Bootstrap Determination of the Co‐Integration Rank in Vector Autoregressive Models

This paper discusses a consistent bootstrap implementation of the likelihood ratio (LR) co‐integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model that obtain under the reduced rank null hypothesis. A full asymptotic theory is provided that shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co‐integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice.     

Bootstrap Testing of Hypotheses on Co‐Integration Relations in Vector Autoregressive Models

It is well known that the finite‐sample properties of tests of hypotheses on the co‐integrating vectors in vector autoregressive models can be quite poor, and that current solutions based on Bartlett‐type corrections or bootstrap based on unrestricted parameter estimators are unsatisfactory, in particular in those cases where also asymptotic χ² tests fail most severely. In this paper, we solve this inference problem by showing the novel result that a bootstrap test where the null hypothesis is imposed on the bootstrap sample is asymptotically valid. That is, not only does it have asymptotically correct size, but, in contrast to what is claimed in existing literature, it is consistent under the alternative. Compared to the theory for bootstrap tests on the co‐integration rank (Cavaliere, Rahbek, and Taylor, 2012), establishing the validity of the bootstrap in the framework of hypotheses on the co‐integrating vectors requires new theoretical developments, including the introduction of multivariate Ornstein–Uhlenbeck processes with random (reduced rank) drift parameters. Finally, as documented by Monte Carlo simulations, the bootstrap test outperforms existing methods.     

On the Failure of the Bootstrap for Matching Estimators

Matching estimators are widely used in empirical economics for the evaluation of programs or treatments. Researchers using matching methods often apply the bootstrap to calculate the standard errors. However, no formal justification has been provided for the use of the bootstrap in this setting. In this article, we show that the standard bootstrap is, in general, not valid for matching estimators, even in the simple case with a single continuous covariate where the estimator is root‐N consistent and asymptotically normally distributed with zero asymptotic bias. Valid inferential methods in this setting are the analytic asymptotic variance estimator of Abadie and Imbens (2006a) as well as certain modifications of the standard bootstrap, like the subsampling methods in Politis and Romano (1994).     

A Genuine Rank‐Dependent Generalization of the Von Neumann‐Morgenstern Expected Utility Theorem

This paper uses “revealed probability trade‐offs” to provide a natural foundation for probability weighting in the famous von Neumann and Morgenstern axiomatic set‐up for expected utility. In particular, it shows that a rank‐dependent preference functional is obtained in this set‐up when the independence axiom is weakened to stochastic dominance and a probability trade‐off consistency condition. In contrast with the existing axiomatizations of rank‐dependent utility, the resulting axioms allow for complete flexibility regarding the outcome space. Consequently, a parameter‐free test/elicitation of rank‐dependent utility becomes possible. The probability‐oriented approach of this paper also provides theoretical foundations for probabilistic attitudes towards risk. It is shown that the preference conditions that characterize the shape of the probability weighting function can be derived from simple probability trade‐off conditions.     

Inference in Arch and Garch Models with Heavy–Tailed Errors

ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.     

Asymptotic Distribution Theory for Nonparametric Entropy Measures of Serial Dependence

Entropy is a classical statistical concept with appealing properties. Establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging. In this paper, we develop an asymptotic theory for a class of kernel‐based smoothed nonparametric entropy measures of serial dependence in a time‐series context. We use this theory to derive the limiting distribution of Granger and Lin's (1994) normalized entropy measure of serial dependence, which was previously not available in the literature. We also apply our theory to construct a new entropy‐based test for serial dependence, providing an alternative to Robinson's (1991) approach. To obtain accurate inferences, we propose and justify a consistent smoothed bootstrap procedure. The naive bootstrap is not consistent for our test. Our test is useful in, for example, testing the random walk hypothesis, evaluating density forecasts, and identifying important lags of a time series. It is asymptotically locally more powerful than Robinson's (1991) test, as is confirmed in our simulation. An application to the daily S&P 500 stock price index illustrates our approach.     

the Block–Block Bootstrap: Improved Asymptotic Refinements

The asymptotic refinements attributable to the block bootstrap for time series are not as large as those of the nonparametric iid bootstrap or the parametric bootstrap. One reason is that the independence between the blocks in the block bootstrap sample does not mimic the dependence structure of the original sample. This is the join‐point problem. In this paper, we propose a method of solving this problem. The idea is not to alter the block bootstrap. Instead, we alter the original sample statistics to which the block bootstrap is applied. We introduce block statistics that possess join‐point features that are similar to those of the block bootstrap versions of these statistics. We refer to the application of the block bootstrap to block statistics as the block–block bootstrap. The asymptotic refinements of the block–block bootstrap are shown to be greater than those obtained with the block bootstrap and close to those obtained with the nonparametric iid bootstrap and parametric bootstrap.     

Bootstrap Methods for Markov Processes

The block bootstrap is the best known bootstrap method for time‐series data when the analyst does not have a parametric model that reduces the data generation process to simple random sampling. However, the errors made by the block bootstrap converge to zero only slightly faster than those made by first‐order asymptotic approximations. This paper describes a bootstrap procedure for data that are generated by a Markov process or a process that can be approximated by a Markov process with sufficient accuracy. The procedure is based on estimating the Markov transition density nonparametrically. Bootstrap samples are obtained by sampling the process implied by the estimated transition density. Conditions are given under which the errors made by the Markov bootstrap converge to zero more rapidly than those made by the block bootstrap.     

Asymptotically Efficient Estimation of Models Defined by Convex Moment Inequalities

This paper examines the efficient estimation of partially identified models defined by moment inequalities that are convex in the parameter of interest. In such a setting, the identified set is itself convex and hence fully characterized by its support function. We provide conditions under which, despite being an infinite dimensional parameter, the support function admits √n‐consistent regular estimators. A semiparametric efficiency bound is then derived for its estimation, and it is shown that any regular estimator attaining it must also minimize a wide class of asymptotic loss functions. In addition, we show that the “plug‐in” estimator is efficient, and devise a consistent bootstrap procedure for estimating its limiting distribution. The setting we examine is related to an incomplete linear model studied in Beresteanu and Molinari (2008) and Bontemps, Magnac, and Maurin (2012), which further enables us to establish the semiparametric efficiency of their proposed estimators for that problem.     

Apparent Overconfidence

It is common for a majority of people to rank themselves as better than average on simple tasks and worse than average on difficult tasks. The literature takes for granted that this apparent misconfidence is problematic. We argue, however, that this behavior is consistent with purely rational Bayesian updaters. In fact, better‐than‐average data alone cannot be used to show overconfidence; we indicate which type of data can be used. Our theory is consistent with empirical patterns found in the literature.     

A Comparison of Three Methods for Calculating Confidence Intervals for the Benchmark Dose

Various methods exist to calculate confidence intervals for the benchmark dose in risk analysis. This study compares the performance of three such methods in fitting nonlinear dose-response models: the delta method, the likelihood-ratio method, and the bootstrap method. A data set from a developmental toxicity test with continuous, ordinal, and quantal dose-response data is used for the comparison of these methods. Nonlinear dose-response models, with various shapes, were fitted to these data. The results indicate that a few thousand runs are generally needed to get stable confidence limits when using the bootstrap method. Further, the bootstrap and the likelihood-ratio method were found to give fairly similar results. The delta method, however, resulted in some cases in different (usually narrower) intervals, and appears unreliable for nonlinear dose-response models. Since the bootstrap method is more time consuming than the likelihood-ratio method, the latter is more attractive for routine dose-response analysis. In the context of a probabilistic risk assessment the bootstrap method has the advantage that it directly links to Monte Carlo analysis.     

A Bootstrap Mechanism for Response Masking in Remote Analysis Systems

National Statistical Agencies and other data custodian agencies hold a wealth of data regarding individuals and organizations, collected from censuses, surveys and administrative sources. In many cases, these data are made available to external researchers, for the investigation of questions of social and economic importance. To enhance access to this information, several national statistical agencies are developing remote analysis systems (RAS) designed to accept queries from a researcher, run them on data held in a secure environment, and then return the results. RAS prevent a researcher from accessing the underlying data, and most rely on manual checking to ensure the responses have acceptably low disclosure risk. However, the need for scalability and consistency will increasingly require automated methods. We propose a RAS output confidentialization procedure based on statistical bootstrapping that automates disclosure control while achieving a provably good balance between disclosure risk and usefulness of the responses. The bootstrap masking mechanism is easy to implement for most statistical queries, yet the characteristics of the bootstrap distribution assure us that it is also effective in providing both useful responses and low disclosure risk. Interestingly, our proposed bootstrap masking mechanism represents an ideal application of Efron's bootstrap—one that takes advantage of all the theoretical properties of the bootstrap, without ever having to construct the bootstrap distribution.     

A Small Sample Correction for the Test of Cointegrating Rank in the Vector Autoregressive Model

With the cointegration formulation of economic long–run relations the test for cointegrating rank has become a useful econometric tool. The limit distribution of the test is often a poor approximation to the finite sample distribution and it is therefore relevant to derive an approximation to the expectation of the likelihood ratio test for cointegration in the vector autoregressive model in order to improve the finite sample properties. The correction factor depends on moments of functions of the random walk, which are tabulated by simulation, and functions of the parameters, which are estimated. From this approximation we propose a correction factor with the purpose of improving the small sample performance of the test. The correction is found explicitly in a number of simple models and its usefulness is illustrated by some simulation experiments.     

Inference in Nonparametric Instrumental Variables With Partial Identification

This paper develops methods for hypothesis testing in a nonparametric instrumental variables setting within a partial identification framework. We construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction. The same test statistic can be employed under identification, in which case the hypothesis is whether the true model satisfies the posited property. An almost sure consistent bootstrap procedure is provided for obtaining critical values. Possible applications include testing for semiparametric specifications as well as building confidence regions for certain functionals on the identified set. As an illustration we obtain confidence intervals for the level and slope of Brazilian fuel Engel curves. A Monte Carlo study examines finite sample performance.     

不完全信息下基于GSP的竞价排名问题研究

本文研究对象是不完全信息下搜索引擎基于GSP拍卖的竞价排名问题。基于静态博弈模型,首先给出了按点击付费情况下广告商的贝叶斯均衡出价函数;其次为了保证均衡存在,给出了关于点击率的充分条件,并分析了点击率与均衡存在的关系;最后分析了点击率变化对搜索引擎收益的影响。结果对不完全信息下广告商的出价策略和搜索引擎的广告位置拍卖策略提供了决策依据。     

基于EM填补和加权秩和比的电力系统黑启动决策方法

电力系统大停电后需要安全可靠的黑启动方案进行电力快速恢复,如何从众多黑启动方案中进行优选是一个重要课题。不同于以往完全信息情况下的研究,本文对电力系统黑启动方案评估问题的研究是基于不完全信息情况下进行的,提出一种EM填补和加权秩和比相结合的黑启动决策方法。首先,采用EM算法填补黑启动方案评价空值,得到完备的黑启动评价矩阵;然后,计算指标间的差异性,利用差异性权重法得到各个指标的权重;最后,采用加权秩和比法确定每个方案的评分值,实现黑启动方案的分级和完全排序。本文方法基于广东电网黑启动数据集进行实验验证,实验结果表明,本文方法可以对不完全信息情况下的黑启动方案进行优劣排序,且具有较高准确性。     

Testing and Characterizing Properties of Nonadditive Measures Through Violations of the Sure‐Thing Principle

In expected utility theory, risk attitudes are modeled entirely in terms of utility. In the rank‐dependent theories, a new dimension is added: chance attitude, modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility. Most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation. Only a few qualitative conditions have been proposed or tested as yet, usually quasi‐concavity or quasi‐convexity in the case of given probabilities. This paper presents a general method of studying qualitative properties of chance attitude such as optimism, pessimism, and the “inverse‐S shape” pattern, both for risk and for uncertainty. These qualitative properties can be characterized by permitting appropriate, relatively simple, violations of the sure‐thing principle. In particular, this paper solves a hitherto open problem: the preference axiomatization of convex (“pessimistic” or “uncertainty averse”) nonadditive measures under uncertainty. The axioms of this paper preserve the central feature of rank‐dependent theories, i.e. the separation of chance attitude and utility.     

A Three‐step Method for Choosing the Number of Bootstrap Repetitions

This paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, confidence regions, hypothesis tests, p‐values, and bias correction. For each of these problems, the paper provides a three‐step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval length, test's critical value, test's p‐value, or bias‐corrected estimate based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B=∞. The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals. Monte Carlo simulations show that the proposed methods work very well.     

Poor (Wo)man's Bootstrap

The bootstrap is a convenient tool for calculating standard errors of the parameter estimates of complicated econometric models. Unfortunately, the fact that these models are complicated often makes the bootstrap extremely slow or even practically infeasible. This paper proposes an alternative to the bootstrap that relies only on the estimation of one‐dimensional parameters. We introduce the idea in the context of M and GMM estimators. A modification of the approach can be used to estimate the variance of two‐step estimators.     

Residual‐Based Block Bootstrap for Unit Root Testing

A nonparametric, residual‐based block bootstrap procedure is proposed in the context of testing for integrated (unit root) time series. The resampling procedure is based on weak assumptions on the dependence structure of the stationary process driving the random walk and successfully generates unit root integrated pseudo‐series retaining the important characteristics of the data. It is more general than previous bootstrap approaches to the unit root problem in that it allows for a very wide class of weakly dependent processes and it is not based on any parametric assumption on the process generating the data. As a consequence the procedure can accurately capture the distribution of many unit root test statistics proposed in the literature. Large sample theory is developed and the asymptotic validity of the block bootstrap‐based unit root testing is shown via a bootstrap functional limit theorem. Applications to some particular test statistics of the unit root hypothesis, i.e., least squares and Dickey‐Fuller type statistics are given. The power properties of our procedure are investigated and compared to those of alternative bootstrap approaches to carry out the unit root test. Some simulations examine the finite sample performance of our procedure.