We found that 1 1 2 1 ˆ t t t t t t t x y x

We found that
1
1
2
1
ˆ
T
t t
t
T
t
t
x y
x






.
Substituting
t t
y Y Y   we obtain
1 1 1 1
1
2 2 2 2
1 1 1 1
( )
ˆ
T T T T
t t t t t t t
t t t t
T T T T
t t t t
t t t t
x y x Y Y xY Y x
x x x x
    
   

   
   
   
But by definition, the sum of the deviations of a variable from its
mean is identically equal to zero,
1
0
T
t
t
x



. Therefore
1
1
2 2 1
1 1
ˆ
.
t
T
t
T
t t
t
T T
t
t t
t t
xY
x
Y
x x
 

 
 
 
 
 
 
 
 
 
 


 
By assumption 2 of the method of least squares, the values of X are a
set of fixed values, which do not change from sample to sample.

ECON 301 (01) - Introduction to Econometrics I March, 2012
METU - Department of Economics
Instructor: Dr. Ozan ERUYGUR e-mail: [email protected]
Lecture Notes
7
Consequently the ratio
2
1
t
T
t
t
x
x


will be constant from sample to sample,
and if we denote the ratio by a
t
we may write the estimator
1
ˆ
 in the
form
1
1
ˆ
T
t t
t
aY 



.
By substituting the value of
0 1
t t t
Y X u      and rearranging the
factors we find
 
1 0 1
1
ˆ
T
t t t
t
a X u   

  

1 0 1
1 1 1
ˆ
T T T
t t t t t
t t t
a a X a u   
  
 

  
Note (and show) that
1
0
T
t
t
a



and
1
1
T
t t
t
a X



.
Therefore, the equation above reduces to
1 1
1
ˆ
T
t t
t
a u  

 

which implies that
1
ˆ
 is a linear estimator because it is a linear
function of Y ; actually it is a weighted average of Y
t
with a
t
serving as
the weights.
Taking expected values yields
1 1
1
ˆ
( ) ( )
T
t t
t
E E a E u  

 
 
 

The significance of the assumption of constant X values is seen in the
above manipulations, in that the operation of taking expected values is
applied to u and Y values but not to X .

ECON 301 (01) - Introduction to Econometrics I March, 2012
METU - Department of Economics
Instructor: Dr. Ozan ERUYGUR e-mail: [email protected]
Lecture Notes
8
Since
1
 (the true population parameter) is constant, we can
write
1 1
( ) E    . Finally using assumption 3 , we have 0
t
E u
 
 
 .
Hence, the equation reduces to
1
1 1
ˆ
ˆ
( )
meanof
E

  
which implies that the mean of OLS estimate
1
ˆ
 is equal to the true
value of the population parameter
1
 .
This implies that the
1
ˆ
 is an unbiased estimator.
B. Variance of
1
ˆ

It can be proved that
2 2
2
1 1 1 1 1
2
1
ˆ ˆ ˆ ˆ
( ) ( )
T
t
t
Var E E E
x

    

   
   
   
    

.
Recall that we established:
1 1
1
ˆ
T
t t
t
a u  

 

which implies that
1 1
1
ˆ
T
t t
t
a u  




.