Bootstrap Unit Root Tests For Heavy Tailed Observations

This article proposes a hybrid bootstrap approach to approximate the augmented Dickey-Fuller test by perturbing both the residual sequence and the minimand of the objective function. Since innovations can be dependent, this allows the inclusion of conditional heteroscedasticity models. The new bootstrap method is also applied to least absolute deviation-based unit root test statistics, which are efficient in handling heavy-tailed time-series data. The asymptotic distributions of resulting bootstrap tests are presented, and Monte Carlo studies demonstrate the usefulness of the proposed tests.

The subject of numerical methods in finance has recently emerged as a new discipline at the intersection of probability theory, finance, and numerical analysis. The methods employed bridge the gap between financial theory and computational practice, and provide solutions for complex problems that are difficult to solve by traditional analytical methods. Although numerical methods in finance have been studied intensively in recent years, many theoretical and practical financial aspects have yet to be explored. This volume presents current research and survey articles focusing on various numerical methods in finance. The book is designed for the academic community and will also serve professional investors.

Heavy tailed data appears frequently in social science, internet traffic, insurance and finance. Statistical inference has been studied for many years, which includes recent bias-reduction estimation for tail index and high quantiles with applications in risk management, empirical likelihood based interval estimation for tail index and high quantiles, hypothesis tests for heavy tails, the choice of sample fraction in tail index and high quantile inference. These results for independent data, dependent data, linear time series and nonlinear time series are scattered in different statistics journals. Inference for Heavy-Tailed Data Analysis puts these methods into a single place with a clear picture on learning and using these techniques. Contains comprehensive coverage of new techniques of heavy tailed data analysis Provides examples of heavy tailed data and its uses Brings together, in a single place, a clear picture on learning and using these techniques

This article presents a family of new tests for unit roots based on M-estimators. Their robustness makes them very appealing when working with distributions that have infinite variance or heavy tails. These tests are completely automatic regardless of the complex distributions of this kind of estimators because the critical values are approximated using bootstrap, no additional parameter has to be estimated and the results obtained are very good in small samples. An exhaustive Monte Carlo study shows the high performance of these tests compared with others proposed in the literature when the variance is infinite.

We evaluate the impact of heavy-tailed innovations on some popular unit root tests. In the context of a near-integrated series driven by linear process shocks, we demonstrate that their limiting distributions are altered under infinite variance vis-à-vis finite variance. Reassuringly, however, simulation results suggest that the impact of heavy-tailed innovations on these tests is relatively small. We use the framework of Amsler and Schmidt ([Amsler C, 2012]) whereby the innovations have local-to-finite variances being generated as a linear combination of draws from a thin-tailed distribution (in the domain of attraction of the Gaussian distribution) and a heavy-tailed distribution (in the normal domain of attraction of a stable law). We also explore the properties of augmented Dickey-Fuller tests that employ Eicker-White standard errors, demonstrating that these can yield significant power improvements over conventional tests.

This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties, but residual-based tests appear more powerful. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. In this article, we study and compare the properties of several bootstrap unit-root tests recently proposed in the literature. The tests are Dickey Fuller (DF) or Augmented DF, based either on residuals from an auto-regression and the use of the block bootstrap or on first-differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences vs. residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties, but residual-based tests appear more powerful.

Heavy tails –extreme events or values more common than expected –emerge everywhere: the economy, natural events, and social and information networks are just a few examples. Yet after decades of progress, they are still treated as mysterious, surprising, and even controversial, primarily because the necessary mathematical models and statistical methods are not widely known. This book, for the first time, provides a rigorous introduction to heavy-tailed distributions accessible to anyone who knows elementary probability. It tackles and tames the zoo of terminology for models and properties, demystifying topics such as the generalized central limit theorem and regular variation. It tracks the natural emergence of heavy-tailed distributions from a wide variety of general processes, building intuition. And it reveals the controversy surrounding heavy tails to be the result of flawed statistics, then equips readers to identify and estimate with confidence. Over 100 exercises complete this engaging package.