Abstract: The extremal index (θ) is the key parameter for extending extreme value theory results from i.i.d. to stationary sequences. One important property of this parameter is that its inverse determines the degree of clustering in the extremes. This article introduces a novel interpretation of the extremal index as a limiting probability characterized by two Poisson processes and a simple family of estimators derived from this new characterization. Unlike most estimators for θ in the literature, this estimator is consistent, asymptotically normal and very stable across partitions of the sample. Further, we show in an extensive simulation study that this estimator outperforms in finite samples the logs, blocks and runs estimation methods. Finally, we apply this new estimator to test for clustering of extremes in monthly time series of unemployment growth and inflation rates and conclude that runs of large unemployment rates are more prolonged than periods of high inflation.
Abstract: Estimation of GARCH models can be simplified by augmenting quasi-maximum likelihood (QML) estimation with variance targeting, which reduces the degree of parameterization and facilitates estimation. We compare the two approaches and investigate, via simulations, how non-normality features of the return distribution affect the quality of estimation of the volatility equation and corresponding value-at-risk predictions. We find that most GARCH coefficients and associated predictions are more precisely estimated when no variance targeting is employed. Bias properties are exacerbated for a heavier-tailed distribution of standardized returns, while the distributional asymmetry has little or moderate impact, these phenomena tending to be more pronounced under variance targeting. Some effects further intensify if one uses ML based on a leptokurtic distribution in place of normal QML. The sample size has also a more favorable effect on estimation precision when no variance targeting is used. Thus, if computational costs are not prohibitive, variance targeting should probably be avoided.
Abstract: This paper introduces a complement statistical test for distinguishing between the predictive accuracy of two sets of forecasts. We propose a non-parametric test founded upon the principles of the Kolmogorov-Smirnov (KS) test, referred to as the KS Predictive Accuracy (KSPA) test. The KSPA test is able to serve two distinct purposes. Initially, the test seeks to determine whether there exists a statistically significant difference between the distribution of forecast errors, and secondly it exploits the principles of stochastic dominance to determine whether the forecasts with the lower error also reports a stochastically smaller error than forecasts from a competing model, and thereby enables distinguishing between the predictive accuracy of forecasts. We perform a simulation study for the size and power of the proposed test and report the results for different noise distributions, sample sizes and forecasting horizons. The simulation results indicate that the KSPA test is correctly sized, and robust in the face of varying forecasting horizons and sample sizes along with significant accuracy gains reported especially in the case of small sample sizes. Real world applications are also considered to illustrate the applicability of the proposed KSPA test in practice.
Abstract: We give a simple explicit formula for turnover reduction when a large number of alphas are traded on the same execution platform and trades are crossed internally. We model turnover reduction via alpha correlations. Then, for a large number of alphas, turnover reduction is related to the largest eigenvalue and the corresponding eigenvector of the alpha correlation matrix.
Abstract: This paper studies the asymptotic normality for the kernel deconvolution estimator when the noise distribution is logarithmic chi-square; both identical and independently distributed observations and strong mixing observations are considered. The dependent case of the result is applied to obtain the pointwise asymptotic distribution of the deconvolution volatility density estimator in discrete-time stochastic volatility models.
Abstract: Several graphical methods for testing univariate composite normality from an i.i.d. sample are presented. They are endowed with correct simultaneous error bounds and yield size-correct tests. As all are based on the empirical CDF, they are also consistent for all alternatives. For one test, called the modified stabilized probability test, or MSP, a highly simplified computational method is derived, which delivers the test statistic and also a highly accurate p-value approximation, essentially instantaneously. The MSP test is demonstrated to have higher power against asymmetric alternatives than the well-known and powerful Jarque-Bera test. A further size-correct test, based on combining two test statistics, is shown to have yet higher power. The methodology employed is fully general and can be applied to any i.i.d. univariate continuous distribution setting.