Asymptotic properties of duration-based VaR backtests

A Population Gini coefficient for the geometric distribution. Derivation from Gini’s own definition

In Appendix A, we show a way of deriving the population Gini coefficient for the geometric distribution directly from Gini’s own definition. Our way is an alternative to the method presented by Dorfmann [20], who, by means of Stieltjes integrals, derived the general representation of the Gini coefficient with the probability function. From this representation, as a special case, he derived the Gini value for the geometric distribution. We show that, for the geometric distribution, the same result may be achieved directly from Gini’s definition, through simple algebraic transformations. It omits the representation of the Gini coefficient with the probability function, attainable through the Stjeltjes or Lebesgue integrals.

Gini’s idea to quantify inequalities is based on differences between pairs of observations. Let us, therefore, assume that D ( 1 ) and D ( 2 ) is a pair of observations coming from the distribution of the variable 𝐷. Then Gini’s definition of the inequality coefficient is half the ratio of the average absolute difference between such pairs of observations to the mean of 𝐷, i.e.

Since

the average absolute difference between pairs of observations may be expressed as^[13]

We, therefore, proceed to evaluate the expectation of the minimum E ⁢ ( min ⁡ { D ( 1 ) , D ( 2 ) } ) . Let us denote by F ⁢ ( d ) the cumulative probability function of the variable 𝐷 at 𝑑, F ⁢ ( d ) = P ⁢ ( D ≤ d ) . Then

For a discrete distribution, where d = 1 , 2 , … , the probability of taking the single value 𝑑 is

We now apply the probability function of the geometric distribution, where, for simplicity of notation, we use the parameter q = 1 - p instead of 𝑝, i.e. F ⁢ ( d ) = 1 - q d ,

Using probabilities (A.4), we have that the unknown expectation of the minimum can be expressed as

To evaluate the sum ∑ d = 1 ∞ d ⁢ q 2 ⁢ d , we use the auxiliary expression A = ∑ d = 1 ∞ q 2 ⁢ d . Since a power series can be differentiated term by term inside its circle of convergence, we obtain

Changing the sequence of operations and evaluating the sum of the series first, we have

The comparison of (A.5) and (A.6) gives

hence the unknown expectation of the minimum is

Substituting this result to equation (A.3) and using the fact that E ⁢ ( D ) = 1 1 - q yields

Therefore, from definition (A.1), the Gini coefficient for the geometric distribution is

Using the parameter 𝑝, this is equivalent to

which was to be shown.

B Partial derivatives of loglikelihood function

In Appendix B, we treat the issue of validity of distributions in (3.1), (3.2), (3.4) and (3.5) in the geometric-VaR test. The sufficient condition for validity of these distributions is that the Fisher information matrix is diagonal, which requires that partial derivatives of relevant loglikelihood functions are zero. These partial derivatives for the loglikelihood log ⁡ L ⁢ ( { D 1 ⁢ ( p ) , … , D N ⁢ ( p ) } | a , b , c ) are derived below.

The loglikelihood function that underlies the geometric-VaR test has the form

hence the second derivatives are

where θ 1 , θ 2 ∈ { a , b , c } and f ⁢ ( D i ⁢ ( p ) ) , S ⁢ ( D i ⁢ ( p ) ) , i = 1 , … , N , are defined by equations (2.2), (2.3) and (2.5).

For simplicity, let us denote

The components of (B.1) are derived based on the first derivatives of f ⁢ ( D i ) , S ⁢ ( D i ) , i = 1 , … , N . For f ⁢ ( D i ) , the first-order derivatives are

In general, these derivatives are not equal to zero. Hence, we proceed to find the approximation of the Fisher information matrix. With the partial derivatives, it can be expressed as

For evaluating the expectation in formula (B.2), we use the Monte Carlo study. Based on the realistic simulations designed in line with the experiments described in Section 4.1, we get the following result:

This result shows that, under the null, the partial derivatives are close to zero. This confirms the practical feasibility of the asymptotic distributions presented in Section 3.1. This conclusion is also in line with the MC size results presented in Section 4.1, where, under the realistic setting, we evidenced via simulations the good fit between the empirical test size in finite samples and the nominal size coming from the asymptotic mixture distributions. These arguments justify the use of the mixture distributions in practical applications.