Dimitrios Pappas, Konstantinos Bisiotis

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In this work we propose an unbiased estimator for a multiple linear regression model of the CAPM in the presence of multicollinearity in the explanatory variables. Multicollinearity is a common problem in empirical Econometrics. The existing methods so far do not deal with cases of perfect multicollinearity. This new optimization method that belongs to the class of unbiased estimators is suitable for cases with strong or perfect multicollinearity, imposes restrictions of the minimizing matrix and produces small standard errors for the estimated parameters. At first we present the theoretical background of our approach and next we derive an expression for the covariance matrix of estimated coeffcients. As an example we estimate the basic linear regression model on Apple Inc expected stock returns and we examine multivariate extensions of this model in the special case of multicollinearity using the proposed method.


CAPM, Data multicollinearity, Moore-Penrose inverse, MDLUE, Multiple linear regression.

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Michael Patrick Allen. Understanding regression analysis. Springer Science & Business

Media, 2004.

A new unbiased estimator 13

Ali Bager, Monica Roman, Meshal Algelidh, and Bahr Mohammed. Addressing multicollinearity

in regression models: A ridge regression application. Journal of Social and

Economic Statistics, 6(1):30{45, 2017.

Jerzy K Baksalary and Friedrich Pukelsheim. On the lowner, minus, and star partial

orderings of nonnegative denite matrices and their squares. Linear Algebra and its

Applications, 151:135{141, 1991.

Adi Ben-Israel and Thomas Nall Eden Greville. Generalized inverses: Theory and

applications. Springer, 2002.

Stephen L. Campbell and Carl D. Meyer. Generalized inverses of linear transformations.

Dover, 1991.

Carsten F Dormann, Jane Elith, Sven Bacher, Carsten Buchmann, Gudrun Carl,

Gabriel Carre, Jaime R Garca Marquez, Bernd Gruber, Bruno Lafourcade, Pedro J

Leitao, et al. Collinearity: A review of methods to deal with it and a simulation study

evaluating their performance. Ecography, 36(1):27{46, 2013.

Yalei Du and Qiujun Lu. Estimate beta coecient of capm based on a fuzzy regression

with interactive coecients. Journal of Applied Mathematics and Physics, 03:664{672,


Michael Friendly and Ernest Kwan. Where's waldo? visualizing collinearity diagnostics.

The American Statistician, 63(1):56{65, 2009.

Otis Lamont Frost. An algorithm for linearly constrained adaptive array processing.

Proceedings of the IEEE, 60(8):926{935, 1972.

N Gake and O Krat. Matrix inequalities in the Lowner ordering. B. Korte (Ed.),

Modern Applied Mathematics, 1982.

Amos Golan, George Judge, and Douglas Miller. Maximum entropy econometrics:

Robust estimation with limited data. John Wiley & Sons, 1997.

Ghadban Khalaf. A comparison between biased and unbiased estimators in ordinary

least squares regression. Journal of Modern Applied Statistical Methods, 12(2):17, 2013.

Ghadban Khalaf and Mohamed Iguernane. Multicollinearity and a ridge parameter

estimation approach. Journal of Modern Applied Statistical Methods, 15:400{410, 11

Olaf Krat. A matrix optimization problem. Linear Algebra and its Applications,

:137 { 142, 1983.

John Lintner. Security prices, risk and maximal gains from diversication. The Journal

of Finance, 20(4):587{615, 1965.

Harry Markowitz. Portfolio selection. The Journal of Finance, 7(1):77{91, 1952.

Harry M. Markowitz. Portfolio selection: Ecient diversication of investments. J.

Wiley, 1959.

Brian D. Marx and Paul H.C. Eilers. Generalized linear regression on sampled signals

and curves: A p-spline approach. Technometrics, 41(1):1{13, 1999.

D. Pappas and A. Perperoglou. Constrained matrix optimization with applications.

Journal of Applied Mathematics and Computing, 40(1):357{369, Oct 2012.

Quirino Paris. Multicollinearity and maximum entropy estimators. Economics Bulletin,

(11):1{9, 2001.

Roger Penrose. A generalized inverse for matrices. In Mathematical proceedings of

the Cambridge Philosophical Society, volume 51, pages 406{413. Cambridge University

Press, 1955.

Silvi Qemo and Eahab Elsaid. Statistical modelling of the capital asset pricing model

(capm). Accounting and Finance Research, 7:146, 03 2018.

C. R. Rao and H Toutenburg. Linear Models: Least Squares and Alternatives. Springer

Series in Statistics. 07 1999.

William F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions

of risk. The Journal of Finance, 19(3):425{442, 1964.

W. Yamaka, A. Autchariyapanitkul, P. Meneejuk, and S. Sriboonchitta. Capital asset

pricing model through quantile regression: An entropy approach. Thai Journal of

Mathematics, page 53{65, 2017.

Y Zakari, SA Yau, and U Usman. Handling multicollinearity; a comparative study of

the prediction performance of some methods based on some probability distributions.

Annals. Computer Science Series, 16(1):15{21, 2018.

DOI: https://doi.org/10.22190/FUMI210406074P


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