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Multicollinearity, LASSO, Ridge Regression, Principal Component Regression

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Multicollinearity, LASSO, Ridge Regression, Principal Component Regression International Journal of Statistics and Applications 2018, 8(4): 167-172 DOI: 10.5923/j.statistics.20180804.02 Regularized Multiple Regression Methods to Deal with Severe Multicollinearity N. Herawati*, K. Nisa, E. Setiawan, Nusyirwan, Tiryono Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Lampung, Bandar Lampung, Indonesia Abstract This study aims to compare the performance of Ordinary Least Square (OLS), Least Absolute Shrinkage and Selection Operator (LASSO), Ridge Regression (RR) and Principal Component Regression (PCR) methods in handling severe multicollinearity among explanatory variables in multiple regression analysis using data simulation. In order to select the best method, a Monte Carlo experiment was carried out, it was set that the simulated data contain severe multicollinearity among all explanatory variables (ρ = 0.99) with different sample sizes (n = 25, 50, 75, 100, 200) and different levels of explanatory variables (p = 4, 6, 8, 10, 20). The performances of the four methods are compared using Average Mean Square Errors (AMSE) and Akaike Information Criterion (AIC). The result shows that PCR has the lowest AMSE among other methods. It indicates that PCR is the most accurate regression coefficients estimator in each sample size and various levels of explanatory variables studied. PCR also performs as the best estimation model since it gives the lowest AIC values compare to OLS, RR, and LASSO. Keywords Multicollinearity, LASSO, Ridge Regression, Principal Component Regression have been developed depending on the sources of 1. Introduction multicollinearity. If the multicollinearity has been created by the data collection, collect additional data over a wider Multicollinearity is a condition that arises in multiple X-subspace. If the choice of the linear model has increased regression analysis when there is a strong correlation or the multicollinearity, simplify the model by using variable relationship between two or more explanatory variables. selection techniques. If an observation or two has induced Multicollinearity can create inaccurate estimates of the the multicollinearity, remove those observations. When regression coefficients, inflate the standard errors of the these steps are not possible, one might try ridge regression regression coefficients, deflate the partial t-tests for the (RR) as an alternative procedure to the OLS method in regression coefficients, give false, nonsignificant, p-values, regression analysis which suggested by [6]. and degrade the predictability of the model [1, 2]. Since Ridge Regression is a technique for analyzing multiple multicollinearity is a serious problem when we need to make regression data that suffer from multicollinearity. By adding inferences or looking for predictive models, it is very a degree of bias to the regression estimates, RR reduces the important to find a best suitable method to deal with standard errors and obtains more accurate regression multicollinearity [3]. coefficients estimation than the OLS. Other techniques, such There are several methods of detecting multicollinearity. as LASSO and principal components regression (PCR), are Some of the common methods are by using pairwise scatter also very common to overcome the multicollinearity. This plots of the explanatory variables, looking at near-perfect study will explore LASSO, RR and PCR regression which relationships, examining the correlation matrix for high performs best as a method for handling multicollinearity correlations and the variance inflation factors (VIF), using problem in multiple regression analysis. eigenvalues of the correlation matrix of the explanatory variables and checking the signs of the regression coefficients [4, 5]. 2. Parameter Estimation in Multiple Several solutions for handling multicollinearity problem Regression * Corresponding author: 2.1. Ordinary Least Squares (OLS) [email protected] (N. Herawati) Published online at http://journal.sapub.org/statistics The multiple linear regression model and its estimation Copyright © 2018 The Author(s). Published by Scientific & Academic Publishing using OLS method allows to estimate the relation between a This work is licensed under the Creative Commons Attribution International dependent variable and a set of explanatory variables. If data License (CC BY). http://creativecommons.org/licenses/by/4.0/ consists of n observations { , } and each observation i 푛 푦푖 푥푖 푖=1 168 N. Herawati et al.: Regularized Multiple Regression Methods to Deal with Severe Multicollinearity includes a scalar response yi and a vector of p explanatory 1 = + (regressors) xij for j=1,...,p, a multiple linear regression , 2 model can be written as = + where is the 푎푟푔푚푖푛 2 for some 훽 ̂휆 0. By Lagrangian� ‖푦 − duality,푿훽‖2 there휆‖훽 ‖is1 one� -to-one vector dependent variable, represents the explanatory 훽0 훽 푛푥1 correspondence between 푛constrained problem variables, is the regression풀 푿휷 coefficients휺 to be풀 estimated, 푛푥푝 휆 ≥ 푿 and the Lagrangian form. For each value of t in the range and represents the errors or residuals. = ‖훽‖1 ≤ 푡 휷푝푥1 where the constraint is active, there is a ( ) is estimated regression coefficients using푂퐿푆 OLS corresponding value of λ that yields the same solution form 휺푝푥1 휷� 1 by minimizing−1 the squared distances between the observed Lagrangian form. Conversely,‖훽‖ ≤ the푡 solution of to the 푿′푿 푿′풀 and the predicted dependent variable [1, 4]. To have problem solves the bound problem with = [17, 18]. unbiased OLS estimation in the model, some assumptions 훽̂휆 Like ridge regression, penalizing the absolute values of should be satisfied. Those assumptions are that the errors 푡 �훽̂휆�1 have an expected value of zero, that the explanatory the coefficients introduces shrinkage towards zero. However, variables are non-random, that the explanatory variables are unlike ridge regression, some of the coefficients are lineary independent, that the disturbance are homoscedastic shrunken all the way to zero; such solutions, with multiple and not autocorrelated. Explanatory variables subject to values that are identically zero, are said to be sparse. The multicollinearity produces imprecise estimate of regression penalty thereby performs a sort of continuous variable coefficients in a multiple regression. There are some selection. regularized methods to deal with such problems, some of c. Principal Component Regression (PCR) them are RR, LASSO and PCR. Many studies on the three Let V=[V ,...,V } be the matrix of size p x p whose methods have been done for decades, however, investigation 1 p columns are the normalized eigenvectors of , and let on RR, LASSO and PCR is still an interesting topic and λ ,..., λ be the corresponding eigenvalues.′ Let attract some authors until recent years, see e.g. [7-12] for 1 p W=[W1,...,Wp]= XV. Then W = XV is the푿 j-푿th sample recent studies on the three methods. j j principal components of X. The regression model can be 2.2. Regularized Methods written as = + = + = where = . Under this formulation, the′ least estimator of is a. Ridge regression (RR) ′ =푌 ( 푿훽 ) 휍 푿푽=푽 훽 휍 . 푾훾 훾 푽 훽 훾 Regression coeficients require X as a centered and And hence, the principal−1 component′ −1 estimator′ of β is scaled matrix, the cross product푂퐿푆 matrix (X’X) is nearly 훾� 푾′푾 푾 풀 휦 푾 풀 defined by = = [19-21]. Calculation of singular when X-columns are휷� highly correlated. It is often the OLS estimates via principal−1 component regression may be case that the matrix X’X is “close” to singular. This 훽� 푽훾� 푽휦 푾′풀 phenomenon is called multicollinearity. In this situation numerically more stable than direct calculation [22]. Severe multicollinearity will be detected as very small eigenvalues. still can be obtained, but it will lead to significant To rid the data of the multicollinearity, principal component changes푂퐿푆 in the coefficients estimates [13]. One way to detect omit the components associated with small eigen values. multicollinearity휷� in the regression data is to use the use the variance inflation factors VIF. The formula of VIF is 2.3. Measurement of Performances (VIF)j=( ) = . 1 To evaluate the performances at the methods studied, 푗 2 Ridge 푉퐼퐹regression1− 푅technique푗 is based on adding a ridge Average Mean Square Error (AMSE) of regression parameter (λ) to the diagonal of X’X matrix forming a new coefficient is measured. The AMSE is defined by matrix (X’X+λI). It’s called ridge regression because the 1 휷� ( ) diagonal of ones in the correlation matrix can be described as = 푚 a ridge [6]. The ridge formula to find the coefficients is 푙 2 = ( + ) , 0 . When =0, the ridge ( ) 퐴푀푆퐸�휷�� ��휷� − 휷� where denotes the estimated푛 푙=1 parameter in the l-th ′ −ퟏ ′ estimator become as the OLS. If all ’s are the same, the simulation.푙 AMSE value close to zero indicates that the slope 훽̂휆 푿 푿 휆푰 푿 풀 휆 ≥ 휆 resulting estimators are called the ordinary ridge estimators and intercept훽̂ are correctly estimated. In addition, Akaike 휆 [14, 15]. It is often convenient to rewrite ridge regression in Information Criterion (AIC) is also used as the performance Lagrangian form: criterion with formula: = 2 2ln ( ) where = + . = , , are the parameter values that maximize , 퐴퐼퐶퐶 푘 − 퐿� 1 2 2 the likelihood function, x = the observed data, n = the number 휆 푎푟푔푚푖푛 2 2 � � � Ridge regression훽̂ has� 2푛the‖푦 −ability푿훽‖ to 휆‖overcome훽‖ � this 퐿of data푝�푥 �points휃 푀� 휃in x, and k = the number of parameters 훽0 훽 multicollinearity by constraining the coefficient estimates, estimated by the model [23, 24]. The best model is indicated hence, it can reduce the estimator’s variance but introduce by the lowest values of AIC. some bias [16]. b. The LASSO 3. Methods The LASSO regression estimates by the optimazation problem: 푂퐿푆 In this study, we consider the true model as = + . 휷� 풀 푿휷 휺 International Journal of Statistics and Applications 2018, 8(4): 167-172 169 We simulate a set of data with sample size n= 25, 50, 75, 100, that sample size affects the value of AMSEs. The higher the 200 contain severe multicolleniarity among all explanatory sample size used, the lower the value of AMSE from each variables (ρ=0.99) using R package with 100 iterations.
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