More Efficient Estimation Strategy for (k-d) Class Estimator in Existence of Multicollinearity and Heteroscedasticity: Some Monte Carlo Simulation Evidence

Article Information

Title: More Efficient Estimation Strategy for (k-d) Class Estimator in Existence of Multicollinearity and Heteroscedasticity: Some Monte Carlo Simulation Evidence

Authors: Waqas Makhdoom, Muhammad Aslam

Journal: Journal Of Statistics

HEC Recognition History

Category	From	To
Y	2024-10-01	2025-12-31
Y	2023-07-01	2024-09-30
Y	1900-01-01	2005-06-30

Publisher: Government College University, Lahore.

Country: Pakistan

Year: 2023

Volume: 27

Issue: 1

Language: English

Keywords: RegressionMulticollinearityEstimationHeteroscedasticityKernel Estimator

Abstract

The typical linear regression model does this to have some sort of heteroscedasticity in the error terms and linear correlation in the regressors. The ordinary least squares estimates are significantly impacted by each of these issues. When these assumptions violated in any multiple linear regression model then ordinary least square estimator happen to unstable and no longer remain best linear unbiased estimator. Therefore, in attempt to tackle the issue of Multicollinearity the rigid, Liu and (k-d) regression exist and easily accessible in literature. The adaptive estimator was recommended to obtain an efficient estimator in comparison to the conventional least square estimator to address the problem of heteroscedasticity. This current work suggests the improved method of adaptation for (kd) class estimator to get more efficient results when dealing with multicollinearity and heteroscedasticity occur at same time. All the numerical work is done by using simulation scheme Monte Carlo, with different degrees of collinearity, severity (existence) of heteroscedasticity, and sample size to assess the performance of the suggested estimator. The simulation results provide best performance of adaptive (k-d) class estimator which is our proposed estimator.

Disclaimer: The following sections are produced through an intensive multi-stage analytical pipeline developed by DefinePK, utilizing advanced natural-language processing, domain-specific knowledge models, and automated cross-referencing systems. Despite the rigor and computational effort involved, the generated summary may still contain inaccuracies or incomplete interpretations. Users are strongly advised to verify all critical information against the original paper and its authoritative sources.

Research Objective

To propose an improved adaptive method for the (k-d) class estimator to achieve more efficient results when multicollinearity and heteroscedasticity occur simultaneously in linear regression models.

Methodology

The study employs a Monte Carlo simulation scheme to evaluate the performance of the proposed adaptive (k-d) class estimator (AKDE) against existing estimators (OLSE, ORRE, ARRE, KDE). The simulation involves varying degrees of collinearity, severity of heteroscedasticity, and sample sizes. The performance is assessed using the Estimated Mean Square Error (EMSE) criterion and relative efficiency. Three methods for estimating rigid parameters (k) are used: HKB, GK, and KS.

Methodology Flowchart

                        graph TD
    A["Define Linear Regression Model with Multicollinearity & Heteroscedasticity"] --> B["Generate Data via Monte Carlo Simulation"];
    B --> C["Vary Collinearity, Heteroscedasticity, Sample Size"];
    C --> D["Estimate Parameters using OLSE, ORRE, ARRE, KDE, AKDE"];
    D --> E["Calculate EMSE and Relative Efficiency"];
    E --> F["Compare Estimator Performance"];
    F --> G["Conclude on AKDE's Efficiency"];

Discussion

The research addresses the common issues of multicollinearity and heteroscedasticity in linear regression, which can lead to unstable and inefficient Ordinary Least Squares (OLS) estimates. Existing methods like Ridge Regression and Liu Estimator are discussed as attempts to mitigate multicollinearity. Adaptive estimation is introduced to handle heteroscedasticity. The study extends the adaptive estimation procedure to the (k-d) class estimator, resulting in the proposed AKDE, which is shown to be more efficient through simulation.

Key Findings

The proposed adaptive (k-d) class estimator (AKDE) demonstrates superior performance compared to other estimators, exhibiting a lower EMSE across various levels of multicollinearity and heteroscedasticity. The AKDE is found to be the best option when linear regression models violate the assumptions of multicollinearity and heteroscedasticity simultaneously.

Conclusion

The study successfully proposes an adaptive (k-d) class estimator (AKDE) that is more efficient in the presence of both multicollinearity and heteroscedasticity. The Monte Carlo simulations confirm that the AKDE consistently outperforms existing estimators under various conditions, making it a suitable choice for regression models with these violations.

Fact Check

1. Multicollinearity Introduction: The term multicollinearity was first introduced by Frisch in 1934. (Confirmed in text)
2. OLS Estimator Formula: The Ordinary Least Squares Estimator (OLSE) is defined as $\hat{\beta} = (X^TX)^{-1}X^Ty$. (Confirmed in text)
3. Simulation Repetitions: The number of Monte Carlo simulation repetitions was fixed at R = 5000. (Confirmed in text)

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