5
Whenever X and Z are orthogonal to one another (have zero
collinearity) they do not overlap as in Figure 2 and the red area
disappears. Because there is no red area in this case, regressing Y
on X alone or on Z produces the same estimates of
x
and
z
, as if
Y were regressed on X and Z together.
Whenever X and Z are higly collinear and therefore overlap a lot,
the blue and green areas become very small, implying that when Y
is regressed on X and Z together very little information is used to
estimate
x
and
z
. This causes the variances of these estimates to
be very large. Thus, the impact of multicollinearity is to raise the
variances of the OLS estimates. Perfect collinearity causes the X
and
Z
circles
to
overlap
completely,
the
blue
and
green
area
disappear and estimation is impossible. Multicollinearity will be
discussed in detail later in this semester.
In Figure 1 the blue area represents the variation in
Y
explained by
X
. Thus
R
2
is given as the ratio of the blue area to the entire
Y
circle. In Figure 2 the blue+red+green area represents the variation
5
As is invariably the case in econometrics, the price of obtaining unbiased estimates is higher
variances.
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View Full DocumentECON 301 (01)  Introduction to Econometrics I
April, 2012
METU  Department of Economics
Instructor: Dr. Ozan ERUYGUR
email:
[email protected]
Lecture Notes
15
in Y explained by X and Z together. Thus, the R
2
resulting from the
multiple regression is given by the ratio of the blue+red+green area
to the entire Y circle.
The yellow area represents variation in Y attributable to the error
(disturbance) term, and thus the magnitude of the yellow area
represents the magnitude of
2
, the variance of the error term. This
implies, for example, that if, in the context of Figure 2, Y had been
regressed on only X, omitting Z,
2
would be estimated by the
yellow+green area, an overestimate.
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 Spring '10
 öcal
 Econometrics, Regression Analysis, ........., Dr. Ozan Eruygur

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