Lecture 2. Relaxing the Assumptions of CLRM_0

8/3/2019 Lecture 2. Relaxing the Assumptions of CLRM_0

1/17

RELAXING THE ASSUMPTIONS OFCLRM

Dr. Obid A.Khakimov

Senior lecturer,

Westminster International

University in Tashkent


2/17

ASSUMPTIONS OF CLASSICAL LINEARREGRESSION MODEL

The regression model is linear in parameters

The values of independent variables are fixedin repeated sampling

Conditional mean of residuals is equal to zero For given Xs there is no autocorrelation in the

residuals

Independent variables , Xs , and residuals of

the regression are independent.


3/17

ASSUMPTIONS OF CLASSICAL LINEARREGRESSION MODEL

The number of observations must be greaterthan number of parameters.

There must be sufficient variability in thevalues of variables.

The regression model should be correctlyspecified.

There is no linear relationship amongindependent variables.

Residuals of the regression normallydistributed

),0(~

2

ei Ne


4/17

MULTICOLLINEARITY:

Agenda:The nature of multicollinearity.

Practical consequences.

Detection. Remedial measures to alleviate the problem.


5/17

PERFECT V S LESS THAN


6/17

PERFECT V.S LESS THANPERFECT

kk XXXX1

3

1

32

1

21 .....

+=

ikk eXXXX

11

3

1

32

1

21

1.....

++=

0.....332211 =++ kkXXXX

Perfect multicollinearity is the case when two ore more independvariables Can create perfect linear relationship.

Perfect multicollinearity is the case when two ore more independvariables Can create less than perfect linear relationship.


7/17

MULTIPLE REGRESSION MODEL

ii uXXY ++= 3322

=2

33221

2 )(min iiii XXYu

33221

XXY =

( )2

32

2

3

2

2

323

2

322

)())((

))(()(

=

iiii

iiiiiii

xxxx

xxxyxxy

( )2

32

2

3

2

2

322

2

233

)())((

))(()(

=

iiii

iiiiiii

xxxx

xxxyxxy


8/17

MULTIPLE REGRESSIONMODEL

( )

( )

( )0

0

)(

)(

_

)())(((

))(()(

)())()((

))(()(

2

33

2

2

3

22

3

2

3

32

3

2

33

2

2

33

2

3

2

3

333233

2

=

=

=

=

=

=

aaa

axyaxy

afi

xxx

xxyxxy

xxxx

xxxyxxy

iiii

iii

iiiiii

iiii

iiiiiii

ii XX 32 =

( )2

32

2

3

2

2

323

2

322

)())((

))(()(

=

iiii

iiiiiii

xxxx

xxxyxxy


9/17

OLS ESTIMATION

2

232

23

22

3232

2

2

2

3

2

3

2

2

1 )(

21

)

var(

++=

iiii

iiii

xxxx

xxXXxXxX

n

)1(

)var(2

3,2

2

2

2

2

rx i

=

)1(

)var(2

3,2

2

3

2

3

rx i

=

=

2

3

2

2

2

3,2

2

3,2

32

)1(

),cov(

ii xxr

r

As degree of collinearity approaches to one,the variances of coefficients approaches toinfinity.

Thus, the resence of hi h collinearit will


10/17

PRACTICAL CONSEQUENCES

The OLS is BLUE but large variances andcovariances making process estimationdifficult.

Large variances cause large confidenceintervals and accepting or rejectinghypothesis are biased.

T statistics are biased

Although t-stats are low, R-square mightbe very high.

The sensitivity of estimators and

variances are very high to small changes


11/17

VARIANCE INFLATION FACTOR

VIFxrxrx iii

=

=

=2

2

2

2

3,2

2

2

2

2

3,2

2

2

2

2)1(

1

)1()var(

0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8 1 1.2

Correlation

VIF


12/17

IMPLICATION FOR K VARIABLE MODELS

VIFxRxRx jjjjj

j =

=

=

2

2

22

2

22

2

)1(

1

)1()var(

ikki uXXXXXY ..... 33221100 +++++=

22

33221100.....

j

kki

RR

XXXXXX

=

++++=


13/17

CONFIDENCE INTERVALS AND T-STATISTICS

VIFse kk )(96.1

VIFset

k

kk

)(

0

=

0...:320

====k

H

)/()1(

)1/(

111 2

2

2

2

knR

kR

R

R

k

kn

RSS

ESS

k

knF

=

=

=

Ha: Not all slope coefficients are simultaneously zero

Due to low t-stats we can not reject ourNull Hypothesis

ue to high R square the F-value will be very high and rejection of Ho will be easy


14/17

DETECTION

Multicollinearity is a question of degree.

It is a feature of sample but not population.

How to detect : High R square but low t-stats. High correlation coefficients among the

independent variables.

Auxiliary regression

High VIF

Eigenvalue and condition index.***


15/17

AUXILIARY REGRESSION

22

33221100.....

j

kki

RR

XXXXXX

=

++++=

)1/()1(

)2/(

...,,2

...,,2

32

32

+

=

knR

kRF

ki

ki

xxxx

xxxx

i

Run regression where one X is dependent and other Xs are independent and

Obtain R square

Ho: The Xi variable is not collinear

Df num = k-2

Df denom = n-k+1

k- is the number of explanatory variables including intercept.n- is sample size.

If F stat is higher than F critical then Xi variable is collinear

Rule of thumb: if R square of auxiliary regression is higher

than over R square then it might be troublesome.


16/17

WHAT TO DO ?

Do nothing.

Combining cross section and time series

Transformation of variables (differencing, ratio

transformation) Additional data observations.


17/17

READING

Gujarati D., (2003), Basic Econometrics,

Ch. 10

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Lecture 2. Relaxing the Assumptions of CLRM_0