The Volatility of Realized Volatility

In recent years, with the availability of high-frequency financial market data modeling realized volatility has become a new and innovative research direction. The construction of “observable” or realized volatility series from intra-day transaction data and the use of standard time-series techniques has lead to promising strategies for modeling and predicting (daily) volatility. In this article, we show that the residuals of commonly used time-series models for realized volatility and logarithmic realized variance exhibit non-Gaussianity and volatility clustering. We propose extensions to explicitly account for these properties and assess their relevance for modeling and forecasting realized volatility. In an empirical application for S&P 500 index futures we show that allowing for time-varying volatility of realized volatility and logarithmic realized variance substantially improves the fit as well as predictive performance. Furthermore, the distributional assumption for residuals plays a crucial role in density forecasting.     

Cointegrated systems II

The three invited papers in this special issue of Econometric Reviews on "Cointegrated Systems II" complement the previous special issue of the journal. The paper by Eric Zivot and Peter Phillips provides a comprehensive Bayesian analysis of trend determination in economic time series. Two interesting comments on some aspects of current research involving cointegration and the modelling of dynamic economic relationships are provided by Clive Granger and Denzil Fiebig.     

A Bootstrap Stationarity Test for Predictive Regression Invalidity

In order for predictive regression tests to deliver asymptotically valid inference, account has to be taken of the degree of persistence of the predictors under test. There is also a maintained assumption that any predictability in the variable of interest is purely attributable to the predictors under test. Violation of this assumption by the omission of relevant persistent predictors renders the predictive regression invalid, and potentially also spurious, as both the finite sample and asymptotic size of the predictability tests can be significantly inflated. In response, we propose a predictive regression invalidity test based on a stationarity testing approach. To allow for an unknown degree of persistence in the putative predictors, and for heteroscedasticity in the data, we implement our proposed test using a fixed regressor wild bootstrap procedure. We demonstrate the asymptotic validity of the proposed bootstrap test by proving that the limit distribution of the bootstrap statistic, conditional on the data, is the same as the limit null distribution of the statistic computed on the original data, conditional on the predictor. This corrects a long-standing error in the bootstrap literature whereby it is incorrectly argued that for strongly persistent regressors and test statistics akin to ours the validity of the fixed regressor bootstrap obtains through equivalence to an unconditional limit distribution. Our bootstrap results are therefore of interest in their own right and are likely to have applications beyond the present context. An illustration is given by reexamining the results relating to U.S. stock returns data in Campbell and Yogo (2006 Campbell, J. Y. and Yogo, M. (2006), “Efficient Tests of Stock Return Predictability,” Journal of Financial Economics, 81, 27–60.[Crossref], [Web of Science ®] , [Google Scholar]). Supplementary materials for this article are available online.     

中国居民收入分布专题实证研究——居民收入分布变迁测度及其影响因素分解

基于核密度估计方法,给出居民收入分布变迁测度及其影响因素分解的相对收入分布方法。使用中国健康和营养调查的微观居民收入数据,实证分析中国居民收入分布及其变迁的特征、过程及其影响因素。实证结果显示:中国居民收入分布变迁的速度与经济体制改革进程相对应;经济增长因素是引起居民收入分布变迁的主要因素,在提高总体居民收入水平的同时加剧了居民收入分布的极化程度;收入分配因素对居民收入分布变迁的作用相对较弱,但有一定的减贫或缩小居民收入差距作用。     

公平分配与居民福利——收入差距对中国居民恩格尔系数的影响分析

居民收入水平不同,恩格尔系数也不同,因此收入分配对社会总体恩格尔系数的影响是不确定的。基于马斯洛层次需求理论证明边际食品消费倾向递减,并且当消费函数为线性或近似线性时,收入差距扩大会降低恩格尔系数。基于省际面板数据的实证分析表明,中国居民边际食品消费倾向递减且消费函数为(近似)线性,居民收入差距扩大降低了居民恩格尔系数(基尼系数升高对恩格尔系数下降的贡献率为22%),从而出现"分配越不平等,居民总体生活水平越高"的矛盾现象。因此在使用总体指标评价居民生活水平时,收入分配指标必须得到同等的重视;在经济发展中,公平和效率必须兼顾才能实现居民福利最大化。     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

Discriminant Analysis for the von Mises-Fisher Distribution

The von Mises-Fisher distribution is widely used for modeling directional data. In this article, we derive the discriminant rules based on this distribution to assign objects into pre-existing classes. We determine a distance between two von Mises-Fisher populations and we calculate estimates of the misclassification probabilities. We also analyze the behavior of the distance between two von Mises-Fisher populations and of the estimates of the misclassification probabilities when we modify the parameters of the populations or the samples size or the dimension of the sphere. Finally, we present an example with real spherical data available in the literature.     

On the use of curve fitting to model the error of the Cornish-Fisher expansion of the pearson type-VI distribution

The Cornish-Fisher expansion of the Pearson type VI distribution is known to be reasonably accurate when both degrees of freedom are relatively large (say greater than or equal to 5). However, when either or both degrees of freedom are less than 5, the accuracy of the computed percentage point begins to suffer; in some cases severely. To correct for this, the error surface in the degrees of freedom plane is modeled by least squares curve fitting for selected levels of tail probability (.025, .05, and .10) which can be used to adjust the percentage point obtained from the usual Cornish-Fisher expansion. This adjustment procedure produces a computing algorithm that computes percentage points of the Pearson type VI distribution at the above probability levels, accurate to at least + 1 in 3 digits in approximately 11 milliseconds per subroutine call on an IBM 370/145. This adjusted routine is valid for both degrees of freedom greater than or equal to 1.     

The portfolio choice problem: comparison of certainty equivalence and optimal Bayes portfolios

For the portfolio problem with unknown parameter values, we compare the conventional certainty equivalence portfolio choice with the optimal Bayes portfolio. In the important single risky asset case a diffuse Bayes rule leads to portfolios that differ significantly from those suggested by a certainty equivalence rule which we show are inadmissible relative to a quadratic utility function for the range of parameters we consider. These results are invariant to arbitrary changes in the utility function parameters. We illustrate the results using a simple mutual fund example.     

Parameter measures of skewness

Several different measures of skewness are commonly used in place of γ₁, the third central moment divided by the cube of the standard deviation. The numerical values of these measures are compared in this paper for members of the gamma, lognormal or Weibull family of distributions and shown to vary considerably in most cases even when skewness and kurtosis are moderate.