A new simple test against spurious long memory using temporal aggregation

We have developed a new test against spurious long memory based on the invariance of long memory parameter to aggregation. By using the local Whittle estimator, the statistic takes the supremum among combinations of paired aggregated series. Simulations show that the test performs good in finite sample sizes, and is able to distinguish long memory from spurious processes with excellent power. Moreover, the empirical application gives further evidence that the observed long memory in German stock returns is spurious.     

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.     

Wavelet estimation of the memory parameter for long range dependent random fields

In this paper we study the estimation of the spatial long memory parameter for stationary long range dependent random fields using wavelet methods. We first show the relation between the wavelet coefficients of the random fields and its long memory parameter. Based on this relation, we construct a log-regression wavelet estimator of the long memory parameter. Under some mild regularity assumptions, the asymptotic properties of the estimators are investigated. Finally, a small simulation study illustrates the method.     

A non linear wavelet based estimator for long memory processes

Two wavelet based estimators are considered in this paper for the two parameters that characterize long range dependence processes. The first one is linear and is based on the statistical properties of the coefficients of a discrete wavelet transform of long range dependence processes. The estimator consists in measuring the slope (related to the long memory parameter) and the intercept (related to the variance of the process) of a linear regression after a discrete wavelet transform is performed (Veitch and Abry, 1999). In this paper its properties are reviewed, and analytic evidence is produced that the linear estimator is applicable only when the second parameter is unknown. To overcome this limitation a non linear wavelet based estimator - that takes into account that the intercept depends on the long memory parameter - is proposed here for the cases in which the second parameter is known or the only parameter of interest is the long memory parameter. Under the same hypothesis assumed for the linear estimator, the non linear estimator is shown to be asymptotically more efficient for the long memory parameter. Numerical simulations show that, even for small data sets, the bias is very small and the variance close to optimal. An application to ATM based Internet traffic is presented.Financial support from the Italian Ministry of University and Scientific Research (MIUR), also in the context of the COFIN 2002 ALINWEB (Algorithms for the Internet and the Web) Project, is gratefully acknowledged.     

Comparison of non-parametric and semi-parametric tests in detecting long memory

The first two stages in modelling times series are hypothesis testing and estimation. For long memory time series, the second stage was studied in the paper published in [M. Boutahar et al., Estimation methods of the long memory parameter: monte Carlo analysis and application, J. Appl. Statist. 34(3), pp. 261–301.] in which we have presented some estimation methods of the long memory parameter. The present paper is intended for the first stage, and hence completes the former, by exploring some tests for detecting long memory in time series. We consider two kinds of tests: the non-parametric class and the semi-parametric one. We precise the limiting distribution of the non-parametric tests under the null of short memory and we show that they are consistent against the alternative of long memory. We perform also some Monte Carlo simulations to analyse the size distortion and the power of all proposed tests. We conclude that for large sample size, the two classes are equivalent but for small sample size the non-parametric class is better than the semi-parametric one.     

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.     

Bootstrap approaches for estimation and confidence intervals of long memory processes

In this work, we investigate an alternative bootstrap approach based on a result of Ramsey [F.L. Ramsey, Characterization of the partial autocorrelation function, Ann. Statist. 2 (1974), pp. 1296–1301] and on the Durbin–Levinson algorithm to obtain a surrogate series from linear Gaussian processes with long range dependence. We compare this bootstrap method with other existing procedures in a wide Monte Carlo experiment by estimating, parametrically and semi-parametrically, the memory parameter d. We consider Gaussian and non-Gaussian processes to prove the robustness of the method to deviations from normality. The approach is also useful to estimate confidence intervals for the memory parameter d by improving the coverage level of the interval.     

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.     

Nonparametric beta kernel estimator for long and short memory time series

In this article we introduce a nonparametric estimator of the spectral density by smoothing the periodogram using beta kernel density. The estimator is proved to be bounded for short memory data and diverges at the origin for long memory data. The convergence in probability of the relative error and Monte Carlo simulations show that the proposed estimator automatically adapts to the long- and the short-range dependency of the process. A cross-validation procedure is studied in order to select the nuisance parameter of the estimator. Illustrations on historical as well as most recent returns and absolute returns of the S&P500 index show the performance of the beta kernel estimator. The Canadian Journal of Statistics 48: 582–595; 2020 © 2020 Statistical Society of Canada     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

First‐order bias correction for fractionally integrated time series

Most of the long memory estimators for stationary fractionally integrated time series models are known to experience non‐negligible bias in small and finite samples. Simple moment estimators are also vulnerable to such bias, but can easily be corrected. In this article, the authors propose bias reduction methods for a lag‐one sample autocorrelation‐based moment estimator. In order to reduce the bias of the moment estimator, the authors explicitly obtain the exact bias of lag‐one sample autocorrelation up to the order n⁻¹. An example where the exact first‐order bias can be noticeably more accurate than its asymptotic counterpart, even for large samples, is presented. The authors show via a simulation study that the proposed methods are promising and effective in reducing the bias of the moment estimator with minimal variance inflation. The proposed methods are applied to the northern hemisphere data. The Canadian Journal of Statistics 37: 476–493; 2009 © 2009 Statistical Society of Canada     

Misspecification tests for periodic long memory GARCH models

Distributional theory for Quasi-Maximum Likelihood estimators in long memory conditional heteroskedastic models is not formally defined, even asymptotically. Because of that, this paper analyses the real size and power of the likelihood ratio and the Lagrange multiplier misspecification tests when periodic long memory GARCH models are involved. The performance of these tests is studied by means of Monte Carlo simulations with respect to the class of generalized long memory GARCH models. For this class of models, analytical derivatives are developed. An application to the USD/JPY exchange rate is also provided.     

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.     

Detecting and identifying interventions with the Whittle spectral approach in a long memory panel data model

This article provides a procedure for the detection and identification of outliers in the spectral domain where the Whittle maximum likelihood estimator of the panel data model proposed by Chen [W.D. Chen, Testing for spurious regression in a panel data model with the individual number and time length growing, J. Appl. Stat. 33(88) (2006b), pp. 759–772] is implemented. We extend the approach of Chang and co-workers [I. Chang, G.C. Tiao, and C. Chen, Estimation of time series parameters in the presence of outliers, Technometrics 30 (2) (1988), pp. 193–204] to the spectral domain and through the Whittle approach we can quickly detect and identify the type of outliers. A fixed effects panel data model is used, in which the remainder disturbance is assumed to be a fractional autoregressive integrated moving-average (ARFIMA) process and the likelihood ratio criterion is obtained directly through the modified inverse Fourier transform. This saves much time, especially when the estimated model implements a huge data-set.

Through Monte Carlo experiments, the consistency of the estimator is examined by growing the individual number N and time length T, in which the long memory remainder disturbances are contaminated with two types of outliers: additive outlier and innovation outlier. From the power tests, we see that the estimators are quite successful and powerful.

In the empirical study, we apply the model on Taiwan's computer motherboard industry. Weekly data from 1 January 2000 to 31 October 2006 of nine familiar companies are used. The proposed model has a smaller mean square error and shows more distinctive aggressive properties than the raw data model does.     

Identification of long memory in GARCH models

Abstract: This work extends the analysis of Baillie, Bollerslev and Mikkelsen (1996) and Bollerslev and Mikkelsen (1996) on the estimation and identification problems of the Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastik (FIGARCH) model. We assess the power of different information criteria and tests in identifying the presence of long memory in the conditional variances. The analysis is performed with a Montecarlo simulation study. In detail, the focus on the Akaike, Hannan-Quinn, Shibata and Schwarz information criteria and on the Jarque-Bera test for normality, Box-Pierce test for residual correlation and Engle test for ARCH effects. This study verifies that information criteria clearly distinguish the presence of long memory while tests do not evidence any difference between the fitted long and short memory models. An empirical application is provided; it analyses, on a high frequency dataset, the returns of the FIB30, the future on the MIB30, the Italian stock market index of highly capitalized firms.Massimiliano Caporin: mcaporin@unive.itThis paper was presented at the SIS 2002 Conference (Italian Statistical society annual meeting) held in Milan, University Bicocca, 5-7 June 2002. A short version of this work can be found in the proceedings of the conference     

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

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.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Understanding Estimators of Treatment Effects in Regression Discontinuity Designs

In this paper, we propose two new estimators of treatment effects in regression discontinuity designs. These estimators can aid understanding of the existing estimators such as the local polynomial estimator and the partially linear estimator. The first estimator is the partially polynomial estimator which extends the partially linear estimator by further incorporating derivative differences of the conditional mean of the outcome on the two sides of the discontinuity point. This estimator is related to the local polynomial estimator by a relocalization effect. Unlike the partially linear estimator, this estimator can achieve the optimal rate of convergence even under broader regularity conditions. The second estimator is an instrumental variable estimator in the fuzzy design. This estimator will reduce to the local polynomial estimator if higher order endogeneities are neglected. We study the asymptotic properties of these two estimators and conduct simulation studies to confirm the theoretical analysis.