Chapter 5 Violations of CLRM Assumptions
Chapter 5 Violations of CLRM Assumptions
Chapter 5 Violations of CLRM Assumptions
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Violation of the Assumptions of the CLRM
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Investigating Violations of the
Assumptions of the CLRM
• We will now study these assumptions further, and in particular look at:
- Causes
- Consequences
-How we test for violations
What are the Consequences of violating the assumptions of the CLRM?
In general we could encounter any combination of 3 problems:
- the coefficient estimates are wrong
- the associated standard errors are wrong
- the distribution that we assumed for the
test statistics will be inappropriate
Solutions:
- the assumptions are no longer violated
- we work around the problem so that we
use alternative techniques
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Assumption 2: Var(ut) = 2 <
• We have so far assumed that the variance of the errors is constant, 2 - this
is known as homoscedasticity.
û +
t
• If the errors do not have a
constant variance, we say
that they are heteroscedastic
• Note: heteroscedasticity often occurs
in cross-sectional data than in time series data .
x2t
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Detection of Heteroscedasticity
• Generally, there are many techniques of detection:
– Graphical methods
– Formal tests: There are many of them
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Detection/Tests of heteroscedasticity i. Graphical
method ^2 ^2
u^2 u yes u yes
no heteroscedasticity
^ ^
Y ^
Y
Y
yes yes yes
u^2 u^2 u^2
^
Y ^ ^
Y Y
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The Goldfeld-Quandt Test
H0 : homoscedasticity Var ( ui ) = 2
H1 : heteroscedasticity Var ( ui ) = i2
Goldfeld-Quandt Test procedures:
(1)Order or rank the observations according to the values of Xi,
beginning with the lowest X value.
(2) Omit c central observations, where c is specified a priori, and
divide the remaining (n-c) observations into two groups/sample each
of (n-c)/2 observations. Usually c is taken to be one-sixth of n.
(3) Run the separate regression on two sub-samples and obtain the
respective RSS1 and RSS2.. Each RSS has [(n-c)/2 - k] df
RSS2/df
=
(4) Compute the GQ or -ratio: RSS1/df
(5) Compare the and the Fc, if > Fc (0.05, k, ((n-c)/2)-k)==> reject the H0
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Gujarati(2003)Table 11.3 Re-order data
RSS 2 / df
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RSS1 / df
The GQ Test (Cont’d)
A problem with the test is that the choice of where to split the sample
is that usually2 arbitrary and may crucially affect the outcome of the
test. GQ s12
s2
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Consequences of Using OLS in the Presence of
Heteroscedasticity
• OLS estimation still gives unbiased coefficient estimates, but they are
no longer BLUE.
• Whether the standard errors calculated using the usual formulae are
too big or too small will depend upon the form of the
heteroscedasticity.
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How Do we Deal with Heteroscedasticity?
Generalised least squares(GLS) estimation
• If the form (i.e. the cause) of the heteroscedasticity is known, then we can
use an estimation method which takes this into account (called generalised
least squares, GLS).
• A simple illustration of GLS is as follows: Suppose that the error variance
is related to another variable zt by
var ut 2 zt2
• To remove the heteroscedasticity, divide the regression equation by zt
yt 1 x x
1 2 2t 3 3t vt
zt zt zt zt
ut
where vt is an error term.
zt
ut var ut 2 zt2
• Now var vt var 2
2
2
for known zt.
zt z t z t
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Dealing
with Heteroscedasticity…
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Multicollinearity
• This problem occurs when the explanatory variables are very highly correlated
with each other.
• Perfect multicollinearity:
Cannot estimate all the coefficients
- e.g. suppose x3 = 2x2
and the model is yt = 1 + 2x2t + 3x3t + 4x4t + ut
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Measuring Multicollinearity
Corr x2 x3 x4
x2 - 0.2 0.8
x3 0.2 - 0.3
x4 0.8 0.3 -
• But another problem: if 3 or more variables are linear
- e.g. x2t + x3t = x4t
• Note that high correlation between y and one of the x’s is not
muticollinearity.
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Variance inflation factor(VIF)
• This is the commonly used method to detect the existence of MC.
• Consider the regression model: Yi= β 2x 2i + β 3x 3i +---+ β kx ki + εi
• The VIF of β jˆ is defined as:
• where Rj2 is the coefficient of determination obtained when the Xj
variable is regressed on the remaining explanatory variables
(called auxiliary regression).
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Autocorrelation
• We assumed of the CLRM’s errors that Cov (ui , uj) = 0 for ij, i.e.
This is essentially the same as saying there is no pattern in the errors.
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Positive Autocorrelation
+
û t ût
+
- +
uˆ t 1 Time
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Negative Autocorrelation
+ ût
ût
+
- +
uˆt 1 T
ime
- -
- +
uˆt 1 Time
-
-
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Detecting Autocorrelation:
The Durbin-Watson Test
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The Durbin-Watson Test:
Critical Values
• We can also write
DW 2(1 ) (2)
where is the estimated correlation coefficient. Since is a
correlation, it implies that 1 pˆ 1.
• Rearranging for DW from (2) would give 0DW4.
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The Durbin-Watson Test: Interpreting the Results
• The coefficient estimates derived using OLS are still unbiased, but
they are inefficient, i.e. they are not BLUE, even in large sample
sizes.
• Thus, if the standard error estimates are inappropriate, there exists the
possibility that we could make the wrong inferences.
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“Remedies” for Autocorrelation: Models in First
Difference Form
• One way to sometimes deal with the problem of autocorrelation is to
switch to a model in first differences.
• Denote the first difference of yt, i.e. yt - yt-1 as yt; similarly for the x-
variables, x2t = x2t - x2t-1 etc.
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