Transcript Chapter 4 - Facultypages.morris.umn.edu

CHAPTER 4
REGRESSION DIAGNOSTIC I:
MULTICOLLINEARITY
Damodar Gujarati
Econometrics by Example
MULTICOLLINEARITY
 One of the assumptions of the classical linear
regression (CLRM) is that there is no exact linear
relationship among the regressors.
 If there are one or more such relationships among
the regressors, we call it multicollinearity, or
collinearity for short.
Perfect collinearity: A perfect linear relationship
between the two variables exists.
Imperfect collinearity: The regressors are highly (but
not perfectly) collinear.
Damodar Gujarati
Econometrics by Example
CONSEQUENCES
 If collinearity is not perfect, but high, several
consequences ensue:
 The OLS estimators are still BLUE, but one or more regression
coefficients have large standard errors relative to the values of
the coefficients, thereby making the t ratios small.
 Even though some regression coefficients are statistically
insignificant, the R2 value may be very high.
 Therefore, one may conclude (misleadingly) that the true values
of these coefficients are not different from zero.
 Also, the regression coefficients may be very sensitive to small
changes in the data, especially if the sample is relatively small.
Damodar Gujarati
Econometrics by Example
VARIANCE INFLATION FACTOR
 For the following regression model:
Yi  B1  B2 X 2i  B3 X 3i  ui
It can be shown that:

var(b2 ) 
 x (1  r )
2
2i
and
var(b3 ) 
2

2
23
2
 x (1  r )
2
3i
2
23



2
x
2
2i

VIF
2
x
2
3i
VIF
where σ2 is the variance of the error term ui, and r23 is the
coefficient of correlation between X2 and X3.
Damodar Gujarati
Econometrics by Example
VARIANCE INFLATION FACTOR (CONT.)
1
VIF 
2
1  r23
is the variance-inflating factor.
VIF is a measure of the degree to which the variance of
the OLS estimator is inflated because of collinearity.
Damodar Gujarati
Econometrics by Example
DETECTION OF MULTICOLLINEARITY
 1. High R2 but few significant t ratios
 2. High pair-wise correlations among explanatory
variables or regressors
 3. High partial correlation coefficients
 4. Significant F test for auxiliary regressions
(regressions of each regressor on the remaining
regressors)
 5. High Variance Inflation Factor (VIF) and low
Tolerance Factor (TOL, the inverse of VIF)
Damodar Gujarati
Econometrics by Example
REMEDIAL MEASURES
 What should we do if we detect multicollinearity?
 Nothing, for we often have no control over the data.
 Redefine the model by excluding variables may attenuate
the problem, provided we do not omit relevant variables.
 Principal components analysis: Construct artificial
variables from the regressors such that they are orthogonal
to one another.
These principal components become the regressors in the
model.
Yet the interpretation of the coefficients on the principal
components is not as straightforward.
Damodar Gujarati
Econometrics by Example