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.     

Threshold Autoregression with a Unit Root

This paper develops an asymptotic theory of inference for an unrestricted two‐regime threshold autoregressive (TAR) model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap‐based tests. These tests and distribution theory allow for the joint consideration of nonlinearity (thresholds) and nonstationary (unit roots). Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two‐parameter empirical process that converges weakly to a two‐parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two‐parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short‐run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.     

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.     

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.     

Comment on: Threshold Autoregressions With a Unit Root

In this paper we revisit the results in Caner and Hansen (2001), where the authors obtained novel limiting distributions of Wald type test statistics for testing for the presence of threshold nonlinearities in autoregressive models containing unit roots. Using the same framework, we obtain a new formulation of the limiting distribution of the Wald statistic for testing for threshold effects, correcting an expression that appeared in the main theorem presented by Caner and Hansen. Subsequently, we show that under a particular scenario that excludes stationary regressors such as lagged dependent variables and despite the presence of a unit root, this same limiting random variable takes a familiar form that is free of nuisance parameters and already tabulated in the literature, thus removing the need to use bootstrap based inferences. This is a novel and unusual occurrence in this literature on testing for the presence of nonlinear dynamics.     

Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey–Fuller unit root tests. The second‐order terms in their expansions are of stochastic orders O_p(n^−1/4) and O_p(n^−1/2), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey–Fuller tests. We show in particular that the bootstrap offers asymptotic refinements for the Dickey–Fuller tests, i.e., it corrects their second‐order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second‐order terms, and the errors in rejection probabilities are of order o(n^−1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples.     

Consistent Testing of Cointegrating Relationships

In this paper we investigate methods for testing the existence of a cointegration relationship among the components of a nonstationary fractionally integrated (NFI) vector time series. Our framework generalizes previous studies restricted to unit root integrated processes and permits simultaneous analysis of spurious and cointegrated NFI vectors. We propose a modified F‐statistic, based on a particular studentization, which converges weakly under both hypotheses, despite the fact that OLS estimates are only consistent under cointegration. This statistic leads to a Wald‐type test of cointegration when combined with a narrow band GLS‐type estimate. Our semiparametric methodology allows consistent testing of the spurious regression hypothesis against the alternative of fractional cointegration without prior knowledge on the memory of the original series, their short run properties, the cointegrating vector, or the degree of cointegration. This semiparametric aspect of the modelization does not lead to an asymptotic loss of power, permitting the Wald statistic to diverge faster under the alternative of cointegration than when testing for a hypothesized cointegration vector. In our simulations we show that the method has comparable power to customary procedures under the unit root cointegration setup, and maintains good properties in a general framework where other methods may fail. We illustrate our method testing the cointegration hypothesis of nominal GNP and simple‐sum (M1, M2, M3) monetary aggregates.     

Nearly Efficient Likelihood Ratio Tests of the Unit Root Hypothesis

Seemingly absent from the arsenal of currently available “nearly efficient” testing procedures for the unit root hypothesis, that is, tests whose asymptotic local power functions are virtually indistinguishable from the Gaussian power envelope, is a test admitting a (quasi‐)likelihood ratio interpretation. We study the large sample properties of a quasi‐likelihood ratio unit root test based on a Gaussian likelihood and show that this test is nearly efficient.     

Subsampling Intervals in Autoregressive Models with Linear Time Trend

A new method is proposed for constructing confidence intervals in autoregressive models with linear time trend. Interest focuses on the sum of the autoregressive coefficients because this parameter provides a useful scalar measure of the long‐run persistence properties of an economic time series. Since the type of the limiting distribution of the corresponding OLS estimator, as well as the rate of its convergence, depend in a discontinuous fashion upon whether the true parameter is less than one or equal to one (that is, trend‐stationary case or unit root case), the construction of confidence intervals is notoriously difficult. The crux of our method is to recompute the OLS estimator on smaller blocks of the observed data, according to the general subsampling idea of Politis and Romano (1994a), although some extensions of the standard theory are needed. The method is more general than previous approaches in that it works for arbitrary parameter values, but also because it allows the innovations to be a martingale difference sequence rather than i.i.d. Some simulation studies examine the finite sample performance.     

Generalized Method of Integrated Moments for High‐Frequency Data

We propose a semiparametric two‐step inference procedure for a finite‐dimensional parameter based on moment conditions constructed from high‐frequency data. The population moment conditions take the form of temporally integrated functionals of state‐variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high‐frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second‐step GMM estimation, which requires the correction of a high‐order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens‐type consistent specification test. These infill asymptotic results are based on a novel empirical‐process‐type theory for general integrated functionals of noisy semimartingale processes.     

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.     

金融ARCH模型的贝叶斯检验和模型选择

在金融时间序列分析中,检验ARCH效果和决定合适的阶是ARCH模型的重要研究主题,在贝叶斯框架下,本文使用贝叶斯因子来检验ARCH效果和选择ARCH模型合适的阶。在路径抽样的基础上,提出了计算ARCH模型贝叶斯因子的方法。最后,我们用一个具体的实例来论证了所提方法的有效性。     

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.     

One‐Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root

This paper examines the problem of testing and confidence set construction for one‐dimensional functions of the coefficients in autoregressive (AR(p)) models with potentially persistent time series. The primary example concerns inference on impulse responses. A new asymptotic framework is suggested and some new theoretical properties of known procedures are demonstrated. I show that the likelihood ratio (LR) and LR^± statistics for a linear hypothesis in an AR(p) can be uniformly approximated by a weighted average of local‐to‐unity and normal distributions. The corresponding weights depend on the weight placed on the largest root in the null hypothesis. The suggested approximation is uniform over the set of all linear hypotheses. The same family of distributions approximates the LR and LR^± statistics for tests about impulse responses, and the approximation is uniform over the horizon of the impulse response. I establish the size properties of tests about impulse responses proposed by Inoue and Kilian (2002) and Gospodinov (2004), and theoretically explain some of the empirical findings of Pesavento and Rossi (2007). An adaptation of the grid bootstrap for impulse response functions is suggested and its properties are examined.     

Higher‐Order Improvements of a Computationally Attractive k‐Step Bootstrap for Extremum Estimators

This paper establishes the higher‐order equivalence of the k‐step bootstrap, introduced recently by Davidson and MacKinnon (1999), and the standard bootstrap. The k‐step bootstrap is a very attractive alternative computationally to the standard bootstrap for statistics based on nonlinear extremum estimators, such as generalized method of moment and maximum likelihood estimators. The paper also extends results of Hall and Horowitz (1996) to provide new results regarding the higher‐order improvements of the standard bootstrap and the k‐step bootstrap for extremum estimators (compared to procedures based on first‐order asymptotics). The results of the paper apply to Newton‐Raphson (NR), default NR, line‐search NR, and Gauss‐Newton k‐step bootstrap procedures. The results apply to the nonparametric iid bootstrap and nonoverlapping and overlapping block bootstraps. The results cover symmetric and equal‐tailed two‐sided t tests and confidence intervals, one‐sided t tests and confidence intervals, Wald tests and confidence regions, and J tests of over‐identifying restrictions.     

基于近邻截断的跳跃扩散过程波动函数的设定检验

高频数据环境下的波动函数设定检验容易受到数据中跳跃的影响。为此，本文基于近邻截断（nearest neighbor truncation）方法，利用残差构造部分和（partial sum）过程构造出波动函数参数形式的设定检验方法，并分析了该检验方法在原假设条件下的近似极限性质与自助检验步骤。所提出的波动函数设定检验方法能渐近有效的避免跳跃扩散过程的漂移项与跳跃项的影响。蒙特卡洛模拟结果表明这些检验方法对跳跃的影响具有稳健性，且具有合理的检验水平（size）和检验功效（power）。利用这些检验方法对我国的上海银行间同业拆放利率（Shibor）数据进行实证分析，结果表明本文所提出的跳跃稳健的波动函数检验方法比非跳跃稳健的波动函数检验方法具有更好的区分度。     

基于修剪平均的神经网络集成时序预测方法

由于复杂时序存在结构性断点和异常值等问题，往往导致预测模型训练效果不佳，并可能出现极端预测值的情况。为此，本文提出了基于修剪平均的神经网络集成预测方法。该方法首先从训练数据中生成多组训练集，然后分别训练多个神经网络预测模型，最后将多个神经网络的预测结果使用修剪平均策略进行集成。相较于简单平均策略而言，修剪平均策略不容易受到极值的影响，能够使集成模型获得鲁棒性强的预测效果。在实证研究中，本文构造了两种神经网络集成预测模型，分别为基于修剪平均的自举神经网络集成模型(Trimmed Average based Bootstrap Neural Network Ensemble, TA-BNNE)和基于修剪平均的蒙特卡洛神经网络集成模型(Trimmed Average based Monte Carlo Neural Network Ensemble, TA-MCNNE)，并采用这两种模型对NN3竞赛数据集进行预测，结果表明在常规和复杂数据集上，修剪平均策略比简单平均策略具有更好的预测精度。此外，本文将所提出的集成模型与NN3的前十名模型进行比较，发现两种模型在全部数据集上均超过了第6名，在复杂数据集上的表现均超过了第1名，进一步验证本文所提方法的有效性。     

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.     

AN EXPERIMENTAL COMPARISON OF STATISTICAL AND LINEAR PROGRAMMING APPROACHES TO THE DISCRIMINANT PROBLEM

This paper reports the results of an experimental comparison of three linear programming approaches and the Fisher procedure for the discriminant problem. The linear programming approaches include two formulations proposed by Freed and Glover and a newly proposed mixed-integer, linear goal programming formulation. Ten test problems were generated for each of the 36 cells in the three-factor, full-factorial experimental design. Each test problem consisted of a 30-case estimation sample and a 1,000-case holdout sample. Experimental results indicate that each of the four approaches was statistically preferable in certain cells of the experimental design. Sample-based rules are suggested for selecting an approach based on Hotelling's T² and Box's M statistics. Subject Areas: Statistical Techniques, Linear Statistical Models, and Linear Programming.     

Hysteresis versus Persistence in Unemployment: A Sceptical Note on Unit Root Tests

ABSTRACT: Previous studies of time-series data suggest that the rate of unemployment is best described as a driftless random walk. This is a far from trivial point because the policy implications of an alternative stationary series can be very different. On the basis of simulated data, this paper demonstrates how the rate of unemployment may appear to be difference stationary, yet is actually trend stationary, and that, contrary to conventional wisdom, the low power of the standard unit root test for stationarity is significant with regard to the conduct of policy. Thus, although the series under investigation may in fact offer little scope for successful policy intervention, the null hypothesis of a unit root may be accepted along with its attendant policy implications. Finally, an alternative test based on a null hypothesis of a stationary process is assessed.