Long memory versus structural breaks: An overview

We discuss the increasing literature on misspecifying structural breaks or more general trends as long-range dependence. We consider tests on structural breaks in the long-memory regression model as well as the behaviour of estimators of the memory parameter when structural breaks or trends are in the data but long memory is not. Methods for distinguishing both of these phenomena are proposed. The financial support of Volkswagenstiftung is gratefully acknowledged.     

Some results on random design regression with long memory errors and predictors

This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the mean integrated squared error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that averaged squared error (ASE) is a good approximation of MISE; however, this is not the case for a cross-validation criterion.     

Evidence of spurious regression driven by heavy-tailed observations with structural changes

This article extends the theoretical analysis of spurious relationship and considers the situation where the deterministic components of the processes generating individual series are independent heavy-tailed with structural changes. It shows when those sequences are used in ordinary least-square regression, the convenient t-statistic procedures wrongly indicate that (i) the spurious significance is established when regressing mean-stationary and trend-stationary series with structural changes, (ii) the spurious relationship occurs under broken mean-stationary and difference-stationary sequences, and (iii) the extent of spurious regression becomes stronger between difference-stationary and trend-stationary series in the presence of breaks. The spurious phenomenon is present regardless of the sample size and structural breaks taking place at the same points or not. Simulation experiments confirm our asymptotic results and reveal that the spurious effects are not only sensitive to the relative location of structural changes with the sample, but seriously depend on the stable indexes.     

An Empirical Strategy to Detect Spurious Effects in Long Memory and Occasional-Break Processes

Long-range dependence and structural changes in level are intimely related phenomena and it is very difficult to separate the two effects. In this article, we present an empirical procedure to distinguish between long-memory and occasional-break processes. An extensive Monte Carlo experiment illustrates the performance of the procedure and an application to real data is also included.     

结构突变趋势平稳过程与随机趋势过程的虚假回归研究

 内容提要：在已有研究的基础上,本文更为深入的研究含有结构突变的趋势平稳变量与随机趋势变量间的虚假回归问题。本文推导出OLS估计下DW统计量、F统计量以及R2的极限分布,并且将回归模型扩展到动态情形下,推导出用于Granger因果检验的F统计量的极限分布;采用Monte Carlo模拟方法分析了数据生成过程的各项参数对各统计量有限样本分布的影响;最后,本文分析了在有限样本下,数据生成过程的各项参数对虚假回归及虚假Granger因果关系发生概率的影响。     

Change-in-mean problem for long memory time series models with applications

Test and estimation procedures for detecting a change in the mean are proposed in infinite moving average long memory time series models. The asymptotic properties of the test statistics and the change-point estimators are investigated. The method is illustrated through the analysis of real data sets from econometrics and climatology.     

平稳阈值自回归下的伪回归研究

 研究表明相互独立的平稳阈值自回归(TAR)模型之间的回归存在伪回归,且伪回归的产生与样本容量和随机干扰项的分布无关。通过一系列的MC模拟,不仅证实了理论结论,而且模拟结果还表明当持久性相同时,两机制TAR回归模型比三机制TAR回归模型具有更大的拒绝率,原因在于两机制TAR下,OLS法估计得到的标准误具有更厚的左尾。此外在模拟中也发现当随机干扰项服从TAR模型时,Newey-West(1987)的一致异方差估计法是不适用的。     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

Estimation in long memory time series models

A regression type estimator of the parameter d in fractionally differenced ARMA (p,q) processes is presented. The proposed estimator is shown to be mean square consistent. Its performance is compared with some of the existing estimators via a simulation study.     

Detection of structural breaks in a time-varying heteroskedastic regression model

A Bayesian method for estimating a time-varying regression model subject to the presence of structural breaks is proposed. Heteroskedastic dynamics, via both GARCH and stochastic volatility specifications, and an autoregressive factor, subject to breaks, are added to generalize the standard return prediction model, in order to efficiently estimate and examine the relationship and how it changes over time. A Bayesian computational method is employed to identify the locations of structural breaks, and for estimation and inference, simultaneously accounting for heteroskedasticity and autocorrelation. The proposed methods are illustrated using simulated data. Then, an empirical study of the Taiwan and Hong Kong stock markets, using oil and gas price returns as a state variable, provides strong support for oil prices being an important explanatory variable for stock returns.     

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching,Structural Break,and Nonlinear Data

The potential observational equivalence between various types of nonlinearity and long memory has been recognized by the econometrics community since at least the contribution of Diebold and Inoue (2001 Diebold, F., Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics 105:131–159.[Crossref], [Web of Science ®] , [Google Scholar]). A large literature has developed in an attempt to ascertain whether or not the long memory finding in many economic series is spurious. Yet to date, no study has analyzed the consequences of using long memory methods to test for unit roots when the “truth” derives from regime switching, structural breaks, or other types of mean reverting nonlinearity. In this article, I conduct a comprehensive Monte Carlo analysis to investigate the consequences of using tests designed to have power against fractional integration when the actual data generating process is unknown. I additionally consider the use of tests designed to have power against breaks and threshold nonlinearity. The findings are compelling and demonstrate that the use of long memory as an approximation to nonlinearity yields tests with relatively high power. In contrast, misspecification has severe consequences for tests designed to have power against threshold nonlinearity, and especially for tests designed to have power against breaks.     

Change-of-variance problem for linear processes with long memory

In this paper we investigate the asymptotic properties of the test statistics for detecting change-points in the variance of infinite moving average sequences with long memory. This research is partly supported by NSFC Grants and SRF for ROCS, SEM.     

Time series modelling of two millennia of northern hemisphere temperatures: long memory or shifting trends?

Summary.  The time series properties of the temperature reconstruction of Moberg and co-workers are analysed. It is found that the record appears to exhibit long memory characteristics that can be modelled by an autoregressive fractionally integrated moving average process that is both stationary and mean reverting, so that forecasts will eventually return to a constant underlying level. Recent research has suggested that long memory and shifts in level and trend may be confused with each other, and fitting models with slowly changing trends is found to remove the evidence of long memory. Discriminating between the two models is difficult, however, and the strikingly different forecasts that are implied by the two models point towards some intriguing research questions concerning the stochastic process driving this temperature reconstruction.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Prediction of long memory processes on same-realisation

For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of _Xn+1$X_{n+1}$ based on (_Xn,…,_X1)$(X_n,\dots ,X_1)$. The predictor is obtained by projecting _Xn+1$X_{n+1}$ onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L²$L^2$- sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past.     

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.     

Detecting structural breaks in multivariate financial time series: evidence from hedge fund investment strategies

This paper extends the class of asset-based style factor models with multiple structural breaks to the multivariate setting. We propose a model that allows for the presence of common breaks in a system of factor models for individual hedge fund investment strategies, which share common investment characteristics. We develop a Bayesian approach to inference for the unknown number and positions of the structural breaks, based on a set of filtering recursions similar to those of the forward–backward algorithm. Furthermore, we identify relevant risk factors, common among the series of hedge funds, using a Bayesian model comparison approach. We apply our method to a set of correlated hedge fund strategies, which are mainly characterized by equity related bets. Multiple common breaks are identified, consistent with well-known market events, which reveal evidence for structural changes in the risk exposures as well as in the correlation structure of the analysed series.     

On rapid change points under long memory

Estimation of points of rapid change in the mean function m(t) is considered under long memory residuals, irregularily spaced time points and smoothly changing marginal distributions obtained by local Gaussian subordination. The approach is based on kernel estimation of derivatives of the trend function. An asymptotic expression for the mean squared error is obtained. Limit theorems are derived for derivatives of m and the time points where rapid change occurs. The results are illustrated by an application to measurements of oxygen isotopes trapped in the Greenland ice sheets during the last 20,000 years.     

Bayesian analysis of a linear model involving structural changes in either regression parameters or disturbances precision

ABSTRACT

The present paper considers the Bayesian analysis of a linear regression model involving structural change, which may occur either due to shift in disturbances precision or due to shift in regression parameters. The posterior density for the regression parameter has been derived and posterior odds ratio for testing the hypothesis that structural change is due to shift in disturbances precision against the alternative that the change is due to shift in regression parameters has been obtained. The findings of a numerical simulation have been presented. The proposed model has been applied to RBI data set on corporate sector.     

Finite sample properties of ml and reml estimators in time series regression models with long memory noise

Finite sample properties of ML and REML estimators in time series regression models with fractional ARIMA noise are examined. In particular, theoretical approximations for bias of ML and REML estimators of the noise parameters are developed and their accuracy is assessed through simulations. The impact of noise parameter estimation on performance of t -statistics and likelihood ratio statistics for testing regression parameters is also investigated.