第6章 多重共线性

5.1 the nature of multicollinearity
three variable linear regression model
Yt= β0+ β1X1t+β2X2t +ut E(Yt)= β0+ β1X1t+β2X2t E(Y suppose that X1，X2 are perfectly linearly
CHAPTER 6 MULTICOLLINEARITY
the assumption 6 of multiple regression can’t be satisfied,namely, among the explanatory variables there is more or less linear function relation e.g. The demand function of chicken Y=B0+B1X1i+B2X2i+B3X3i+B4X4i Yi-demand of chicken, X1-PDI 。 X2-the price of chicken，X3，X4-price of pork and ， beef
Ct= β0+ β1Yt+β2Yt-1 + β3Yt-2 +ut
theoretically, OLS estimator is still BLUE， ， but practically： ：
1 ． Even though the variance of OLS estimator is minimum, the estimators have large variance ,making the estimation difficult 2． The confidence interval becomes wider and t ratio ． becomes smaller ,leading to the acceptance of the “zero null hypothesis” and some regressors are statistically insignificant 3. R2 , the overall measure of goodness of fit, can be very high 4． The ols estimator and their standard error can be ． sensitive to small changes in the data。 。 5．The sign of regression coefficient maybe wrong ． 6. It is difficult to separate the influence of every regressor on the regressand
4．Auxiliary regression Auxiliary regression: regress each Xi on the remaining X variables and compute the corresponding Ri2 regressorX1，X2，…，Xk， ， X1=f1（X2,X3, …,Xk） X2=f2（X1,X3, …,Xk） Xj=fj（X1,…,Xj-1, Xj+1,…Xk） Xk=fk（X1,X2,…Xk1） RULE: Among the corresponding R12 ， R22 ， … ， Rk2 ， the greatest one and statistically significant ,namely, Rj2 indicates that Xj is the main factor leading to mulicollearity
5.eigenvalues(本征值） and condition index（条 本征值）
件指数） 件指数）
CI =
max imum min imum
eigenvalue eigenvalue
=
k
CI<10, slight multicollinearity Ci is between 10~30，moderate ， mulicollinearity CI>30，severe or strong multicolliearity ，
Y=f（X1,X2,…Xk）, R2 of origin regression （ regression Y1=f1 （ X2,X3,…,Xk ） , R12 of excluding X1 Y2=f2 （ X1,X3,…,Xk ） , R22 of regression excluding X2 Yi=fi（X1, …,Xi-1,Xi+1…,Xk） Yh=fk （ X1,X2,…,Xk-1 ） , RK2 of regression excluding Xk RULE: the Xi corresponding to the greatest Ri2 maybe the regressor leading multicollinearity。 to the most severe multicollinearity。
6、variance inflation factor(方差膨胀因子)
∧ δ2 1 δ2 ) = 2 • VIF j Var = 2 • ( 2 β j Σx Σx j 1− R j j 1 , VIF j = 2 1− R j
is called variance inflation factor Ri is corresponding coefficient of determination in the regression of regressor Xi on the remaining regressors in the model. IF VIF of a variable exceeds 10, the variable is said be highly collinear.
correlated, therefore their coefficient of correlation is 1
r2 =
(Σx1i
x2i )2
Σx1i Σx2i
2
2
=1
Σx ⋅ Σx − Σx1i ⋅ Σx2i
2 1i 2 2i
(Hale Waihona Puke )2=0see the estimator of parameters of regression ∧ (Σ y i ⋅ x 1 i ) ⋅ Σ x 22 i − Σ y i ⋅ x 2 i ⋅ Σ x 1 i ⋅ x 2 i = 2 β1 Σ x 12i ⋅ Σ x 2 i − (Σ x 1 i ⋅ x 2 i )2 ∧ (Σ y i ⋅ x 2 i ) ⋅ Σ x 12i − Σ y i ⋅ x 1 i ⋅ Σ x 1 i ⋅ x 2 i = 2 β2 Σ x 12i ⋅ Σ x 2 i − (Σ x 1 i ⋅ x 2 i )2
1、Do nothing 、 2、use of priori information 、
priori information come from previous empirical work and relevant theory Y=AKαLβ lnY=lnA+αlnk+βLnL+µ suppose constant returns to scale,namely,α+β=1， ， substitute β=1- α into double-log model
5.4 REMEDIAL MEASURES OF MULTICOLLINEARITY
Keep these rules in your mind: Multicollinearity is a question of degree and not of kind Multicollinearity is a feature of the sample and not of the population Whether the multicollinearity need to be corrected is decided by the purpose of regression:to forecast or to estimate the accurate regression coefficient.
2 ∧ δ 2 Σx2i Var = β 2 2 1 Σx1i Σx2i − Σx1i x2i
(
)
2
2 ∧ δ 2 Σx1i Var = β 2 2 2 Σx1i Σx2i − Σx1i ⋅ x2i
(
)
2
Because denominator is zero, the regression coefficients are indeterminate and standard error are infinite
5.2 the source and practical consequence of multicollinearity
5.2.1 the source of multicollinearity Multicollinearity may be due to the following factors 1. Data of regressors share a common time trend. 2 ． The inherent relationship among the crosseconomic variable especially in crosssection data 3 ． Model specification, e.g. polynomial model, or as following