چهارچوب کاهش خود بخود دنباله رفتار شرطی و نتایج بر روی حکومت اسپرد بازدهی اوراق قرضه

Previous evidence in empirical finance indicates the potential usefulness of modeling time variation particularly in the tails of speculative return distributions. Based on results from extreme value theory, the present paper proposes a fixed changepoint Pareto-type autoregressive conditional tail (ARCT) model. Regression-based parameter estimation of the unobservable time-varying tail index is carried out via classical Kalman filtering. A model application highlights the tail index dynamics for daily changes in Government bond yield spreads between the U.S. Dollar and the Euro zone.

The forecasting of large price movements in economics and finance is surely a difficult, while at the same time, central issue in financial management. This explains why the prediction of risk—frequently associated with the prediction of return volatility—has been the subject of a vast number of papers within the empirical finance literature. Despite huge advances in the area during the last two decades—including the modeling of generalized autoregressive conditional heteroskedasticity (GARCH)—several authors recently put some doubt on the applicability of available volatility models for risk management and for extreme price movements in particular. As such, Shephard (1996, p. 21) notes that “GARCH models cannot deal with the extremely large movements in financial markets, even though they are good models of changing variance.” Danielsson and Morimoto (2000, p. 15) suggest that “the wild swings observed in the GARCH VaR [Value-at-Risk] predictions are more of an artifact of the GARCH model rather than the underlying data.” Engle (2000, p. 2) notes “it is not clear whether the tails have the same dynamic behavior as the rest of the distribution as would be assumed by GARCH style models.” Considering higher conditional moments, Rockinger and Jondeau (2002, p. 140) conclude that it seems that “there is little evidence that skewness and kurtosis are dependent on past returns. One possible reason for this finding is that these moments are driven by extreme realizations that occur only infrequently.” At the same time, empirical findings suggest the modeling of time variation in the conditional tails of return distributions. Quintos, Fan, and Phillips (2001, p. 634) summarize earlier work by noting that “there is a consensus from past empirical research that the tail behavior of certain financial series are time varying.” Along with the authors mentioned above, Christoffersen and Diebold (2000, p. 21) point out that “it seems […] that all models miss the really big movements […], and ultimately the really big movements are the most important for risk management. This suggests the desirability of directly modeling the extreme tails of return densities […].” Given the above statements among others, the present paper aims at going in the indicated direction by proposing a model based on autoregressive conditional tail (ARCT) behavior. The route taken follows the one directed by extreme value theory, i.e., focus is put on the tail of the conditional return distribution function. Instead of modeling conditional return variance, taking volatility clustering into account or modeling conditional quantiles as was done previously1, we suggest modeling a time-varying conditional tail index. The so-called tail index is known to be a key parameter which characterizes tail behavior in extreme value theory. As supported by theoretical arguments and empirical evidence, the assumption is made that the time-varying distributions are fat tailed throughout; that is, they all belong to the maximum domain of attraction of the Fréchet-type extreme value distribution. Consequently, a Pareto-type ARCT model results. Introducing fixed changepoints which are evenly spread through time, model estimation is based on a regression equation which is derived from the empirical distribution function. Although other estimation techniques found wider application in extreme value statistics, the static regression approach is known for long; recently, related statistical results were outlined, for example, in Datta and McCormick (1998). Van den Goorbergh (1999) provides a financial application. We choose the approach for three reasons. First, it follows immediately from the assumption of fat-tailedness; second, it has a simple graphical implication; and lastly, it allows us to introduce a dynamic state space formulation for the unobservable time-varying tail index. Parameter estimation can then be carried out via standard Kalman filtering. By now, no framework was proposed which includes a conditional time-series model for the dynamics of the tail index. While evidence on time-varying volatility is vast, approaches to time-varying tails were considered in a few financial studies so far. Tsay (1999) applies the time-varying inhomogeneous Poisson model by Smith (1989) in a study of S&P 500 index returns and reports a tendency for increasingly fat tails during the 1962–1997 period. Smith (2000) proposes a random changepoint model with time-varying parameters for S&P 500 index returns. Quintos et al. (2001) derive a statistical test of tail index stationarity based on a rolling series of so-called Hill estimators. Their analysis documents structural breaks in the tail index of emerging stock index returns during the 1997 Asian crisis. A related study is by Galbraith and Zernov (2004); using daily U.S. stock market returns during the period 1960–2002, they document increased tail fatness for the subperiod starting in the mid-1980s. In contrast to the other studies, Chavez-Demoulin, Davison, and McNeil (2003) do not model time-varying tails but use extreme value theory in a setting that allows for time variation in the arrival intensity and the distribution of excesses of a given high threshold. The use of internal risk models according to the Basel Capital Adequacy Directive II requires competitive models which ensure safety of the financial system while at the same time avoid costly over-allocation of capital. Hence, the second part of the paper is devoted to a first application of ARCT behavior to financial time series. In particular, we study changes in Government bond yield spreads. Such changes are relevant to financial risk management as they have a direct impact on the pricing of foreign exchange currency forward rate agreements (FRAs) and currency options (see, e.g., Hull, 2000 for a survey on these derivative instruments). Even mature Government bond markets can be subject to spread changes of substantial magnitude. We study daily changes in yield spreads between U.S. and German Government mid-maturity swap rates from October 1, 1997 to June 30, 2003. The sample period contains several crisis periods, such as the October 1997 Asian crisis, the August–September 1998 Russian crisis,2 as well as September 11, 2001. In addition, the introduction of the Euro in January 1999 is contained. As Germany—given European Monetary Union convergence—started to represent market conditions for the Euro-Zone well before mid-1997 (see, e.g., Stracca, 1999), the results do in fact represent an analysis of yield spreads for the U.S. Dollar versus the Euro-Zone. We analyze in-sample tail index dynamics and discuss tail index prediction. The findings indicate substantial tail index variation without trend. The dynamics are consistent with stationarity, tail index predictability under mean reversion and provide some suggestive evidence of cyclic behavior. The paper precedes as follows. Section 2 proposes a model of ARCT behavior and a regression-based approach to model estimation is outlined here. The empirical application for Government bond yield spreads is given in Section 3. Section 4 concludes.

The modeling of potential extreme market events can have an important impact on the calculation of minimum capital risk requirements (see, e.g., Clare et al., 2002). It appears that approaches to time-varying extreme financial risk may provide a potentially useful tool to extend our understanding of market stress and may also contribute to the set of available risk prediction methods. This paper outlines a straightforward model of ARCT behavior. Estimation and application to financial risk modeling yields results on the tail index dynamics of yield spreads. Future work in the area may be devoted to generalize and robustify the model and its estimation as well as to investigate its applicability and predictive performance.