3. THE PARTITIONED REGRESSION MODEL
Consider taking a regression equation in the form of
(1)
y
= [
X
1
X
2
]
β
1
β
2
+
ε
=
X
1
β
1
+
X
2
β
2
+
ε.
Here, [
X
1
, X
2
] =
X
and [
β
1
, β
2
] =
β
are obtained by partitioning the matrix
X
and vector
β
of the equation
y
=
Xβ
+
ε
in a conformable manner. The normal
equations
X Xβ
=
X y
can be partitioned likewise. Writing the equations
without the surrounding matrix braces gives
X
1
X
1
β
1
+
X
1
X
2
β
2
=
X
1
y,
(2)
X
2
X
1
β
1
+
X
2
X
2
β
2
=
X
2
y.
(3)
From (2), we get the equation
X
1
X
1
β
1
=
X
1
(
y
−
X
2
β
2
), which gives an expres
sion for the leading subvector of
ˆ
β
:
(4)
ˆ
β
1
= (
X
1
X
1
)
−
1
X
1
(
y
−
X
2
ˆ
β
2
)
.
To obtain an expression for
ˆ
β
2
, we must eliminate
β
1
from equation (3). For
this purpose, we multiply equation (2) by
X
2
X
1
(
X
1
X
1
)
−
1
to give
(5)
X
2
X
1
β
1
+
X
2
X
1
(
X
1
X
1
)
−
1
X
1
X
2
β
2
=
X
2
X
1
(
X
1
X
1
)
−
1
X
1
y.
When the latter is taken from equation (3), we get
(6)
X
2
X
2
−
X
2
X
1
(
X
1
X
1
)
−
1
X
1
X
2
β
2
=
X
2
y
−
X
2
X
1
(
X
1
X
1
)
−
1
X
1
y.
On defining
(7)
P
1
=
X
1
(
X
1
X
1
)
−
1
X
1
,
can we rewrite (6) as
(8)
X
2
(
I
−
P
1
)
X
2
β
2
=
X
2
(
I
−
P
1
)
y,
whence
(9)
ˆ
β
2
=
X
2
(
I
−
P
1
)
X
2
−
1
X
2
(
I
−
P
1
)
y.
1
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TOPICS IN ECONOMETRICS
Now let us investigate the effect that conditions of orthogonality amongst
the regressors have upon the ordinary leastsquares estimates of the regression
parameters. Consider a partitioned regression model, which can be written as
(10)
y
= [
X
1
, X
2
]
β
1
β
2
+
ε
=
X
1
β
1
+
X
2
β
2
+
ε.
It can be assumed that the variables in this equation are in deviation form.
Imagine that the columns of
X
1
are orthogonal to the columns of
X
2
such that
X
1
X
2
= 0. This is the same as assuming that the empirical correlation between
variables in
X
1
and variables in
X
2
is zero.
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 Fall '11
 D.S.G.Pollock
 Least Squares, Linear Regression, Regression Analysis, Errors and residuals in statistics, X1, Linear least squares

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