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5A.4
Derivations of Equations (5.10.2) and (5.10.6)
Variance of Mean Prediction
Given
X
i
=
X
0
, the true mean prediction
E
(
Y
0
|
X
0
) is given by
E
(
Y
0
|
X
0
)
=
β
1
+
β
2
X
0
(1)
5
For proof, see Robert V. Hogg and Allen T. Craig,
Introduction to Mathematical Statistics,
2d ed.,
Macmillan, New York, 1965, p. 144.
6
For proof, see J. Johnston,
Econometric Methods,
3d ed., McGraw-Hill, New York, 1984, pp. 181–182.
(Knowledge of matrix algebra is required to follow the proof.)
guj75772_ch05.qxd 07/08/2008 12:46 PM Page 145
146
Part One
Single-Equation Regression Models
We estimate Eq. (1) from
ˆ
Y
0
= ˆ
β
1
+ ˆ
β
2
X
0
(2)
Taking the expectation of Eq. (2), given
X
0
, we get
E
(
ˆ
Y
0
)
=
E
(
ˆ
β
1
)
+
E
(
ˆ
β
2
)
X
0
=
β
1
+
β
2
X
0
because
ˆ
β
1
and
ˆ
β
2
are unbiased estimators. Therefore,
E
(
ˆ
Y
0
)
=
E
(
Y
0
|
X
0
)
=
β
1
+
β
2
X
0
(3)
That is,
ˆ
Y
0
is an unbiased predictor of
E
(
Y
0
|
X
0
).
Now using the property that var (
a
+
b
)
=
var (
a
)
+
var (
b
)
+
2 cov (
a
,
b
) , we obtain
var (
ˆ
Y
0
)
=
var (
ˆ
β
1
)
+
var (
ˆ
β
2
)
X
2
0
+
2 cov (
ˆ
β
1
ˆ
β
2
)
X
0
(4)
Using the formulas for variances and covariance of
ˆ
β
1
and
ˆ
β
2
given in Eqs. (3.3.1), (3.3.3), and
(3.3.9) and manipulating terms, we obtain
var (
ˆ
Y
0
)
=
σ
2
1
n
+
(
X
0
− ¯
X
)
2
x
2
i
=
(5.10.2)
Variance of Individual Prediction
We want to predict an individual
Y
corresponding to
X
=
X
0
; that is, we want to obtain
Y
0
=
β
1
+
β
2
X
0
+
u
0
(5)
We predict this as
ˆ
Y
0
= ˆ
β
1
+ ˆ
β
2
X
0
(6)
The prediction error,
Y
0
− ˆ
Y
0
, is
Y
0
− ˆ
Y
0
=
β
1
+
β
2
X
0
+
u
0
−
(
ˆ
β
1
+ ˆ
β
2
X
0
)
=
(
β
1
− ˆ
β
1
)
+
(
β
2
− ˆ
β
2
)
X
0
+
u
0
(7)
Therefore,
E
(
Y
0
− ˆ
Y
0
)
=
E
(
β
1
− ˆ
β
1
)
+
E
(
β
2
− ˆ
β
2
)
X
0
−
E
(
u
0
)
=
0
because
ˆ
β
1
,
ˆ
β
2
are unbiased,
X
0
is a fixed number, and
E
(
u
0
) is zero by assumption.
Squaring Eq. (7) on both sides and taking expectations, we get var (
Y
0
− ˆ
Y
0
)
=
var (
ˆ
β
1
)
+
X
2
0
var (
ˆ
β
2
)
+
2
X
0
cov (
β
1
,
β
2
)
+
var (
u
0
)
.
Using the variance and covariance formulas
for
ˆ
β
1
and
ˆ
β
2
given earlier, and noting that var (
u
0
)
=
σ
2
, we obtain
var (
Y
0
− ˆ
Y
0
)
=
σ
2
1
+
1
n
+
(
X
0
− ¯
X
)
2
x
2
i
=
(5.10.6)
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147
Some aspects of linear regression analysis can be easily introduced within the framework
of the two-variable linear regression model that we have been discussing so far. First we
consider the case of
regression through the origin,
that is, a situation where the inter-
cept term,
β
1
, is absent from the model. Then we consider the question of the
units of
measurement,
that is, how the
Y
and
X
variables are measured and whether a change in the
units of measurement affects the regression results. Finally, we consider the question of the
functional form
of the linear regression model. So far we have considered models that
are linear in the parameters as well as in the variables. But recall that the regression theory
developed in the previous chapters requires only that the parameters be linear; the variables
may or may not enter linearly in the model. By considering models that are linear in the
parameters but not necessarily in the variables, we show in this chapter how the two-
variable models can deal with some interesting practical problems.
Once the ideas introduced in this chapter are grasped, their extension to multiple
regression models is quite straightforward, as we shall show in Chapters 7 and 8.
6.1
Regression through the Origin
There are occasions when the two-variable population regression function (PRF) assumes
the following form:
(6.1.1)
In this model the intercept term is absent or zero, hence the name
regression through the
origin.
As an illustration, consider the capital asset pricing model (CAPM) of modern portfolio
theory, which, in its risk-premium form, may be expressed as
1
(ER
i
−
r
f
)
=
β
i
(ER
m
−
r
f
)
(6.1.2)
Y
i
=
β
2
X
i
+
u
i
Chapter
6
Extensions of the
Two-Variable Linear
Regression Model
1
See Haim Levy and Marshall Sarnat,
Portfolio and Investment Selection: Theory and Practice
, Prentice-
Hall International, Englewood Cliffs, NJ, 1984, Chap. 14.
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148
Part One
Single-Equation Regression Models
where ER
i
=
expected rate of return on security
i
ER
m
=
expected rate of return on the market portfolio as represented by, say, the
S&P 500 composite stock index
r
f
=
risk-free rate of return, say, the return on 90-day Treasury bills
β
i
=
the Beta coefficient, a measure of systematic risk, i.e., risk that cannot be
eliminated through diversification. Also, a measure of the extent to which
the
i
th security’s rate of return moves with the market. A
β
i
>
1 implies a
volatile or aggressive security, whereas a
β
i
<
1 suggests a defensive secu-
rity. (
Note:
Do not confuse this
β
i
with the slope coefficient of the two-
variable regression,
β
2
.)
If capital markets work efficiently, then CAPM postulates that security
i
’s expected risk
premium (
=
ER
i
−
r
f
) is equal to that security’s
β
coefficient times the expected market
risk premium (
=
ER
m
−
r
f
). If the CAPM holds, we have the situation depicted in Fig-
ure 6.1. The line shown in the figure is known as the
security market line
(SML).
For empirical purposes, Equation 6.1.2 is often expressed as
R
i
−
r
f
=
β
i
(
R
m
−
r
f
)
+
u
i
(6.1.3)
or
R
i
−
r
f
=
α
i
+
β
i
(
R
m
−
r
f
)
+
u
i
(6.1.4)
The latter model is known as the
Market Model.
2
If CAPM holds,
α
i
is expected to be
zero. (See Figure 6.2.)
In passing, note that in Equation 6.1.4 the dependent variable,
Y
, is (
R
i
−
r
f
) and the
explanatory variable,
X
, is
β
i
, the volatility coefficient, and
not
(
R
m
−
r
f
). Therefore, to run
regression Eq. (6.1.4), one must first estimate
β
i
, which is usually derived from the
characteristic line,
as described in Exercise 5.5. (For further details, see Exercise 8.28.)
As this example shows, sometimes the underlying theory dictates that the intercept
term be absent from the model. Other instances where the zero-intercept model may be
appropriate are Milton Friedman’s permanent income hypothesis, which states that perma-
nent consumption is proportional to permanent income; cost analysis theory, where it is
1
ER
i
–
r
f
Security market line
0
β
i
ER
i
–
r
f
FIGURE 6.1
Systematic risk.
2
See, for instance, Diana R. Harrington,
Modern Portfolio Theory and the Capital Asset Pricing Model: A
User’s Guide
, Prentice Hall, Englewood Cliffs, NJ, 1983, p. 71.
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Chapter 6
Extensions of the Two-Variable Linear Regression Model
149
postulated that the variable cost of production is proportional to output; and some versions
of monetarist theory that state that the rate of change of prices (i.e., the rate of inflation) is
proportional to the rate of change of the money supply.
How do we estimate models like Eq. (6.1.1), and what special problems do they pose? To
answer these questions, let us first write the sample regression function (SRF) of Eq. (6.1.1),
namely,
Y
i
= ˆ
β
2
X
i
+ ˆ
u
i
(6.1.5)
Now applying the ordinary least squares (OLS) method to Eq. (6.1.5), we obtain the fol-
lowing formulas for
ˆ
β
2
and its variance (proofs are given in Appendix 6A, Section 6A.1):
ˆ
β
2
=
X
i
Y
i
X
2
i
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