Problem set 2 - the derive of OLS estimator PDF

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the derive of OLS estimator...

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The University of California, Berkeley Economics Department Scott Alan Carson, Ph.D. Problem set 2 1. Using scalar notation where α is the y-intercept and β is the slope coefficient, derive OLS estimators for and . Answer: The starting point for deriving the formulas for OLS estimators is that we first need to set up the minimization problem, which is: N 2 −  −  xi )

 (yi min   i =1

,

Then, as learned in calculus, if we want to solve this minimization problem, we need to involve taking the derivative and setting the partial derivative equal to N

zero. So I use J to denote

(y i −  −  i

 x i )2 , and this gives us:

=1

J  J

N

−2( y i  i

=

−  −  xi ) = 0

=1

N

−2 xi( yi  i

=



−  −  xi ) = 0

=1

Now the task is to solve the equation showed above. After using some algebra N

methods, we can get

(y i −   i

− x i ) = 0 .

Besides, we know that

=1

N

yi  i =1

N

= N y ,  xi = N x

, then this leaves us with:

i= 1

N = N y − N  x  = y − x

Now, when it comes to considering about  , we can repeat the steps used before, and we can get: N

(x iy i − (y  i

−  x )x i − x 2i ) = 0

=1

N

x iy i  i =1

N

The same, we let

yi  i =1

will give us:

N

N

N

i =1

i =1

i =1

2 − y  xi +  x  xi −   xi = 0

N

= N y ,  xi = N x i= 1

, then we can solve the equation and it

N

x iy i  i

− N xy

=1

 =

N 2

xi  i

− Nx

2

=1

N

And if we let

x iy i  i =1

N

− N xy =

N

 (x i i =1

2

− x )(y i − y ),  (x i − x ) = i= 1

N 2

 xi i

−Nx

2

, we

=1

can get: N

 =

( xi  i =1

N

− x )( y i − y ) 2

 ( xi − x ) i =1

Finally, through equation  = y −  x , we can calculate the exact value of  ....