**MULTICOLLINEARITY**

*Nature of Multicollinearity*

*Problem*

Many finance/ economic variables, especially time series variables are
closely related with each other.

**Example;**Population and GDP are closely related, i.e highly correlated.

In multiple regression models,
a regression coefficient measures the

**partial effect**of that individual variable on Y when all other X variables in the model are fixed.
However, when two explanatory
variables move closely together, we cannot assume that one is fixed while the
other is changing. Because when one changes, the other one also changes as they
are closely related. In such a case it is

**difficult to isolate the partial effect****of a single X variable.****This is the problem of Multicollinearity****Definition;**

*Multicollinearity***originally it meant the existence of a “perfect,” or exact, linear relationship among some or all explanatory variables of a regression model.**

- 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.
- Independent variables show no statistical significance when conducting the basic significance test
- It is not a mistake in the model specification, but due to the nature of the data at hand

**PERFECT MULTICOLLINEARITY**

Perfect
multicollinearity occurs when there is a perfect linear correlation between two
or more independent variables

When
independent variable takes a constant value in all observations

*Imperfect Multicollinearity*

Although perfect multicollinearity is theoretically possible,
in practice imperfect multicollinearity is what we commonly observed.

Typical examples of perfect multicollinearity are when
the researcher makes a mistake(including the same variable twice or forgetting
to omit a default category for a series of dummy variables)

**SEVERE MULTICOLLINEARITY**

The
OLS method cannot produce parameter estimates.

A
certain degree of correlation (multicollinearity) between the independent
variables is normal and expected in most cases.

*Symptoms of Multicollinearity*
The
symptoms of a multicollinearity problem

§
independent
variable(s) considered critical in explaining the model’s dependent variable
are not statistically significant according to the tests

§
High
R

^{2}, highly significant F-test, but few or no statistically significant t tests
§
Parameter
estimates drastically change values and become statistically significant when
excluding some independent variables from the regression

*Detecting Multicollinearity*- ·
*Few significant t-ratios but a high R*^{2}and a*collective significance of the variables* - ·
*High pairwise correlation between the explanatory variables* - ·
*Examination of partial correlations* - ·
*Estimate auxiliary regressions* - ·
*Estimate variance inflation factor (VIF)*

i.
A
simple test for multicollinearity is to conduct “artificial” regressions
between each independent variable (as the “dependent” variable) and the
remaining independent variables

ii.
Variance
Inflation Factors (VIF

_{j}) are calculated as:
iii.
VIF

_{j}= 2, for example, means that variance is twice what it would be if X_{j}, was not affected by multicollinearity
iv.
A
VIF

_{j}>10 is clear evidence that the estimation of B_{j}is being affected by `multicollinearity

*Addressing Multicollinearity*
1.
Although
it is useful to be aware of the presence of multicollinearity, it is not easy
to remedy severe (non-perfect) multicollinearity

2.
If
possible, adding observations or taking a new sample might help lessen
multicollinearity

3.
Exclude
the independent variables that appear to be causing the problem

4.
Modifying
the model specification sometimes help, for example:

·
using real instead of nominal economic data

·
using a reciprocal instead of a polynomial
specification on a given independent variable