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
2
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(y t ) = 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. 2
var(y i ) = var(ui ) = σ i
• Then
5
Graphical Illustration of
Heteroscedasticity
6
Graphical Illustration of
Heteroscedasticity
Density
Y
σ1
2
σ 22
σ 32
X
7
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
8
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.
9
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 0

Y2
2
var(u )  E (uu ')  

0

   2

10
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
11
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
12
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.
13
Generalized Least Squares
• Then;
 X0i
Yi
= β0 
σi
 σi

 Xi   u i 
 + β1   +  

 σi   σi 
Y*i = β 0 X*0i + β1 X*i + u*i
where
X0i
is equal to 1 for each i.
14
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
15
An Example
• Let us assume that we have a model as;
Yt = β0 + β1 Xt + ut
and we know there is a relation for error
terms as;
var  ut  = σ = σ Xt
2
t
2
16
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 = β 0 X*0i + β1 X*i + u*i
17
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
*
18
An Example
• Let assume we have the following model;
Yt = β0 + β1 X1t + β2 X2t + β 3 X3t + .. + β4 X4t + u t
and,
 2i   2 x i
Then,
1
T
xi
19
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 
20
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
21
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   
22
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  



1
1
2
1
2
1
1
 
Var     x  x 
1
I
1
1
1
1
1
1
1
1
2
1
1
1

Var     x  x   x  x  x  x 
2
1

1
23
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
24
Detecting Heteroscedasticity
1. Goldfeld-Quandt Test
This method applicable where one
assumes the heteroscedastic variance is
positively related one of variables.
var  ut  = σ = σ Xt
2
t
2
As in our previous example;
25
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
26
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
27
Detecting Heteroscedasticity
2. Breusch-Pagan-Godfrey(BPG)
BPG assumes that the error variance
described as;
σ i = f (α1 + α 2 Z 2i + α 3 Z 3i + ... + α m Z mi )
2
where Z’s are some functions of nonstochastic variables.
28
Detecting Heteroscedasticity
• The BPG proceeds as follows;
2
2
ˆ
σ
=
Σu
– Step1:Run OLS and obtain residuals
i /n
2
2
ˆ
p
=
u
/
σ
– Step2:Obtain
i
i
– Step3: Generate series of p’s as;
pi = α1 + α 2 Z2i + α 3 Z3i + ... + α m Zmi
– Step5: Regress
2
– Step4:
ESS fromwhere
previous
step
and calculate;
Θ asy χObtain
Θ
=
1/2(ESS)
m-1
29
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
30
Detecting Heteroscedasticity
3. White Test
White test has no assumptions and easy
to apply.
Therefore it is commonly used test for the
heteroscedasticity.
31
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 * R asy χ df
2
2
– Step3:Test the following;
In the large
samples
32
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
33
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 = σ 2 X2t
– Error variance is proportional to
var  ut  = σ = σ
2
t
– Log transformation
2
X 2t
 E(Yi ) 
E  Y 
2
2
i
34
END
End of
lecture
35