INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 49
∑
...[2.13] We will find that
RSS is minimised when
( )
( )
…[2.14] And
̅
̅
…[2.15] The derivation
of the expressions for b1
and
b2
will follow the same procedure as the derivation in the preceding example, and you can compare the general version with the examples at each step. We will begin by expressing the square of the residual in observation
iregarding
b1
,
b2
and
the data on X and
Y:
(
̂
)
(
)
…[2.16] Summing overall the
nobservations, we can write
RSS as
(
)
(
)
∑
∑
∑
∑
∑
…[2.17]
Note that RSS is effectively a quadratic expression in
b1
and
b2
, with numerical coefficients determined by the data on
X and
Y in the sample.
We can influence the size of RSS only through our choice of
b1
and
b2
. The data on
X and
Y, which determine the locations of the observations in the scatter diagram and are fixed once we have taken the sample. This equation [2.17] is the generalized version of the equations. The first
order conditions fora minimum,
…[2.18] Yield the following equations
