7djdud5y2fia6ch 6 multicollinearity&heterosced (1)

8/3/2019 7dJDuD5Y2Fia6Ch 6 Multicollinearity&Heterosced (1)

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Gujarati(2003): Chapter 10 and 11


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Multicollinearity

One of the assumptions of the CNLRMrequires that THERE IS NO EXACTLINEAR RELATIONSHIP between any of

the explanatory variables

i.e non of the X variables are perfectlycorrelated. If so, we say there is perfect

multicollinearity.


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The nature ofmulticollinearity

Ifmulticollinearity is perfect, the regression coefficients of

the Xi variables, is, are indeterminate and their standarderrors, Se(i)s, are infinite.

In general:When there are some functional relationships

among independent variables, that is iXi = 0or 1X1+ 2X2 + 3X3++ iXi = 0Such as 1X1+ 2X2= 0 X1= -2X2


4/23

The nature of multicollinearity

If all the Xs are uncorrelated, there is ABSENCE

OF MULTICOLLINEARITY.

Cases in between are described by variousdegress of multicollinearity.

High degree of multicollinearity occurs when anX is highly correlated with another X or with alinear combination of other Xs.


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The nature of multicollinearity

So, we study

- nature of multicollinearity

- consequencirs of multicollinearity- detection and remedies


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Multicollinearity

NOTE:1) Multicollinearity is a question of degree notof kind. It is not a matter of irs absence of

presence. Its degree is important.

2) It is a feature of the sample, not of thepopulation. It is due to the assumption of

nonstochastic Xs. We do not test formulticollinearity but we can measure itsdegree in a particular sample.


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Multicollinearity

If the X variables in a multiplr regressionmodel are highly correlated, we say thatdegree of multicollinearity is HIGH.

Then,variances of estimated coefficientsare large, t-values are low and soestimates ate highly unreliable.


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Consequences of imperfect multicollinearity

5. The OLS estimators and their standard errors

can be sensitive to small change in the data.

Can be

detected from

regression

results

1. Although the estimated coefficients are BLUE,

OLS estimators have large variances and

covariances, making the estimation with

less accuracy.

2. The estimation confidence intervals tend to bemuch wider, leading to accept the zero null

hypothesis more readily.

3. The t-statistics of coefficients tend to be

statistically insignificant.4. The R2 can be very high.


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Detection of high degree of multicollinearitySymptoms

Using the feature of imperfect (or high degree of)multicollinerity summarised under item 5, we candelete or add a few observations from the data set andexamine the sensitivity of the coefficient estimates and

their standard errors to the changes. i.e are thecoefficient estimates and their standar errors changesubstantially?

Based on large , t-values are low leading to

unprecise estimates. This is the reason why we mayget high significant F-test bu t-tests may beinsignificant.

)(. es

2R


10/23

Difficulty in detecting high degree ofmulticollinearity

1) However, although we may have significant t-ratios and significant F-test, still it is possible tohave high degree of multicollinearity.

2) High variances (so standard errors) andcovariances of estimated coefficients may alsooccur due to small variation in Xs i.e due large

error variances


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In a three-variable case:

From the formula for

uXXY 33221

2

)1()()(

2

23

2

2

2

2

32

2

3

2

2

22

3

2rxxxxx

xVar

uu


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where is obtained from the OLS regression of

on as

It can be seen that high variance of depends on 3factors

1) high error variance,

2) small dispersion in Xs,3) correlation between ,

2

23r

2X 3212 XX3X

2

2

u

2

ix

32 & XX 23r


13/23

Remedial Measures

1. Utilise a priori information

2. Combining cross-sectional and time-series data

3. Dropping a variable(s) and re-specify the regression

4. Transformation of variables:

(i) First-difference form

(ii) Ratio transformation

5. Additional or new data

6. Reducing collinearity in polynomial regression

7. Do nothing (if the objective is only for prediction)

'33221

uXXY

uXXY 33221

uXX 32221

1.0

uXX )1.0( 3221 uZ

21

23 1.0 given

')()1

(3

3

2

2

3

1

3

uX

X

XX

Y

'33

2

221uXXY '2

33221uXXY


14/23

Insignificant

Significant

Since GDP and GNPare highly related

Other examples: CPI WPI;

CD rate TB rate

M2 M3


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Heteroscedasticity

One of the assumptions of the CNLRM was that errorswere homoscedastic (equally spread)

22

2

)(

)(

uE

uVar


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The probability density function for Yi at two

levels of family income, X

i , are identical.

Homoscedasticity Case

.

.

x1ix11=80 x13=10

0

f(Yi)

income

Var(ui) = E(ui2)= 2

.

x12=90


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.

..

..

..

...

.

..

..

..

. .. .... .

..

. .

. ...

.

...

..

.

.

xi

yi

0

Homoscedastic pattern of errors

The scattered points spread out quite equally


18/23

The variance of Yi increases as family income,

Xi, increases.

Heteroscedasticity Case

.

x1x11

x12

f(Yi)

x13

.

.

income

Var(ui) = E(ui2)= i2


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Heteroscedastic pattern of errors.

.

.

. .. .

..

..

....

..

..

.

..

.

..

...

.....

..

.

..

... .. . .

.

xt

yt

0The scattered points spread out quite unequally


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Definition of Heteroscedasticity:

Two-variable regression: Y = 1 + 2 X + ui2 = = ki Y =ki (1 + 2 X + ui)

xyx2^

2 = 2 + ki ui E(2) = 2 unbiased^ ^

if12

= 22

= 32

= i.e.,homoscedasticity

2

xi2

Var (2)=^

if12

22

32

i.e.,heteroscedasticityVar (2) =^

Var(ui) = E(ui2) = i22

1iiXk 0ik

^

xi2i2

( xi2)2

=


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Consequences of heteroscedasticity

1. OLS estimators are stilllinear and unbiased

2. Var( i )s are not minimum.^

=>not the best=>not efficiency=>not BLUE

4. 2 = is biased, E(2) 2ui

2^^

n-k^

5. t and F statistics are unreliable. Y = 0 + 1 X + u

Cannot be min.

SEE = RSS = u 2^ ^

3. Var ( 2) = instead of Var( 2) =x ii2

(xi2)2

^ ^ 2x2

Two-variablecase


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Detection of heteroscedasticity

1.Graphical method :

plot the estimated residual ( ui ) or squared (ui2 ) against the

predicted dependent Variable (Yi) or any independent

variable(Xi).

^ ^^

Observe the graph whether there is a systemic pattern as:

Y^

u 2^

Yes,

heteroscedasticity exists


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Detection of heteroscedasticity: Graphical method

Y

u2^no heteroscedasticity

Y

u2^ yes

Y

u2^ yes

Y^

u2^yes

Y^

u2^yes

Y^

u2^yes

7djdud5y2fia6ch 6 multicollinearity&heterosced (1)

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