An asset can have a higher unconditional average return than predicted by
the unconditional CAPM, if its beta moves with the market risk premium.
One way to test a conditional model is to parameterize the variables that
shift betas over time.
For example, we might write
°
imt
=
°
i
0
+
°
i
1
z
t
:
(3.32)
The conditional model can then be written as
E
t
R
e
i;t
+1
=
°
i
0
E
t
R
e
m;t
+1
+
°
i
1
E
t
z
t
R
e
m;t
+1
(3.33)
and now we can take unconditional expectations to get
E
R
e
i;t
+1
=
°
i
0
E
R
e
m;t
+1
+
°
i
1
E
z
t
R
e
m;t
+1
:
(3.34)
61

CHAPTER 3.
STATIC EQUILIBRIUM ASSET PRICING
This is a multifactor model, where the factors are the excess market return
and the excess market return scaled by the state variable
z
t
.
The scaled ex-
cess market return can be interpreted as the return on a dynamic investment
strategy that invests more aggressively in the market when
z
t
is high.
3.3
Empirical Evidence
3.3.1
Test methodology
In practice the CAPM will never hold exactly.
We need a statistical test
to tell whether deviations from the model (mean-variance ine¢ ciency of the
market portfolio, or equivalently nonzero alphas) are statistically signi±cant.
The two leading approaches are time-series and cross-sectional.
At a
deep level, they are much more similar than they appear to be at ±rst.
Time-series approach
The time-series approach starts from the regression
R
e
it
=
²
i
+
°
im
R
e
mt
+
"
it
;
(3.35)
where
R
e
it
=
R
it
°
R
ft
and
R
e
mt
=
R
mt
°
R
ft
.
The null hypothesis is
that
²
i
= 0
.
This is a simple parameter restriction for any one asset; the
challenge is to test it jointly for a set of
N
assets.
An asymptotic test is as follows.
De±ne
²
as the
N
-vector of inter-
cepts
²
i
, and
´
as the variance-covariance matrix of the regression residuals
"
it
.
(Note that this is di∕erent from the matrix
³
, which is the variance-
covariance matrix of the raw returns rather than the residuals.)
Then as
the sample size
T
increases, asymptotically
T
2
4
1 +
R
e
m
»
(
R
e
mt
)
!
2
3
5
°
1
b
²
0
b
´
°
1
b
²
²
À
2
N
:
(3.36)
To see the intuition, suppose there were no market return in the model.
Then the vector
²
would be a vector of sample mean excess returns, with
variance-covariance matrix
(1
=T
)³
.
Thus the quadratic form
b
²
0
b
´
°
1
b
²
is a
sum of squared intercepts, divided by its variance-covariance matrix, which
has a
À
2
N
distribution.
The term in square brackets is a correction for the
presence of the market return in the model.
Uncertainty about the betas
a∕ects the alphas, and more so when the market has a high expected return
relative to its variance.
62

CHAPTER 3.
STATIC EQUILIBRIUM ASSET PRICING
A ±nite-sample test makes a further correction for the fact that the
variance-covariance matrix
´
must be estimated.
Under the assumption
that the
"
it
are serially uncorrelated, homoskedastic, and normal, we have
²
T
°
N
°
1
N
³
2
4
1 +
R
e
m
»
(
R
e
mt
)
!