Lecture 2. Relaxing the Assumptions of CLRM_0
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Transcript of Lecture 2. Relaxing the Assumptions of CLRM_0
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8/3/2019 Lecture 2. Relaxing the Assumptions of CLRM_0
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RELAXING THE ASSUMPTIONS OFCLRM
Dr. Obid A.Khakimov
Senior lecturer,
Westminster International
University in Tashkent
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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.
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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
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MULTICOLLINEARITY:
Agenda:The nature of multicollinearity.
Practical consequences.
Detection. Remedial measures to alleviate the problem.
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PERFECT V S LESS THAN
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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.
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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
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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
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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
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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
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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
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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
=
++++=
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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
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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.***
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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.
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WHAT TO DO ?
Do nothing.
Combining cross section and time series
Transformation of variables (differencing, ratio
transformation) Additional data observations.
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READING
Gujarati D., (2003), Basic Econometrics,
Ch. 10