Submitted by: Submitted by ralikuwalu
Views: 10
Words: 284
Pages: 2
Category: Business and Industry
Date Submitted: 07/03/2016 10:32 AM
EQUATION SET 1
Suppose you want to estimate the Population Regression Function (PRF).
Y = + X +u
i
1
2 i
i
you must note the above PRF is not directly observable.
As such we have estimate PRF from SRF
ˆ
ˆ
ˆ
Y = + X +u
i
1
2 i
i
ˆ ˆ
Y =Y u
i
i
i
ˆ
From above you can find an expression for u
i
ˆ
ˆ
u Y Y
i
i
ˆ
ˆ
where u is Residual,Y is Actual Y values , Y is estimated Yvalues.
i
i
Further manipulation gives us the
ˆ
u
2
i
ˆ 2
(Y1 Yi )
2
ˆ
ˆ
(Y 1 2 X i )
So our message from above is :
ˆ
u
i
2
ˆ ˆ
f (1 , 2 )
As such sum of residual squared is some function of the two estimators.
ˆ ˆ
OLS chooses the two estimators( , ) in such a way
1
2
2
ˆ
that for a given sample, ui is small as possible.
We obtain the estimator using the above idea.
Knowing how to derive the estimators requires knowledge of
differential calculus.
Ordinary Least Squares Estimators
n X iYi X i Yi
ˆ
2
2
2
n X i ( X i )
(X
ˆ
2 =
i
X ) (Yi Y )
(X
x y
x
i
2
i
i
X)
2
ˆ
This formula is the estimator for 2 .
i
Use this formula to find estimated value of 2 .
2
ˆ
1
X Y X X Y
n X ( X )
i
i
2
i
i
i
i
2
i
ˆ
ˆ
ˆ
1 = Y- 2 X This formula is the estimator for 1 .
Use this formula to find estimated value of 1 .