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Perfect multicollinearity is the violation of
Assumption 6
(no explanatory variable is a
perfect linear function of any other explanatory variables). Perfect (or Exact) Multicollinearity If two or more independent variables have an exact linear relationship between them thenwe have perfect multicollinearity. Examples
: including the same information twice (weight in pounds and weight in
kilograms), not using dummy variables correctly (falling into the dummyvariable trap), etc.
Here is an example of perfect multicollinearity in a model with two explanatoryvariables:
^
ܺଵ^
ଵ
ܺଶ^
ଶ
ଵ
^
ܺଵ^
ଶ
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Consequence
: OLS cannot generate estimates of regression coefficients (error message).
Why?
OLS cannot estimate the marginal effect of
ଵ^
on
while holding
ଶ^
constant
because
ଶ^
moves exactly when
ଵ^
moves!
Solution:
Easy - Drop one of the variables!
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The Consequences of Multicollinearity
Imperfect multicollinearity does not violate Assumption 6. Therefore the Gauss-Markov Theorem tells us that the OLS estimators are BLUE.So then why do we care about multicollinearity?
The variances and the standard errors of the regression coefficient estimates willincrease. This means lower
t -statistics.
The overall fit of the regression equation will be largely unaffected bymulticollinearity. This also means that forecasting and prediction will be largelyunaffected.
Regression coefficients will be sensitive to specifications. Regression coefficientscan change substantially when variables are added or dropped.
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The Detection of MulticollinearityHigh Correlation Coefficients Pairwise correlations among independent variables might be high (in absolute value).Rule of thumb: If the correlation > 0.8 then severe multicollinearity may be present. High
with low
t
-Statistic Values
Possible for individual regression coefficients to be insignificant but for the overall fit ofthe equation to be high. High Variance Inflation Factors (VIFs) A VIF measures the extent to which multicollinearity has increased the variance of anestimated coefficient. It looks at the extent to which an explanatory variable can beexplained by all the other explanatory variables in the equation.
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Remedies for Multicollinearity No single solution exists that will eliminate multicollinearity. Certain approaches may beuseful:
Do Nothing^ Live with what you have. 2.
Drop a Redundant Variable^ If a variable is redundant, it should have never been included in the model in thefirst place. So dropping it actually is just correcting for a specification error. Useeconomic theory to guide your choice of which variable to drop. 3.
Transform the Multicollinear Variables^ Sometimes you can reduce multicollinearity by re-specifying the model, forinstance, create a combination of the multicollinear variables. As an example, ratherthan including the variables GDP and population in the model, includeGDP/population (GDP per capita) instead.
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Increase the Sample Size^ Increasing the sample size improves the precision of an estimator and reduces theadverse effects of multicollinearity. Usually adding data though is not feasible.
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What if a researcher were interested in the individual effect of each of theseexpenditures?vote share
= i
^
spendingଵ
ଶ^
advertising +
ଷ^
surveys +
ସ^
office +
ହ^
salaries +
^
other +
^
incumbency
male
ଽ^
Liberal
ଵ
Conservative
ଵଵ
BQ
ଵଶ
NDP
Even if you correct the model there still may be an imperfect multicollinearity betweenthe components of campaign expenditures.
