prediction using estimators of linear regression model with

Journal of Economics and Sustainable Development ISSN 2222-1700 (Paper) ISSN 2222-2855 (Online)Vol.3, No.13, 2012

Prediction Using Estimators of Linear Regression Model with Aurocorrelated Error Terms and Correlated Stochastic Uniform

Kayode Ayinde

1. Department of Statistics, Ladoke Akintola University of Technology

P

2. Department of

*E-mail:

Abstract

Prediction remains one of the fundamental reasons for regression analysis. However, the Classical Linear Regression Model is formulated under some assumptions which are not always satisfied especially economic and social sciences leading to the development of many estimatorsexamine the performances of the Ordinary Least Square estimator (OLS), CochraneMaximum Likelihood estimator (ML) and the estimators based on Principal Cprediction of linear regression model under the violations of assumption of non independent regressors and error terms. With stochastic uniform variables as regressors, Monte experiments were conducted over the levels of autocorrelation(multicollinearity -λ ) and sample sizes, and best estimators for prediction pgoodness of fit statistics of the estimators. Resultsmulticollinearity over the levels of autocorrelation have a convex are concave – like. Also, as the level of multicollinearity inestimators perform much better at all the levels of autocorrelationthe sample size is small (n=10), the performances of the COR and ML estimators are generally bethe same, even though at low level of autocorrelation the PC estimator either performs better than or competes with the best estimator when ≤λestimator is best except when the autocorrelation level is low and PR2 estimator is best. Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator competes with the best estimator in all the levels of multicollinearity.

.Keywords: Prediction, Estimators, Linear Regression Model, Autocorrelation

1.0 Introduction

Linear regression model is probably the most widely used statistical techrelationship problems among variables. It helps to explain observations of a dependent variable,values of one or more independent variables, prediction of its values often becomes very essential and necessary. However, the linear regression model is formulated under some basic assumptions. Among these assumptions are regressors being assumed to be (non-stochastic) and independent; and that themethods of estimation of the parameter model have been developed.

The assumption of non-stochastic regressors is not always satisfied especially in business, economic and social sciences because their regressors are often generated by stochastic process beyond their control. Many authors including Neter and Wasserman (1974), Fomby et al. (1984), Maddala (2002) have given situations and instances where these assumptions may be violated and alOrdinary Least Square (OLS) estimator when used to estimate the model parameters.stochastic and independent of the error terms; the OLS estimator is still unbiased and has minimum variance even though it is not Best Linear Unbiased Estimator (BLUE). Also,valid if the error terms are further assumed to be normal. However, modification is required in the area of confidence interval and power of the test

The violation of the assumption of independent regressors leads to multcollinearitybusiness and economics data. With strongly interrelated regressors, interpretation given to the regression coefficients may no longer be valid because the assumption under which the regression model is built has been violated. Although the estimates of the regression coefficients provided by the OLS estimator is still unbiased as long as multicollinearity is not perfect,

Journal of Economics and Sustainable Development 2855 (Online)

119


Regressors

Kayode Ayinde1* K. A. Olasemi2 O.E.Olubusoye2

Department of Statistics, Ladoke Akintola University of Technology

P. M. B. 4000, Ogbomoso, Oyo State,, Nigeria.

Department of Statistics, University of Ibadan, Oyo State, Nigeria.

mail: [email protected], or [email protected]

Prediction remains one of the fundamental reasons for regression analysis. However, the Classical Linear Regression Model is formulated under some assumptions which are not always satisfied especially

ading to the development of many estimators. This work, therefore, attempts toof the Ordinary Least Square estimator (OLS), Cochrane-Orcutt estimator (COR),

Maximum Likelihood estimator (ML) and the estimators based on Principal Component analysis (PC) in prediction of linear regression model under the violations of assumption of non – independent regressors and error terms. With stochastic uniform variables as regressors, Monte

cted over the levels of autocorrelation )(ρ , correlation between regressors ) and sample sizes, and best estimators for prediction purposes are identified using

he estimators. Results show that the performances of COR and ML at each level of multicollinearity over the levels of autocorrelation have a convex – like pattern while that of OLS, PR1 and PR2

like. Also, as the level of multicollinearity increases the estimators especially the COR and ML perform much better at all the levels of autocorrelation. Furthermore, results show that

the sample size is small (n=10), the performances of the COR and ML estimators are generally bethe same, even though at low level of autocorrelation the PC estimator either performs better than or competes

49.0−≤ and 6.0≥λ . When the sample size is small (n =10), the COR mator is best except when the autocorrelation level is low and 4.0−≤λ or 2.0≥λ . At these instances, the

Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator h the best estimator in all the levels of multicollinearity.

Prediction, Estimators, Linear Regression Model, Autocorrelation, Multicollinearity

Linear regression model is probably the most widely used statistical technique for solving functional relationship problems among variables. It helps to explain observations of a dependent variable,values of one or more independent variables, X1, X2,...,Xp. In an attempt to explain the dependent variable,

ction of its values often becomes very essential and necessary. However, the linear regression model is formulated under some basic assumptions. Among these assumptions are regressors being assumed to be

and independent; and that the error terms are assumed to be independent. Consequently, various methods of estimation of the parameter model have been developed.

stochastic regressors is not always satisfied especially in business, economic cause their regressors are often generated by stochastic process beyond their control.

Many authors including Neter and Wasserman (1974), Fomby et al. (1984), Maddala (2002) have given situations and instances where these assumptions may be violated and also discussed their consequences on the Ordinary Least Square (OLS) estimator when used to estimate the model parameters.stochastic and independent of the error terms; the OLS estimator is still unbiased and has minimum variance

Unbiased Estimator (BLUE). Also, the traditional hypothesis testing remains valid if the error terms are further assumed to be normal. However, modification is required in the area of

and power of the test and the power of the test calculated for each sample.

The violation of the assumption of independent regressors leads to multcollinearitybusiness and economics data. With strongly interrelated regressors, interpretation given to the regression oefficients may no longer be valid because the assumption under which the regression model is built has been

violated. Although the estimates of the regression coefficients provided by the OLS estimator is still unbiased as perfect, the regression coefficients have large sampling errors which affect both

www.iiste.org


Department of Statistics, Ladoke Akintola University of Technology

University of Ibadan, Oyo State, Nigeria.

[email protected]

Prediction remains one of the fundamental reasons for regression analysis. However, the Classical Linear Regression Model is formulated under some assumptions which are not always satisfied especially in business,

. This work, therefore, attempts to Orcutt estimator (COR),

omponent analysis (PC) in stochastic regressors,

independent regressors and error terms. With stochastic uniform variables as regressors, Monte - Carlo , correlation between regressors

urposes are identified using the the performances of COR and ML at each level of

like pattern while that of OLS, PR1 and PR2 especially the COR and ML

show that except when the sample size is small (n=10), the performances of the COR and ML estimators are generally best and almost the same, even though at low level of autocorrelation the PC estimator either performs better than or competes

. When the sample size is small (n =10), the COR . At these instances, the

Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator

Multicollinearity

nique for solving functional relationship problems among variables. It helps to explain observations of a dependent variable,y, with observed

In an attempt to explain the dependent variable, ction of its values often becomes very essential and necessary. However, the linear regression model is

formulated under some basic assumptions. Among these assumptions are regressors being assumed to be fixed error terms are assumed to be independent. Consequently, various

stochastic regressors is not always satisfied especially in business, economic cause their regressors are often generated by stochastic process beyond their control.

Many authors including Neter and Wasserman (1974), Fomby et al. (1984), Maddala (2002) have given so discussed their consequences on the

Ordinary Least Square (OLS) estimator when used to estimate the model parameters. When regressors are stochastic and independent of the error terms; the OLS estimator is still unbiased and has minimum variance

the traditional hypothesis testing remains valid if the error terms are further assumed to be normal. However, modification is required in the area of

mple.

The violation of the assumption of independent regressors leads to multcollinearity as found in business and economics data. With strongly interrelated regressors, interpretation given to the regression oefficients may no longer be valid because the assumption under which the regression model is built has been

violated. Although the estimates of the regression coefficients provided by the OLS estimator is still unbiased as have large sampling errors which affect both


the inference and forecasting that is based on the model (Chartterjee et al., 2000). Various methods have been developed to estimate the model parameters when muitcollineariinclude Ridge Regression estimator developed by Hoerl (1962) and Hoerl and Kennard (1970), Estimator based on Principal Component Regression suggested by Massy (1965), Marquardt (1970) and Bock, Yancey and Judg(1973), Naes and Marten (1988), and method of Partial Least Squares developed by Hermon Wold in the 1960s (Helland, 1988, Helland, 1990, Phatak and Jony 1997).

When all the assumptions of the Classical Linear Regression Model hold except that the erronot homoscedastic (i.e. E (U1U) = σGeneralized Least Squares (GLS) Model. Aitken (1935) has shown that the GLS estimator (X1Ω

1X )-1 X1Ω

1Y is efficient among the class of linear unbiased estimators of matrix of β given as V(β) = σ2(X1

Ω

is often estimated by^

Ω to have what is known

When the assumption of independencethe problem of autocorrelation arises. parameter estimation of the linear regression model when the error term follows autoregressive of orders one. The OLS estimator is inefficient even though unbiased. Its predicted values are also inefficient and the sampling variances of the autocorrelated error termsinvalid (Johnston, 1984; Fomby et al., 1984; Chartterjee, 2000; Maddala, 2002). To compensate for the lost of efficiency, several feasible GLS estimators have been developed. These includeCochrane and Orcutt (1949), Paris and Winstern (1954), Hildreth and Lu (1960), Durbin (1960), Theil (1971), the Maximum Likelihood and the Maximum Likelihood Grid (Beach and Mackinnon, 1978), and Thornton (1982). Chipman (1979), Kramer (1980), Kleiber (2001), Iyaniwura and Nwabueze (2004), Nwabueze ( 2005a, b, c), Ayinde and Ipinyomi (2007) and many other authors have not only observed the asymptotic equivalence of these estimators but have also noted that that their performancesregressor used. Rao and Griliches (1969) did one of the earliest Monteproperties of several two-stage regression methods in the context of autocorrelation error. Otherdone on these estimators and the violations of the assumptions of classical linear regression model include thatof Ayinde and Oyejola (2007), Ayinde (2007a,b), Ayinde and Olaomi (2008), Ayinde (2008), and Ayinde and Iyaniwura (2008).

In spite of these several works on these estimators, none has actually been done on their prediction. Very fundamentally, prediction is one of the basic reasons for regression analysis. Therefore, this paper does not only examine the predictive potential of somassumption of regression model making the model much closer to reality.

2.0 Materials and Methods

Consider the linear regression model is of the form:

ttt XXXY +++= 322110 ββββWhere , 0(~ε Ntcorrelated.

For Monte-Carlo simulation study, the parameters of equation (1) were specified and fixed as= 1.8 and β3 = 0.6. The levels of multicollinearity among the independent variables were sixteen (16) and specified as: )()( 1312 == xx λλλis twenty-one (21) and are specified as: experiment was replicated in 1000 times (R =1000) under Six (6) levels of sample sizes (n =10, 15, 20, 30, 50, 100).

The correlated stochastic uniform regressors were generated by first using the equations provided by Ayinde (2007) and Ayinde and Adegboye (2010) to generate normally distributed random variables with specified intercorrelation. With P= the equations give:

X1 = µ1 + σ1Z1

X2 = µ2 + ρ12 σ2Z1 + Z2

X3 = µ3 + ρ13 σ3Z1 + Z2 +

where m22 = , m23 = inter-correlation matrix has to be positive definite and hence, the correlations among the independent variable were taken as prescribed earlier). In the study, we assumed X


120

the inference and forecasting that is based on the model (Chartterjee et al., 2000). Various methods have been developed to estimate the model parameters when muitcollinearity is present in a data set. These estimators include Ridge Regression estimator developed by Hoerl (1962) and Hoerl and Kennard (1970), Estimator based on Principal Component Regression suggested by Massy (1965), Marquardt (1970) and Bock, Yancey and Judg(1973), Naes and Marten (1988), and method of Partial Least Squares developed by Hermon Wold in the 1960s (Helland, 1988, Helland, 1990, Phatak and Jony 1997).

When all the assumptions of the Classical Linear Regression Model hold except that the erro) = σ2In) but are heteroscedastic (i.e. E(U1U) = σ2

Ω), the resulting model is the Generalized Least Squares (GLS) Model. Aitken (1935) has shown that the GLS estimator

ent among the class of linear unbiased estimators of β with variance Ω

1X)-1 where Ω is assumed to be known. However, Ω is not always known, it to have what is known as Feasible GLS estimator.

of independence of error terms is violated as it is often found in time series data, the problem of autocorrelation arises. Several authors have worked on this violation especially in terms of the

stimation of the linear regression model when the error term follows autoregressive of orders one. The OLS estimator is inefficient even though unbiased. Its predicted values are also inefficient and the sampling variances of the autocorrelated error terms are known to be underestimated causing the t and the F tests to be invalid (Johnston, 1984; Fomby et al., 1984; Chartterjee, 2000; Maddala, 2002). To compensate for the lost of efficiency, several feasible GLS estimators have been developed. These include the estimator provided by Cochrane and Orcutt (1949), Paris and Winstern (1954), Hildreth and Lu (1960), Durbin (1960), Theil (1971), the Maximum Likelihood and the Maximum Likelihood Grid (Beach and Mackinnon, 1978), and Thornton

Kramer (1980), Kleiber (2001), Iyaniwura and Nwabueze (2004), Nwabueze ( 2005a, b, c), Ayinde and Ipinyomi (2007) and many other authors have not only observed the asymptotic equivalence of these estimators but have also noted that that their performances and efficiency depend on the structure of the regressor used. Rao and Griliches (1969) did one of the earliest Monte-Carlo investigations on the small sample

stage regression methods in the context of autocorrelation error. Otherdone on these estimators and the violations of the assumptions of classical linear regression model include that

Ayinde and Oyejola (2007), Ayinde (2007a,b), Ayinde and Olaomi (2008), Ayinde (2008), and Ayinde and

ite of these several works on these estimators, none has actually been done on their prediction. Very fundamentally, prediction is one of the basic reasons for regression analysis. Therefore, this paper does not only examine the predictive potential of some of these estimators but also does it under some violations of assumption of regression model making the model much closer to reality.

Consider the linear regression model is of the form:

tt uX +3

),0 2σ , t = 1, 2, 3,...n and ~0,1) i = 1, 2, 3 are stochastic and

simulation study, the parameters of equation (1) were specified and fixed as= 0.6. The levels of multicollinearity among the independent variables were sixteen (16) and

.99.0,9.0,8.0,...,3.0,4.0,49.0)( 23 −−−=xλ The levels of autocorrelation one (21) and are specified as: 99.0,9.0,8.0,...,8.0,9.0,99.0 −−−=ρ

experiment was replicated in 1000 times (R =1000) under Six (6) levels of sample sizes (n =10, 15, 20, 30, 50,

The correlated stochastic uniform regressors were generated by first using the equations provided by Ayinde Ayinde and Adegboye (2010) to generate normally distributed random variables with specified

the equations give:

1

2 (2)

Z3

and n33 = m33 - ; and Zi N (0, 1) i = 1, 2, 3. (The positive definite and hence, the correlations among the independent variable

were taken as prescribed earlier). In the study, we assumed Xi N (0, 1), i = 1, 2, 3. We further utilized the

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the inference and forecasting that is based on the model (Chartterjee et al., 2000). Various methods have been ty is present in a data set. These estimators

include Ridge Regression estimator developed by Hoerl (1962) and Hoerl and Kennard (1970), Estimator based on Principal Component Regression suggested by Massy (1965), Marquardt (1970) and Bock, Yancey and Judge (1973), Naes and Marten (1988), and method of Partial Least Squares developed by Hermon Wold in the 1960s

When all the assumptions of the Classical Linear Regression Model hold except that the error terms are Ω), the resulting model is the

Generalized Least Squares (GLS) Model. Aitken (1935) has shown that the GLS estimator β of β given as β = β with variance – covariance

Ω is not always known, it

of error terms is violated as it is often found in time series data, Several authors have worked on this violation especially in terms of the

stimation of the linear regression model when the error term follows autoregressive of orders one. The OLS estimator is inefficient even though unbiased. Its predicted values are also inefficient and the sampling

are known to be underestimated causing the t and the F tests to be invalid (Johnston, 1984; Fomby et al., 1984; Chartterjee, 2000; Maddala, 2002). To compensate for the lost of

the estimator provided by Cochrane and Orcutt (1949), Paris and Winstern (1954), Hildreth and Lu (1960), Durbin (1960), Theil (1971), the Maximum Likelihood and the Maximum Likelihood Grid (Beach and Mackinnon, 1978), and Thornton

Kramer (1980), Kleiber (2001), Iyaniwura and Nwabueze (2004), Nwabueze ( 2005a, b, c), Ayinde and Ipinyomi (2007) and many other authors have not only observed the asymptotic equivalence

and efficiency depend on the structure of the Carlo investigations on the small sample

stage regression methods in the context of autocorrelation error. Other recent works done on these estimators and the violations of the assumptions of classical linear regression model include that

Ayinde and Oyejola (2007), Ayinde (2007a,b), Ayinde and Olaomi (2008), Ayinde (2008), and Ayinde and

ite of these several works on these estimators, none has actually been done on their prediction. Very fundamentally, prediction is one of the basic reasons for regression analysis. Therefore, this paper does not

e of these estimators but also does it under some violations of

(1)

) i = 1, 2, 3 are stochastic and

simulation study, the parameters of equation (1) were specified and fixed as β0 = 4, β1 = 2.5, β2 = 0.6. The levels of multicollinearity among the independent variables were sixteen (16) and

The levels of autocorrelation .99 Furthermore, the

experiment was replicated in 1000 times (R =1000) under Six (6) levels of sample sizes (n =10, 15, 20, 30, 50,

The correlated stochastic uniform regressors were generated by first using the equations provided by Ayinde Ayinde and Adegboye (2010) to generate normally distributed random variables with specified

N (0, 1) i = 1, 2, 3. (The positive definite and hence, the correlations among the independent variable

We further utilized the


properties of random variables that cumulative distribution functiowithout affecting the correlation among the variables (Schumann, 2009) to generate

The error terms were generated using one of the distributional properties of the autocorrela(0, )) and the AR(1) equation as follows:

u1=

ut = ρut-1 + εt t = 2,3,4,…n

Since some of these estimators have now been incorporated into the Time Series Processor (TSP 5.0, 2005) software, a computer program was written usDetermination of the model )(

2_

Restimator, Maximum Likelihood estimator and the estimator based on Principal ComponenThe Adjusted Coefficient of Determination of the model was averaged over the numbers of replications. i.e.

The two possible PCs (PC1 and PC2) of the Priseparate Adjusted Coefficient of Determination. An estimator is best if its Adjusted Coefficient of Determination is closest to unity.

3.0 Results and Discussion

The full summary of the simulated resuand autocorrelation is contained in the work of Olasemi (2011). The graphical representations of the results when n=10, 15, 20, 30, 50 and 100 are respectively presented in Figure 1


121

properties of random variables that cumulative distribution function of Normal distribution produces U (0, 1) without affecting the correlation among the variables (Schumann, 2009) to generate

The error terms were generated using one of the distributional properties of the autocorrela) and the AR(1) equation as follows:

(3)

t = 2,3,4,…n (4)

Since some of these estimators have now been incorporated into the Time Series Processor (TSP 5.0, 2005) software, a computer program was written using the software to estimate the Adjusted Coefficient of

) the Ordinary Least Square (OLS) estimator, Cochrane Oestimator, Maximum Likelihood estimator and the estimator based on Principal ComponenThe Adjusted Coefficient of Determination of the model was averaged over the numbers of replications. i.e.

∑=

=

=R

i

iRRR

1

2_1 (5)

The two possible PCs (PC1 and PC2) of the Principal Component Analysis were used. Each provides its separate Adjusted Coefficient of Determination. An estimator is best if its Adjusted Coefficient of Determination

The full summary of the simulated results of each estimator at different level of sample size, muticollinearity, and autocorrelation is contained in the work of Olasemi (2011). The graphical representations of the results when n=10, 15, 20, 30, 50 and 100 are respectively presented in Figure 1, 2, 3, 4, 5 and 6.

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n of Normal distribution produces U (0, 1) .

The error terms were generated using one of the distributional properties of the autocorrelated error terms (ut N

(3)

Since some of these estimators have now been incorporated into the Time Series Processor (TSP 5.0, 2005) ing the software to estimate the Adjusted Coefficient of

, Cochrane Orcutt (COR) estimator, Maximum Likelihood estimator and the estimator based on Principal Component Analysis (PRN). The Adjusted Coefficient of Determination of the model was averaged over the numbers of replications. i.e.

(5)

ncipal Component Analysis were used. Each provides its separate Adjusted Coefficient of Determination. An estimator is best if its Adjusted Coefficient of Determination

lts of each estimator at different level of sample size, muticollinearity, and autocorrelation is contained in the work of Olasemi (2011). The graphical representations of the results

, 2, 3, 4, 5 and 6.



122

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From these figures, it can be generally observed that the performances of COR and ML at each level of multicollinearity over the levels of autocorrelation have a convex are concave – like. Also, as the level of multicollinearity increases the estimators perform much better at all the levels of autocorrelation even though the values of their averaged adjusted coefficient of determination are not high at low and moderate levels of estimators are much better at all the levels of autocorrelation. Furthermore except when the sample size is small (n=10), the performances of the COR and ML estimators are gelow level of autocorrelation the PR2 estimator either performs better than or competes with the best estimator when 49.0−≤λ and 6.0≥λ . When the sample size is small (n =10), twhen the autocorrelation level is low and Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator cestimator in all the levels of multicollinearity.

Very specifically in term of identification of the best estimator, Table 1, 2, 3, 4, 5 and 6 respectively summarize the best estimator for prediction at all the levels of autocorrelat10, 15, 20, 30, 50, 100.

Table 1: The Best Estimator for Prediction at different level of Multicollinearity and Autocorrelation when n=10. ρ

-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR COR COR -0.2 PC2 COR COR ML -0.1 PC2 PC2 ML ML

0 PC2 PC2 ML ML 0.1 PC2 PC2 ML ML 0.2 PC2 PC2 ML ML 0.3 PC2 PC2 ML ML 0.4 PC2 PC2 ML ML 0.5 PC2 COR COR ML 0.6 COR COR COR COR COR0.7 COR COR COR COR COR0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR

From Table 1 when n = 10, COR estimator is best for prediction at4.0−≤ρ and 7.0≥ρ . When −

estimator is only best when 2.0 ≤−Table 2: The Best Estimator for Prediction at different level when n=15.


123

From these figures, it can be generally observed that the performances of COR and ML at each level of multicollinearity over the levels of autocorrelation have a convex – like pattern while that of OLS, PR1 and PR2

like. Also, as the level of multicollinearity increases the estimators perform much better at all the levels of autocorrelation even though the values of their averaged adjusted coefficient of determination are not high at low and moderate levels of autocorrelation except when 6.0≥λ . At these instances, COR and ML estimators are much better at all the levels of autocorrelation. Furthermore except when the sample size is small (n=10), the performances of the COR and ML estimators are generally best and almost the same, even though at low level of autocorrelation the PR2 estimator either performs better than or competes with the best estimator

. When the sample size is small (n =10), the COR estimator is best except when the autocorrelation level is low and 4.0−≤λ or 2.0≥λ . At these instances, the PR2 estimator is best. Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator cestimator in all the levels of multicollinearity.

Very specifically in term of identification of the best estimator, Table 1, 2, 3, 4, 5 and 6 respectively summarize the best estimator for prediction at all the levels of autocorrelation and multicollinearity when the sample size is

The Best Estimator for Prediction at different level of Multicollinearity and Autocorrelation

λ -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR ML ML ML ML ML COR PC2 PC2 ML ML ML ML PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML PC2 PC2 PC2 PC2 PC2 ML ML ML ML PC2 PC2 PC2 PC2 COR ML ML COR COR COR COR PR2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

From Table 1 when n = 10, COR estimator is best for prediction at all levels of multicollinearity especially when 5.02.0 ≤≤− ρ and 4.0−≤λ or 2.0≥λ , the PR2 is best. The ML

5.0≤≤ ρ and 1.03.0 ≤≤− λ .


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From these figures, it can be generally observed that the performances of COR and ML at each level of

like pattern while that of OLS, PR1 and PR2 like. Also, as the level of multicollinearity increases the estimators perform much better at all the

levels of autocorrelation even though the values of their averaged adjusted coefficient of determination are not . At these instances, COR and ML

estimators are much better at all the levels of autocorrelation. Furthermore except when the sample size is small nerally best and almost the same, even though at

low level of autocorrelation the PR2 estimator either performs better than or competes with the best estimator he COR estimator is best except

. At these instances, the PR2 estimator is best. Moreover, at low level of autocorrelation in all the sample sizes, the OLS estimator competes with the best

Very specifically in term of identification of the best estimator, Table 1, 2, 3, 4, 5 and 6 respectively summarize ion and multicollinearity when the sample size is


0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PR2 PR2 PR2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

all levels of multicollinearity especially when , the PR2 is best. The ML

Autocorrelation


ρ

-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR COR COR COR-0.2 COR COR COR COR COR-0.1 COR COR COR COR COR

0 PC2 COR COR COR COR0.1 PC2 COR COR COR COR0.2 PC2 COR COR COR COR0.3 PC2 COR COR COR COR0.4 COR COR COR COR COR0.5 COR COR COR COR COR0.6 COR COR COR COR COR0.7 COR COR COR COR COR0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR

From the Table 2, it can be seen that the COR estimator is genemulticollinearity and autocorrelation except occasionally when autocorrelation level is low and multicollinearityis best .

Table 3: The Best Estimator for Prediction at different level of Multicollinearity and when n=20.

ρ

-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR COR COR COR-0.2 COR COR COR ML -0.1 COR ML ML ML

0 PC2 ML ML ML 0.1 PC2 ML ML ML 0.2 PC2 ML ML ML 0.3 COR ML ML ML 0.4 COR COR COR COR COR0.5 COR COR COR COR COR0.6 COR COR COR COR COR0.7 COR COR COR COR COR0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR

When n = 20 as revealed in Table 3, the COR estimator is best except atthe best estimator is PC2 when λsparsely COR even though the two of them compete very favorably well. (See figure 3).


-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR ML ML -0.2 ML ML ML ML


124

λ

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR PC2 COR COR COR COR COR COR COR PC2 COR COR COR COR COR COR COR PC2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

From the Table 2, it can be seen that the COR estimator is generally best for prediction at all levels of multicollinearity and autocorrelation except 3.00 ≤≤ ρ when 49.0−≤λ or 6.0≥λoccasionally when autocorrelation level is low and multicollinearity level is very high or tends to unity, the PR2

The Best Estimator for Prediction at different level of Multicollinearity and

λ -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

When n = 20 as revealed in Table 3, the COR estimator is best except at 02.0 ≤≤− ρ49.0−≤λ or 7.0≥λ ; otherwise, the best estimator is often ML and

sparsely COR even though the two of them compete very favorably well. (See figure 3).

r Prediction at different level of Multicollinearity and Autocorrelation when

λ

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML

www.iiste.org

0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR PC2 COR PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 COR COR COR PC2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

rally best for prediction at all levels of

At these instances and level is very high or tends to unity, the PR2

Autocorrelation

0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR ML ML ML ML PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 ML PC2 PC2 PC2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

3.0 . At these instances, ; otherwise, the best estimator is often ML and

r Prediction at different level of Multicollinearity and Autocorrelation when

0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR ML ML COR COR ML ML ML ML


-0.1 ML ML ML ML 0 ML ML ML ML

0.1 ML ML ML ML 0.2 ML ML ML ML 0.3 ML ML ML ML 0.4 COR COR COR COR COR0.5 COR COR COR COR COR0.6 COR COR COR COR COR0.7 COR COR COR COR COR0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR

From Table 4, it can be observed that the best estimator for prediction when n = 30 is generally COR except when the 3.0≤ρ At these instances, the estimator based on using PC2 is best when

7.0≥λ ; otherwise, the best estimator is often ML and sparsely COR even though the two of them still compete very favorably well. (See figure 4).

Table 5: The Best Estimator for Prediction at different level of Multicollinearity when n=50. ρ

-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR COR COR COR-0.2 COR COR COR COR COR-0.1 ML COR COR COR COR0 ML ML COR COR COR0.1 ML COR COR COR COR0.2 ML COR COR COR COR0.3 COR COR COR COR COR0.4 COR COR COR COR COR0.5 COR COR COR COR COR0.6 COR COR COR COR COR0.7 COR COR COR COR COR0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR

From Table 5 when n= 50, the best estimator for prediction is still COR except when and when 2.01.0 ≤≤− ρ andλbest at the latter.


-0.49 -0.4 -0.3 -0.2

-0.99 COR COR COR COR COR-0.9 COR COR COR COR COR-0.8 COR COR COR COR COR-0.7 COR COR COR COR COR0.-6 COR COR COR COR COR-0.5 COR COR COR COR COR-0.4 COR COR COR COR COR-0.3 COR COR COR COR COR-0.2 COR COR COR COR COR-0.1 COR COR COR COR COR

0 COR COR COR COR COR0.1 COR COR COR COR COR0.2 COR COR COR COR COR0.3 COR COR COR COR COR0.4 COR COR COR COR COR0.5 COR COR COR COR COR0.6 COR COR COR COR COR0.7 COR COR COR COR CO0.8 COR COR COR COR COR0.9 COR COR COR COR COR0.99 COR COR COR COR COR


125

ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML ML COR ML ML COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

From Table 4, it can be observed that the best estimator for prediction when n = 30 is generally COR except At these instances, the estimator based on using PC2 is best when

; otherwise, the best estimator is often ML and sparsely COR even though the two of them still compete very favorably well. (See figure 4).

The Best Estimator for Prediction at different level of Multicollinearity and

λ

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

From Table 5 when n= 50, the best estimator for prediction is still COR except when ρ49.0−≤ . At the former estimator based on using PC2 is best while ML is

The Best Estimator for Prediction at different level of Multicollinearity and Autocorrelation when

λ

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

www.iiste.org

PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 PC2 ML ML PC2 PC2 ML ML ML COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

From Table 4, it can be observed that the best estimator for prediction when n = 30 is generally COR except At these instances, the estimator based on using PC2 is best when 2.01.0 ≤≤ ρ and

; otherwise, the best estimator is often ML and sparsely COR even though the two of them still

Autocorrelation

0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR PC2 COR COR PC2 PC2 COR COR PC2 PC2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR

1.0≤ρ and 9.0≥λ . At the former estimator based on using PC2 is best while ML is

The Best Estimator for Prediction at different level of Multicollinearity and Autocorrelation when

0.7 0.8 0.9 0.99

COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR PC2 PC2 COR COR COR PC2 COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR COR


When the sample size is large (n =100), the best estimator is generally COR except when 9.0≥λ . At these instances, the estimator based on

4.0 Conclusions

The effect of two major problems, MuCOR, ML and PC estimators of linear regression model has been jointly examined in this paper. the pattern of performances of COR and ML at each level of multicollineato be generally and evidently convex especially when generally concave. Moreover, the COR and ML estimators perform equivalently and betterperformances become much better as multicollinearity increases. The COR estimator is generally the best estimator for prediction except at high level of multicollinearity and low levels of autocorrelation. At these instances, the PC estimator is either best or competes with the COR estimator. is small (n=10) and multicollinearity level is not high, the OLS estimator is best at low level of autocorrelation whereas the ML is best at moderate levels of autocorrelation

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