Rewriting these equations, one obtains the normal equations:
(5.7)
(5.7)
Solving the normal equations simultaneously for aand byields the least
squares regression estimates:
(5.8)
(5.8)
(5.8)
(5.8)
An estimation of the regression line’s coefficients and “goodness of fit”
also can be found in terms of expressing the dependent and independent
variables in terms of deviations from their means, their sample moments.
The sample moments will be denoted by M.
Mxxx
NX X
XX i
i
N
i
N
ii
i
N
i
N
==−
=−
= =
= =
∑ ∑
∑ ∑
1
22
1 1
1 1
2
()
bˆ
XY X Y
NX X
ii
i
N
i
i
N
i
i
N
i
N
i
i
N
=
−
−
===
==
∑∑∑
∑∑
111
1
2
11
2
aˆ
XY XY
NX X
i
N
i
i
N
ii
i
N
i
N
i
i
N
=
−
−
== =
==
∑∑ ∑
∑∑
1
2
11 1
1
2
11
2
1 1 1
1
2
1 1 1
YabX
YX a X b X
ii
i
N
i
N
i
N
ii i
i
N
i
N
i
N
=+
=+
= = =
= = =
∑ ∑ ∑
∑ ∑ ∑