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.

A testable version of linear factor models is

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

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

where is a statistic with 1 degree of freedom in the numerator and degrees of freedom in the denominator.

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

Let and, then

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

where is Kummer’s confluent hypergeometric function which is defined as

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

where is defined in (5). Therefore, the density function of the average F statistic, , under the null hypothesis is obtained by the following;

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.

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,

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

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

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.

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.