Some Improved Generalised Liu-Type Estimators for Linear Regression Models in the Presence of Multicollinearity and Heteroscedasticity

1. Introduction

The Ordinary Least Squares (OLS) estimator, while optimal under the classical linear regression assumptions, is highly sensitive to violations such as multicollinearity and heteroscedasticity. Multicollinearity inflates the variance of coefficient estimates, leading to unstable and unreliable results. Heteroscedasticity, on the other hand, results in inefficient estimators and invalid statistical inferences due to incorrect estimation of standard errors. To address multicollinearity, biased estimation techniques like Ridge Regression (Hoerl & Kennard, 1970) and the Liu Estimator (Liu, 1993) were introduced. Ridge regression imposes a penalty on the size of regression coefficients, while the Liu estimator incorporates an additional shrinkage parameter that enhances the bias-variance trade-off, often resulting in lower mean squared error (MSE). More recently, the Kibria-Lukman estimator (Kibria and Lukman, 2020) added further modifications to improve performance under multicollinearity among several other estimators. Despite these advancements, most biased estimators, including Ridge and Liu, assume homoscedasticity. White (1980) proposed heteroscedasticity-consistent standard errors to permit valid inference under heteroscedasticity. However, these adjustments do not enhance the efficiency of coefficient estimation. Techniques such as Generalized Least Squares (GLS) and Weighted Least Squares (WLS) have also been used, though they require prior knowledge of the error variance structure, which is often unavailable in practice. To extend generalised Liu-type estimators to heteroscedastic settings, several researchers have introduced modifications. For instance, Yang and Wang (2012) developed a heteroscedasticity-adjusted Liu-type estimator using a combination of weighting and shrinkage strategies. However, many such models rely on restrictive assumptions about the error structure. Recognizing this limitation, recent efforts have focused on designing estimators that can simultaneously handle both multicollinearity and heteroscedasticity. Ahmed et al. (2016) and Akdeniz et al. (2020) proposed hybrid estimators incorporating elements of Ridge, Liu, and WLS. Although these estimators show promise, they often suffer from challenges such as complex parameter selection and limited generalizability. Arman and Aktas (2018) introduced adaptive estimators capable of dynamically switching among Ridge, Liu, and Lasso methods depending on the degree of multicollinearity and heteroscedasticity. A significant body of research has also focused on refining Liu-type estimators. Sclove (1987) emphasized the importance of optimal parameter selection, while Zhou and Wan (2005) developed pretest and Stein-type Liu estimators that improve estimator performance under model uncertainty. Saleh and Morshedi (2006) demonstrated that pretest and shrinkage Liu-type estimators can outperform traditional Ridge and OLS under specific conditions. Further, El-Denif and Ahmed (2015) proposed a class of weighted Liu and Ridge-type shrinkage estimators suitable for heteroscedastic and multicollinear data. Other innovations include Arashi et al. (2014) worked on double shrinkage Liu-type estimator that combines Ridge and Liu penalties, and Abdel kader et al. (2019) developed composite estimator that integrates Ridge, Liu, and pretest elements for enhanced robustness. Comparative studies by Ibrahim and Kibria (2012) and Asar and Genç (2011) have validated the superior performance of modified Liu-type estimators in high multicollinearity environments. Simulation-based evaluations, such as those by McDonald and Galarneau (1975), further reinforce the practical advantages of biased estimators in real-life applications. Nevertheless, many of these estimators were developed with a primary focus on multicollinearity, often overlooking the simultaneous presence of heteroscedasticity, which is common in applied data sets. Consequently, several of the existing methods remain suboptimal in achieving minimum MSE under both conditions. This gap highlights the need for improved generalised Liu-type estimators that maintain efficiency by optimizing the biasing parameter d in the presence of heteroscedasticity. This study seeks to address this gap by developing new or improved generalised Liu-type estimators designed to perform robustly in the presence of both multicollinearity and heteroscedasticity. The proposed estimators will be theoretically analyzed and empirically validated through simulation studies to evaluate their efficiency and accuracy relative to existing methods.

2. The Classical Regression

Regression analysis is a fundamental statistical technique used for modeling the relationship between a dependent variable and one or more independent variables. As one of the most widely applied tools in statistical inference, regression enables researchers to understand, interpret, and predict outcomes across various fields such as economics, engineering, medicine, social sciences, and natural sciences. Regression analysis aims to establish a mathematical equation that best describes how changes in explanatory variables influence a response variable. In its simplest form, simple linear regression involves a single independent variable and assumes a linear relationship between variables. When multiple predictors are involved, the model is extended to multiple linear regression. The matrix form of a multiple linear regression model is given by:

Consider the linear regression model:

$$Y=X\beta +ϵ$$

(1)

where X is an (n$\times p)$ matrix with full rank, Y is a (n$\times 1$) vector of dependent variable., $\beta$ is a (p$\times 1)$ vector of unknown parameters, and $ϵ$ is the error term such that E($ϵ)=0$ and $E(ϵ{ϵ}^{\prime }$)=${\sigma }^{2}I$.

One notable biased estimation technique for estimating parameter in (1) is the Liu regression estimator, which, despite being a biased estimator, incorporates shrinkage parameters k and d to improve estimation accuracy, particularly under multicollinearity. Over time, various forms and modifications of Liu-type estimators have been proposed in the literature, primarily focusing on the selection and optimization of these shrinkage parameters. Researchers such as Hoerl and Kennard (1970), Hoerl et al. (1975), Kibria (2003), and Muniz et al. (2009) contributed significantly to the development and refinement of the shrinkage parameter k. On the other hand, scholars including Shakallioglu and Kaçıranlar (2006) have explored and introduced alternative approaches involving the shrinkage parameter d, expanding the utility and adaptability of the Liu estimator in various regression contexts.

3. Improved Generalised Liu-Type Estimator

The improvement in generalised Liu type estimator is achieved by the removal of these three quantities, ${\displaystyle \frac{1}{n}}$. ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$ and ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$ respectively from the biased parameter d, such that -$\infty <$d $<\infty$, the quantity proposed by Shakalliogu and Kaciranla (2006) stated below as d-optimum ($d_{opt})$in (2) is being adopted in this research work for improvement.

$$d_{opt}={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2({\mathsf{\lambda }}_j+{\widehat{k}}_{mjn})}}{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\mathsf{\lambda }}_j{\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2{({\mathsf{\lambda }}_j+{\widehat{k}}_{mjn})}^2}}}$$

(2)

The three (3) improved versions of the optimum d is given in (3), (4) and (5) below

$$\mathrm{where} \mathrm{the} \mathrm{improved} d_1=d_{opt\left(1\right)}={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2({\mathsf{\lambda }}_j+{\widehat{k^{*}}}_{mjn})}}{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\mathsf{\lambda }}_j{\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2{({\mathsf{\lambda }}_j+{\widehat{k^{*}}}_{mjn})}^2}}}$$

(3)

and $k^{*}$ is the improved k with the quantity ${\displaystyle \frac{1}{n}}$

$$\mathrm{The} \mathrm{improved} d_2=d_{opt\left(2\right)}={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2({\mathsf{\lambda }}_j+{\widehat{k^{**}}}_{mjn})}}{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\mathsf{\lambda }}_j{\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2{({\mathsf{\lambda }}_j+{\widehat{k^{**}}}_{mjn})}^2}}}$$

(4)

and $k^{**}$ is the improved k with the quantity ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$

$$\mathrm{The} \mathrm{improved} d_3=d_{opt\left(3\right)}={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2({\mathsf{\lambda }}_j+{\widehat{k^{***}}}_{mjn})}}{{\displaystyle {\sum }_{j=1}^p}\frac{\left\{{\mathsf{\lambda }}_j({\mathsf{\lambda }}_j{\widehat{\alpha }}_j^2-{\widehat{\sigma }}^2)\right\}}{{\left(\mathsf{\lambda }+1\right)}^2{({\mathsf{\lambda }}_j+{\widehat{k^{***}}}_{mjn})}^2}}}$$

(5)

and $k^{***}$ is the improved k with the quantity ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$

3.1. Improved Generalized Two Parameter Liu Estimator

Following Hoerl and Kennard, (1970) and Liu, (1993), (2003) and (2004), the generalized Liu-type estimator (GLTE) which is a generalization for Liu-type estimator, is defined as:

$${\widehat{\beta }}_{GLTE}= ({𝑋}^{\ast }\prime {𝑋}^{\ast }+ 𝐾){-}^1 ({𝑋}^{\ast }\prime {𝑋}^{\ast }-𝐷) {\widehat{\beta }}_{OLS}$$

(6)

Equation (6) is improved by replacing k with $k_i$ and D with $D_i$, hence, equation (6) results into

$${\widehat{\beta }}_{IGLTE}({𝑋}^{\ast }\prime {𝑋}^{\ast }+k_i){-}^1 ({𝑋}^{\ast }\prime {𝑋}^{\ast }-D_i) {\widehat{\beta }}_{OLS}$$

(7)

where: $k_i$ = diag(k_i1, k_i2, … , k_ip) , k_ij ≥ 0, $D_i$ = diag(d_i1, d_i2, … , d_ip), 0 <d_ij< 1 , −∞ <k_ij< ∞ , j = 1,2, … , p . i=1,2,3 to produce the three (3) forms of possible improvements in (8), (9) and (10) respectively.

$${\widehat{\beta }}_{IGLTE\left(1\right)}=({𝑋}^{\ast }\prime {𝑋}^{\ast }+k_1){-}^1 ({𝑋}^{\ast }\prime {𝑋}^{\ast }-D_1) {\widehat{\beta }}_{OLS}$$

(8)

$${\widehat{\beta }}_{IGLTE\left(2\right)}==({𝑋}^{\ast }\prime {𝑋}^{\ast }+k_2){-}^1 ({𝑋}^{\ast }\prime {𝑋}^{\ast }-D_2) {\widehat{\beta }}_{OLS}$$

(9)

$${\widehat{\beta }}_{IGLTE\left(3\right)}==({𝑋}^{\ast }\prime {𝑋}^{\ast }+k_3){-}^1 ({𝑋}^{\ast }\prime {𝑋}^{\ast }-D_3) {\widehat{\beta }}_{OLS}$$

(10)

3.2. MSE of the Improved Generalized Two Parameter Liu Estimator

$${\widehat{\beta }}_{IGLTE}=\left({X^{*}}^{\text{'}}{X}^{*}+K_i^{\text{'}}\right)\left({X^{*}}^{\text{'}}{X}^{*}-D_i\right){\widehat{\beta }}_{ols}$$

(11)

The canonical form of equation (1) is written as

$${\widehat{\alpha }}_{IGLTF}={\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right){\widehat{\alpha }}_{ols}$$

(12)

Where ${\mathsf{\Lambda }}^{*}=T^{\text{'}}{X^{*}}^{\text{'}}{X}^{*}T=diag\left({\lambda }_1^{*}, {\lambda }_2^{*}, \dots , {\lambda }_p^{*}\right)$ such that

$${\lambda }_1^{*}\ge {\lambda }_2^{*}, \dots \ge {\lambda }_p^{*} are the ordereed eigen values of {X^{*}}^{-1}{X}^{*}$$

Properties of  ${\widehat{\mathit{\alpha }}}_{\mathit{I}\mathit{G}\mathit{L}\mathit{T}\mathit{E}}$

$$E\left[{\widehat{\alpha }}_{IGLTE}\right]=E[({{\mathsf{\Lambda }}^{\mathrm{*}}+K_i)}^{-1}\left(\mathsf{\Lambda }-D_i\right){\widehat{\alpha }}_{ols}]$$

(13)

$$={\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}\left(\mathsf{\Lambda }-D_i\right){\alpha }_{ols}$$

(14)

$$Bias \left({\widehat{\alpha }}_{IGLTE}\right)=E\left[{\widehat{\mathsf{\alpha }}}_{IGLTE}\right]-{\alpha }_{ols}$$

(15)

$$={\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right){\alpha }_{ols}-{\alpha }_{ols}$$

(16)

$$=\left[{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right)-I\right]{\alpha }_{ols}$$

(17)

$$Bia{s}^2\left({\widehat{\alpha }}_{IGLTE}\right)={\left[{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right)-I\right]}^2{\alpha }_{ols}^2$$

(18)

$$={\left[{\displaystyle \frac{({\mathsf{\Lambda }}^{*}-D_i}{({\mathsf{\Lambda }}^{*}+K_i}}-I\right]}^2{\alpha }_{ols}^2$$

(19)

$$={\left[{\displaystyle \frac{\left({\mathsf{\Lambda }}^{*}-D_i\right)-({\mathsf{\Lambda }}^{*}+K_i)}{({\mathsf{\Lambda }}^{*}+K_i)}}\right]}^2{\alpha }_{ols}^2$$

(20)

$$={\left[{\displaystyle \frac{\left(D_{i }+K_i\right)}{\left({\mathsf{\Lambda }}^{*}+K_i\right)}}\right]}^2{\alpha }_{ols}^2$$

(21)

$$={\displaystyle \frac{{\left(D_i+K_i\right)}^2}{{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^2}}{\alpha }_{ols}^2$$

(22)

$$Var \left({\widehat{\alpha }}_{IGLTE}\right)={\mathsf{\sigma }}^2{\left({\mathsf{\Lambda }}^{\mathrm{*}}+K_i\right)}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right){{\mathsf{\Lambda }}^{*}}^{-1}\left({\mathsf{\Lambda }}^{*}-D_i\right){\left({\mathsf{\Lambda }}^{*}+K_i\right)}^{-1}$$

(23)

$$={\sigma }^2\left[{\displaystyle \frac{({\mathsf{\Lambda }}^{*}-D_i^2}{{\mathsf{\Lambda }}^{*2}{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^2}}\right]$$

(24)

$$MSEM\left({\widehat{\alpha }}_{IGLTE}\right)=Var\left({\widehat{\alpha }}_{IGLTE}\right)+Bia{s}^2\left({\widehat{\alpha }}_{IGLTE}\right)$$

(25)

$$={\sigma }^2\left[{\displaystyle \frac{{\left({\mathsf{\Lambda }}^{*}-D_i\right)}^2}{{\mathsf{\Lambda }}^{*2}{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^2}}\right]+{\displaystyle \frac{{\left(D_i+K_i\right)}^2{{\alpha }_i^2}_{ols}}{{\left({\mathsf{\Lambda }}^{*}+K_i\right)}^2}}$$

(26)

$$MSE\left({\widehat{\alpha }}_{IGLTE}\right)=trace\left(MSEM\left({\widehat{\alpha }}_{IGLTE}\right)\right)$$

(27)

$$={\displaystyle {\sum }_{i=1}^p}diag\left(MSEM\left({\widehat{\alpha }}_{IGLTE}\right)\right)$$

(28)

$$={\sigma }^2{\displaystyle {\sum }_{i=1}^p}{\displaystyle \frac{{\left({\lambda }_i^{*}-d_i\right)}^2}{{\lambda }_i^2{\left({\lambda }_i^{*}+k_i\right)}^2}}+{\displaystyle {\sum }_{i=1}^p}{\displaystyle \frac{{\left(d_i+k_i\right)}^2}{{\left({\lambda }_i^{*}+k_i\right)}^2}}{\alpha }_{ols}^2$$

(29)

3.3. The Simulation Procedure

A linear regression model of the form:

$$Y_t={\beta }_0+{\beta }_1{X}_{1t}+{\beta }_2{X}_{2t}+\dots +{\beta }_p{X}_{pt}+U_t t=\mathrm{1,2},\dots ,n ;p=3$$

(30)

Such that $U_t~N\left(0,{\sigma }^2\right)$ was considered. ${\beta }_0$ = 0, ${\beta }_1$= 0.67679354, ${\beta }_2$=0.06697104, ${\beta }_3$= 0.73312031 were employed. The model was studied with fixed regressors, $X_{jt}~N\left(0,{\sigma }^2\right)$ i=1,2,…,p; t=1,2,...,n.

The heteroscedasticity problem was introduced into the model using equation (31).

$$U_t=\mathrm{Z}\sigma \sqrt{E{\left(Y_t\right)}^{2\mathsf{\delta }}}$$

(31)

where $U_t$ is the error term and (δ = 0, 1, 1.5 and 2) is the strength of heteroscedasticity variance considered. The simulation study of the regression model in (30) without the intercept term was employed. Data was generated by conducting Monte-Carlo experiment with replications one thousand times (R=1000), varying sample sizes n=10, 20, 30, 50, 75 and 100. The error term $U_t$ is generated using normal distribution with mean zero and variance ${\sigma }^2;$that is, $U_t~N\left(0,{\sigma }^2\right).$The values of error variance ${\sigma }^2$ was assumed 1, 5 and 10 as it is widely used in literatures by researchers. The equation used by McDonald and Galarneau (1975), Wichern and Churchill (1978), Gibbons (1981) and Kibria (2003) was employed to generate the explanatory variables in this study: The model equation for the estimation of ${ X}_{ij}$ is given as:

$${ X}_{ij}=(1-{\rho }^2{)}^{{\displaystyle \frac{1}{2}}}{Z}_j+\rho Z_{jp},\mathrm{i}=1,2,3,\dots ,\mathrm{n.} \mathrm{j}=1,2,\dots \mathrm{p}.$$

(32)

where $Z_j$ is independent standard normal distribution with mean zero and unit variance, $\rho$is the correlation between any two explanatory variables and p is the number of explanatory variables. The values of $\rho$ were taken as 0.7, 0.75 0.8, 0.85 0.9, 0.95 and 0.99 respectively and p = 3.Meanwhile, equation (30) was used to generate response variable

such that ${\beta }_0$ is the intercept term it will be randomly allotted and the other regression coefficients are chosen as follows: ${\beta }_1$= 0.67679354, ${\beta }_2$=0.06697104, ${\beta }_3$= 0.73312031. The parameter values were chosen such that ${\beta }^{\text{'}}\beta$=1 which is a common restriction in simulation studies like this (Muniz, et al. 2010). The experiment was replicated one thousand times (R=1000) with these sample sizes; n = 10, 20, 30, 50, 75 and 100 respectively.

In matrix form, (30) can be written as:

$$Y=X\beta +U$$

(33)

where Y is (n x 1) vector of observation of the dependent variable,

X is (n x (p +1)) vector of observation of the regressors,

${\beta }_{,}$is a ((p +1) x 1) vector of unknown parameters to be estimated and

U is (n x 1) observation of the error term.

Where $U$ are independent normal pseudo-random numbers with mean 0 and variance σ². In this study, β₀ is taken to be zero and different values of σ were considered.

In order to compare the performance of Improved Liu estimator with other existing Liu amidst other biased estimators, a criterion for measuring the goodness of an estimator is required. For this purpose, the (MSE) criteria was used to measure the goodness of an estimator.

At each replication for specified sample size, multicollinearity and error variance level, the performance of the existing and improved Liu parameter estimation techniques were examined. The MSE of the existing Liu Estimator is defined as

$$MSE\left(\widehat{\beta }\right)=V\left(\widehat{\beta }\right)+(B\left(\widehat{\beta }\right){)}^2$$

(34)

Such that $V\left(\widehat{\beta }\right)$ is the variance of $\widehat{\beta }$, $(B\left(\widehat{\beta }\right){)}^2$ is the square of the biased quantity $\widehat{\beta }$

$$V\left(\widehat{\beta }\right)={\sigma }^2\left\{{\displaystyle \frac{1}{n}}+{\displaystyle {\sum }_{j=1}^p}{\displaystyle \frac{{\mathsf{\lambda }}_j{(k+n-1)}^2}{{{(\mathsf{\lambda }}_j+k)}^2{(n-1)}^2}}\right\}$$

(35)

$$(B\left(\widehat{\beta }\right){)}^2={\displaystyle {\sum }_{j-1}^p}{\displaystyle \frac{k^2{\alpha }_j^2{{(\mathsf{\lambda }}_j-n+1)}^2}{{{(\mathsf{\lambda }}_j+k)}^2{(n-1)}^2}}$$

(36)

$$MSE\left(\widehat{\beta }\right)={\sigma }^2\left\{{\displaystyle \frac{1}{n}}+{\displaystyle {\sum }_{j=1}^p}{\displaystyle \frac{{\mathsf{\lambda }}_j{(k+n-1)}^2}{{{(\mathsf{\lambda }}_j+k)}^2{(n-1)}^2}}\right\}+{\displaystyle {\sum }_{j-1}^p}{\displaystyle \frac{k^2{\alpha }_j^2{{(\mathsf{\lambda }}_j-n+1)}^2}{{{(\mathsf{\lambda }}_j+k)}^2{(n-1)}^2}}$$

(37)

The quantities of the variance and bias are given as (35) and (36) respectively, while the MSE is given in equation (37) where k is the ridge (Shrinkage) parameter which has been proposed and estimated by several researchers including Hoerl and Kennard (1970), ${\widehat{\sigma }}^2$ is the sample variance, $\widehat{k}$ is the estimator of the ridge (Shrinkage) parameter, scale ${\mathsf{\lambda }}_j$ is the eigen value and ${\alpha }_j^2$ is the square of the ith element of the vector $Q^{\text{'}}\beta$, where k and d are the improved ridge (Shrinkage) and Liu parameter with quantities ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$, ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$ and ${\displaystyle \frac{1}{n}}$, which is greater than zero, k$>$0 where -$\infty <$d $<\infty$, k and d are shrinkage parameters in the Liu and Liu type estimators.

This was done for all replications at different combinations of sample sizes, degree of heteroscedasticity, levels of multicollinearity and error variances. The MSE of the best six (6) estimators at each parameter point were tabulated respectively, the MSE of existing and improved Liu estimators were then compared, see Durogade and Kashid (2010) among others. In other to compare the performance of the existing Liu estimators with the improved versions of Liu, the sum of the mean rank across the sample sizes was used to sort the MSE of the estimators at the Six (6) Levels of Multicollinearity, three (3) levels of error variances, Four (4) levels of Heteroscedasticity in the six (6) specified sample sizes.

4. Results and Discussion

4.1. Results

Results are presented, and discussions are made in line with the objectives of this paper.

Table 1. Sample of Mean Squared Error Estimates of Gen_Liu_Type Estimator at ρ = 0.7, δ = 1, σ = 1, n = 50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	2.219244808	2.216942159	2.218444346	2.219228793
Gen_Liu_Type_2	2.628644411	2.625412394	2.627520782	2.628621929
Gen_Liu_Type_3	4.241889478	4.234036357	4.239158516	4.241834828
Gen_Liu_Type_4	20.74370928	20.63647402	20.7063585	20.74296123
Gen_Liu_Type_5	7.211848699	7.192329794	7.205058168	7.211712783
Gen_Liu_Type_6	2.40088728	2.398185508	2.399948036	2.400868487
Gen_Liu_Type_7	2.050705428	2.04875309	2.050026767	2.05069185
Gen_Liu_Type_8	2.491013106	2.488106275	2.490002556	2.490992887
Gen_Liu_Type_9	2.045950827	2.044007567	2.045275323	2.045937313
Gen_Liu_Type_10	2.98621548	2.982082279	2.984778449	2.986186726
Gen_Liu_Type_11	2.093479865	2.091440436	2.092770924	2.093465681

Table 1. Sample of Mean Squared Error Estimates of Gen_Liu_Type Estimator at ρ = 0.7, δ = 1, σ = 1, n = 50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	2.219244808	2.216942159	2.218444346	2.219228793
Gen_Liu_Type_2	2.628644411	2.625412394	2.627520782	2.628621929
Gen_Liu_Type_3	4.241889478	4.234036357	4.239158516	4.241834828
Gen_Liu_Type_4	20.74370928	20.63647402	20.7063585	20.74296123
Gen_Liu_Type_5	7.211848699	7.192329794	7.205058168	7.211712783
Gen_Liu_Type_6	2.40088728	2.398185508	2.399948036	2.400868487
Gen_Liu_Type_7	2.050705428	2.04875309	2.050026767	2.05069185
Gen_Liu_Type_8	2.491013106	2.488106275	2.490002556	2.490992887
Gen_Liu_Type_9	2.045950827	2.044007567	2.045275323	2.045937313
Gen_Liu_Type_10	2.98621548	2.982082279	2.984778449	2.986186726
Gen_Liu_Type_11	2.093479865	2.091440436	2.092770924	2.093465681

Table 2. Ranking of Sample Mean Squared Error Estimates of Gen_Liu_Type Estimator at ρ = 0.7, δ = 1, σ = 1, n = 50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	4	1	2	3
Gen_Liu_Type_2	4	1	2	3
Gen_Liu_Type_3	4	1	2	3
Gen_Liu_Type_4	4	1	2	3
Gen_Liu_Type_5	4	1	2	3
Gen_Liu_Type_6	4	1	2	3
Gen_Liu_Type_7	4	1	2	3
Gen_Liu_Type_8	4	1	2	3
Gen_Liu_Type_9	4	1	2	3
Gen_Liu_Type_10	4	1	2	3
Gen_Liu_Type_11	4	1	2	3
Total	44	11	22	33

Table 2. Ranking of Sample Mean Squared Error Estimates of Gen_Liu_Type Estimator at ρ = 0.7, δ = 1, σ = 1, n = 50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	4	1	2	3
Gen_Liu_Type_2	4	1	2	3
Gen_Liu_Type_3	4	1	2	3
Gen_Liu_Type_4	4	1	2	3
Gen_Liu_Type_5	4	1	2	3
Gen_Liu_Type_6	4	1	2	3
Gen_Liu_Type_7	4	1	2	3
Gen_Liu_Type_8	4	1	2	3
Gen_Liu_Type_9	4	1	2	3
Gen_Liu_Type_10	4	1	2	3
Gen_Liu_Type_11	4	1	2	3
Total	44	11	22	33

Interpretation of results using ranking approach

Using ranking, the estimates of the MSE offered by Gen_Liu estimators viz: Gen_Liu_type_1, Gen_Liu_ Type _2, Gen_Liu_ Type _3, Gen_Liu_Type _4, Gen_Liu_ Type _5, Gen_Liu_ Type _6, Gen_Liu_ Type _7, Gen_Liu_ Type _8, Gen_Liu_ Type _9, Gen_Liu_ Type _10 and Gen_Liu_ Type _11 are best under improvement k₁d₁. No significant improvement was observed under original k and d, k₂d₂ and likewise, k₃d₃.

Summary of Tables, When rho ( ρ)=0.7, Delta (δ)=1, Sigma(σ) =1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Liu Estimators.

Interpretation of results using ranking approach

From Table 3, the best estimators for the Gen_Liu_Type estimators are improved versions $k_1{d}_1$ and $k_2{d}_2$ with improvements${\displaystyle \frac{1}{n}} and {\displaystyle \frac{1}{\mathrm{m}\mathrm{a}\mathrm{x}\left(vifs\right)}}$ in small sample situations and when n=50 as it offered the minimum mean square error in most cases. The original version of the estimators are best when $n=30, 75 and 100.$

When rho ( $\rho )=0.8$, Delta ($\delta )=1,$ Sigma ($\sigma )$ =1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Generalised Liu-type Estimators.

Table 4. Mean Squared Error Estimates of Gen_Liu_Type Estimator at Delta=1, Sigma=1, n=50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	0.3210601	0.320982	0.3209321	0.32106
Gen_Liu_Type_2	0.3244	0.3243228	0.3242735	0.3244
Gen_Liu_Type_3	0.326097	0.3260205	0.3259717	0.32609
Gen_Liu_Type_4	0.3279235	0.3278478	0.3277995	0.32792
Gen_Liu_Type_5	0.3275558	0.3274799	0.3274315	0.32755
Gen_Liu_Type_6	0.3221564	0.3220787	0.3220291	0.32215
Gen_Liu_Type_7	0.318476	0.3183975	0.3183474	0.31847
Gen_Liu_Type_8	0.3224905	0.322413	0.3223635	0.32249
Gen_Liu_Type_9	0.3184121	0.3183336	0.3182835	0.31841
Gen_Liu_Type_10	0.3243517	0.3242747	0.3242255	0.32435
Gen_Liu_Type_11	0.3194118	0.3193335	0.3192835	0.31941

Table 4. Mean Squared Error Estimates of Gen_Liu_Type Estimator at Delta=1, Sigma=1, n=50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	0.3210601	0.320982	0.3209321	0.32106
Gen_Liu_Type_2	0.3244	0.3243228	0.3242735	0.3244
Gen_Liu_Type_3	0.326097	0.3260205	0.3259717	0.32609
Gen_Liu_Type_4	0.3279235	0.3278478	0.3277995	0.32792
Gen_Liu_Type_5	0.3275558	0.3274799	0.3274315	0.32755
Gen_Liu_Type_6	0.3221564	0.3220787	0.3220291	0.32215
Gen_Liu_Type_7	0.318476	0.3183975	0.3183474	0.31847
Gen_Liu_Type_8	0.3224905	0.322413	0.3223635	0.32249
Gen_Liu_Type_9	0.3184121	0.3183336	0.3182835	0.31841
Gen_Liu_Type_10	0.3243517	0.3242747	0.3242255	0.32435
Gen_Liu_Type_11	0.3194118	0.3193335	0.3192835	0.31941

Table 5. Ranking of Mean Squared Error Estimates of Gen_Liu_Type Estimator at Delta=1, Sigma=1, n=50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	4	2	1	3
Gen_Liu_Type_2	4	2	1	3
Gen_Liu_Type_3	4	2	1	3
Gen_Liu_Type_4	4	2	1	3
Gen_Liu_Type_5	4	2	1	3
Gen_Liu_Type_6	4	2	1	3
Gen_Liu_Type_7	4	2	1	3
Gen_Liu_Type_8	4	2	1	3
Gen_Liu_Type_9	4	2	1	3
Gen_Liu_Type_10	4	2	1	3
Gen_Liu_Type_11	4	2	1	3
Total	44	22	11	33

Table 5. Ranking of Mean Squared Error Estimates of Gen_Liu_Type Estimator at Delta=1, Sigma=1, n=50.

Estimators	Original	${\mathit{k}}_1{\mathit{d}}_1$	${\mathit{k}}_2{\mathit{d}}_2$	${\mathit{k}}_3{\mathit{d}}_3$
Gen_Liu_Type_1	4	2	1	3
Gen_Liu_Type_2	4	2	1	3
Gen_Liu_Type_3	4	2	1	3
Gen_Liu_Type_4	4	2	1	3
Gen_Liu_Type_5	4	2	1	3
Gen_Liu_Type_6	4	2	1	3
Gen_Liu_Type_7	4	2	1	3
Gen_Liu_Type_8	4	2	1	3
Gen_Liu_Type_9	4	2	1	3
Gen_Liu_Type_10	4	2	1	3
Gen_Liu_Type_11	4	2	1	3
Total	44	22	11	33

Interpretation of results using ranking approach

Using ranking, none of the estimates of MSE offered by Gen_Liu_Type estimators is best with the original, likewise under improvement${ d}_1$ and $d_3$. However, all estimators are best under improvement $d_2.$Summary of Tables, When rho (ρ) = 0.8, Delta (δ) = 1, Sigma (σ) = 1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Liu Estimators

From Table 6, the best estimator for Gen_Liu_Type estimator is the improved version $d_2$ with ${\displaystyle \frac{1}{\mathrm{m}\mathrm{a}\mathrm{x}\left(vif\right)}}$ when the sample size is large as it offered the least mean square error estimates most of the times. When the sample size is small, the original version of the estimators offered the least mean square error estimates, although the improved version $k_3{d}_3$ with improvement ${\displaystyle \frac{1}{n\left(\mathrm{m}\mathrm{a}\mathrm{x}\right(vifs)}}$ is the best when $n =10$.

Summary of Tables, When rho (ρ) = 0.85, Delta (δ) = 1, Sigma (σ) = 1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Gen_Liu_Type Estimators.

From Table 7, the best estimator for Gen_Liu_Type estimator is the improved version $k_2{d}_2$ with ${\displaystyle \frac{1}{\mathrm{m}\mathrm{a}\mathrm{x}\left(vifs\right)}}$ when the sample size is small. But when the sample size is 30, the original version of the estimators offered the least MSE almost all the time. The same in the case of sample sizes 75 and 100.

Summary of Tables, When rho ($\mathit{\rho })=0.9,\mathit{D}\mathit{e}\mathit{l}\mathit{t}\mathit{a}\left(\mathit{\delta }=1\right),\mathit{s}\mathit{i}\mathit{g}\mathit{m}\mathit{a}\left(\mathit{\sigma }\right)=1$, varying samples $\mathit{n}=10,20,30,50,70,100$ respectively using various Generalised Liu Type Estimators.

From Table 8, the best estimator for Generalised Liu Type estimator is the original nature of the estimators as it offered the minimum MSE estimates most of the times. Although when $n=10$, the improved version ${k_{3}d}_3$ with ${\displaystyle \frac{1}{n\left(\max \left(vifs\right)\right)}}$ is the best. When $n=50$, the improved version $k_2{d}_2$ with ${\displaystyle \frac{1}{\mathrm{m}\mathrm{a}\mathrm{x}\left(vifs\right)}}$ is the best as it offered the minimum MSE estimates.

Summary of Tables, When rho ( $\mathit{\rho })=0.95$, Delta ($\mathit{\delta })$=1, Sigma($\mathit{\sigma })$ =1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Generalised Liu Type Estimators.

From Table 9, the best estimator for the Generalised_Liu_Type is the improved version $k_2{d}_2$ with ${\displaystyle \frac{1}{\mathrm{m}\mathrm{a}\mathrm{x}\left(vifs\right)}}$ as it offered the minimum MSE estimates. In small sample situations, improved version $k_3{d}_3$ offered the minimum MSE estimates when $n=10 and 20$. In large sample situations, the original nature of the estimators offered the minimum MSE estimates when $n=75 and 100$.

Summary of Tables, When rho ($\mathit{\rho })=0.99$, Delta ($\mathit{\delta })=1$, Sigma ($\mathit{\sigma })$ =1, varying sample sizes n=10, 20, 30, 50, 70, 100 respectively using various Generalised Liu-Type Estimators.

From Table 10, the best estimator for Generalised Liu_Type estimator is original nature of the estimators as it offered the minimum MSE estimates especially in large sample situations. Although the improved version $k_3{d}_3$ with ${\displaystyle \frac{1}{n\left(\max \left(vifs\right)\right)}}$ is best when $n=10$. The improved version $k_1{d}_1$ with ${\displaystyle \frac{1}{n}}$ is best when $n=20$. Generally, the original nature of the estimator is the best.

4.2. Application to Real Life Situations

Results of the application to Real life Data set are presented and discussions are made based on the results obtained.

Interpretation of results using ranking approach

Using ranking, from Table 11, the estimates of MSE offered by Liu estimators viz: Gen Liu_7, Gen Liu_9 Liu_10 and Gen Liu_11 are best with the original d. Gen Liu_3, Gen Liu_4, and Gen Liu_8 are best under improvement d₁. Gen Liu_1, and Gen Liu_2, are best under improvement d₂, Gen Liu _6 is the best under improvement d₃. Both improvement d₂ and d₃ has the least sum of ranks follow by the Original. Improvement d₂ is the best.

Table 11. Ranking of MSE of Generalised Liu type Estimators of Portland Cement Data Set at $\mathit{\rho }\ge 0.7$, $\mathit{\delta }$= 0, $\mathit{\sigma }$ = 1, n=13.

Estimators	Original	d1	d2	d3
Gen_Liu_type_1	3	4	1	2
Gen_Liu_type 2	3	4	1	2
Gen_Liu_type 3	4	1	2	3
Gen_Liu_type 4	4	1	2	3
Gen_Liu_type 5	3	4	1	2
Gen_Liu_type 6	4	2	3	1
Gen_Liu_type 7	1	4	3	2
Gen_Liu_type 8	4	1	2	3
Gen_Liu_type 9	1	4	3	2
Gen_Liu_type 10	1	4	3	2
Gen_Liu_type 11	1	4	3	2
Total	29	33	24	24

Table 11. Ranking of MSE of Generalised Liu type Estimators of Portland Cement Data Set at $\mathit{\rho }\ge 0.7$, $\mathit{\delta }$= 0, $\mathit{\sigma }$ = 1, n=13.

Estimators	Original	d1	d2	d3
Gen_Liu_type_1	3	4	1	2
Gen_Liu_type 2	3	4	1	2
Gen_Liu_type 3	4	1	2	3
Gen_Liu_type 4	4	1	2	3
Gen_Liu_type 5	3	4	1	2
Gen_Liu_type 6	4	2	3	1
Gen_Liu_type 7	1	4	3	2
Gen_Liu_type 8	4	1	2	3
Gen_Liu_type 9	1	4	3	2
Gen_Liu_type 10	1	4	3	2
Gen_Liu_type 11	1	4	3	2
Total	29	33	24	24

4.3. Discussion of Results

The performance of the original and improved generalised Liu-type estimators was assessed using theMSE criterion. Since a smaller MSE indicates a more efficient estimator, the ranking procedure assigned the best rank to the estimator with the minimum MSE under each experimental condition. The discussion therefore focuses on the relative performance of the original generalised Liu-type estimators and the three proposed improved versions, namely (k₁d₁), (k₂d₂), and (k₃d₃). For the initial simulation condition, the ranking results showed that all eleven generalised Liu-type estimators, namely Gen_Liu_Type_1 to Gen_Liu_Type_11, performed best under the first improved parameter combination, (k₁d₁). This result indicates that the modification based on (1/n) produced the lowest MSE values across the estimators considered. Conversely, the original parameter combination and the second and third improved combinations, (k₂d₂) and (k₃d₃) did not show substantial improvement under this condition. This suggests that the first improvement is particularly effective when the level of multicollinearity is relatively moderate.

The results summarized in Table 3 further revealed that the relative performance of the estimators depended on sample size. The improved versions (k₁d₁) and (k₂d₂) based on the adjustment factors (1/n) and ${\displaystyle \frac{1}{\left(\mathrm{Max}\left({VIF}_s\right)\right)}}$ respectively, performed better in small-sample situations and at (n=50). These improved estimators produced the minimum MSE values in most cases, indicating that shrinkage parameter modification is beneficial when the available sample size is limited. However, at (n=30), (n=75), and (n=100), the original generalised Liu-type estimators performed better in several cases. This implies that, as the sample size increases, the advantage of introducing additional shrinkage may decline under some conditions. The ranking results presented for the next simulation setting showed that the second improved parameter combination, (k₂d₂) consistently produced the lowest MSE values for all the generalised Liu-type estimators. Neither the original estimator nor the first and third improved forms, (k₁d₁) and (k₃d₃) ranked best in this setting. This finding provides strong evidence that the improvement based on ${\displaystyle \frac{1}{\left(\mathrm{Max}\left({VIF}_s\right)\right)}}$ can be highly effective in controlling the effect of multicollinearity, particularly when the degree of correlation among the regressors is sufficiently pronounced.

At ($\rho$= 0.80), the results showed a mixed pattern. The improved version (k₂d₂), based on ${\displaystyle \frac{1}{\left(\mathrm{Max}\left({VIF}_s\right)\right)}}$performed best in large-sample situations, where it produced the lowest MSE values most frequently. In contrast, the original generalised Liu-type estimators performed better in most small-sample situations. Nevertheless, at (n=10), the third improved version, (k₃d₃) based on ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$was the best-performing estimator. This result suggests that the third improvement may be useful when the sample size is extremely small, because it combines the influence of sample size and multicollinearity in selecting the shrinkage parameters. For the condition corresponding to $\rho$= 0.85, the improved version (k₂d₂) performed best in small samples. This confirms the usefulness of the ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$adjustment in situations where multicollinearity is relatively high and the sample size is limited. However, when (n=30), (n=75), and (n=100), the original generalised Liu-type estimators produced the lowest MSE values in most cases. This indicates that the original estimators may remain competitive when the sample size is moderate or large, even in the presence of substantial multicollinearity.

At $\rho$=0.90, the original generalised Liu-type estimators generally performed best, as they produced the minimum MSE values in most of the sample-size settings. However, the improved version (k₃d₃) was best when (n=10), while (k₂d₂) was best when (n=50). These results show that, under high multicollinearity, the performance of the improved estimators is sensitive to sample size. The third improved version appears to be more appropriate for very small samples, whereas the second improved version may be preferable for moderate sample sizes. The results at $\rho$=0.95 showed that the second improved version, (k₂d₂) generally provided the minimum MSE values across the estimators. This finding suggests that the ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$ modification is particularly useful when multicollinearity becomes severe. In small-sample situations, especially at (n=10) and (n=20), the third improved version, (k₃d₃), performed better. However, in large-sample situations, particularly at (n=75) and (n=100), the original generalised Liu-type estimators produced lower MSE values. Therefore, while the proposed improvements are advantageous under severe multicollinearity, the original estimator may still be preferred when the sample size is sufficiently large. At the highest level of multicollinearity, $\rho$=0.99, the original generalised Liu-type estimators generally performed best, especially in large samples. This result indicates that excessive shrinkage may not always be beneficial when the explanatory variables are extremely highly correlated and the sample size is large. Nevertheless, the improved version (k₃d₃) performed best when (n=10), while the first improved version, (k₁d₁), performed best when (n=20). These findings reinforce the importance of considering sample size when selecting an appropriate improved generalised Liu-type estimator.

The empirical application to the Portland Cement data set also supported the simulation findings. The ranking results showed that Gen_Liu_Type_7, Gen_Liu_Type_9, Gen_Liu_Type_10, and Gen_Liu_Type_11 performed best under the original Liu parameter. Gen_Liu_Type_3, Gen_Liu_Type_4, and Gen_Liu_Type_8 performed best under the first improvement, (d₁), while Gen_Liu_Type_1 and Gen_Liu_Type_2 performed best under the second improvement, (d₂). In addition, Gen_Liu_Type_6 performed best under the third improvement, (d₃). Based on the total ranking scores, the second and third improvements had the lowest overall ranks, with the second improvement, (d₂), emerging as the best overall choice. This empirical evidence confirms that the proposed modifications can improve the performance of generalised Liu-type estimators in practical data sets affected by multicollinearity.

Overall, the findings demonstrate that no single estimator is uniformly superior under all combinations of sample size and multicollinearity. However, the improved generalised Liu-type estimators, particularly the second improved form (k₂d₂) frequently produced lower MSE values and better ranks across several simulation and real-data conditions. The first improvement, (k₁d₁), was particularly effective under moderate multicollinearity, while the third improvement, (k₃d₃) was useful in very small samples. The original generalised Liu-type estimators remained competitive and were often preferred under very high multicollinearity and larger sample sizes. Therefore, the selection of an appropriate generalised Liu-type estimator should be based on the degree of multicollinearity, the sample size, and the relative MSE performance of the competing estimators.

5. Summary, Conclusion and Recommendations

5.1. Summary

This study investigated improved generalized Liu-type estimators for linear regression models affected by multicollinearity and heteroscedasticity. The study was motivated by the limitations of the Ordinary Least Squares estimator, which may produce unstable coefficient estimates and inflated variances when explanatory variables are highly correlated. Heteroscedasticity further reduces the efficiency of estimation and may affect the reliability of statistical inference. The study improved the existing generalized Liu-type estimator through modifications of the shrinkage parameters (k) and (d). Three improved parameter combinations, denoted by (k₁d₁), (k₂d₂) and (k₃d₃) were developed. The first improvement was based on (1/n), the second was based on ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$, and the third was based on ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$. The theoretical properties of the proposed estimators were examined using bias, variance, and mean square error expressions.

A Monte Carlo simulation experiment was conducted with 1,000 replications. The study considered sample sizes of (n=10, 20, 30, 50, 75,) and (100), levels of multicollinearity ranging from ($\rho$=0.70) to $\rho$=0.99, different error variances, and varying levels of heteroscedasticity. The original and improved generalised Liu-type estimators were compared using the mean square error criterion and a ranking approach. The simulation results showed that the improved estimators generally produced lower mean square errors than the original estimators in several experimental conditions. The first improvement, (k₁d₁), performed well under moderate multicollinearity. The second improvement, (k₂d₂) frequently ranked best across several levels of multicollinearity, particularly under moderate-to-severe multicollinearity. The third improvement, (k₃d₃) was particularly useful in very small samples, especially when (n=10).

The results also showed that the original generalised Liu-type estimators remained competitive in some situations. In particular, the original estimators often performed better when the sample size was large and the degree of multicollinearity was extremely high, especially at ($\rho$=0.95) and ($\rho$=0.99). This indicates that no single estimator was uniformly superior under all conditions. The empirical application using the Portland Cement data set supported the simulation findings. The ranking results showed that the second improvement, d₂, had the best overall performance, while the third improvement, d₃ also performed favourably. The results confirmed that the proposed improvements can enhance the efficiency of generalised Liu-type estimators in practical regression applications involving multicollinearity.

5.2. Conclusions

This study developed improved generalised Liu-type estimators for linear regression models affected by multicollinearity and heteroscedasticity. The proposed estimators were obtained by modifying the shrinkage parameters (k) and (d) in the existing generalised Liu-type estimator. Their performances were evaluated using theoretical mean square error properties, Monte Carlo simulation, and an empirical application. The findings established that the proposed improved estimators can provide lower mean square errors than the original generalised Liu-type estimators under many combinations of sample size and multicollinearity. Among the proposed estimators, the second improvement, k₂d₂, based on ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$, was the most consistent performer and frequently produced the minimum mean square error. The first improvement (k₁d₁), was effective under moderate multicollinearity, while the third improvement, (k₃d₃) was more useful in very small samples.

However, the study also established that the original generalised Liu-type estimators may be preferable in some large-sample situations, particularly under extremely high multicollinearity. Therefore, the choice of estimator should not be made independently of the sample size and severity of multicollinearity. Hence, the study concludes that the improved generalised Liu-type estimators are valuable alternatives to the original estimator for regression analysis involving multicollinearity and heteroscedasticity. The proposed estimators can improve coefficient stability and reduce estimation error by achieving a more favourable balance between bias and variance.

5.3. Recommendations

Based on the findings of this study, the following recommendations are made:

• The improved generalised Liu-type estimators should be considered when regression data exhibit multicollinearity and heteroscedasticity, particularly where the Ordinary Least Squares estimator produces unstable coefficient estimates.
• The second improved estimator, k₂d₂, based on ${\displaystyle \frac{1}{\mathrm{Max}\left({VIF}_j\right)}}$, is recommended as the preferred alternative in many practical situations because it frequently produced the lowest mean square error across the simulation and empirical results.
• The third improved estimator, (k₃d₃), based on ${\displaystyle \frac{1}{n(\mathrm{Max}\left({VIF}_j\right))}}$, should be considered when the sample size is very small, especially when (n=10), since it performed favourably in several small-sample situations.
• The first improved estimator, (k₁d₁), based on (1/n), may be considered under moderate multicollinearity because it performed well in several simulation settings.
• The original generalised Liu-type estimator should still be considered when the sample size is large and multicollinearity is extremely severe, particularly at correlation levels close to ($\rho$=0.95) and ($\rho$=0.99).
• Researchers should conduct multicollinearity diagnostics, such as the variance inflation factor, before selecting an estimator, since the severity of multicollinearity influences the relative performance of the original and improved estimators.
• Future studies should investigate the performance of the proposed estimators under non-normal error distributions, outliers, autocorrelation, missing observations, and measurement errors.
• Further research should develop data-driven or adaptive procedures for selecting the shrinkage parameters automatically, particularly for high-dimensional regression models.