ANCOVA: A global test based on a robust measure of location or quantiles when there is curvature

For two independent groups, let $M_j(x)$ be some conditional measure of location for the $j$th group associated with some random variable $Y$, given that some covariate $X=x$. When $M_j(x)$ is a robust measure of location, or even some conditional quantile of $Y$, given $X$, methods have been proposed and studied that are aimed at testing $H_0$: $M_1(x)=M_2(x)$ that deal with curvature in a flexible manner. In addition, methods have been studied where the goal is to control the probability of one or more Type I errors when testing $H_0$ for each $x \in \{x_1, \ldots, x_p\}$. This paper suggests a method for testing the global hypothesis $H_0$: $M_1(x)=M_2(x)$ for $\forall x \in \{x_1, \ldots, x_p\}$ when using a robust or quantile location estimator. An obvious advantage of testing $p$ hypotheses, rather than the global hypothesis, is that it can provide information about where regression lines differ and by how much. But the paper summarizes three general reasons to suspect that testing the global hypothesis can have more power. 2 Data from the Well Elderly 2 study illustrate that testing the global hypothesis can make a practical difference.Comment: 23 pp 2 Figure

Value at risk models in finance

The main objective of this paper is to survey and evaluate the performance of the most popular univariate VaR methodologies, paying particular attention to their underlying assumptions and to their logical flaws. In the process, we show that the Historical Simulation method and its variants can be considered as special cases of the CAViaR framework developed by Engle and Manganelli (1999). We also provide two original methodological contributions. The first one introduces the extreme value theory into the CAViaR model. The second one concerns the estimation of the expected shortfall (the expected loss, given that the return exceeded the VaR) using a regression technique. The performance of the models surveyed in the paper is evaluated using a Monte Carlo simulation. We generate data using GARCH processes with different distributions and compare the estimated quantiles to the true ones. The results show that CAViaR models perform best with heavy-tailed DGP. JEL Classification: C22, G22CAViaR, extreme value theory, Value at Risk

Pseudo-nonstationarity in the scaling exponents of finite-interval time series

The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series

Time Variation in the Tail Behaviour of Bund Futures Returns

The literature on the tail behaviour of asset prices focuses mainly on the foreign exchange and stock markets, with only a few papers dealing with bonds or bond futures. The present paper addresses this omission. We focus on three questions: (i) Are heavy tails a relevant feature of the distribution of BUND futures returns? (ii) Is the tail behaviour constant over time? (iii) If it is not, can we use the tail index as an indicator for financial market risk and does it add value in addition to classical indicators? The answers to these questions are (i) yes, (ii) no, and (iii) yes. We find significant heaviness of the tails of the Bund future returns. The tail index is on average around 3, implying the nonexistence of the forth moments. With the aid of a recently developed test for changes in the tail behaviour we identify several breaks in the degree of heaviness of the return tails. Interestingly, the tails of the return distribution do not move in parallel to realised volatility. This suggests that the tails of futures returns contain information for risk management that complements those gained from more standard statistical measures. -- Die Literatur über Extreme der Renditeverteilung hat sich bisher überwiegend mit Wechselkursen und Aktienkursen befasst. Die Kurse von Rentenwerten oder Terminkontrakten auf Rentenwerte haben hingegen bisher kaum Beach- tung erfahren. Das vorliegende Arbeitspapier versucht diese Lücke zu schließen. Unser Augenmerk gilt dabei insbesondere drei Fragen: (i) Haben die Ren- diteverteilungen von Terminkontrakten auf Bundeswertpapiere "fat tails"? (ii) Ist die Wahrscheinlichkeit extremer Kursbewegungen im Zeitablauf kon- stant? (iii) Kann ein Tail-Index Informationen Äuber den Grad von Marktun- sicherheit liefern, die klassische Indikatoren wie die VolatilitÄat nicht liefern können? Die Antworten zu diesen drei Fragen sind (i) ja, (ii) nein und (iii) ja. Wir finden ein signifikantes "fat tails" Phänomen in der Renditeverteilung von BUND Future Kontrakten. Ein Tail-Index von circa 3 impliziert, dass das vierte und alle höheren Momente der Verteilung nicht existieren. Mit Hilfe kürzlich entwickelter Tests finden wir Brüche der Tail-Stärke der Ren- diteverteilungen. Interessanterweise bewegt sich der Tail-Index nicht immer in die gleiche Richtung wie die Volatilität. Dies lässt vermuten, dass die Betrachtung der Tails dem Risikomanagement Informationen liefert, die mit herkömmlichen Verfahren nicht gewonnen werden kÄonnen.

Asymptotic Properties of the Partition Function and Applications in Tail Index Inference of Heavy-Tailed Data

The so-called partition function is a sample moment statistic based on blocks of data and it is often used in the context of multifractal processes. It will be shown that its behaviour is strongly influenced by the tail of the distribution underlying the data either in i.i.d. and weakly dependent cases. These results will be exploited to develop graphical and estimation methods for the tail index of a distribution. The performance of the tools proposed is analyzed and compared with other methods by means of simulations and examples.Comment: 31 pages, 5 figure