Descriptive statistics of gross 1220 in-sample returns and 757 out-of-sample returns (in brackets)

In the estimation of volatilities from the in-sample returns EViews 4.0 and an ADModel-Builder- based program of Otter Research Ltd for DVEC-GARCH modelling were helpful; however, in the majority of computations the utilization of MS ® Excel 2000 dominated. Besides the traditional option of the normal distribution, in specifying the distribution of the innovative term in GARCH and EGARCH equations the generalized error distribution (GED) and the Student’s t-distribution were also experimented. This said, the innovations in MGARCH’s specifications were opted for to comply with the conservative choice of the normal distribution. The constructed GARCH models possessed desirable econometric properties, with the exception of a few cases in which location υ•-term was found on the

verge of significance. As suggestive of the graphs presented in Figure 1, the models captured the dynamics of volatility after their own fashion, and yet some general patterns (at least pairwise) are traceable.

Subsequently, it was possible to carry out the procedure outlined above, i. e. to construct the standardized returns zΠ [•], to estimate the parameters of the distributions under consideration and compute

the respective α-quantile, and eventually to piece the components qα and σΠ [•] together so as to form

5

For a clarification of possible misunderstanding as to the specification, the representation of the non-central t-distribution is to be found e. g. in Goorbergh (1999), and that of the generalized Pareto distribution in Embrechts et al. (1997) or McNeil (1999). 6

Kupiec’s test bases on the idea that the frequency of exceedances of an adequate model answers to the significance level α:. In this case, statistics LR = –2lg[(1 – α)n–x αx] + 2lg[(1 – x/n)n–x (x/n)x], where x is the number of exceedances over n forecasts, follows an asymptotical χ2(1) distribution. The absolute percentage exceedance measures the magnitude of exceedances over a given period of n predictions and is specified, for the purpose of this article, by the formula APE = Σ|εi|, where εi denotes the

Constant & Nonweighted EWMA

GARCH(1,1) (Student) EGARCH(1,1) (Student) UNIVARIATE MODELS OF VOLATILITY

Constant Nonweighted

EWMA DVEC-MGARCH(1,1) MULTIVARIATE MODELS OF VOLATILITY

Figure 1 In-sample volatilities estimates for (most of) the individual models

out-of-sample estimates of value at risk. The fitting of distributions to zΠ was facilitated chiefly through

Xtremes 3.01.7, 8

All in all, a collection of the 13 volatility models (of which 9 univariate and 4 multivariate) and 4 distributional forms was considered. The results for each model comprising the corresponding quantile q1–α for the distribution of zΠ, the average number of failures of the value at risk forecasts over 250 trading

days and the value of APE indicator are presented in Scheme 3.

Even though the diversity of the results is striking, they appear to be differentiated both by the volatility specification and by the distributional form. When inspecting the performance of the individual volatility specifications, one sees that their propensity to fail varies roughly and their predictive capability is influenced by the magnitude of the quantile. The possibly lower predictive capability of a volatility specification is generally compensated by the higher quantile number. Of the volatility models in focus, the best performance is recorded with the multivariate GARCH specification, and, furthermore and surprisingly, with the either model of constant volatility. The other models, which in fact are founded on the daily updating volatility scheme, seem to be comparatively proner to fail as to their capability to predict the loss occurred. Another ascertainment may be induced when comparing the performance of the single distributions. Save the generalized Pareto distribution, the t-distribution manifest the satisfactory eligibility to cope with heavy tails and its estimates are relatively satisfactory. As for the generalized Pareto distribution, its utility comes when losses exceed a pre-determined threshold and their occurrence is observed rare.9 At large, the use of the non-central t-distribution was not very successful.

7

The parameters (µ, σ2) of the normal distribution were estimated in an unbiased form and the degrees of freedom ν of the Student t-distribution were picked up so that the theoretical dispersion ν/(ν – 2) (of course, for ν > 2) might correspond to the actual dispersion of zΠ . In some cases the t-distribution proved itself unfit to the data (when the dispersion was less than 0) or ν was very large (in that event the t-distribution was virtually identical with the normal distribution). For the other two distributions the method of maximum likelihood was employed. Specifically, the parameters (ξ, ν, β) of the generalized Pareto distribution were received from the re-signed positive standardized returns zΠ [•], and the computed (1 – α)-quantile was converted back to the negative part of the number axis by adding a negative sign to it, becoming the α-quantile. The selection of thresholds was opted automatic and left to the program.

8

The application of the maximum likelihood method requires the data be independent and identically distributed. The iid property of the series was tested by the BDS test for independence as included in EViews 4.0. The test indicated that most series are prone to evince dependence. The outcome correspondent with independence was paradoxically received in the case of constant and nonweighted volatility models (both for the univariate and multivariate specification), and with DVEC-MGARCH(1, 1) specifi- cation. Then, the standardized returns of GARCH(1, 1) with normal innovations bordered on independence. The estimated parameters in the unfavourable cases may thus be liable to produce distorted results.

9

On no account is this meant to challenge the significance of the generalized Pareto distribution in modelling; however, one should beware that its foundations are in extreme value theory and their usage and interpretation should be adjusted accordingly.

MODEL OF VOLATILITY zΠ ∼ N(µ, σ

2_{) zΠ ∼ T(ν) zΠ ∼ nctT(µ, ξ, ν) zΠ ∼ GPD(ξ, ν, β)}