• Multicollinearity occurs when two or more
independent variables in a regression model
are highly correlated to each other.
• Standard error of the OLS parameter estimate
will be higher if the corresponding independent
variable is more highly correlated to the other
independent variables in the model.
Multicollinearity is a statistical phenomenon in
which two or more predictor variables in
a multiple regression model are
highly correlated, meaning that one can be
linearly predicted from the others with a non-
trivial degree of accuracy.
THE NATURE OF MULTICOLLINEARITY
Multicollinearity originally it meant the existence of a “perfect,” or
exact, linear relationship among some or all explanatory variables of a
regression model. For the k-variable regression involving explanatory
variable X1, X2, . . . , Xk (where X1 = 1 for all observations to allow for
the intercept term), an exact linear relationship is said to exist if the
following condition is satisfied:
λ1X1 + λ2X2 +· · ·+λkXk = 0 (10.1.1)
where λ1, λ2, . . . , λk are constants such that not all of them are zero
Today, however, the term multicollinearity is used to include the case
where the X variables are intercorrelated but not perfectly so, as
λ1X1 + λ2X2 +· · ·+λ2Xk + vi = 0 (10.1.2)
where vi is a stochastic error term.
To see the difference between perfect and less than perfect
multicollinearity, assume, for example, that λ2 ≠ 0. Then,
(10.1.1) can be written as:
which shows how X2 is exactly linearly related to other
variables. In this situation, the coefficient of correlation
between the variable X2 and the linear combination on the
right side of (10.1.3) is bound to be unity.
Similarly, if λ2 ≠ 0, Eq. (10.1.2) can be written as:
which shows that X2 is not an exact linear combination of
other X’s because it is also determined by the stochastic
error term vi.
• Perfect multicollinearity occurs when there is a
perfect linear correlation between two or more
• When independent variable takes a constant
value in all observations.
Symptoms of Multicollinearity
• The symptoms of a multicollinearity problem
1. independent variable(s) considered critical
in explaining the model’s dependent
variable are not statistically significant
according to the tests
Symptoms of Multicollinearity
2. High R2, highly significant F-test, but few or
no statistically significant t tests
3. Parameter estimates drastically change
values and become statistically significant
when excluding some independent
variables from the regression
CONSEQUENCES OF MULTICOLLINEARITY
In cases of near or high multicollinearity, one is
likely to encounter the following consequences:
1. Although BLUE, the OLS estimators have large
variances and covariance, making precise
2. The confidence intervals tend to be much wider,
leading to the acceptance of the “zero null
hypothesis” (i.e., the true population coefficient is
zero) more readily.
1. Large Variances and Co variances of OLS Estimators
The variances and co variances of βˆ2 and βˆ3 are given by
It is apparent from (7.4.12) and (7.4.15) that as r23 tends toward 1,
that is, as co linearity increases, the variances of the two estimators
increase and in the limit when r23 = 1, they are infinite. It is equally
clear from (7.4.17) that as r23 increases toward 1, the covariance of
the two estimators also increases in absolute value.
2. Wider Confidence Intervals
Because of the large standard errors, the
confidence intervals for the relevant
population parameters tend to be larger.
Therefore, in cases of high multicollinearity,
the sample data may be compatible with a
diverse set of hypotheses. Hence, the
probability of accepting a false hypothesis
(i.e., type II error) increases.
3) t ratio of one or more coefficients tends to be
• In cases of high collinearity, as the estimated
standard errors increased, the t values
i.e. t= βˆ2 /S.E(βˆ2 )
Therefore, in such cases, one will increasingly
accept the null hypothesis that the relevant
true population value is zero
4) Even the t ratio is insignificant, R2 can be very high.
Since R2 is very high, we reject the null
hypothesis i.e H0: β1 =β2 = Βk = 0; due to the
significant relationship between the variables.
But since t ratios are small again we reject H0.
Therefore there is a contradictory conclusion
in the presence of multicollinearity.
5) The OLS estimators and their standard errors can be
sensitive to small changes in the data.
As long as multicollinearity is not perfect,
estimation of the regression coefficients is
possible but the OLS estimates and their
standard errors are very sensitive to even the
slightest change in the data. If we change
even only one observation, there is dramatic
change in the values.
Detection of Multicollinearity
• Multicollinearity cannot be tested; only the degree of
multicollinearity can be detected.
• Multicollinearity is a question of degree and not of kind.
The meaningful distinction is not between the presence
and the absence of multicollinearity, but between its
• Multicollinearity is a feature of the sample and not of the
population. Therefore, we do not “test for
multicollinearity” but we measure its degree in any
A. The Farrar Test
• Computation of F-ratio to test the location of
• Computation of t-ratio to test the pattern of
• Computation of chi-square to test the presence of
multicollinearity in a function with several
B. Bunch Analysis
a. Coefficient of determination, R2 , in the
presence of multicollinearity, R2 is high.
b. In the presence of muticollinearity in the
data, the partial correlation coefficients, r12
c. The high standard error of the parameters
shows the existence of multicollinearity.
Unsatisfactory Indicator of
• A large standard error may be due to the
various reasons not because of the presence of
• A very high rxiyi only is neither sufficient but
nor necessary condition for the existence of
• R2 may be high to rxiyi and the result may be
highly imprecise and significant.
Other methods of remedying multicollinearity
Additional or new data
Transformation of variables
Dropping a variable(s) and specification bias
Combining cross-sectional and time series data
A priori information
• Do nothing
– If you are only interested in prediction, multicollinearity
is not an issue.
– t-stats may be deflated, but still significant, hence
multicollinearity is not significant.
– The cure is often worse than the disease.
• Drop one or more of the multicollinear variables.
– This solution can introduce specification bias. WHY?
– In an effort to avoid specification bias a researcher can
introduce multicollinearity, hence it would be
appropriate to drop a variable.
• Transform the multicollinear variables.
– Form a linear combination of the multicollinear variables.
– Transform the equation into first differences or logs.
• Increase the sample size.
– The issue of micronumerosity.
– Micronumerosity is the problem of (n) not exceeding (k). The
symptoms are similar to the issue of multicollinearity (lack of
variation in the independent variables).
• A solution to each problem is to increase the sample size.
This will solve the problem of micronumerosity but not
necessarily the problem of multicollinearity.
• Estimate a linear model with a double-logged model.
– Such a model may be theoretically appealing since one is
estimating the elasticities.
– Logging the variables compresses the scales in which the variables
are measured hence reducing the problem of heteroskedasticity.
• In the end, you may be unable to resolve the problem of
heteroskedasticity. In this case, report White’s
heteroskedasticity-consistent variances and standard errors
and note the substantial efforts you made to reach this
• Multicollinearity is a statistical phenomenon in
which there exists a perfect or exact relationship
between the predictor variables.
• When there is a perfect or exact relationship
between the predictor variables, it is difficult to
come up with reliable estimates of their individual
• The presence of multicollinearity can cause
serious problems with the estimation of β and the
• When multicollinearity is present in the data,
ordinary least square estimators are
• If goal is to understand how the various X
variables impact Y, then multicollinearity is a
big problem. Thus, it is very essential to detect
and solve the issue of multicollinearity before
estimating the parameter based on fitted
• Detection of multicollinearity can be done by
examining the correlation matrix or by using
• Remedial measures help to solve the problem