Journal of Multidisciplinary Applied Natural Science
Open Access Journal
Scopus CiteScore 2024
4.8
Calculated on 05 May, 2025
SJR 2024
0.31
Powered by scimagojr.com
Open Access Journal
4.8
Calculated on 05 May, 2025
0.31
Powered by scimagojr.com
https://doi.org/10.47352/jmans.2774-3047.372
Tarek Mahmoud Omara
This study extends the seemingly unrelated regression (SUR) model through introducing the ridge (RTSUR) and Liu-Type (LTSUR) estimators as biased estimation techniques to address the problem of multicollinearity in the SUR Tobit (SURT) model. This study theoretically evaluates the superiority of the proposed estimators based on the mean square error (MSE) criterion. The results for the theoretically study showing that, the Liu-Type estimator outperforms other estimators under many conditions. A simulation study was conducted to compare the estimators under various factors. The results of simulation show that, the maximum likelihood (MLE) estimator is the worst estimator at all factors and the LTSUR and RTSUR estimators perform better at high levels of multicollinearity and censoring. The LTSUR still achieved a significant superiority over the RTSUR. In addition, when the number of observations in the equations increases, the performance of the LTSUR and RTSUR estimators improves. Moreover, the simulation mean squared errors (SMSE) values for LTSUR estimator converge as number of observation and censored level increase. To study the behavior of the proposed estimators on real data, we used weather data from Cairo city to examine their influence on pollution levels of carbon monoxide (CO), sulfur dioxide (SO2), and nitrogen dioxide (NO2). The results from the real data were consistent with the findings from the simulation study.
[1] A. Zellner. (1962). "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias". Journal of the American Statistical Association. 57 (298): 348-368. 10.2307/2281644.
DOI: https://doi.org/10.1080/01621459.1962.10480664[2] Y. Li and H. Yang. (2012). "A New Liu-Type Estimator in Linear Model". Statistical Papers. 53 : 427-437. 10.1007/s00362-010-0349-y.
DOI: https://doi.org/10.1007/s00362-010-0349-y[3] V. K. Srivastava and D. E. Giles. (1987). "Seemingly Unrelated Regression Equations Models: Estimation and Inference". Marcel Dekker, New York.
[4] L. Firinguetti. (1997). "Ridge Regression in the Context of a System of Seemingly Unrelated Regression Equations". Journal of Statistical Computation and Simulation. 56 (2): 145-162. 10.1080/00949659708811785.
DOI: https://doi.org/10.1080/00949659708811785[5] B. M. G. Kibria. (2003). "Performance of Some New Ridge Regression Estimators". Communications in Statistics: Simulation and Computation. 32 (2): 419-435. 10.1081/SAC-120017499.
DOI: https://doi.org/10.1081/SAC-120017499[6] T. Omara. (2025). "Robust Lasso Estimator for the Liu-Type Regression Model and Its Applications". Statistics, Optimization and Information Computing. 13 (5): 1819-1831. 10.19139/soic-2310-5070-2106.
DOI: https://doi.org/10.19139/soic-2310-5070-2106[7] T. M. Omara. (2018). "Stochastic Restricted Liu-Type Estimator for SUR Model". Pakistan Journal of Statistics and Operation Research. 14 (4): 903-911. 10.18187/pjsor.v14i4.2302.
DOI: https://doi.org/10.18187/pjsor.v14i4.2302[8] T. M. Omara. (2019). "Modification of Liu-Type Estimator for Two SUR Model". Advances and Applications in Statistics. 55 (1): 47-66. 10.17654/AS055010047.
DOI: https://doi.org/10.17654/AS055010047[9] Z. Y. Algamal. (2018). "A New Method for Choosing the Biasing Parameter in Ridge Estimator for Generalized Linear Model". Chemometrics and Intelligent Laboratory Systems. 183 : 96-101. 10.1016/j.chemolab.2018.10.014.
DOI: https://doi.org/10.1016/j.chemolab.2018.10.014[10] Z. Y. Algamal. (2019). "Performance of Ridge Estimator in Inverse Gaussian Regression Model". Communications in Statistics: Theory and Methods. 48 (15): 92-100. 10.1080/03610926.2018.1481977.
DOI: https://doi.org/10.1080/03610926.2018.1481977[11] Z. Y. Algamal. (2020). "Shrinkage Parameter Selection via Modified Cross-Validation Approach for Ridge Regression Model". Communications in Statistics: Simulation and Computation. 49 (7): 82-99. 10.1080/03610918.2018.1508704.
DOI: https://doi.org/10.1080/03610918.2018.1508704[12] M. Abonazel, Z. Algamal, F. Awwad, and A. Taha. (2022). "A New Two-Parameter Estimator for Beta Regression Model: Method, Simulation, and Application". Frontiers in Applied Mathematics and Statistics. 7 : 1-10. 10.3389/fams.2021.780322.
DOI: https://doi.org/10.3389/fams.2021.780322[13] T. Omara. (2025). "Penalized Estimators for Modified Log-Bilal Regression: Simulations and Applications". Statistics, Optimization and Information Computing. 14 (3): 1566-1583. 10.19139/soic-2310-5070-2513.
DOI: https://doi.org/10.19139/soic-2310-5070-2513[14] M. A. Alkhamisi and G. Shukur. (2008). "Developing Ridge Parameters for SUR Model". Communications in Statistics: Theory and Methods. 37 (4): 544-564. 10.1080/03610920701469152.
DOI: https://doi.org/10.1080/03610920701469152[15] A. El-Houssainy, S. Rady, A. Abdel-Aziz, N. Morad, and T. Omara. (2011). "Robust Cross-Validation in SUR Ridge Estimators and SUR Robust Ridge Estimators". Journal of Statistical Theory and Applications. 10 (1): 37-54.
[16] K. Liu. (2003). "Using Liu-Type Estimator to Combat Collinearity". Communications in Statistics: Theory and Methods. 32 (5): 1009-1020. 10.1081/STA-120019959.
DOI: https://doi.org/10.1081/STA-120019959[17] T. M. Omara. (2019). "Modifying Two-Parameter Ridge Liu Estimator Based on Ridge Estimation". Pakistan Journal of Statistics and Operation Research. 15 (4): 881-890. 10.18187/pjsor.v15i4.2575.
DOI: https://doi.org/10.18187/pjsor.v15i3.2575[18] J. Wu. (2014). "Improved Liu-Type Estimator of Parameters in Two Seemingly Unrelated Regressions". Mathematical Problems in Engineering. 10.1155/2014/679835.
DOI: https://doi.org/10.1155/2014/679835[19] T. Omara. (2021). "Robust Liu-Type Estimator for SUR Model". Statistics, Optimization and Information Computing. 9 (3): 607-617. 10.19139/soic-2310-5070-985.
DOI: https://doi.org/10.19139/soic-2310-5070-985[20] J. S. Long. (1997). "Regression Models for Categorical and Limited Dependent Variables". Sage Publications, Thousand Oaks, CA.
[21] J. Tobin. (1958). "Estimation of Relationships for Limited Dependent Variables". Econometrica. 26 24-36. 10.2307/1907382.
DOI: https://doi.org/10.2307/1907382[22] M. Iqbal, J. Ma, Z. Mushtaq, N. Ahmad, M. Z. Yousaf, B. Tarawneh, W. Khan, M. Pushkarna, and I. Zaitsev. (2025). "Energy Efficiency Evaluation of Construction Projects Using Data Envelopment Analysis and Tobit Regression". Scientific Reports. 15 11444. 10.1038/s41598-025-90671-3.
DOI: https://doi.org/10.1038/s41598-025-90671-3[23] A. N. Moluno and L. C. Eme. (2025). "Efficiency and Profitability Analysis of Agro-Production in the Niger Delta: A Data Envelopment and Tobit Regression Approach". Malaysian Journal of Sustainable Agriculture. 9 (2): 115-120. 10.26480/mjsa.02.2025.115.120.
DOI: https://doi.org/10.26480/mjsa.02.2025.115.120[24] D. Zhao, H. Huang, Y. Zhang, and S. Li. (2025). "Study on the Efficiency of Health Resource Allocation in the Western Region of China Based on Three-Stage DEA and Tobit Regression Analysis". BMC Health Services Research. 25 : 480. 10.1186/s12913-025-12630-y.
DOI: https://doi.org/10.1186/s12913-025-12630-y[25] J. Sun, A. B. Rosli, and A. Daud. (2025). "Efficiency Analysis of Listed Pharmaceutical Companies in China: A Method Combining Three-Stage DEA with Undesirable Output, PCA, and Tobit Regression". PLoS One. 20 (8): e0329767. 10.1371/journal.pone.0329767.
DOI: https://doi.org/10.1371/journal.pone.0329767[26] T. Mai. (2025). "High-Dimensional Bayesian Tobit Regression for Censored Response with Horseshoe Prior". arXiv Preprint. 10.1002/sta4.70008.
DOI: https://doi.org/10.1002/sta4.70008[27] V. R. Dănăilă and C. Buiu. (2024). "A Deep Learning Approach to Censored Regression". Pattern Analysis and Applications. 27 (1). 10.1007/s10044-024-01216-9.
DOI: https://doi.org/10.1007/s10044-024-01216-9[28] G. Khalaf, K. Månsson, P. Sjölander, and G. Shukur. (2014). "A Tobit Ridge Regression Estimator". Communications in Statistics: Theory and Methods. 43 : 131-140. 10.1080/03610926.2012.655881.
DOI: https://doi.org/10.1080/03610926.2012.655881[29] D. Aydın, Ö. İşçi, and E. Yılmaz. (2020). "Optimum Shrinkage Parameter Selection for Ridge Type Estimator of Tobit Model". Journal of Statistical Computation and Simulation. 91 (5): 952-975. 10.1080/00949655.2020.1838523.
DOI: https://doi.org/10.1080/00949655.2020.1838523[30] F. Alhusseini and M. Odah. (2016). "Principal Component Regression for Tobit Model and Purchases of Gold". Bucharest, Romania.
[31] S. Toker, N. Özbay, G. Şiray, and I. Yenilmez. (2021). "Tobit Liu Estimation of Censored Regression Model: An Application to Mroz Data and a Monte Carlo Simulation Study". Journal of Statistical Computation and Simulation. 91 (6): 1061-1091. 10.1080/00949655.2020.1828416.
DOI: https://doi.org/10.1080/00949655.2020.1828416[32] T. M. Omara. (2023). "Almost Unbiased Liu-Type Estimator for Tobit Regression and Its Application". Communications in Statistics: Simulation and Computation. 1234-1249. 10.1080/03610918.2023.2257006.
DOI: https://doi.org/10.1080/03610918.2023.2257006[33] H. Huang. (1999). "Estimation of the SUR Tobit Model via the MCECM Algorithm". Economics Letters. 64 (1): 25-30. 10.1016/S0165-1765(99)00075-0.
DOI: https://doi.org/10.1016/S0165-1765(99)00075-0[34] W. A. Kamakura and M. Wedel. (2001). "Exploratory Tobit Factor Analysis for Multivariate Censored Data". Multivariate Behavioral Research. 36 (1): 53-82. 10.1207/S15327906MBR3601_03.
DOI: https://doi.org/10.1207/S15327906MBR3601_03[35] C. Huang, F. Sloan, and K. Adamache. (1987). "Estimation of Seemingly Unrelated Tobit Regressions via the EM Algorithm". Journal of Business and Economic Statistics. 5 (3): 425-430. 10.2307/1391618.
DOI: https://doi.org/10.1080/07350015.1987.10509607[36] H. Huang. (2001). "Bayesian Analysis of the SUR Tobit Model". Applied Economics Letters. 8 (9): 617-622. 10.1080/13504850010026069.
DOI: https://doi.org/10.1080/13504850010026069[37] N. Baranchuk and S. Chib. (2008). "Assessing the Role of Option Grants to CEOs: How Important Is Heterogeneity?". Journal of Empirical Finance. 15 (2): 145-166. 10.1016/j.jempfin.2006.10.004.
DOI: https://doi.org/10.1016/j.jempfin.2006.10.004[38] M. Taylor and D. Phaneuf. (2009). "Bayesian Estimation of the Impacts of Food Safety Information on Household Demand for Meat and Poultry". Milwaukee, USA.
[39] R. Farebrother. (1976). "Further Results on the Mean Square Error of Ridge Regression". Journal of the Royal Statistical Society: Series B (Statistical Methodology). 38 (3): 248-250. 10.1111/j.2517-6161.1976.tb01588.x.
DOI: https://doi.org/10.1111/j.2517-6161.1976.tb01588.x[40] Y. Liu, Y. Zhou, and J. Lu. (2020). "Exploring the Relationship Between Air Pollution and Meteorological Conditions in China Under Environmental Governance". Scientific Reports. 10 : 119-129. 10.1038/s41598-020-71338-7.
DOI: https://doi.org/10.1038/s41598-020-71338-7