Chapter 10 Multicollinearity: What Happens if Explanatory Variables are Correlated
Chapter 10 Multicollinearity: What Happens if Explanatory Variables are Correlated
r One of the CLRM assumptions is there is no perfect multicollinearity--no exact linear relationships among explanatory variables, XS, in a multiple regression In practice, one rarely encounters perfect multicollinearity, but cases of near or very high multicollinearity where explanatory variables are approximately linearly related frequently arise in many applications
One of the CLRM assumptions is: there is no perfect multicollinearity—no exact linear relationships among explanatory variables, Xs, in a multiple regression. In practice, one rarely encounters perfect multicollinearity, but cases of near or very high multicollinearity where explanatory variables are approximately linearly related frequently arise in many applications
The objects of this chapter. OThe Nature of multicollinearity o ls multicollinearity really a problem? o The theoretical consequences of multicollinearity; o How to detect multicollinearity? o The remedial measures which can be used to eliminate multicollinearity
The objects of this chapter: ●The Nature of multicollinearity; ● Is multicollinearity really a problem? ● The theoretical consequences of multicollinearity; ● How to detect multicollinearity? ● The remedial measures which can be used to eliminate multicollinearity
10. 1: The Nature of Multicollinearity The Case of Perfect Multicollinearity rIn cases of perfect linear relationship or perfect multicollinearity among explanatory variables, we cannot obtain unique estimates of all parameters. And since we cannot obtain their unique estimates, we cannot draw any statistical inferences i.e hypothesis testing) about them from a given sample
10.1: The Nature of Multicollinearity: The Case of Perfect Multicollinearity In cases of perfect linear relationship or perfect multicollinearity among explanatory variables, we cannot obtain unique estimates of all parameters. And since we cannot obtain their unique estimates, we cannot draw any statistical inferences (i.e., hypothesis testing) about them from a given sample
Y=A1+A2×2+A3×3+g Transformation:×31=300-2×2 Y=A1+A2X2+A3(300-2×2)+ =(A1+300A3)+(A2-2A3)×2+u1 =C1+C2×2+ui Estimation get the OlS estimators C1=A1+300A3,C2=A2-2A3 So from the estimators of Cl, c2, we can not get the estimators of A, A2 and A3
Yi=A1+A2X2i+A3X3i+μi Transformation: X3i =300－2X2i Yi=A1+A2X2i+A3 ( 300－2X2i ) +μi =(A1+300 A3 )+(A2 －2 A3 ) X2i +μi = C1 + C2 X2i +μi Estimation: get the OLS estimators C1 =A1+300 A3 , C2 =A2 －2 A3, So from the estimators of C1 , C2 , we can not get the estimators of A1 , A2 and A3
That is in cases of perfect multicollinearity, estimation and hypothesis testing about individual regression coefficients in a multiple regression are not possible. We can just obtain estimates of a near combination of the original coefficients, but not each of them individuall
That is : in cases of perfect multicollinearity, estimation and hypothesis testing about individual regression coefficients in a multiple regression are not possible. We can just obtain estimates of a linear combination of the original coefficients, but not each of them individually
10.2 The Case of Near, or Imperfect or High Multicollinearity wHen we talk about multicollinearity we usually refer it to imperfect multicollinearity X:=B, +BX:+e
10.2 The Case of Near, or Imperfect, or High Multicollinearity When we talk about multicollinearity, we usually refer it to imperfect multicollinearity. X3i=B1+B2X2i+ei
r If there are just two explanatory variables. the coefficient of correlation r can be used as a measure of the degree or strength of collinearity. But if more than two explanatory variables are involved as we will show later the coefficient of correlation may not be an adequate measure of collinearity
If there are just two explanatory variables, the coefficient of correlation r can be used as a measure of the degree or strength of collinearity. But if more than two explanatory variables are involved, as we will show later, the coefficient of correlation may not be an adequate measure of collinearity
10.3 Theoretical Consequences of Multicollinearity r Note: we consider only the case of imperfect multicollinearity r When collinearity is not perfect, OLS estimators still remain BLUE even though one or more of the partial regression coefficients in a multiple regression can be individually statistically insignificant
10.3 Theoretical Consequences of Multicollinearity Note: we consider only the case of imperfect multicollinearity When collinearity is not perfect, OLS estimators still remain BLUE even though one or more of the partial regression coefficients in a multiple regression can be individually statistically insignificant
r1. OLS estimators are unbiased But unbiasedness is a repeated sampling property. In reality, we rarely have the luxury of replicating samples. r2. LS estimators have minimum variance But this does not mean however that the variance of an OLS estimator will be small in any given sample, minimum variance does not mean that every numerical value of the variance will be small 3. Multicollinearity is essentially a sample (regression)phenomenon
1. OLS estimators are unbiased. But unbiasedness is a repeated sampling property. In reality, we rarely have the luxury of replicating samples. 2. OLS estimators have minimum variance. But this does not mean, however, that the variance of an OLS estimator will be small in any given sample, minimum variance does not mean that every numerical value of the variance will be small. 3.Multicollinearity is essentially a sample (regression)phenomenon