Transcript ECONOMETRICS By Prof. Burak Saltoglu

EC 532
Advanced Econometrics
Lecture 1: Heteroscedasticity
Prof. Burak Saltoglu
Outline
•
•
•
•
•
What is Heteroscedasticty
Graphical Illustration of Heteroscedasticity
Reasons for Heteroscedastic errors
Consequneces of Heteroscedasticity
Generalized Least Squares
– GLS in Matrix Notation
• Testing Heteroscedasticity
• Remedies
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Consequneces of
Heteroscedasticity
• As we know
heteroscedastic
error
 E u u under
E u u 
 



terms,


E uu   0
E u 
 




E
u




1
1
1
2
2
2
2
N
12
E uu    0
 
0
 22


 
 2N 
3
What is Heteroscedasticty
• Or more specifically for time series;
var(yt ) = var(ut ) = σ
2
means that the variance of disturbances
do not change over time.
4
What is Heteroscedasticty
• The violation of this assumption is called
as heteroscedasticity.
• In the case that the variances of all
disturbances are not same, we say that
the heteroscedasticity exists.
var(yi ) = var(ui ) = σi
2
• Then
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Graphical Illustration of
Heteroscedasticity
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Graphical Illustration of
Heteroscedasticity
Density
Y
σ
2
1
σ 22
σ 32
X
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Some Reasons to
Heteroscedasticity
• The Error-Learning Models
• Improvement of data collecting
(As data collecting techniques improve variances tend to
reduce)
• Presence of outliers
• Misspecification of model
• Volatility clustering and news effect
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The Consequneces of
Heteroscedasticity for OLS
• At the presence of heteroscedasticity;
– OLS estimators are still linear and unbiased
estimators, but they are no longer the best.
(BLUE)
– The standard errors computed for the OLS
are incorrect, then inference might be
misleading.
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Heteroscedasticity
effect of income to household
is it safe to assume that variability of
consumption
2
2



Yi
i
is stable for all income
levels?
suppose, variability of consumption
increases with income in a relation such
that
 Y1

var(u )  E (uu ')   2 

0
Y2
0

   2

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Consequences
of
Heteroscedasticity
Properties of OLS Estimators: Assume an
regression
y  X u
with E(u)  0 and E(uu ')   2
    ( X ' X )1 X ' u
E( )  
(1)Unbiasedness still holds since
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Consequences of Heteroscedasticity
OLS standard errors, which would be
derived from σ2(X’X)-1 are incorrect since
Var (  )  E[(    )(    ) ']
 E[( X ' X ) 1 X ' uu ' X ( X ' X ) 1 ]
  2 ( X ' X ) 1 X ' X ( X ' X ) 1
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Generalized Least Squares
• As we discussed the variance of
observations might be different. But the
OLS does not take into account the
possibility of different variances.
• The method of GLS is OLS on the
transformed variables that satisfies the
assumptions.
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Generalized Least Squares
• Then;
 X0i
Yi
= β0 
σi
 σi

 Xi
 + β1 

 σi
  ui 
+ 
  σi 
Y*i = β0 X*0i +β1X*i +u*i
where
X 0i
is equal to 1 for each i.
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Generalized Least Squares
• So the variance is;
var  u
*
 = E u 
*
i
i
2
 ui 
= E 
 σi 
1
2
var  u i  = 2 E  u i 
σi
2
*
var  u
*
i

Now the residual
is
homoscedastic
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An Example
• Let us assume that we have a model as;
Yt = β0 + β1Xt + ut
and we know there is a relation for error
terms as;
var ut  = σ = σ Xt
2
t
2
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An Example
• Now, for this case we can define the
transformed form as;
 1 
 Xi   u i 
Yi
= β0 
 + β1 
+

 X 
 X   X 
Xi
i 
i  
i 


Y i = β0 X +β1X +u
*
*
0i
*
i
*
i
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An Example
• The variance;
var  u
*
 = E u 
*
i
i
2
 ui 
= E

 X 
i 

2
1
2
var  u i  =
E  ui 
Xi
*
1 2
var  u i  =
σ Xi = σ 2
Xi
*
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An Example
• Let assume we have the following model;
Yt = β0 + β1X1t + β2 X2t + β3 X3t + .. + β4 X4t + ut
and,
 2i   2 x i
Then,
1
T
xi
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An Example



2
2
T x i T   



1
x1
0
0




0
1
xN




x
0


 1
 0  0 
    x N  




1
x1
0



0

 

0 
1 
x N 
1 0 
T 2 x i T   2 0  0 
   1 
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GLS in Matrix Notation
• If  is a symmetric and positive semi-definite; then there
exists a non-singular matrix P such that;
  PP 
P 1P 1  I
If we set;

 1  P  1P 1   P 1  P 1
T  P 1
 1  T T
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Properties of GLS
TY   Tx   Tu

ˆ  
Tx
Tx
  



1

Tx
  TY

ˆ   xT Tx  xT Ty
1
1
1
ˆ
   x x  x1Y
1
1
1
1
1
ˆ
   x x   x x     x x  x1u
     x  1 x  1 x  1E u

E   
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GLS in Matrix Notation

 




Var      E    E  




1

1
Var    x  1 x  x  1 u  x  1 x  x  1 u
      
 


Var      x  x  x  uu  x x  x  


Var      x  x  x  
 x  x  x  



Var      x  x   x  x  x  x  
Var     x  x 
1
1
2
1
2
1
1
1
1
1
1
1
1
1
I
1
1
2
2
1
1
1
1
1
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GLS in Matrix Notation
• When we faced with heteroscedasticity if
we can find an nxn nonsingular
transformation matrix T such that;
TT  I
TY   Tx   Tu
then we multiply everything by T,
E Tuu T   T 2T
E Tuu T    2 TT
2


E  Tuu T  = σ
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Detecting Heteroscedasticity
1. Goldfeld-Quandt Test
This method applicable where one
assumes the heteroscedastic variance is
positively related one of variables.
var ut  = σt2 = σ2 Xt
As in our previous example;
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Detecting Heteroscedasticity
• Goldfeld-Quandt test proceed following steps;
–
–
–
–
Step1: Order the observations (lowest to highest)
Step2: Omit c central observations
Step3: Fit separate OLS regressions
Step4: Compute;
n - c

where, df =
-k
2
RSS 2 /df
λ=
RSS1 /df
k is the number of
estimated
parameters
including the
intercept
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Detecting Heteroscedasticity
λ
follows an F-distribution and the null
hypothesis of the test is that the residual is
homoscedastic.
λ
Therefore if the is greater than the critical F
value at the chosen significance level, we can
reject the null and say the residual is
heteroscedastic
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Detecting Heteroscedasticity
2. Breusch-Pagan-Godfrey(BPG)
BPG assumes that the error variance
described as;
σi = f (α1 +α2Z2i +α3Z3i + ... +αmZmi )
2
where Z’s are some functions of nonstochastic variables.
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Detecting Heteroscedasticity
• The BPG proceeds as follows;
2
2
ˆ
σ
=
Σu
– Step1:Run OLS and obtain residuals
i /n
pi = uˆ i 2 / σ2
– Step2:Obtain
– Step3: Generate series of p’s as;
pi = α1 + α2Z2i + α3Z3i + ... + αmZmi
– Step5: Regress
2
– Step4:
ESS from where
previous
step
and calculate;
Θ asy χObtain
Θ
=
1/2(ESS)
m-1
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Detecting Heteroscedasticity
• The null hypothesis
homoscedastic.
that
the
residual
is
• Therefore if our test statistic exceeds the critical
value at the chosen significance level, we can
reject the null hypothesis and we have sufficient
evidence to say there is heteroscedasticity
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Detecting Heteroscedasticity
3. White Test
White test has no assumptions and easy
to apply.
Therefore it is commonly used test for the
heteroscedasticity.
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Detecting Heteroscedasticity
• White test proceed following steps;
– Step1:Obtain residuals
– Step2:Run auxiliary regression and obtain Rsquared
uˆ i 2 = α1 + α 2 X2i + α 3 X3i + α 4 X2i 2
+α 5 X2i 2 + α 6 X2i X3i + v i
n * R2asy χdf 2
– Step3:Test the following;
In the large
samples
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Detecting Heteroscedasticity
• The null hypothesis again claims that there
is no heteroscedasticity
• Therefore if our test statistic exceeds the
critical value at the chosen significance
level, we can reject the null and have
sufficient evidence to say there is
heteroscedasticity
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How to Deal with
Heteroscedasticity
• WLS (Weighted Least Squares)
• White’s Heteroscedasticity consistent variances and
standard errors
• Plausible assumption about heteroscedasticity pattern
Xt
– Error variance is proportional
to
var ut  = σt2 = σ2 Xt
– Error variance is proportional to
var ut  = σt2 = σ2 X2t
– Error variance is proportional to
X
2
t
 E(Yi ) 
var  u t  = σ t2 = σ 2  E  Yi  
– Log transformation
2
2
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END
End of
lecture
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