INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 48
...[2.10] And
...[2.11] And so we have
And
Solving these two equations, we obtain
b1
= 1 and
b2
= 2, and
hence the regression equation ̂
Just to check that we have come to the right conclusion, we shall calculate the residuals
e1
= 3 –
b1
–
b2
= 3 – 1 – 2 = 0
e2
= 5 –
b1
– 2
b2
= 5 – 1 – 4 = 0 Thus both residuals are equal to 0, implying that the line passes exactly through both points.
2.1.3.3.1 Least Squares Regression with One Explanatory Variable We shall now consider the general case where there are
n observations
on two variables X and
Y and supposing
Y to depend on
X; we will fit the equation
̂
...[2.12] The fitted value of the dependent variable in observation
i.
will be (
b1
+
b2
Xi) and the residual will be (
Yi–
b1
–
b2
Xi).
We wish to choose b1
and
b2 so as to minimize the residual sum of the squares
RSS given by
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
