Some Exact Results for an Asset Pricing Test Based on the Average F Distribution

We provide some exact results for an asset pricing theory test statistic based on the average F distribution. This test is preferred to existing procedures because it deals with the case of more assets than data points. The case mentioned is the practical one that asset managers routinely have to consider.


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.

Definition of the Average F Distribution
A testable version of linear factor models is (1) where is a  vector of excess returns for where and  and  are the maximum likelihood estimators of  and  , respectively.Under the classical assump- tion hat asset returns are multivariate normal conditional on factors, the average F statistic is distributed as where degrees of freedom in the denominator.

 
Ψ  is Tricomi's confluent hyper- geometric functi quation (5) was first formulated by [5].Tricomi's confluent hypergeometric function is where is Kummer's confluent hyper See [6] for a detailed explanation of variou hy uation ( 7) is a non-positiv s ty e in (7) pes of teger, pergeometric functions and their applications to economic theory.
If b in Eq 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   , j t n  be defined as the characteristic where     is defined in (5).Therefore, the density functio the average F statistic S , n of   pdf y , under the null hypothesis is obtained by the ll fo owing; where y is a variable distributed as the average of the N different  

1, F n distributions This mean of F-distrib tions can d when the variance-covariance matrix
Σ is a diagonal matrix.

t Distribution of Average
ube use

The Exac
where the of pdf   S when can be found in [8].Y and Y be the two independent lows e asso iated ran is given by Equation (10).More ge- o make much progress with  uation (11) in obtaining closed form solutions, we note the following.From known moments of the t distribution, it is possible to calculate the moments of S for any N , where they exist.
Proposition 1.The moments  so that the highest order term, for any , is i .
, and thus    The literature on density functions of linear c nation of Beta distributions is rather sparse.[9] present expressions fo hen 2 N  .Thus using their results we can arrive at an expression for   2 pdf y which is complex and de- pends upon hypergeometric functions.Extensions for 2 N  t appear to be derived as yet.

Conclusion e developments on the aver do no
We provide som age F test dissimulation of the statistic is straightforing of the functional form is invaluable We leave a full solution of the problem for future study.

[ 1 ]
J. Affleck-Graves and B. McDonald, "Multivariate Tests of Asset Prici wer of Alternative Statistics," Jo uantitative Analy-tribution.Although ward, an understand in terms of appreciation of the properties of the test statistic.
Note that all F -distributions in quation (3) have the same degrees f freedom, and S is thus distributed as the sample mean of N indepe dent and identically distributed F distributi s.Let