Understand the types and major causes of business failure and the use of voluntary settlements to sustain or liquidate the failed firm. A firm may fail because it has negative or low returns, is insolvent, or is bankrupt. The major causes of business failure are mismanagement, downturns in economic activity, and corporate maturity. Voluntary settlements are initiated by the debtor and can result in sustaining the firm via an extension, a composition, creditor control of the firm, or a combination of these strategies. If creditors do not agree to a plan to sustain a firm, they may recommend voluntary liquidation, which requirements and costs of bankruptcy proceedings

1. VIOLATION OF CLRM ASSUMPTIONS:
MULTICOLLINEARITY
Chapter 8
1

2. MULTICOLLINEARITY: NATURE
 No exact linear relationship or no collinearity means that a variable,
say 𝑋2, can not be expressed as an exact linear function of another
variable, say 𝑋3.
 Thus if we can express that
𝑋2 = 4𝑋3
then the two variables and are collinear, for there is an exact linear
relationship between them.
 Consider the following 3-variable mathematical equation:
𝑌 = 𝛽1 + 𝛽2𝑋2 + 𝛽3𝑋3
or, 𝑌 = 𝛽1 + 𝛽2(4𝑋3) + 𝛽3𝑋3 = 𝛽1 + 4𝛽2𝑋3 + 𝛽3𝑋3
= 𝛽1 +𝑋3(4𝛽2 + 𝛽3)
 Now, assume 𝐴 = 4𝛽2 + 𝛽3, we get 𝑌 = 𝛽1 + A𝑋3
 Note that we have one equation with two unknowns, we need at
least two equations to get the values of two unknowns. 2

3. MULTICOLLINEARITY: NATURE
 In presence of multicollinearity, we can not disentangle the
individual effect of 𝑋2 and 𝑋3 on Y.
 Exact linear relationship: For a k-variable regression model
involving explanatory variables 𝑋1, 𝑋2, 𝑋3 … 𝑋𝑘 ( 𝑋1 is the
intercept) an exact linear relationship is said to exist if the
following condition is satisfied
𝛾1𝑋1 + 𝛾2𝑋2 + ⋯ + 𝛾𝑘𝑋𝑘 = 0
 Where 𝛾1, 𝛾2, … 𝛾𝑘 are constants and not all of them are zero
simultaneously.
 Near-exact linear relationship:
𝛾1𝑋1 + 𝛾2𝑋2 + ⋯ + 𝛾𝑘𝑋𝑘 + 𝑢𝑖 = 0
Where 𝛾1, 𝛾2, … 𝛾𝑘 are constants and not all of them are zero
simultaneously and 𝑢𝑖 is a random error term. 3

4. MULTICOLLINEARITY: NATURE
 Point to remember
 Multicollinearity as we have defined above refers only to
linear relationship among X variables of a regression
model. But Xs may be nonlinearly correlated as
 Variables and are functionally related with but
the relationship in nonlinear thus the model does not
violate the assumption of multicollinearity.
4
i
i
i
i
i u
X
X
X
Y 



 3
4
2
3
2
1 



2
i
X 3
i
X i
X

5. MULTICOLLINEARITY: EXAMPLE
5
𝑋2𝑖 𝑋3𝑖 𝑋4𝑖
3 9 13 (+4)
5 15 16 (+1)
6 18 21 (+3)
8 24 25 (+1)
10 30 30 (+0)

6. THEORETICAL CONSEQUENCES OF MULTICOLLINEARITY
 CASE 1: PERFECT LINEAR RELATIONSHIP BETWEEN VARIABLES
 If multicollinearity is perfect (exact linear relationship), the
partial regression coefficients of the X variables are
indeterminate and their standard errors are infinite.
 If 𝑋2 and 𝑋3 are perfectly collinear, there is no way that
𝑋2 can be kept constant when 𝑋3 changes. As long as we
fail to separate individual effect of 𝑋2 and 𝑋3 on Y, we can
not get a unique solution for individual regression
coefficients.
6

7. THEORETICAL CONSEQUENCES OF MULTICOLLINEARITY
 Consider the following 3-variable regression model:

7

8. THEORETICAL CONSEQUENCES OF MULTICOLLINEARITY
 CASE 2: NEAR PERFECT LINEAR RELATIONSHIP BETWEEN VARIABLES
 If multicollinearity is less than perfect, the regression
coefficients, although determinate, posses large standard
errors (in relation to the coefficients themselves), which means
that the coefficients can not be estimated with great precision.
 The effect of multicollinearity is to make it hard to obtain
estimates of coefficients with small standard error.
 However, having a small number of observations also has
similar effect (small sample size). For this reason,
multicollinearity is considered as a small sample phenomenon.
Some Economists also use separate term for multicollinearity,
like micronumerosity instead of multicollinearity.
8

9. PRACTICAL CONSEQUENCES OF MULTICOLLINEARITY
In case near perfect or high multicollinearity, one is likely to face following
problems:
1. Although BLUE, the OLS estimates have large variances and covariances
making precise estimation difficult.
2. Because of 1, the confidence intervals for parameter estimates become
large leading to accepting the zero 𝐻0 more readily.
3. Also because of 1, the t- statistics of one or more coefficients tends to be
statistically insignificant.
4. Although the t - statistics of one or more coefficients is statistically
insignificant, 𝑅2
, the overall measure of goodness-of-fit, can be very high.
5. The OLS estimators and their standard errors can be sensitive to small
changes in the data. 9

10. DETECTION OF MULTICOLLINEARITY
 High R2 but few significant t ratios.
 “Classic” symptom of multicollinearity. If R2 is high, say, in excess of 0.8, the F
test in most cases will reject the null hypothesis that the partial slope
coefficients are jointly or simultaneously equal to zero. But individual t tests
will show that none or very few partial slope coefficients are statistically
different from zero.
 Subsidiary, or auxiliary, regressions.
 One way of finding out which X variable is highly collinear with other X
variables in the model is to regress each X variable on the remaining X
variables and to compute the corresponding R2.
𝑌 = 𝛽1 + 𝛽2𝑋2 + ⋯ 𝛽7𝑋7 + 𝑢
 Consider the regression of Y on X2, X3, X4, X5, X6, and X7—six explanatory
variables.
 If this regression shows that the R2 is high but very few X coefficients are
individually statistically significant, we then look for the “culprit,” the variable(s)
that may be a perfect or near perfect linear combination of the other X’s. 10

11. DETECTION OF MULTICOLLINEARITY
11
𝐹 =
R2
/(k − 1)
(1 − R2)/(n − k)
𝑋2 on remaining Xs

12. DETECTION OF MULTICOLLINEARITY
 The variance inflation factor
𝑉𝐼𝐹 =
1
(1 − 𝑅2)
 The larger the value of VIFj, the more “troublesome” or
collinear the variable Xj.
 it may be noted that the inverse of the VIF is called tolerance
(TOL). That is,
 One could use TOLj as a measure of multicollinearity in view
of its intimate connection with VIFj. The closer is TOLj to
zero, the greater the degree of collinearity of that variable
with the other regressors. 12

13. REMEDIAL MEASURES
 What can be done if multicollinearity is serious? We have
two choices:
(1) do nothing or (2) follow some rules of thumb.
 Dropping a Variable(s) from the Model
 Acquiring Additional Data or a New Sample
 Rethinking the Model
 Prior Information about Some Parameters
 Transformation of Variables
 Combining time series and cross-sectional data,
factor or principal component analysis and ridge
regression. 13