Econometrics

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 .