The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

(6.1.6)
var (
ˆ
β
2
)
=
σ
2
X
2
i
(6.1.7)
where 
σ
2
is estimated by
ˆ
σ
2
=
ˆ
u
2
i
n
−
1
(6.1.8)
It is interesting to compare these formulas with those obtained when the intercept term is
included in the model:
ˆ
β
2
=
x
i
y
i
x
2
i
(3.1.6)
var (
ˆ
β
2
)
=
σ
2
x
2
i
(3.3.1)
ˆ
σ
2
=
ˆ
u
2
i
n
−
2
(3.3.5)
β
i
Systematic risk
Security risk premium
0
R
i
–
r
f
FIGURE 6.2
The Market Model
of Portfolio Theory
(assuming 
α
i
=
0).
guj75772_ch06.qxd 07/08/2008 07:00 PM Page 149

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150
Part One
Single-Equation Regression Models
The differences between the two sets of formulas should be obvious: In the model with the
intercept term absent, we use 
raw
sums of squares and cross products but in the intercept-
present model, we use adjusted (from mean) sums of squares and cross products. Second,
the df for computing 
ˆ
σ
2
is (
n
−
1) in the first case and (
n
−
2) in the second case. (Why?)
Although the interceptless or zero intercept model may be appropriate on occasions,
there are some features of this model that need to be noted. First, 
ˆ
u
i
, which is always
zero for the model with the intercept term (the conventional model), need not be zero when
that term is absent. In short, 
ˆ
u
i
need not be zero for the regression through the origin.
Second, 
r
2
, the coefficient of determination introduced in Chapter 3, which is always non-
negative for the conventional model, can on occasions turn out to be 
negative
for the inter-
ceptless model! This anomalous result arises because the 
r
2
introduced in Chapter 3
explicitly assumes that the intercept is included in the model. Therefore, the conventionally
computed 
r
2
may not be appropriate for regression-through-the-origin models.
3