Edgeworth Corrections for Realized Volatility

The quality of the asymptotic normality of realized volatility can be poor if sampling does not occur at very high frequencies. In this article we consider an alternative approximation to the finite sample distribution of realized volatility based on Edgeworth expansions. In particular, we show how confidence intervals for integrated volatility can be constructed using these Edgeworth expansions. The Monte Carlo study we conduct shows that the intervals based on the Edgeworth corrections have improved properties relatively to the conventional intervals based on the normal approximation. Contrary to the bootstrap, the Edgeworth approach is an analytical approach that is easily implemented, without requiring any resampling of one's data. A comparison between the bootstrap and the Edgeworth expansion shows that the bootstrap outperforms the Edgeworth corrected intervals. Thus, if we are willing to incur in the additional computational cost involved in computing bootstrap intervals, these are preferred over the Edgeworth intervals. Nevertheless, if we are not willing to incur in this additional cost, our results suggest that Edgeworth corrected intervals should replace the conventional intervals based on the first order normal approximation.     

Monte carlo approximation to edgeworth expansions

Since the 1930s, empirical Edgeworth expansions have been employed to develop techniques for approximate, nonparametric statistical inference. The introduction of bootstrap methods has increased the potential usefulness of Edgeworth approximations. In particular, a recent paper by Lee & Young introduced a novel approach to approximating bootstrap distribution functions, using first an empirical Edgeworth expansion and then a more traditional bootstrap approximation to the remainder. In principle, either direct calculation or computer algebra could be used to compute the Edgeworth component, but both methods would often be difficult to implement in practice, not least because of the sheer algebraic complexity of a general Edgeworth expansion. In the present paper we show that a simple but nonstandard Monte Carlo technique is a competitive alternative. It exploits properties of Edgeworth expansions, in particular their parity and the degrees of their polynomial terms, to develop particularly accurate approximations.     

Second-order and resampling approximation of finite population U-statistics based on stratified samples

We construct one-term Edgeworth expansions to distributions of U statistics and Studentized U-statistics, based on stratified samples drawn without replacement. Replacing the cumulants defining the expansions by consistent jackknife estimators, we obtain empirical Edgeworth expansions. The expansions provide second-order approximations that improve upon the normal approximation. Theoretical results are illustrated by a simulation study where we compare various approximations to the distribution of the commonly used Gini's mean difference estimator.     

Stationary bootstrapping for realized covariations of high frequency financial data

This paper studies the stationary bootstrap applicability for realized covariations of high frequency asynchronous financial data. The stationary bootstrap method, which is characterized by a block-bootstrap with random block length, is applied to estimate the integrated covariations. The bootstrap realized covariance, bootstrap realized regression coefficient and bootstrap realized correlation coefficient are proposed, and the validity of the stationary bootstrapping for them is established both for large sample and for finite sample. Consistencies of bootstrap distributions are established, which provide us valid stationary bootstrap confidence intervals. The bootstrap confidence intervals do not require a consistent estimator of a nuisance parameter arising from nonsynchronous unequally spaced sampling while those based on a normal asymptotic theory require a consistent estimator. A Monte-Carlo comparison reveals that the proposed stationary bootstrap confidence intervals have better coverage probabilities than those based on normal approximation.     

Asymptotic approximations to the distribution of Kendall's sample τ

This paper provides a saddlepoint approximation to the distribution of the sample version of Kendall's τ, which is a measure of association between two samples. The saddlepoint approximation is compared with the Edgeworth and the normal approximations, and with the bootstrap resampling distribution. A numerical study shows that with small sample sizes the saddlepoint approximation outperforms both the normal and the Edgeworth approximations. This paper gives also an analytical comparison between approximated and exact cumulants of the sample Kendall's τ when the two samples are independent.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

Second-order approximations of finite population L-statistics

We construct Edgeworth and empirical Edgeworth approximations to distribution functions of finite population L-statistics and compare their accuracy with that of the normal approximation and the bootstrap approximation in a simulation study.     

An L 2 Comparison Between the Bootstrap and the Empirical Edgeworth Expansion

The Bootstrap estimate for studentized statistics is more accurate than both the normal approximation and the two-term empirical Edgeworth expansion. In this article, it will be shown that the three-term empirical Edgeworth expansion for studentized statistics compares well with the bootstrap. It is also shown that the three-term Edgeworth expansion is superior to the bootstrap in some cases, using more efficient estimators than sample moments in the Edgeworth expansion, such as using maximum likelihood estimators in the one-parameter exponential family.     

Data-based interval estimation of classification error rates

Leave-one-out and 632 bootstrap are popular data-based methods of estimating the true error rate of a classification rule, but practical applications almost exclusively quote only point estimates. Interval estimation would provide better assessment of the future performance of the rule, but little has been published on this topic. We first review general-purpose jackknife and bootstrap methodology that can be used in conjunction with leave-one-out estimates to provide prediction intervals for true error rates of classification rules. Monte Carlo simulation is then used to investigate coverage rates of the resulting intervals for normal data, but the results are disappointing; standard intervals show considerable overinclusion, intervals based on Edgeworth approximations or random weighting do not perform well, and while a bootstrap approach provides intervals with coverage rates closer to the nominal ones there is still marked underinclusion. We then turn to intervals constructed from 632 bootstrap estimates, and show that much better results are obtained. Although there is now some overinclusion, particularly for large training samples, the actual coverage rates are sufficiently close to the nominal rates for the method to be recommended. An application to real data illustrates the considerable variability that can arise in practical estimation of error rates.     

On bootstrap confidence limits for possibly skew distributed statistics

Statistics for which confidence limits or tests are calculated by bootstrap techniques frequently have asymmetric distributions. Approaches based only on boot-strapped variance are inadequatein such cases. In a Mte. Carlo study with a markedly skew X²-distributed statistic an approach by Edgeworth expansions using bootstrapped estimates of variance and skewness of the statistic's distribution performed well with respect to size and power and is proposed for variaus applications.     

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.     

Editorial collaborators

Confidence intervals obtained by bootstrap methods and normal approximation are compared, based on output data from terminating and steady-state simulations. Bootstrap intervals are equal or better than normal approximation intervals in actual probability coverages. Furthermore, bootstrap methods capture the skewness in the distribution of outputs and, therefore, are more desirable than normal approximation.     

Block Bootstrapping for Kernel Density Estimators under ψ-Weak Dependence

Block bootstrap methods are applied to kernel-type density estimator and its derivatives for ψ-weakly dependent processes. Nonparametric density estimation is discussed via moving block bootstrap (MBB) and disjoint block bootstrap (DBB). Asymptotic validity is proved for MBB and DBB. A Monte-Carlo experiment compares confidence intervals based on MBB and DBB with an existing method based on normal approximation (NA) in terms of serial correlation, dynamic asymmetry, and conditional heteroscedasticity. The experiment shows that, in cases of substantial serial correlation, MBB and DBB perform better than NA and, in the other cases, MBB and DBB perform as good as NA.     

CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN TWO PARTIAL AUCS

As new diagnostic tests are developed and marketed, it is very important to be able to compare the accuracy of a given two continuous‐scale diagnostic tests. An effective method to evaluate the difference between the diagnostic accuracy of two tests is to compare partial areas under the receiver operating characteristic curves (AUCs). In this paper, we review existing parametric methods. Then, we propose a new semiparametric method and a new nonparametric method to investigate the difference between two partial AUCs. For the difference between two partial AUCs under each method, we derive a normal approximation, define an empirical log‐likelihood ratio, and show that the empirical log‐likelihood ratio follows a scaled chi‐square distribution. We construct five confidence intervals for the difference based on normal approximation, bootstrap, and empirical likelihood methods. Finally, extensive simulation studies are conducted to compare the finite‐sample performances of these intervals, and a real example is used as an application of our recommended intervals. The simulation results indicate that the proposed hybrid bootstrap and empirical likelihood intervals outperform other existing intervals in most cases.     

Weighted Bootstrap Confidence Intervals for Generalized Behrens-Fisher Problems

In this article, the weighted bootstrap difference between two-sample means for generalized Behrens-Fisher problems is investigated along with its strong consistency. Moreover, the one-order accurate weighted bootstrap approximation to the sample distribution of sample difference is also established and hence based on it the weighted bootstrap intervals for the population difference is constructed. Simulation studies show that the weighted bootstrap interval performs better than other intervals we considered in some cases.     

Small sample approximations for spacing statistics

Edgeworth expansions as well as saddle-point methods are used to approximate the distributions of some spacing statistics for small to moderate sample sizes. By comparing with the exact values when available, it is shown that a particular form of Edgeworth expansion produces extremely good results even for fairly small sample sizes. However, this expansion suffers from negative tail probabilities and an accurate approximation without this disadvantage, is shown to be the one based on saddle-point method. Finally, quantiles of some spacing statistics whose exact distributions are not known, are tabulated, making them available in a variety of testing contexts.     

ON STUDENTIZING AND BLOCKING METHODS FOR IMPLEMENTING THE BOOTSTRAP WITH DEPENDENT DATA

Gōtze & Kūnsch (1990) announced that a certain version of the bootstrap percentile-t method, and the blocking method, can be used to improve on the normal approximation to the distribution of a Studentized statistic computed from dependent data. This paper shows that this result depends fundamentally on the method of Studentization. Indeed, if the percentile-t method is implemented naively, for dependent data, then it does not improve by an order of magnitude on the much simpler normal approximation despite all the computational effort that is required to implement it. On the other hand, if the variance estimator used for the percentile-t bootstrap is adjusted appropriately, then percentile-t can improve substantially on the normal approximation.     

Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

Let ( X , Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non-parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975 ), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.     

Multivariate Stochastic Volatility Model With Realized Volatilities and Pairwise Realized Correlations

Abstract

Although stochastic volatility and GARCH (generalized autoregressive conditional heteroscedasticity) models have successfully described the volatility dynamics of univariate asset returns, extending them to the multivariate models with dynamic correlations has been difficult due to several major problems. First, there are too many parameters to estimate if available data are only daily returns, which results in unstable estimates. One solution to this problem is to incorporate additional observations based on intraday asset returns, such as realized covariances. Second, since multivariate asset returns are not synchronously traded, we have to use the largest time intervals such that all asset returns are observed to compute the realized covariance matrices. However, in this study, we fail to make full use of the available intraday informations when there are less frequently traded assets. Third, it is not straightforward to guarantee that the estimated (and the realized) covariance matrices are positive definite.

Our contributions are the following: (1) we obtain the stable parameter estimates for the dynamic correlation models using the realized measures, (2) we make full use of intraday informations by using pairwise realized correlations, (3) the covariance matrices are guaranteed to be positive definite, (4) we avoid the arbitrariness of the ordering of asset returns, (5) we propose the flexible correlation structure model (e.g., such as setting some correlations to be zero if necessary), and (6) the parsimonious specification for the leverage effect is proposed. Our proposed models are applied to the daily returns of nine U.S. stocks with their realized volatilities and pairwise realized correlations and are shown to outperform the existing models with respect to portfolio performances.     

Empirical likelihood for the varying‐coefficient single‐index model

In this article the author investigates the application of the empirical‐likelihood‐based inference for the parameters of varying‐coefficient single‐index model (VCSIM). Unlike the usual cases, if there is no bias correction the asymptotic distribution of the empirical likelihood ratio cannot achieve the standard chi‐squared distribution. To this end, a bias‐corrected empirical likelihood method is employed to construct the confidence regions (intervals) of regression parameters, which have two advantages, compared with those based on normal approximation, that is, (1) they do not impose prior constraints on the shape of the regions; (2) they do not require the construction of a pivotal quantity and the regions are range preserving and transformation respecting. A simulation study is undertaken to compare the empirical likelihood with the normal approximation in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. A real data example is given to illustrate the proposed approach. The Canadian Journal of Statistics 38: 434–452; 2010 © 2010 Statistical Society of Canada