Some Exact Results for an Asset Pricing Test Based on the Average F Distribution ()
1. Introduction
The idea of the average F test was first introduced to the literature by [1] as a means of testing asset pricing theories in linear factor models. Recently [2] developed the idea further by focusing on the average pricing error, extending the multivariate F test of [3]. They show that the average F test can be applied to thousands of individual stocks rather than a smaller number of portfolios and thus does not suffer from the information loss or the data snooping biases. In addition, the test is robust to ellipticity. More importantly, [2] demonstrate that the power of average F test continues to increase as the number of stocks increases.
One drawback of the average F test is that [2] did not provide the closed form solution for the average F density function. Despite the fact that the average F statistic has been used in other areas of econometrics, e.g., [4] in the study of structural breaks of unknown timing in regression models, the functional form of the average F distribution remains unknown.
In this study we propose a few analytical developments for the average F distribution. Although the complete functional form is not provided, our results might be useful toward further research in the future.
2. Definition of the Average F Distribution
A testable version of linear factor models is
(1)
where 
is a 
vector of excess returns for 
assets and 
is a 
vector of factor portfolio returns, 
is a vector of intercepts, 
is an 
matrix of factor sensitivities, and
is a 
vector of idiosyncratic errorswhose covariance matrix is
. For the null hypothesis 
tested against the alternative hypothesis
, the average 
-test statistic is defined as
(2)
where
and 
and 
are the maximum likelihood estimators of 
and
, respectively. Under the classical assumption that asset returns are multivariate normal conditional on factors, the average F statistic is distributed as
(3)
where 
is a 
statistic with 1 degree of freedom in the numerator and 
degrees of freedom in the denominator.
3. Characteristic Function of the Average F Distribution
The distribution function of the average 
statistic is unknown. Note that all 
-distributions in Equation (3) have the same degrees of freedom, and 
is thus distributed as the sample mean of 
independent and identically distributed 
distributions. Let 
be a variable distributed as
, where
, and denote its probability density function as
. Then the characteristic function of the 
distribution can be derived as follows
(4)
Let 
and
, then
(5)
where 
is the gamma function, 
is the imaginary number, and 
is Tricomi’s confluent hypergeometric function. Equation (5) was first formulated by [5]. Tricomi’s confluent hypergeometric function is
(6)
where 
is Kummer’s confluent hypergeometric function which is defined as
(7)
See [6] for a detailed explanation of various types of hypergeometric functions and their applications to economic theory.
If b in Equation (7) is a non-positive integer, 
and thus
is not defined. Note that 
is a positive integer as it represents the degrees of freedom in the denominator of the 
distribution; thus, we need
and 
in Equation (6) to be positive integers. However, since n is a positive integer, both
and 
cannot be kept to be positive integers. More generallywhen 
we have a definition referred to as the “logarithmic case” alternative to Tricomi’s confluent hypergeometric function in (6). See [6] and [7] (Vol. 1, pp. 260-262 and Vol. 2, p. 9) for discussions on the logarithmic case.
Let 
be defined as the characteristic function of the 
independent 
variable. Then, the characteristic function of 
is
(8)
where 
is defined in (5). Therefore, the density function of the average F statistic
, 
, under the null hypothesis is obtained by the following;
(9)
where y is a variable distributed as the average of the 
different 
distributions
This mean of F-distributions can be used when the variance-covariance matrix 
is a diagonal matrix.
4. The Exact Distribution of Average F Test for Small N
When
, we have 
Using the result that 
is the square of
, i.e., a 
distribution with 
degrees of freedom, we see that the 
of 
is given by
, letting
,
(10)
where the 
of 
can be found in [8].
To find the 
of 
when
, 
, we proceed as follows. Let 
be the associated random variable, and 
and 
be the two independent 
variables. Then we have
Therefore 
where 
is given by Equation (10). More generally, by induction, it follows that
(11)
Although it is hard to make much progress with Equation (11) in obtaining closed form solutions, we note the following. From known moments of the 
distribution, it is possible to calculate the moments of 
for any
, where they exist.
Proposition 1. The moments of S exist for
.
Proof. Let 
Then 
so that the highest order term, for any
, is 
Now from Equation (10),
and thus 
which exists if 
Proposition 2 For
, 
can be represented as a scale Beta type II function.
Proof. For 
given by Equation (10), let
Then 
and simple change of variable shows that 
is a Beta 
random variable.
Since Proposition 2 establishes that 
is a scaled Beta, we now have a representation of 
Denoting 
to reflect the dependence on
, it follows from Proposition 2 that
(12)
where 
denotes a type II beta with parameters 
and 
and the 
outside
reflects the scale factors. Thus Equation (12) establishes that 
can be represented as a linear combination of Beta type II distributions.
The literature on density functions of linear combination of Beta distributions is rather sparse. [9] present expressions for linear combinations of Beta distributions when
. Thus using their results we can arrive at an expression for 
which is complex and depends upon hypergeometric functions. Extensions for 
do not appear to be derived as yet.
5. Conclusion
We provide some developments on the average F test distribution. Although simulation of the statistic is straightforward, an understanding of the functional form is invaluable in terms of appreciation of the properties of the test statistic. We leave a full solution of the problem for future study.