.128

Bias of∆ -0.000 0.018 2.442 0.046 1.599 -0.159 0.1589 -0.159 1.766 0.231 1.382 -0.151

Variation ∆ 375.861 1.492 289.019 1.322 285.223 1.347 283.756 1.343 279.659 0.930 288.149 1.329


𝝈𝝐

=

5

�� 4.958 4.928 5.107 4.992 2.222 5.544 5.106 5.023 5.525 4.965 5.969 4.355

∆ 0.834 0.670 53.154 2.179 68.921 1.951 56.915 2.290 51.308 2.603 51.893 2.495

Bias of �� -0.042 -0.072 0.207 -0.008 52.821 0.829 56.926 2.291 51.584 2.604 52.834 2.911

Variation

of �� 65549.822 1699.397 49026.125 1104.163 99403.568 2870.227 25173.098 504.122 21818.178 416.498 21911.188 440.965

Bias of∆ -9.166 -0.329 43.154 1.179 58.921 0.951 46.915 1.290 41.308 1.603 41.893 1.495

Variation ∆ 22716.709 28.612 12003.865 16.836 17932.175 23.880 11631.592 15.943 12018.537 14.738 12099.407 16.327

%negative

variance 58.43% 52.02% 0% 0% 23.59 34.16 17.97% 7.42% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 4.756 5.117 2.169 5.539 8.456 4.399 4.920 5.009 4.601 5.081 11950.182 2117.064

∆ -5.114 0.493 110.711 3.779 162.094 3.967 136.802 4.698 105.703 4.122 21005.81 680.833

Bias of �� -0.244 0.117 -2.831 0.539 3.456 -0.601 -0.079 0.009 -0.399 0.081 11945.182 2112.064

Variation

of �� 66180.821 1667.105 73405.722 2399.038 192667.681 5756.227 4227.460 162.986 1404.661 52.319 349400889598 266405627

Bias of∆ -15.114 -0.507 100.711 2.779 152.094 2.967 126.802 3.698 95.702 3.122 20995.808 679.833

Variation ∆ 92615.276 89.361 44360.843 44.647 58261.801 58.079 40842.794 39.599 44614.617 40.817 999140174892 28399851

%negative

variance 57.71% 54.76% 0% 0% 17.87% 29.03% 10.5% 8.14% 0% 0% 0% 0%

N = T =

10

𝝈𝝐 =

1

�� 4.998 4.995 4.998 4.995 4.999 4.994 4.999 4.994 4.998 4.995 4.997 4.993

∆ 10.005 1.008 10.005 1.008 10.005 1.007 10.005 1.007 10.005 1.007 10.005 1.007

Bias of �� -0.002 -0.005 -0.002 -0.005 -0.001 -0.006 -0.000 -0.006 -0.002 -0.005 -0.003 -0.007

Variation

of �� 1.126 0.106 1.126 0.106 1.139 0.107 1.156 0.108 1.126 0.106 1.129 0.118

Bias of∆ 0.005 0.008 0.005 0.008 0.005 0.007 0.005 0.007 0.005 0.007 0.005 0.007

Variation ∆ 27.540 0.245 27.539 0.245 27.532 0.245 27.532 0.245 27.532 0.245 27.539 0.245

%negative

variance 0.04% 0% 0% 0% 0% 0.01% 0% 0.01% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 7.036 4.593 4.904 5.014 4.544 5.082 4.905 5.015 4.760 5.042 40.364 9.359

∆ 9.639 0.982 13.597 1.036 22.982 0.842 21.277 0.913 13.122 1.228 44.348 2.067

Bias of �� 2.036 -0.407 -0.096 0.014 -0.456 0.082 -0.095 0.015 -0.239 0.042 35.364 4.359

Variation

of �� 56797.744 2247.801 1450.777 55.129 482.679 18.692 1498.749 57.756 384.994 14.842 9115.353 119.157

Bias of∆ -0.361 -0.018 3.597 0.035 12.982 -0.158 11.277 -0.087 3.122 0.228 34.348 1.067

Variation ∆ 489.029 1.124 334.525 0.987 246.004 1.052 241.109 0.953 324.375 0.631 6736.187 4.151

%negative

variance 37.58% 16.95% 0% 0% 0.21% 16.68% 0% 9.19% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 2.019 5.527 4.278 5.123 -1.281 6.389 5.477 4.892 4.909 5.003 41506.528 2615.932

∆ 9.319 0.986 30.135 1.521 80.142 1.514 57.755 2.399 28.416 1.837 53391.14 602.671

Bias of �� -2.981 0.527 -0.722 0.123 -6.281 1.389 0.477 -0.108 -0.091 0.003 41501.528

2610.932

Variation

of �� 42098.527 1449.086 8012.731 280.192 358948.87 18144.04 2155.271 78.972 217.859 5.158 23170464851 19584387

Bias of∆ -0.681 -0.014 20.135 0.521 70.142 0.514 47.755 1.399 18.416 0.837 53381.136 601.671

Variation ∆ 4394.206 6.668 2311.897 4.252 2301.606 4.749 2019.998 3.407 2295.829 3.164 40594028775 1107469

%negative

variance 49.71% 39.46% 0% 0% 0.7% 23.05% 0.24% 5.8% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

15

𝝈𝝐 = 1

�� 5.004 4.999 5.004 4.999 5.004 4.999 5.004 4.999 5.004 4.999 5.004 4.999

∆ 10.006 0.999 10.006 0.999 10.006 0.999 10.006 0.999 10.006 0.999 10.006 0.999

Bias of �� 0.004 -0.001 0.004 -0.001 0.004 -0.001 0.004 -0.001 0.004 -0.001 0.004 -0.001

Variation

of �� 0.727 0.067 0.727 0.067 0.727 0.067 0.727 0.067 0.727 0.067 0.727 0.067

Bias of∆ 0.006 -0.000 0.006 -0.000 0.006 -0.000 0.006 -0.000 0.006 -0.000 0.006 -0.000

Variation ∆ 16.421 0.149 16.421 0.149 16.421 0.149 16.421 0.149 16.421 0.149 16.421 0.149


𝝈𝝐 =

5

�� 4.773 5.055 4.996 5.006 4.936 5.017 4.925 5.022 4.981 5.008 100.479 9.782

∆ 9.889 0.998 10.661 1.005 15.647 0.879 15.002 0.912 10.608 1.075 84.661 1.955

Bias of �� -0.227 0.055 -0.005 0.006 -0.064 0.017 -0.075 0.022 -0.019 0.008 95.479 4.782

Variation

of �� 3046.032 116.873 40.165 1.395 27.226 1.324 38.566 2.365 7.244 0.393 83855.459 138.117

Bias of∆ -0.111 -0.002 0.661 0.005 5.647 -0.121 5.002 -0.089 0.608 0.075 74.661 0.955

Variation ∆ 122.583 0.466 101.723 0.451 100.406 0.533 93.712 0.493 98.985 0.345 51408.168 4.852

%negative

variance 18.64% 4.52% 0% 0% 0% 8.28% 0% 5.54% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 5.007 4.999 4.793 5.037 4.509 5.092 4.956 5.007 4.982 5.001 84805.38 5839.10

∆ 10.149 1.014 17.969 1.168 63.640 1.053 57.648 1.323 17.264 1.418 94449.02 1244.769

Bias of �� 0.007 -0.001 -0.207 0.037 -0.491 0.092 -0.044 0.007 -0.018 0.001 84800.38 5834.10

Variation

of �� 3912.271 138.359 704.283 25.123 575.278 22.368 20.156 0.909 5.639 0.242 26655708488 58088742

Bias of∆ 0.149 0.014 7.969 0.168 53.640 0.053 47.648 0.323 7.264 0.418 94439.017 1243.769

Variation ∆ 915.515 1.907 567.209 1.536 1815.486 1.625 1526.727 1.613 554.138 1.152 30681549209 2501753

%negative

variance 42.039% 25.375% 0% 0% 0% 18.383% 0% 11.799% 0% 0% 0% 0%

N = T =

20

𝝈𝝐 =

1

�� 5.007 5.0009 5.007 5.0009 5.007 5.0009 5.007 5.0009 5.007 5.0009 5.007 5.0009

∆ 9.942 1.003 9.942 1.003 9.942 1.003 9.942 1.003 9.942 1.003 9.942 1.003

Bias of �� 0.007 0.0009 0.007 0.0009 0.007 0.0009 0.007 0.0009 0.007 0.0009 0.007 0.0009

Variation

of �� 0.507 0.052 0.507 0.052 0.507 0.052 0.507 0.052 0.507 0.052 0.507 0.052

Bias of∆ -0.057 0.003 -0.057 0.003 -0.057 0.003 -0.057 0.003 -0.057 0.003 -0.057 0.003

Variation ∆ 11.378 0.107 11.378 0.107 11.378 0.107 11.378 0.107 11.378 0.107 11.378 0.107

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 5.141 4.973 5.078 4.984 5.216 4.955 5.027 4.993 5.019 4.995 29.768 5.925

∆ 10.014 1.003 10.204 1.003 12.273 0.939 12.094 0.948 10.270 1.018 29.194 1.186

Bias of �� 0.141 -0.026 0.078 -0.015 0.216 -0.044 0.027 -0.006 0.019 -0.004 24.768 0.925

Variation

of �� 56.642 1.800 12.752 0.500 251.964 10.286 1.564 0.089 1.241 0.076 11783.019 13.825

Bias of∆ 0.014 0.003 0.204 0.003 2.273 -0.060 2.094 -0.051 0.270 0.018 19.194 0.186

Variation ∆ 60.629 0.232 56.077 0.232 62.426 0.288 59.245 0.275 54.624 0.211 6859.306 0.642

%negative

variance 8.114% 0.371% 0% 0% 0% 3.457% 0% 2.385% 0% 0% 0% 0%

𝝈𝝐 =

10 �� 5.196 4.959 5.163 4.970 5.523 4.903 4.695 5.070 5.023 4.998 221966.45 5139.51

∆ 10.193 1.020 14.329 1.071 53.169 0.909 49.580 1.074 14.177 1.250 231207.123 1035.703

Bias of �� 0.196 -0.040 0.163 -0.029 0.523 -0.096 -0.304 0.070 0.023 -0.001 221961.45 5134.51

Variation

of �� 1161.808 49.628 55.619 2.111 938.508 35.310 541.203 24.080 3.161 0.145 156552817279 50276565

Bias of∆ 0.193 0.020 4.329 0.071 43.169 -0.090 39.580 0.074 4.177 0.250 231197.123 1034.703

Variation ∆ 489.722 1.027 314.927 0.892 1623.762 1.037 1398.915 0.981 308.220 0.584 177216288388 2053979

%negative

variance 35.728% 14.328% 0% 0% 0% 15.657% 0% 11.242% 0% 0% 0% 0%




SWAMY R W Z

MINQUE1 MINQUE2 IAUE1 IAUE2

𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T = 5

𝝈𝝐 = 1

�� 0.784 1.067 0.798 1.056 17.419 -1.681 0.188 1.103 0.232 1.149 -0.023 0.709

∆ 0.996 4.959 1.708 4.959 2.037 2.820 1.945 2.822 2.140 3.244 1.721 2.827

Bias of �� -0.216 0.067 -0.202 0.056 16.419 -2.681 -0.812 0.103 -0.768 0.149 -1.023 -0.291

Variation

of �� 6978.418 265.869 8072.326 293.388 2176060.20 52456.47 6831.249 265.171 5772.118 233.951 5771.543 233.910

Bias of∆ -0.004 -0.040 0.708 -0.040 1.037 -2.179 0.945 -2.178 1.140 -1.756 0.721 -2.173

Variation ∆ 10.403 12.788 6.689 12.787 7.169 13.289 6.189 13.278 5.458 11.146 6.645 13.253

%negative variance 43.65% 0.1% 0% 0% 14.35% 17.28% 9.88% 8% 0% 0% 0% 0%

𝝈𝝐

=

5

�� -8.532 2.681 -0.767 1.324 8.478 -0.494 0.602 1.086 0.872 1.033 2.311 1.354

∆ -8.851 4.701 60.336 5.889 71.451 4.310 63.746 4.578 57l899 4.814 65.173 5.152

Bias of �� -7.532 1.681 -1.767 0.324 7.478 -1.494 -0.398 0.086 -0.128 0.033 1.331 0.354

Variation

of �� 427016.31 13359.49 6409.330 231.502 382936.08 16025.21 374.403 15.368 149.516 6.519 792.656 22.613

Bias of∆ -9.851 -0.299 59.336 0.889 70.451 -0.689 62.747 -0.422 565.899 -0.186 64.173 0.152

Variation ∆ 29987.926 81.406 14307.072 61.987 14317.955 62.806 13651.791 58.811 14170.992 56.796 23942.568 62.322

%negative

variance 59.69% 33.2% 0% 0% 19.09% 30.99% 14.01% 5.75% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 1.695 0.886 -33.672 7.225 -7.292 2.522 1.495 0.913 1.211 0.968 94.038 70.330

∆ -28.973 3.815 195.154 9.457 262.075 8.321 214.986 9.649 190.717 9.051 1017.729 118.670

Bias of �� 0.695 -0.114 -34.672 6.225 -8.292 1.522 0.495 -0.087 0.211 -0.032 93.038 69.330

Variation

of �� 18112.328 506.164 10813135.5 356181.5 314150.06 10065.42 3475.557 105.124 399.126 10.616 2996315.6 678493.6

Bias of∆ -29.973 -1.185 194.154 4.457 261.075 3.321 213.986 4.649 189.717 4.051 1016.729 113.670

Variation ∆ 352317.988 479.053 178591.585 292.279 275702.967 407.389 172626.330 279.014 179022.366 286.344 152629107 1409920

%negative

variance 61.6% 49.54% 0% 0% 24.01% 34.3% 18.05% 7.18% 0.06% 0% 0% 0%


𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

50

𝝈𝝐 = 1

�� 5.0002 4.996 5.0002 4.996 5.0002 4.996 5.0002 4.996 5.0002 4.996 5.0002 4.996

∆ 9.986 1.004 9.986 1.004 9.986 1.004 9.986 1.004 9.986 1.004 9.986 1.004

Bias of �� 0.0002 -0.003 0.0002 -0.003 0.0002 -0.003 0.0002 -0.003 0.0002 -0.003 0.0002 -0.003

Variation

of �� 0.214 0.020 0.214 0.020 0.214 0.020 0.214 0.020 0.214 0.020 0.214 0.020

Bias of∆ -0.013 0.004 -0.013 0.004 -0.013 0.004 -0.013 0.004 -0.013 0.004 -0.013 0.004

Variation ∆ 16.644 0.168 16.644 0.168 16.644 0.168 16.644 0.168 16.644 0.168 16.644 0.168


𝝈𝝐 =

5

�� 5.017 5.0003 5.017 5.0003 5.017 5.0003 5.017 5.0003 5.017 5.0003 5.017 5.0003

∆ 10.292 1.024 10.292 1.024 10.292 1.024 10.292 1.024 10.292 1.024 10.292 1.024

Bias of �� 0.017 0.0003 0.017 0.0003 0.017 0.0003 0.017 0.0003 0.017 0.0003 0.017 0.0003

Variation

of �� 0.301 0.022 0.301 0.022 0.301 0.022 0.301 0.022 0.301 0.022 0.301 0.022

Bias of∆ 0.292 0.024 0.292 0.024 0.292 0.024 0.292 0.024 0.292 0.024 0.292 0.024

Variation ∆ 7.193 0.067 7.193 0.067 7.193 0.067 7.193 0.067 7.193 0.067 7.193 0.067

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 5.060 4.985 5.059 4.985 5.057 4.986 5.206 4.974 5.058 4.985 66460.380 496.900

∆ 9.973 0.993 10.086 0.993 13.788 0.962 13.748 0.964 10.123 1.0001 57519.522 93.926

Bias of �� 0.060 -0.014 0.059 -0.014 0.057 -0.013 0.206 -0.025 0.058 -0.014 66455.380 491.900

Variation

of �� 0.603 0.035 0.602 0.035 0.600 0.035 12.355 0.237 0.601 0.035 129413933150 6504362

Bias of∆ -0.026 -0.006 0.086 -0.006 3.788 -0.037 3.748 -0.035 0.123 0.0001 57509.522 92.926

Variation ∆ 6.437 0.052 6.409 0.052 24.873 0.051 24.664 0.051 6.409 0.053 5.175𝑒9 1.200𝑒4

%negative

variance 3.693% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

10

𝝈𝝐 = 1

�� 1.009 0.985 1.096 0.970 1.139 0.951 2.162 0.726 1.001 0.986 0.894 0.829

∆ 0.997 4.989 1.042 4.988 1.189 4.238 1.160 4.240 1.196 4.386 1.043 4.239

Bias of �� 0.009 -0.015 0.096 -0.029 0.139 -0.049 1.162 -0.274 0.001 -0.014 -0.106 -0.171

Variation

of �� 4.171 0.597 100.147 3.623 500.345 28.105 15416.010 768.742 0.579 0.518 0.621 0.577

Bias of∆ -0.003 -0.012 0.042 -0.012 0.189 -0.761 0.160 -0.758 0.196 -0.614 0.043 -0.761

Variation ∆ 1.080 5.628 0.967 5.628 0.799 8.038 0.816 8.027 0.778 6.909 0.966 8.040

%negative variance 15.18% 0% 0% 0% 0.06% 5.82% 0% 3.01% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 0.008 1.188 0.467 1.093 0.807 1.039 0.736 1.041 0.904 1.010 6.575 1.841

∆ 0.827 5.025 8739 5.029 21.573 2.646 19.341 2.732 9.256 3.124 34.511 3.931

Bias of �� -0.992 0.288 -0.533 0.093 -0.193 0.039 -0.264 0.041 -0.096 0.010 5.575 0.841

Variation

of �� 13510.282 519.757 2746.641 108.491 1005.785 38.101 2254.118 80.582 26.198 1.365 301.015 6.954

Bias of∆ -0.173 0.025 7.739 0.029 20.573 -2.354 18.341 -2.268 8.256 -1.876 33.511 -1.069

Variation ∆ 464.134 9.769 225.210 9.733 195.770 12.681 175.587 12.238 215.741 10.368 3709.283 10.127

%negative

variance 54.45% 0.97% 0% 0% 0.59% 21.6% 0.15% 12.56% 0% 0% 0% 0%

𝝈𝝐 =

10

�� -8.209 2.652 -0.422 1.287 0.307 1.122 2.269 0.745 1.095 0.969 6516.748 1335.972

∆ -0.684 4.926 27.549 5.082 81.585 3.294 72.509 3.677 27.508 3.634 41582.47 1628.939

Bias of �� -9.209 1.652 -1.422 0.287 -0.693 0.122 1.269 -0.255 0.095 -0.031 6515.748 1334.972

Variation

of �� 357987.84 11851.81 15554.002 695.089 1025.717 38.081 10213.152 384.109 378.546 11.966 324371655 6123064

Bias of∆ -1.684 -0.074 26.549 0.082 80.585 -1.706 71.509 -1.323 26.508 -1.366 41581.471 1623.939

Variation ∆ 5215.377 22.009 2506.748 20.143 2409.734 22.207 2116.426 20.101 2463.606 19.463 8954740995 6073156

%negative

variance 57.25% 12.53% 0% 0% 0.35% 22.47% 0.03% 12.25% 0% 0% 0% 0%

N = T =

15

𝝈𝝐 =

1

�� 0.995 0.998 0.996 0.997 0.991 0.996 0.995 0.997 0.996 0.997 0.972 0.963

∆ 0.998 5.015 1.002 5.015 1.034 4.846 1.028 4.847 1.036 4.879 1.002 4.846

Bias of �� -0.005 -0.002 -0.004 -0.003 -0.009 -0.004 -0.005 -0.003 -0.004 -0.003 -0.028 -0.037

Variation

of �� 0.149 0.335 0.123 0.334 11.163 0.665 0.227 0.338 0.122 0.334 0.136 0.358

Bias of∆ -0.002 0.015 0.002 0.015 0.034 -0.154 0.028 -0.153 0.036 -0.121 0.002 -0.154

Variation ∆ 0.437 3.622 0.427 3.622 0.394 4.296 0.397 4.293 0.392 4.014 0.427 4.297

%negative

variance 3.371% 0% 0% 0% 0% 1.086% 0% 0.586% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 1.272 0.953 1.144 0.976 1.980 0.798 0.792 1.052 1.117 0.981 100.363 2.864

∆ 0.880 5.029 3.675 5.029 15.755 2.601 14.284 2.677 4.185 3.057 382.601 4.763

Bias of �� 0.272 -0.047 0.144 -0.024 0.980 -0.202 -0.208 0.052 0.117 -0.019 99.363 1.864

Variation

of �� 283.105 10.163 52.427 1.722 1610.709 73.327 855.091 42.525 47.531 1.547 50986.224 13.223

Bias of∆ -0.119 0.029 2.675 0.029 14.755 -2.399 13.284 -2.323 3.185 -1.943 381.601 -0.237

Variation ∆ 51.413 8.469 28.904 8.469 123.248 7.935 104.762 7.759 27.193 6.927 4.537967e+05 8.901121e+00

%negative

variance 50.912% 0% 0% 0% 0% 17.830 0% 12.892% 0% 0% 0% 0%

𝝈𝝐 =

10 �� -0.532 1.298 1.891 0.826 1.112 0.964 0.578 1.099 0.909 1.022 33791.06 2417.51

∆ 0.693 4.965 12.209 4.980 64.526 2.883 58.057 3.213 12.692 3.276 178926.357 2637.287

Bias of �� -1.532 0.298 0.891 -0.174 0.112 -0.036 -0.422 0.099 -0.091 0.022 33790.06 2416.51

Variation

of �� 5881.109 219.431 10631.766 423.454 678.102 38.688 690.866 34.804 148.043 5.407 4629160902 12014951

Bias of∆ -0.307 -0.035 11.209 -0.019 63.526 -2.117 57.057 -1.787 11.692 -1.724 178925.357 2632.287

Variation ∆ 910.095 9.447 407.998 9.316 1664.037 13.008 1399.474 11.682 395.059 10.699 102755061124 10281354

%negative

variance 54% 2.657% 0% 0% 0% 20.886% 0% 13.614% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

20

𝝈𝝐 = 1

�� 1.003 1.009 1.003 1.009 1.005 1.008 0.961 1.018 1.003 1.009 1.0008 1.0005

∆ 0.994 4.991 0.994 4.991 0.998 4.972 0.997 4.972 0.998 4.975 0.994 4.972

Bias of �� 0.003 0.009 0.003 0.009 0.005 0.008 -0.038 0.018 0.003 0.009 0.0008 0.005

Variation

of �� 0.075 0.248 0.075 0.248 0.087 0.249 9.583 0.707 0.075 0.248 0.077 0.251

Bias of∆ -0.005 -0.005 -0.005 -0.008 -0.001 -0.027 -0.002 -0.027 -0.001 -0.024 -0.005 -0.027

Variation ∆ 0.241 2.620 0.240 2.620 0.237 2.701 0.237 2.701 0.237 2.669 0.240 2.701

%negative variance 0.371 % 0 % 0% 0% 0 % 0.114 % 0 % 0.085 % 0 % 0 % 0 % 0 %

𝝈𝝐

=

5

�� 1.130 0.985 0.997 1.012 -31.092 7.579 0.669 1.061 0.997 1.012 38.285 1.669

∆ 0.866 4.973 2.429 4.973 14.061 2.684 12.898 2.736 2.905 3.133 145.204 3.726

Bias of �� 0.130 -0.0145 -0.002 0.012 -32.092 6.579 -0.330 0.061 -0.002 0.012 37.285 0.669

Variation

of �� 138.036 4.948 0.794 0.276 7096232.4 298019.5 1163.453 32.641 0.758 0.275 4227.075 1.790

Bias of∆ -0.133 -0.026 1.429 -0.026 13.061 -2.315 11.898 -2.263 1.905 -1.866 144.204 -1.273

Variation ∆ 26.662 3.074 13.868 3.074 113.286 8.053 96.516 7.793 11.804 5.888 52573.214 4.104

%negative

variance 47.642 % 0 % 0% 0% 0 % 18.785 0 % 13.4 % 0 % 0 % 0 % 0 %

𝝈𝝐

=

10

�� 3.501 0.546 0.995 0.995 1.557 0.883 0.913 1.010 1.014 0.991 21388.233 1460.862

∆ 0.906 5.015 7.913 5.015 59.313 2.688 54.608 2.914 8.437 3.132 110669.983 1478.983

Bias of �� 2.5018 -0.453 -0.004 -0.004 0.557 -0.116 -0.086 0.010 0.014 -0.008 21387.233 1459.862

Variation

of �� 45139.172 1439.135 4.097 0.399 1242.303 50.201 110.307 4.049 2.747 0.346 1203281340 3555518

Bias of∆ -0.093 0.0150 6.913 0.015 58.313 -2.311 53.608 -2.085 7.437 -1.867 110668.983 1473.983

Variation ∆ 363.200 5.042 165.201 5.041 1856.567 9.903 1612.585 9.010 157.164 7.682 27732470542 2730336

%negative

variance 52.557 % 0.0428 % 0% 0% 0 % 20.7 % 0 % 14.485 % 0 % 0 % 0 % 0%

N = T =

50

𝝈𝝐 =

1

�� 1.003 0.992 1.003 0.992 1.003 0.992 1.003 0.992 1.003 0.992 1.003 0.992

∆ 1.006 5.000 1.006 5.000 1.006 5.000 1.006 5.000 1.006 5.000 1.006 5.000

Bias of �� 0.003 -0.007 0.003 -0.007 0.003 -0.007 0.003 -0.007 0.003 -0.007 0.003 -0.007

Variation

of �� 0.024 0.103 0.024 0.103 0.024 0.103 0.024 0.103 0.024 0.103 0.024 0.103

Bias of∆ 0.006 0.000 0.006 0.000 0.006 0.000 0.006 0.000 0.006 0.000 0.006 0.000

Variation ∆ 0.057 0.056 0.057 1.056 0.057 1.056 0.057 1.056 0.057 1.056 0.057 1.056

%negative

variance 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 1.001 0.994 1.001 0.994 -1.106 1.388 1.426 0.912 1.001 0.995 1.001 0.995

∆ 1.000 5.028 1.133 5.028 6.392 3.970 6.249 3.978 1.343 4.179 1.343 4.179

Bias of �� 0.001 -0.005 0.001 -0.005 -2.106 0.388 0.426 -0.087 0.001 -0.004 0.001 -0.004

Variation

of �� 0.110 0.101 0.109 0.101 4201.232 146.093 76.427 2.771 0.109 0.101 0.109 0.101

Bias of∆ 2.171𝑒−5 0.028 0.133 0.028 5.392 -1.029 5.249 -1.021 0.343 -0.820 0.343 -0.820

Variation ∆ 1.570 1.148 1.158 1.148 93.763 5.093 89.497 5.036 0.847 3.601 0.847 3.601

%negative

variance 21% 0% 0% 0% 0% 9.6% 0% 8% 0% 0% 0% 0%

𝝈𝝐 =

10 �� 1.047 0.988 1.047 0.988 1.121 0.975 0.117 1.200 1.046 0.988 1.046 0.988

∆ 1.002 5.052 2.242 5.052 43.811 3.024 42.410 3.096 2.659 3.440 2.659 3.440

Bias of �� 0.047 -0.011 0.047 -0.011 0.121 -0.024 -0.882 0.200 0.046 -0.011 0.046 -0.011

Variation

of �� 0.423 0.111 0.398 0.111 7.766 0.364 1163.976 47.642 0.397 0.111 0.397 0.111

Bias of∆ 0.002 0.052 1.242 0.052 42.811 -1.975 41.410 -1.903 1.659 -1.559 1.659 -1.559

Variation ∆ 18.580 1.333 9.020 1.333 22.82.727 7.240 2178.037 6.941 7.396 4.959 7.396 4.959

%negative

variance 41.4% 0% 0% 0% 0% 21.7% 0% 16.8% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T = 5

𝝈𝝐 = 1

�� 7.615 10.560 9.588 10.092 10.114 9.999 10.408 9.931 10.212 9.979 6.309 3.601

∆ 9.599 -0.011 10.231 0.069 6.482 0.078 6.318 0.082 6.652 0.681 6.052 0.085

Bias of �� -2.385 0.560 -0.412 0.092 0.114 -0.000 0.408 -0.69 0.212 -0.021 -3.691 -6.399

Variation

of �� 23747.446 1282.242 4300.622 135.109 1462.088 44.905 453.421 12.860 285.539 2.239 295.664 26.698

Bias of∆ -0.401 -0.011 0.232 0.069 -3.518 0.078 -3.682 0.082 -3.348 0.681 -3.948 0.085

Variation ∆ 141.207 0.041 125.148 0.019 118.554 0.020 118.709 0.018 114.657 0.171 121.665 0.017


𝝈𝝐

=

5

�� 13.856 9.365 11.914 9.705 12.197 9.737 10.894 9.912 10.705 9.960 32.724 20.059

∆ 0.044 -0.284 55.204 1.727 64.302 1.991 60.057 2.085 53.472 2.312 65.683 3.459

Bias of �� 3.856 -0.635 1.914 -0.295 2.197 -0.263 0.894 -0.088 0.705 -0.039 22.724 10.059

Variation

of �� 84877.103 2390.529 39278.504 1320.559 48653.911 1102.417 6313.202 63.894 5592.819 20.966 41342.409 4142.664

Bias of∆ -9.956 -0.284 45.204 1.727 54.302 1.992 50.057 2.085 43.472 2.312 55.683 3.459

Variation ∆ 23183.807 25.487 11815.255 12.048 12037.706 12.801 11322.968 11.056 11857.737 10.204 24972.939 66.329

%negative

variance 57.48% 59.6% 0% 0% 22.28% 23.18% 13.31% 7.42% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 32.545 5.968 16.325 8.876 13.782 9.510 10.754 9.959 10.419 10.037 14290.438 8111.947

∆ -28.786 -1.089 202.966 6.939 244.385 8.033 227.719 8.404 199.257 7.425 13111.43 1376.92

Bias of �� 22.545 -4.032 6.325 -1.124 3.782 -0.489 0.754 -0.040 0.419 0.037 14280.438 8101.947

Variation

of �� 5980799.3 260540.2 305368.16 11442.11 168788.528 4908.427 22430.02 1312.31 20073.156 1179.716 13570036160 1825606951

Bias of∆ -38.786 -1.089 192.966 6.939 234.385 8.033 217.719 8.404 189.257 7.425 13101.43 1376.92

Variation ∆ 353051.347 409.610 171699.425 194.263 175649.570 206.222 163696.456 178.343 172419.601 186.857 15359361643 68279481

%negative

variance 59.81% 59.55% 0% 0% 22.24% 23.1% 13% 7.29% 0% 0% 0% 0%

N = T =

10

𝝈𝝐 =

1

�� 9.868 10.024 10.041 9.992 9.988 10.001 9.964 10.007 9.992 10.001 6.363 4.143

∆ 10.062 -3.631e-05 10.062 0.012 5.302 0.014 5.222 0.017 5.345 0.561 4.789 0.017

Bias of �� -0.132 0.024 0.041 0.008 -0.012 0.001 -0.036 0.006 -0.008 0.001 -3.637 -5.857

Variation

of �� 842.753 33.142 19.540 0.578 42.989 1.582 15.266 0.417 7.605 0.126 17.598 27.326

Bias of∆ 0.061 3.631e-05 0.062 0.012 -4.698 0.014 -4.778 0.017 -4.655 0.561 -5.210 0.017

Variation ∆ 29.189 9.35e-04 29.174 4.344e-04 38.26 .4 80e-04 38.964 0.000 37.814 0.228 42.876 5.008e-04

%negative

variance 0.16% 55.76% 0% 0% 0.28% 21.39% 0.2% 12.76% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 9.333 10.091 10.545 9.901 9.477 10.108 9.738 10.056 9.25 10.039 34.404 16.465

∆ 9.932 -0.001 14.665 0.297 25.844 0.346 23.669 0.434 12.521 0.876 23.569 1.495

Bias of �� -0.667 0.091 0.545 -0.099 -0.523 0.108 -0.262 0.056 -0.175 0.039 24.404 6.465

Variation

of �� 7666.998 335.578 4595.949 165.068 633.662 25.851 379.376 15.596 264.809 11.278 3072.953 256.880

Bias of∆ -0.068 -0.001 4.665 0.297 15.844 0.346 13.669 0.434 2.521 0.876 13.569 1.495

Variation ∆ 578.375 0.584 373.991 0.272 236.612 0.304 245.706 0.0264 374.037 0.170 726.736 2.572

%negative

variance 38.17% 55.75% 0% 0% 0.3% 23.03% 0.2% 13.73% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 5.953 10.800 13.293 9.314 9.649 10.052 10.209 9.937 10.383 0.903 19248.268 9351.227

∆ 9.643 -0.004 37.028 1.187 90.929 1.397 82.077 1.754 34.920 1.744 12091.39 1117.366

Bias of �� 4.047 0.800 3.293 -0.686 -0.351 0.052 0.209 -0.063 0.384 -0.097 19238.268 9341.227

Variation

of �� 131627.599 4573.942 29128.773 1173.811 1841.826 77.354 807.742 34.735 328.339 16.612 1565714833 219644006

Bias of∆ -0.357 -0.004 27.028 2.287 80.929 1.397 72.077 1.754 24.920 1.744 12081.394 1117.366

Variation ∆ 6967.775 9.351 3542.718 4.345 2607.445 4.904 2538.349 4.238 3569.485 3.243 665060845 3340543

%negative

variance 50.58% 55.76% 0% 0% 0.3% 23.53% 0.21% 13.99% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

15

𝝈𝝐 = 1

�� 10.071 9.982 9.989 10.001 9.966 10.006 9.990 10.002 9.987 10.002 5.979 4.149

∆ 9.944 -0.000 9.944 0.005 5.040 0.006 4.985 0.009 5.057 0.551 4.499 0.006

Bias of �� 0.071 -0.018 -0.010 0.001 -0.034 0.006 -0.009 0.002 -0.013 0.002 -4.021 -5.851

Variation

of �� 137.107 5.675 2.365 0.076 8.928 0.355 4.155 0.131 1.114 0.022 13.413 27.126

Bias of∆ 0.056 -0.000 -0.056 0.005 -4.959 0.006 -5.015 0.009 -4.942 0.551 -5.501 0.006

Variation ∆ 16.576 1.986e-04 16.576 9.10e-05 28.486 1.123e-04 28.943 1.077e-04 28.319 0.231 33.096 1.007

%negative variance 0% 55.84 0% 0% 0% 22.61% 0% 14.54% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 10.496 9.876 10.667 9.846 10.453 9.885 10.630 9.853 10.561 9.865 66.794 18.963

∆ 9.715 -0.005 10.788 0.134 21.652 0.157 20.208 0.221 8.072 0.701 31.276 1.539

Bias of �� 0.496 -0.124 0.667 -0.154 0.453 -0.115 0.630 -0.147 0.561 -0.135 56.794 8.963

Variation

of �� 9221.823 442.783 5530.883 278.438 5478.876 276.591 5455.286 275.594 5367.169 272.546 12032.041 462.230

Bias of∆ -0.285 -0.005 0.788 0.134 11.652 0.157 10.208 0.221 -1.928 0.701 21.276 1.539

Variation ∆ 149.988 0.124 119.788 0.057 79.295 0.071 77.164 0.068 123.629 0.142 1009.874 2.187

%negative

variance 22.31% 55.84% 0% 0% 0% 23.64% 0% 15.3% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 10.596 9.899 8.834 10.218 9.839 10.025 10.176 9.965 10.023 9.992 45572.36 13920.18

∆ 9.209 -0.019 19.717 0.535 76.195 0.627 70.102 0.896 17.805 1.109 25275.93 1489.561

Bias of �� 0.596 -0.101 -1.166 0.218 -0.160 0.025 0.176 -0.035 0.023 -0.008 45562.36 13910.18

Variation

of �� 6052.616 233.660 9665.184 347.718 497.735 17.907 404.664 14.357 34.549 1.576 4053009339 223964037

Bias of∆ -0.792 -0.019 9.717 0.535 66.195 0.627 60.102 0.896 7.805 1.109 25265.926 1489.561

Variation ∆ 1490.563 1.987 823.832 0.910 1331.139 1.151 1167.574 1.102 827.039 0.534 1324714240 2650099

%negative

variance 45.15% 55.84% 0% 0% 0% 24.67% 0% 15.79% 0% 0% 0% 0%

N = T =

20

𝝈𝝐 =

1

�� 9.990 10.001 9.993 10.0004 10.029 9.993 9.982 10.003 9.993 10.0004 5.775 4.220

∆ 10.05 1.524𝑒−5 10.053 0.002 5.100 0.003 5.055 0.005 5.105 0.546 4.550 0.002

Bias of �� 0.009 0.001 -0.006 0.0004 0.029 -0.006 -0.017 0.003 -0.006 0.0004 -4.224 -5.779

Variation

of �� 6.789 0.273 0.546 0.001 2.555 0.078 7.649 0.313 0.544 0.001 14.478 26.792

Bias of∆ 0.053 0.00001 0.053 0.002 -4.899 0.003 -4.944 0.005 -4.894 0.546 -5.449 0.002

Variation ∆ 12.087 5.613𝑒−5 12.087 2.492𝑒−5 26.599 3.498𝑒−5 26.976 4.416𝑒−5 26.547 0.235 31.349 2.519𝑒−5

%negative

variance 0% 55.5% 0% 0% 0% 24.471% 0% 16.557% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 9.685 10.062 9.978 10.003 10.140 9.972 9.849 10.031 9.993 10.0005 136.814 19.158

∆ 10.131 0.0003 10.357 0.073 20.331 0.083 19.195 0.134 6.977 0.623 56.042 1.458

Bias of �� -0.314 0.062 -0.021 0.003 0.140 -0.027 -0.150 0.031 -0.006 0.0005 126.814 9.158

Variation

of �� 717.974 28.753 41.381 1.541 40.630 1.555 50.623 2.296 1.420 0.033 28103.589 139.783

Bias of∆ 0.131 0.0003 0.357 0.073 10.331 0.083 9.195 0.134 -3.022 0.623 46.042 1.458

Variation ∆ 69.300 0.035 63.742 0.015 61.570 0.021 57.207 0.027 74.578 0.173 3956.074 1.924

%negative

variance 8.871% 55.5% 0% 0% 0% 24.814% 0% 16.757% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 9.802 10.044 10.011 9.997 10.119 9.977 9.699 10.063 9.978 10.003 101729.75 14621.23

∆ 10.280 0.001 14.967 0.294 70.544 0.332 65.822 0.543 12.899 0.862 53068.826 1481.897

Bias of �� -0.197 0.044 0.011 -0.002 0.119 -0.022 -0.300 0.063 -0.021 0.003 101719.75 14611.23

Variation

of �� 5667.787 224.860 16.039 0.561 225.126 8.775 185.417 8.411 3.583 0.111 16533359114 210848071

Bias of∆ 0.280 0.001 4.967 0.294 60.544 0.332 55.822 0.543 2.899 0.862 53058.826 1481.897

Variation ∆ 575.550 0.561 364.800 0.249 1452.747 0.357 1276.926 0.453 370.941 0.154 4769889275 2203940

%negative

variance 36.385% 55.5% 0% 0% 0% 26% 0% 17.642% 0% 0% 0% 0%





𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T = 5

𝝈𝝐 = 1

�� 5.079 4.977 1.149 5.832 5.492 4.916 5.347 4.884 4.985 5.004 3.063 1.863

∆ -0.397 0.976 1.965 0.982 2.356 0.472 2.193 0.476 2.523 1.066 1.932 0.479

Bias of �� 0.079 -0.023 -3.851 0.832 0.492 -0.084 0.347 -0.116 -0.015 0.004 -1.937 -3.137

Variation

of �� 613.956 21.066 217081.88 9123.07 8178.796 258.791 2256.190 198.561 109.696 3.394 111.916 9.515

Bias of∆ -0.397 -0.024 1.965 -0.018 2.356 -0.528 2.193 -0.524 2.523 0.066 1.932 -0.521

Variation ∆ 34.569 0.729 16.464 0.717 16.516 0.651 15.321 0.646 14.046 0.338 16.069 0.643

%negative variance 60.38% 5.71% 0% 0% %21.32 22.24 %12.57 %7.02 %0 %0 %0 %0

𝝈𝝐

=

5

�� 11.610 4.064 8.735 4.191 7.088 4.659 5.740 4.819 5.299 4.958 16.219 10.026

∆ -10.207 0.687 48.931 2.353 58.641 2.424 54.381 2.519 47.729 2.749 59.818 3.896

Bias of �� 6.610 -0.936 3.735 -0.809 2.088 -0.341 0.740 -0.181 0.299 -0.042 11.219 5.026

Variation

of �� 355983.736 9690.219 197638.796 8030.697 48939.471 1305.639 3653.827 286.838 482.052 15.080 9030.502 1044.520

Bias of∆ -10.207 -0.313 48.931 1.353 58.641 1.424 54.381 1.519 47.729 1.749 59.818 2.896

Variation ∆ 21496.707 30.748 10193.414 17.176 10481.720 17.827 9716.825 15.988 10181.562 14.927 23174.759 69.618

%negative

variance 60.44% 51.96% 0% 0% 22.55% 23.41% 13.55% 7.42% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 4.274 5.566 -6.222 7.115 9.196 4.363 5.835 4.863 5.305 5.023 7099.283 4066.495

∆ -39.504 -0.146 196.583 7.515 238.113 8.463 221.381 8.841 192.757 7.866 13145.19 1380.224

Bias of �� -0.726 0.566 -11.222 2.115 4.196 -0.637 0.835 -0.137 0.305 0.023 7094.283 4061.495

Variation

of �� 40957.38 2879.85 808381.54 27746.52 160557.846 4128.049 8826.749 480.950 3529.257 68.224 3078169239 454146401

Bias of∆ -39.504 -1.146 196.583 6.515 238.113 7.463 221.381 7.841 192.757 6.866 13145.189 1379.244

Variation ∆ 345745.105 428.146 164687.750 211.972 168937.061 223.825 156870.383 195.600 165269.923 204.898 15345841886 68173474

%negative

variance 60.37% 57.55% 0% 0% 22.26% 23.25% 13.02% 7.43% 0% 0% 0% 0%


𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

50

𝝈𝝐 = 1

�� 10.014 10.0004 10.014 10.0003 10.016 10.00006 9.835 10.0377 10.015 10.0001 0.031 4.464

∆ 9.751 -0.00008 9.751 0.0005 5.076 0.0006 5.060 0.001 5.077 0.535 4.537 0.0005

Bias of �� 0.014 0.0004 0.014 0.0003 0.016 5.575𝑒−5 -0.164 0.037 0.015 0.0001 -4.757 5.535

Variation

of �� 0.209 0.0001 0.208 0.0001 0.210 0.0001 20.150 0.880 0.208 0.0001 19.426 25.927

Bias of∆ -0.248 -0.00008 -0.248 -0.0005 -4.923 0.0006 -4.939 0.001 -4.922 0.535 -5.462 0.0005

Variation ∆ 5.797 1.275𝑒−7 5.797 6.355𝑒−8 2.839 1.436𝑒−7 2.838 7.491𝑒−7 2.839 0.031 2.825 6.355𝑒−8

%negative variance 0% 54.233% 0% 0% 0% 24.713% 0% 20.137% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 10.012 10.002 10.023 10.0003 10.044 9.993 9.276 10.158 10.027 9.999 2832.960 20.414

∆ 9.464 -0.002 9.464 0.012 18.359 0.015 17.941 0.037 5.341 0.549 828.504 1.481

Bias of �� 0.012 0.002 0.023 0.0003 0.044 -0.006 -0.723 0.158 0.027 -0.0004 2822.960 10.414

Variation

of �� 0.358 0.005 0.304 0.003 0.554 0.023 538.872 23.574 0.308 0.003 8729769.997 109.230

Bias of∆ -0.535 -0.002 -0.535 0.012 8.359 0.015 7.941 0.037 -4.658 0.549 818.504 1.481

Variation ∆ 5.251 6.910𝑒−5 5.251 3.373𝑒−5 20.91 7.910𝑒−5 19.94 4.279𝑒−4 2.995 0.028 7.860𝑒4 0.221

%negative

variance 0% 54.475% 0% 0% 0% 25.575% 0% 20.204% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 10.035 9.996 10.027 9.998 10.027 9.996 10.189 9.982 10.031 9.997 10.031 9.997

∆ 9.845 0.0003 9.912 0.057 59.440 0.061 57.817 0.146 6.688 0.587 6.688 0.587

Bias of �� 0.035 -0.003 0.027 -0.001 0.027 -0.003 0.189 -0.017 0.031 -0.002 0.031 -0.002

Variation

of �� 0.632 0.015 0.057 0.014 1.253 0.058 2069.557 89.195 0.600 0.014 0.600 0.014

Bias of∆ -0.154 0.0003 -0.087 0.057 49.440 0.061 47.817 0.146 -3.311 0.587 -3.311 0.587

Variation ∆ 24.862 0.004 24.755 0.002 1.021𝑒3 5.483𝑒−3 971.849 0.035 19.071 0.102 19.071 0.102

%negative

variance 3.934% 53.156% 0% 0% 0% 26.532% 0% 21.043% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

10

𝝈𝝐 = 1

�� 5.016 4.999 6.527 4.717 5.073 4.983 4.838 5.033 5.009 5.002 3.186 2.056

∆ -0.003 1.009 0.309 1.009 0.829 0.461 0.747 0.464 0.872 1.011 0.314 0.464

Bias of �� 0.016 -0.001 1.527 -0.283 0.073 -0.017 -0.162 0.033 0.009 0.002 -1.814 -2.944

Variation

of �� 107.367 3.909 22448.893 786.046 1078.146 47.441 362.494 14.623 24.783 0.830 27.404 7.558

Bias of∆ -0.003 0.009 0.309 0.009 0.829 -0.539 0.747 -0.536 0.872 0.011 0.314 -0.536

Variation ∆ 0.634 0.248 0.289 0.248 0.258 0.384 0.239 0.381 0.189 0.119 0.286 0.382


𝝈𝝐 =

5

�� 10.318 4.089 25.563 1.106 9.423 4.193 9.341 4.211 9.576 4.167 21.609 7.341

∆ -0.066 1.011 7.724 1.109 21.065 0.802 18.936 0.888 8.094 1.316 18.926 1.928

Bias of �� 5.318 -0.910 20.563 -3.894 4.423 -0.807 4.341 -0.789 4.576 -0.833 16.609 2.341

Variation

of �� 221544.680 7303.003 2251268.39 80001.79 211644.706 7062.807 210890.063 7029.873 210532.962 7015.654 211125.454 7082.889

Bias of∆ -0.066 0.011 7.724 0.096 21.065 -0.198 18.936 -0.112 8.094 0.316 18.926 0.928

Variation ∆ 396.195 1.475 180.957 1.240 158.091 1.331 146.911 1.214 172.123 0.741 623.601 2.602

%negative

variance 56.09% 20.54% 0% 0% 0.29% 22.55% 0.22% 13.29% 0% 0% 0% 0%

𝝈𝝐

=

10

�� -2.026 6.255 5.251 4.943 4.535 5.085 4.633 5.065 4.963 5.002 9501.623 4633.263

∆ -0.277 1.010 30.886 1.865 85.905 1.874 77.126 2.228 30.302 2.213 11926.7 1107.536

Bias of �� -7.026 1.255 0.251 -0.057 -0.465 0.085 -0.367 0.065 -0.037 0.002 9496.623 4628.263

Variation

of �� 496876.6 17539.5 1416.989 54.699 1955.485 78.465 750.000 28.335 142.503 5.020 383190940 54779790

Bias of∆ -0.277 0.010 30.886 0.865 85.905 0.874 77.126 1.228 30.302 1.213 11926.699 1106.536

Variation ∆ 6338.535 12.319 2895.139 7.412 2446.048 7.881 2292.884 6.897 2877.509 5.923 647998985 3310072

%negative

variance 56.1% 44.12% 0% 0% 0.3% 23.35% 0.23% 13.83% 0% 0% 0% 0%

N = T =

15

𝝈𝝐 =

1

�� 5.333 4.936 4.996 4.999 4.954 5.008 4.879 5.023 4.997 4.999 3.017 2.108

∆ -0.005 0.999 0.136 0.999 0.672 0.452 0.617 0.455 0.688 0.990 0.137 0.451

Bias of �� 0.333 -0.064 -0.004 -0.001 -0.046 0.008 -0.121 0.023 -0.003 -0.001 -1.983 -2.892

Variation

of �� 1089.503 38.026 0.371 0.078 403.579 16.305 87.698 3.256 0.091 0.068 3.248 6.819

Bias of∆ -0.005 -0.001 0.136 -0.001 0.672 -0.548 0.617 -0.545 0.688 -0.009 0.137 -0.549

Variation ∆ 0.126 0.154 0.055 0.154 0.167 0.319 0.142 0.318 0.152 0.072 0.055 0.320

%negative

variance 55.18% 0% 0% 0% 0% 22.15% 0% 14.32% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 0.399 5.848 5.235 4.952 4.902 5.017 4.943 5.011 5.010 4.996 32.042 9.427

∆ -0.123 1.001 3.411 1.009 16.939 0.603 15.538 0.665 3.943 1.122 26.233 1.938

Bias of �� -4.601 0.848 0.235 -0.048 -0.098 0.017 -0.057 0.011 0.010 -0.004 27.042 4.427

Variation

of �� 179706.656 6169.358 404.529 15.905 457.071 19.425 83.583 3.251 3.887 0.197 1674.441 47.264

Bias of∆ -0.123 0.001 3.411 0.009 16.939 -0.397 15.538 -0.335 3.943 0.122 26.233 0.938

Variation ∆ 78.461 0.514 34.382 0.496 104.023 0.644 88.531 0.587 30.831 0.263 1091.579 1.609

%negative

variance 55.18% 4.88% 0% 0% 0% 22.43% 0% 14.5% 0% 0% 0% 0%

𝝈𝝐 =

10

�� -2.431 6.362 5.039 4.983 4.742 5.043 4.995 5.000 4.954 5.004 22002.368 6779.398

∆ -0.657 0.993 13.576 1.251 70.147 1.075 64.236 1.337 13.729 1.539 24391.63 1452.068

Bias of �� -7.431 1.362 0.039 -0.017 -0.258 0.043 -0.005 0.000 -0.046 0.004 21997.368 6774.398

Variation

of �� 177217.189 7069.555 342.657 13.229 828.592 34.454 285.781 10.473 6.221 0.286 984401503 56427535

Bias of∆ -0.657 -0.007 13.576 0.251 70.147 0.075 64.236 0.337 13.729 0.539 24391.627 1451.068

Variation ∆ 1091.849 2.833 499.329 2.114 1955.753 2.300 1658.239 2.126 487.421 1.571 1172761594 2549327

%negative

variance 55.444% 31.199% 0% 0% 0% 23.622% 0% 15.156% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

20

𝝈𝝐 = 1

�� 5.001 4.998 4.997 4.999 5.080 4.9833 5.230 4.952 4.997 4.999 2.914 2.145

∆ 0.001 1.003 0.081 1.003 0.625 0.456 0.581 0.458 4.997 0.991 0.081 0.455

Bias of �� 0.001 -0.001 -0.002 -0.0007 0.080 -0.016 0.230 -0.047 -0.002 -0.0007 -2.085 -2.854

Variation

of �� 7.115 0.271 0.050 0.052 72.912 2.847 411.837 16.795 0.048 0.052 3.599 6.736

Bias of∆ 0.001 0.003 0.081 0.003 0.625 -0.543 0.581 -0.541 0.629 -0.008 0.081 -0.544

Variation ∆ 0.043 0.113 0.019 0.113 0.186 0.302 0.162 0.300 0.177 0.053 0.019 0.303

%negative variance 54.8% 0% 0% 0% 0% 24.171% 0% 16.557% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 5.600 4.890 5.019 4.995 5.178 4.964 5.170 4.965 4.997 4.999 67.386 9.513

∆ 0.028 1.0005 2.042 1.0008 15.645 0.528 14.546 0.577 2.588 1.054 50.810 1.878

Bias of �� 0.600 -0.109 0.019 -0.004 0.178 -0.035 0.170 -0.034 -0.002 -0.0004 62.386 4.513

Variation

of �� 3275.317 109.839 3.829 0.179 101.369 3.943 348.718 14.471 0.895 0.081 7064.270 35.208

Bias of∆ 0.028 0.0005 2.042 0.0008 15.645 -0.471 14.546 -0.422 2.588 0.054 50.810 0.878

Variation ∆ 27.084 0.250 12.121 0.249 116.448 0.442 101.705 0.404 10.128 0.128 4311.114 1.161

%negative

variance 54.8% 0.8% 0% 0% 0% 24.214% 0% 16.6% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 5.045 4.886 5.088 4.532 4.941 5.292 4.985 5.073 17.042 4.988 7053.785 49116.405

∆ 0.997 0.114 1.059 8.171 0.763 64.013 0.967 59.472 1.274 8.604 1429.831 51203.923

Bias of �� -0.113 0.045 -0.467 0.088 0.292 -0.058 0.073 -0.014 -0.011 0.001 49111.405 7048.785

Variation

of �� 548.834 19.126 931.221 34.401 207.648 8.087 309.286 13.336 3.060 0.157 4130473558 53314982

Bias of∆ 0.114 -0.002 8.171 0.059 64.013 -0.236 59.472 -0.032 8.604 0.274 51203.923 1428.831

Variation ∆ 433.351 1.084 193.945 0.919 1834.232 1.143 1603.609 1.057 185.390 0.516 4736818021 2221692

%negative

variance 54.8% 16.171% 0% 0% 0% 25.014% 0% 17.042% 0% 0% 0% 0%

N = T =

50

𝝈𝝐 =

1

�� 5.003 5.001 5.002 5.001 4.959 5.014 5.994 4.798 5.003 5.001 5.003 5.001

∆ -0.0002 0.995 0.0161 0.995 0.540 0.477 0.523 0.478 0.540 0.997 0.540 0.997

Bias of �� 0.003 0.001 0.002 0.001 -0.040 0.014 0.994 -0.201 0.003 0.001 0.003 0.001

Variation

of �� 0.004 0.021 0.003 0.021 2.207 0.195 2052.725 84.557 0.003 0.021 0.003 0.021

Bias of∆ -0.0002 -0.004 0.0161 -0.004 0.540 -0.522 0.523 -0.521 0.540 -0.002 0.540 -0.002

Variation ∆ 0.0002 0.123 0.0001 0.123 0.067 0.064

0.063 0.064 0.067 0.121 0.067 0.121

%negative

variance 52.387% 0% 0% 0% 0% 27.496% 0% 19.247% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 5.003 4.999 5.009 4.997 4.963 5.010 3.231 5.395 5.009 4.997 5.009 4.997

∆ -0.007 1.004 0.402 1.004 13.744 0.484 13.336 0.505 0.936 1.013 0.936 1.013

Bias of �� 0.0003 -0.0008 0.009 -0.002 -0.036 0.010 -1.768 0.395 0.009 -0.002 0.009 -0.002

Variation

of �� 0.158 0.026 0.096 0.024 1.413 0.119 8953.042 402.322 0.095 0.024 0.095 0.024

Bias of∆ -0.007 0.004 0.402 0.004 13.744 -0.515 13.336 -0.494 0.936 0.013 0.936 0.013

Variation ∆ 0.152 0.136 0.080 0.136 46.061 0.073 43.726 0.074 0.144 0.135 0.144 0.135

%negative

variance 53.395% 0% 0% 0% 0% 27.163% 0% 19.707% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 5.020 5.0002 5.009 5.001 4.978 5.009 4.414 5.161 5.010 5.001 5.010 5.001

∆ 0.032 1.004 1.614 1.004 53.578 0.538 51.820 0.629 2.134 1.050 2.134 1.050

Bias of �� 0.020 0.0002 0.009 0.001 -0.021 0.009 -0.585 0.161 0.010 0.001 0.010 0.001

Variation

of �� 0.992 0.050 0.387 0.033 1.029 0.066 8206.602 362.642 0.384 0.033 0.384 0.033

Bias of∆ 0.032 0.004 1.614 0.004 53.578 -0.461 51.820 -0.370 2.134 0.050 2.134 0.050

Variation ∆ 3.935 0.216 2.028 0.216 1122.120 0.147 1059.365 0.174 2.037 0.221 2.037 0.221

%negative

variance 51.967% 0% 0% 0% 0% 26.184% 0% 19.116% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N

=

T

=

5

𝝈𝝐

=

1

�� -0.333 0.049 -0.849 0.118 -0.294 0.056 0.022 0.003 -0.071 0.011 -0.071 0.011

∆ -0.468 -0.013 1.913 0.068 2.330 0.079 2.173 0.083 2.521 0.701 1.901 0.086

Bias of �� -0.333 0.049 -0.849 0.118 -0.294 0.056 0.022 0.003 -0.071 0.011 -0.071 0.011

Variation

of ��

619.942 22.299 5523.281 109.694 570.983 21.123 101.693 2.479 54.838 1.273 54.481 1.259

Bias of∆ -0.468 -0.013 1.913 0.068 2.330 0.079 2.173 0.083 2.521 0.701 1.901 0.086

Variation ∆ 34.586 0.041 16.515 0.020 16.993 0.021 15.747 0.018 14.410 0.164 16.512 0.018

%negativ

e variance

61.03% 59.98% 0% 0% 22.43% 23.17% 12.65% 7.18% 0% 0% 0% 0%

𝝈𝝐

=

5

�� -1.474 0.220 -4.394 0.619 -1.489 0.285 0.124 0.011 -0.369 0.059 -0.290 0.067

∆ -11.834 -0.326 47.679 1.686 58.016 1.979 54.065 2.064 47.510 2.289 59.772 3.458

Bias of �� -1.474 0.220 -4.394 0.619 -1.489 0.285 0.124 0.011 -0.369 0.059 -0.290 0.067

Variation

of ��

15159.38

1

551.565 138032.4

06 2740.094 14257.348 527.267 2515.730 60.848 1353.533 31.008 1584.113 32.436

Bias of∆ -11.834 -0.326 47.679 1.686 58.016 1.979 54.065 2.064 47.510 2.289 59.772 3.458

Variation ∆ 21640.51 25.901 10371.49

9 12.528 10678.721 13.229 9904.022 11.547

10357.05

4 10.709 22080.311 60.377

%negativ

e variance

61.07% 60.06% 0% 0% 22.53% 23.17% 13.16% 7.12% 0% 0% 0% 0%

𝝈𝝐

=

1

0

�� -3.328 0.486 -8.488 1.181 -2.942 0.562 0.218 0.027 -0.719 0.114 78.886 9.487

∆ -46.706 -1.272 191.331 6.774 233.109 7.931 217.399 8.269 188.806 7.285 13339.49 1381.005

Bias of �� -3.328 0.486 -8.488 1.181 -2.942 0.562 0.218 0.027 -0.719 0.114 78.886 9.487

Variation

of ��

61994.19

8

2229.949 552328.1 10969.4 57098.255 2112.271 10169.359 247.976 5486.307 127.348 112255008 1888581

Bias of∆ -46.706 -1.272 191.331 6.774 233.109 7.931 217.399 8.269 188.806 7.285 13339.487 1381.005

Variation ∆ 345946.2 414.999 165204.3

2 200.223 169979.762 211.582 157523.903 184.643

165680.5

3 192.885 14445024787 57729448

%negativ

e variance

61.03% 59.98% 0% 0% 22.43% 23.17% 12.65% 7.18% 0% 0% 0% 0%

N

=

T

=

1

0

𝝈𝝐

=

1

�� -2.346 0.446 -0.293 0.055 0.049 -0.014 -0.045 0.005 -0.069 0.010 -0.066 0.009

∆ 0.001 0.000 0.316 0.012 0.868 0.015 0.779 0.018 0.921 0.614 0.313 0.017

Bias of �� -2.346 0.446 -0.293 0.055 0.049 -0.014 -0.045 0.005 -0.069 0.010 -0.066 0.009

Variation

of ��

56899.91

0

2099.181 911.401 32.095 199.411 6.347 91.822 2.002 78.274 1.426 78.207 1.424

Bias of∆ 0.001 0.000 0.316 0.012 0.868 0.015 0.779 0.018 0.921 0.614 0.313 0.017

Variation ∆ 0.659 0.001 0.306 0.000 0.247 0.001 0.234 0.000 0.171 0.218 0.306 0.001

%negativ

e variance

56.07% 55.41% 0% 0% 0.45% 22.96% 0.25% 13.45% 0% 0% 0% 0%

𝝈𝝐

=

5

�� -11.737 2.233 -1.465 0.273 0.243 -0.068 -0.223 0.026 -0.343 0.050 -0.349 0.051

∆ 0.033 0.009 7.906 0.305 21.709 0.364 19.476 0.453 8.291 0.897 18.966 1.497

Bias of �� -11.737 2.233 -1.465 0.273 0.243 -0.068 -0.223 0.026 -0.343 0.050 -0.349 0.051

Variation

of ��

1422497.

26

52479.49 22785.02

9 802.371 4985.278 158.667 2295.539 50.052 1957.071 35.666 1963.282 35.724

Bias of∆ 0.033 0.009 7.906 0.305 21.709 0.364 19.476 0.453 8.291 0.897 18.966 1.497

Variation ∆ 411.675 0.601 191.399 0.284 154.525 0.315 145.987 0.271 184.014 0.162 577.803 2.506

%negativ

e variance

56.07% 55.42% 0% 0% 0.45% 22.97% 0.24% 13.45% 0% 0% 0% 0%

𝝈𝝐

=

1

0

�� -23.463 4.464 -2.931 0.547 0.486 -0.137 -0.447 0.053 -0.686 0.100 0.105 0.800

∆ 0.135 0.035 31.623 1.218 86.824 1.454 77.894 1.813 31.322 1.783 11486.04 1085.071

Bias of �� -23.463 4.464 -2.931 0.547 0.486 -0.137 -0.447 0.053 -0.686 0.100 0.105 0.800

Variation

of ��

5689991.

0

209918.1 91140.10

7 3209.484 19941.109 634.669 9182.155 200.207 7828.250 142.659 1012998.54 83348.46

Bias of∆ 0.135 0.035 31.623 1.218 86.824 1.454 77.894 1.813 31.322 1.783 11486.041 1085.071

Variation ∆ 6586.211 9.613 3061.993 4.537 2473.295 5.042 2336.589 4.341 3053.393 3.393 586801868 3245085

%negativ

e variance

56.07% 55.41% 0% 0% 0.45% 22.96% 0.25% 13.45% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

15

𝝈𝝐 = 1

�� 0.246 -0.068 -0.025 0.005 0.018 -0.033 -0.059 0.013 0.000 0.000 -0.001 0.000

∆ 0.002 0.000 0.141 0.006 0.733 0.007 0.672 0.009 0.752 0.606 0.137 0.006

Bias of �� 0.246 -0.069 -0.025 0.005 0.018 -0.033 -0.059 0.013 0.000 0.000 -0.001 0.000

Variation

of �� 693.584 35.516 3.724 0.130 3.344 0.126 11.325 0.484 1.447 0.047 1.417 0.046

Bias of∆ 0.002 0.000 0.142 0.006 0.733 0.007 0.672 0.009 0.752 0.606 0.137 0.006

Variation ∆ 0.129 0.000 0.006 9.516e-05 0.146 0.000 0.126 0.000 0.129 0.221 0.059 0.000

%negative variance 55.0.43% 55.371% 0% 0% 0% 25.529% 0% 15.943% 0% 0% 0% 0%

𝝈𝝐

=

5

�� 1.232 -0.344 -0.123 0.024 0.089 -0.017 -0.295 0.064 0.003 0.001 0.04 -3.95e-05

∆ 0.054 0.004 3.543 0.139 18.313 0.163 16.805 0.229 4.019 0.736 28.035 1.613

Bias of �� 1.232 -0.344 -0.123 0.024 0.089 -0.017 -0.295 0.064 0.003 0.001 0.04 -3.95e-05

Variation

of �� 17339.608 887.893 93.092 3.252 83.595 3.154 283.117 12.097 36.238 1.173 47.949 1.203

Bias of∆ 0.054 0.004 3.543 0.139 18.313 0.163 16.805 0.229 4.019 0.736 28.035 1.613

Variation ∆ 81.072 0.127 36.989 0.059 91.523 0.074 78.532 0.069 33.480 0.126 1057.620 2.181

%negative

variance 55.043% 55.371% 0% 0% 0% 25.529% 0% 15.943% 0% 0% 0% 0%

𝝈𝝐

=

10

�� 2.465 -0.687 -0.247 0.047 0.179 -0.034 -0.590 0.128 0.006 0.001 19.393 -0.898

∆ 0.216 0.017 14.173 0.559 73.252 0.065 67.219 0.919 14.230 1.140 25233.726 1504.641

Bias of �� 2.465 -0.687 -0.247 0.047 0.179 -0.034 -0.590 0.128 0.006 0.001 19.393 -0.898

Variation

of �� 69358.43 3551.57 327.366 13.007 334.379 12.614 1132.470 48.387 144.929 4.692 2686886.23 48315.47

Bias of∆ 0.216 0.017 14.173 0.559 73.252 0.651 67.219 0.919 14.230 1.140 25233.726 1504.641

Variation ∆ 1297.158 2.024 591.826 0.952 1464.372 1.191 1256.510 1.108 582.429 0.542 1245036621 2657338

%negative

variance 55.043% 55.371% 0% 0% 0% 25.529% 0% 15.943% 0% 0% 0% 0%

N = T =

20

𝝈𝝐 =

1

�� 0.049 -0.013 -0.017 0.003 -0.003 0.0006 0.058 -0.010 -0.006 0.001 -0.007 0.001

∆ 1.996𝑒−4 2.562𝑒−5 0.081 0.002 0.677 0.003 0.628 0.005 0.685 0.597 0.078 0.002

Bias of �� 0.049 -0.013 -0.017 0.003 -0.003 0.0006 0.058 -0.010 -0.006 0.001 -0.007 0.001

Variation

of �� 63.855 2.414 1.215 0.032 10.129 0.397 48.294 1.802 0.718 0.013 0.702 0.013

Bias of∆ 1.996𝑒−4 2.562𝑒−5 0.081 0.002 0.677 0.003 0.628 0.005 0.685 0.597 0.078 0.002

Variation ∆ 0.042 5.523𝑒−5 0.018 2.428𝑒−5 0.169 3.561𝑒−5 0.148 4.462𝑒−5 0.161 0.227 0.018 2.481𝑒−5

%negative

variance 54.185% 54.957% 0% 0% 0% 26.771% 0% 17.828% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 0.247 -0.065 -0.086 0.017 -0.015 0.003 0.292 -0.054 -0.033 0.007 0.029 0.006

∆ 0.004 0.0006 2.039 0.073 16.928 0.084 15.715 0.138 2.562 0.665 54.074 1.548

Bias of �� 0.247 -0.065 -0.086 0.017 -0.015 0.003 0.292 -0.054 -0.033 0.007 0.029 0.006

Variation

of �� 1596.384 60.361 30.383 0.811 253.243 9.930 1207.373 45.065 17.963 0.345 51.457 0.360

Bias of∆ 0.004 0.0006 2.039 0.073 16.928 0.084 15.715 0.138 2.562 0.665 54.075 1.548

Variation ∆ 26.470 0.034 11.460 0.015 106.157 0.022 92.814 0.027 9.483 0.161 3859.753 1.813

%negative

variance 54.185% 54.957% 0% 0% 0% 26.771% 0% 17.828% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 0.494 -0.130 -0.173 0.035 -0.030 0.006 0.584 -0.109 -0.065 0.014 27.433 -1.927

∆ 0.019 0.002 8.159 0.293 67.712 0.336 62.863 0.554 8.427 0.877 53377.486 1513.367

Bias of �� 0.494 -0.130 -0.173 0.035 -0.030 0.006 0.584 -0.109 -0.065 0.014 27.433 -1.927

Variation

of �� 6385.538 241.447 121.533 3.246 1012.974 39.723 4829.494 180.261 71.868 1.380 9788718.25 29223.41

Bias of∆ 0.019 0.002 8.159 0.293 67.712 0.336 62.863 0.554 8.427 0.877 53377.486 1513.367

Variation ∆ 423.530 0.552 183.367 0.242 1698.522 0.356 1485.037 0.446 176.642 0.139 4247977718 2104146

%negative

variance 54.185% 54.957% 0% 0% 0% 26.771% 0% 17.828% 0% 0% 0% 0%




𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏 𝜷𝟎 𝜷𝟏

N = T =

50

𝝈𝝐 = 1

�� 0.002 -0.000 -0.001 0.000 0.021 -0.005 -0.003 0.000 -0.001 0.000 -0.001 0.000

∆ -1.580𝑒−3 -4.687𝑒−5 0.015 0.000 0.632 0.000 0.611 0.001 0.634 0.615 0.634 0.615

Bias of �� 0.002 0.000 -0.001 0.000 0.021 -0.005 -0.003 0.000 -0.001 0.000 -0.001 0.000

Variation

of �� 0.016 0.000 0.003 0.000 3.836 0.169 0.121 0.005 0.003 0.000 0.003 0.000

Bias of∆ -1.580𝑒−3 -4.687𝑒−5 0.015 0.000 0.632 0.000 0.611 0.001 0.634 0.615 0.634 0.615

Variation ∆ 3.548𝑒−4 4.614𝑒−7 1.749𝑒−4 2.288𝑒−7 0.117 6.370𝑒−7 0.111 3.873𝑒−6 0.118 0.117 0.118 0.117

%negative variance 54.864% 53.693% 0% 0% 0% 29.279% 0% 21.891% 0% 0% 0% 0%

𝝈𝝐 =

5

�� 0.010 -0.001 -0.011 0.002 0.117 -0.028 -0.006 0.001 -0.012 0.002 -0.012 0.002

∆ -0.032 -0.000 0.013 0.383 15.692 0.015 15.176 0.041 0.978 0.622 0.978 0.622

Bias of �� 0.010 -0.001 -0.011 0.002 0.117 -0.028 -0.006 0.001 -0.012 0.002 -0.012 0.002

Variation

of �� 0.450 0.018 0.096 0.003 105.933 4.673 2.786 0.116 0.095 0.003 0.095 0.003

Bias of∆ -0.032 -0.000 0.383 0.013 15.692 0.015 15.176 0.041 0.978 0.622 0.978 0.622

Variation ∆ 0.201 0.000 0.100 0.000 67.21 3.579𝑒−4 63.411 0.002 0.189 0.106 0.189 0.106

%negative

variance 54.427% 53.034% 0% 0% 0% 29.054% 0% 21.592% 0% 0% 0% 0%

𝝈𝝐 =

10

�� 0.032 -0.005 -0.026 0.005 0.267 -0.064 0.079 -0.015 -0.027 0.006 -0.027 0.006

∆ -0.186 -0.005 1.521 0.055 63.025 0.062 60.838 0.174 2.061 0.662 2.061 0.662

Bias of �� 0.032 -0.005 -0.026 0.005 0.267 -0.064 0.079 -0.015 -0.027 0.006 -0.027 0.006

Variation

of �� 2.079 0.084 0.386 0.013 513.889 22.681 6.077 0.243 0.385 0.013 0.385 0.013

Bias of∆ -0.186 -0.005 1.521 0.055 63.025 0.062 60.838 0.174 2.061 0.662 2.061 0.662

Variation ∆ 2.699 0.003 1.336 0.001 911.3 4.750𝑒−3 856.001 0.028 1.359 0.090 1.359 0.090

%negative

variance 55.380% 53.083% 0% 0% 0% 28.657% 0% 20.798% 0% 0% 0% 0%


Fig (B.1) 2D graph comparing different methods of estimating variance

component applying on RCR model according to sample size

a) Comparing the bias in estimating 𝛽°

between 5 methods

according to the sample size

b) Comparing the bias in estimating 𝛽1

between 5 methods


c) Comparing the variation in estimating 𝛽°

between 5 methods according to the sample size

d) Comparing the variation in estimating 𝛽1

between 5

methods according to the sample size

e) Comparing the bias in estimating ∆° between 5 methods


f) Comparing the bias in estimating ∆1


g) Comparing the variation in estimating ∆°


h) Comparing the variation in estimating ∆1


i) Comparing the negative value in estimating ∆°


j) Comparing the negative value in estimating ∆1



Fig (B.2) 2D graph comparing different methods of estimating variance

component applying on RCR model according to error variance

Received: October 10, 2017; Published: October 30, 2017

a) Comparing the bias in estimating 𝛽°

between 5 methods

according to the error variance

b) Comparing the bias in estimating 𝛽1

between 5 methods


c) Comparing the variation in estimating 𝛽°

between 5 methods


d) Comparing the variation in estimating 𝛽1

between 5

methods according to the error variance

e) Comparing the bias in estimating ∆°

between 5 methods according to the error variance

f) Comparing the bias in estimating ∆1

between 5 methods


g) Comparing the variation in estimating ∆°

between 5 methods according to the error variance

h) Comparing the variation in estimating ∆1

between 5


i) Comparing the negative value in estimating ∆°

between 4


j) Comparing the negative value in estimating ∆1

between

4 methods according to the error variance

Documents

Comparing MINQUE and IAUE Estimates of Variance …Comparing MINQUE and IAUE estimates of variance components 2729 variance is used, and also when other methods are used for the same