Transcript STOCHASTIC REGRESSORS - Department of Economics

STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
Until now we have assumed that the explanatory variables in a regression model are
nonstochastic, that is, that they do not have random components. Relaxing this assumption
does not in itself undermine the OLS regression technique.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
OLS estimators remain unbiased, efficient, and consistent, provided that the random
components of the explanatory are distributed independently of the disturbance term and
provided that the model is correctly specified.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
We will take efficiency on trust but we will demonstrate unbiasedness and consistency.
After decomposing the slope coefficient in the usual way, we will focus on the error term
and show that it has expected value 0.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
1
( xi  x )( ui  u )
Cov( x , u) n
1  x x 

  i
( ui  u )
Var( x )
Var( x )
n  Var( x ) 
1
  f ( xi )( ui  u )
n
To do this, we restructure the error term as (1/n) times the sum of a series of terms, each of
which is the product of a function of x and a function of u. We start by writing out the
covariance term in full.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
1
( xi  x )( ui  u )
Cov( x , u) n
1  x x 

  i
( ui  u )
Var( x )
Var( x )
n  Var( x ) 
1
  f ( xi )( ui  u )
n
We bring the Var(x) term into the summation expression as shown.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
1
( xi  x )( ui  u )
Cov( x , u) n
1  x x 

  i
( ui  u )
Var( x )
Var( x )
n  Var( x ) 
1
  f ( xi )( ui  u )
n
xi  x
f ( xi ) 
Var( x )
Thus each term within the summation is the product of a function of x and a function of u.
The function of x is shown above. The function of u is just (ui-u).
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , y ) Cov( x ,[  x  u])
Cov( x , u)
b


Var( x )
Var( x )
Var( x )
1
( xi  x )( ui  u )
Cov( x , u) n
1  x x 

  i
( ui  u )
Var( x )
Var( x )
n  Var( x ) 
1
  f ( xi )( ui  u )
n
xi  x
f ( xi ) 
Var( x )
Cov( x , u) 
1
 1
E

E
f
(
x
)(
u

u
)

 
   E  f ( xi )( ui  u )  0
i
i
n
 n
 Var( x ) 
The expected value of the error term is equal to the expected value of this expression. We
can take 1/n out as a common factor and we can reverse the summation and expectation
operators. (The expected value of the sum is the sum of the expectations.)
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INDEPENDENCE OF TWO RANDOM VARIABLES
Two random variables x and y are said to be
independent if
E{f(x)g(y)} = E{f(x)} E{g(y)}
for any functions f(x) and g(y)
E  f ( xi )( ui  u )  E  f ( xi )E ( ui  u )
 E  f ( xi )E ( ui )  E ( u )  E  f ( xi ) [0  0]
Cov( x , u) 
1
 1
E

E
f
(
x
)(
u

u
)

 
   E  f ( xi )( ui  u )  0
i
i
n
 n
 Var( x ) 
Provided that x and u are distributed independently (this condition is crucial - here is a
reminder from the Review chapter), we can decompose the expected value of each term into
the product of expected values.
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INDEPENDENCE OF TWO RANDOM VARIABLES
Two random variables x and y are said to be
independent if
E{f(x)g(y)} = E{f(x)} E{g(y)}
for any functions f(x) and g(y)
E  f ( xi )( ui  u )  E  f ( xi )E ( ui  u )
 E  f ( xi )E ( ui )  E ( u )  E  f ( xi ) [0  0]
Cov( x , u) 
1
 1
E

E
f
(
x
)(
u

u
)

 
   E  f ( xi )( ui  u )  0
i
i
n
 n
 Var( x ) 
The expected value of the u component is 0, so the expected value of each term in the
summation is 0, and so the expected value of the error term is 0. Thus b is an unbiased
estimator of .
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , u)
b 
Var( x )
plim Cov( x , u)
plim b   
plim Var( x )
 x ,u
0
 2  2 
x
x
Consistency is demonstrated equally easily. The limiting values of the numerator and
denominator of the error term are the population covariance and variance, respectively.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , u)
b 
Var( x )
plim Cov( x , u)
plim b   
plim Var( x )
 x ,u
0
 2  2 
x
x
 x ,u  E( x   x )( u   u )
 E ( x   x ) E (u   u )  0  0
If x and u are independent, the population covariance decomposes as the product of two
expectations.
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STOCHASTIC REGRESSORS
y    x  u
Cov( x , u)
b 
Var( x )
plim Cov( x , u)
plim b   
plim Var( x )
 x ,u
0
 2  2 
x
x
 x ,u  E( x   x )( u   u )
 E ( x   x ) E (u   u )  0  0
Both of these expectations are zero. Hence plim b is equal to .
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Copyright Christopher Dougherty 2000. This slideshow may be freely copied for personal
use.