Estimation of structural mean breaks for long-memory data sets

In long-memory data sets such as the realized volatilities of financial assets, a sequential test is developed for the detection of structural mean breaks. The long memory, if any, is adjusted by fitting an HAR (heterogeneous autoregressive) model to the data sets and taking the residuals. Our test consists of applying the sequential test of Bai and Perron [Estimating and testing linear models with multiple structural changes. Econometrica. 1998;66:47–78] to the residuals. The large-sample validity of the proposed test is investigated in terms of the consistency of the estimated number of breaks and the asymptotic null distribution of the proposed test. A finite-sample Monte-Carlo experiment reveals that the proposed test tends to produce an unbiased break time estimate, while the usual sequential test of Bai and Perron tends to produce biased break times in the case of long memory. The experiment also reveals that the proposed test has a more stable size than the Bai and Perron test. The proposed test is applied to two realized volatility data sets of the S&P index and the Korea won-US dollar exchange rate for the past 7 years and finds 2 or 3 breaks, while the Bai and Perron test finds 8 or more breaks.     

Semiparametric Whittle estimation of a cyclical long-memory time series based on generalised exponential models

This paper considers a semiparametric estimation of the memory parameter in a cyclical long-memory time series, which exhibits a strong dependence on cyclical behaviour, using the Whittle likelihood based on generalised exponential (GEXP) models. The proposed estimation is included in the so-called broadband or global method and uses information from the spectral density at all frequencies. We establish the consistency and the asymptotic normality of the estimated memory parameter for a linear process and thus do not require Gaussianity. A simulation study conducted using Monte Carlo experiments shows that the proposed estimation works well compared to other existing semiparametric estimations. Moreover, we provide an empirical application of the proposed estimation, applying it to the growth rate of Japan's industrial production index and detecting its cyclical persistence.     

Estimation of the frequency in cyclical long-memory series

Time series with cyclical long memory are characterized by a spectral pole at some frequency ω between 0 and π such that the series has a persistent cycle of period 2π/ω, implying a quasi-periodic behaviour that slightly evolves with time. Accurate estimation of ω is needed for a precise determination of the characteristic of the series (e.g. for business cycle determination or signal estimation). We propose a simple iterative algorithm of estimation of ω based on the maximizer of the periodogram evaluated at an increasingly finer grid of frequencies and compare its performance with more usual methods of estimation restricted to Fourier frequencies. We also apply this technique to the estimation of the frequency of the sunspot index and the business cycle of the differenced unemployment level of the USA.     

Indirect inference for fractional time series models

In this paper we present an indirect estimation procedure for (ARFIMA) fractional time series models.The estimation method is based on an ‘incorrect’criterion which does not directly provide a consistent estimator of the parameters of interest,but leads to correct inference by using simulations.

The main steps are the following. First,we consider an auxiliary model which can be easily estimated.Specifically,we choose the finite lag Autoregressive model.Then, this is estimated on the observations and simulated values drawn from the ARFIMA model associated with a given value of the parameters of interest.Finally,the latter is calibrated in order to obtain close values of the two estimators of the auxiliary parameters.

In this article,we describe the estimation procedure and compare the performance of the indirect estimator with some alternative estimators based on the likelihood function by a Monte Carlo study.     

Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.     

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.     

On parameter estimation for locally stationary long-memory processes

We consider parameter estimation for time-dependent locally stationary long-memory processes. The asymptotic distribution of an estimator based on the local infinite autoregressive representation is derived, and asymptotic formulas for the mean squared error of the estimator, and the asymptotically optimal bandwidth are obtained. In spite of long memory, the optimal bandwidth turns out to be of the order n^-1/5$n^{-1/5}$ and inversely proportional to the square of the second derivative of d. In this sense, local estimation of d is comparable to regression smoothing with iid residuals.     

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.     

Tests for structural breaks in memory parameters of long-memory heterogeneous autoregressive models

Tests for structural breaks in the coefficients of the long-memory heterogeneous autoregressive (HAR) models are developed. The tests are based on the partial sum process of the normalized efficient score vector. The tests have the nice property of identifying the parameters of the daily, weekly, and monthly regressors in which breaks occur. Limiting null distributions of the proposed tests are proven to be derived from standard Brownian bridges. A finite sample Monte-Carlo experiment shows reasonable size and power properties of the proposed tests. The proposed method is illustrated by a real data analysis.     

Residual empirical processes for nearly unstable long-memory time series

This paper studies the goodness-of-fit test of the residual empirical process of a nearly unstable long-memory time series. Chan and Ling (2008) showed that the usual limit distribution of the Kolmogorov–Smirnov test statistics does not hold for an unstable autoregressive model. A key question of interest is what happens when this model has a near unit root, that is, when it is nearly unstable. In this paper, it is established that the statistics proposed by Chan and Ling can be generalized to encompass nearly unstable long-memory models. In particular, the limit distribution is expressed as a functional of an Ornstein–Uhlenbeck process that is driven by a fractional Brownian motion. Simulation studies demonstrate that the limit distribution of the statistic possesses desirable finite sample properties and power.     

Local asymptotic normality for long-memory process with strong mixing noises

Abstract

We will establish the local asymptotic normality (LAN) for fractional autoregressive long memory model in the case of strong mixing noises. This opens the way in future work to construct an adaptive estimator and construct optimal tests for the parameters. To check the feasibility and validity of our theoretical results a simulations study is considered.     

Invertibility of non-linear time series models

Sufficient conditions for invertibility of non-linear time series models are available in the literature only for a few special cases. In this paper a practical and general method for checking invertibility is presented. Briefly stated, it consists of feeding independent and identically distributed innovations into the non-linear model and then observing whether the model blows up or not. Using this idea invertibility conditions are derived for several recently proposed non-linear moving average models. Finally, the method is applied to a number of bilinear models fitted to economic time series.     

Structural breaks in panel data: Large number of panels and short length time series

The detection of (structural) breaks or the so called change point problem has drawn increasing attention from the theoretical, applied economic and financial fields. Much of the existing research concentrates on the detection of change points and asymptotic properties of their estimators in panels when N, the number of panels, as well as T, the number of observations in each panel are large. In this paper we pursue a different approach, i.e., we consider the asymptotic properties when N→∞ while keeping T fixed. This situation is typically related to large (firm-level) data containing financial information about an immense number of firms/stocks across a limited number of years/quarters/months. We propose a general approach for testing for break(s) in this setup. In particular, we obtain the asymptotic behavior of test statistics. We also propose a wild bootstrap procedure that could be used to generate the critical values of the test statistics. The theoretical approach is supplemented by numerous simulations and by an empirical illustration. We demonstrate that the testing procedure works well in the framework of the four factors CAPM model. In particular, we estimate the breaks in the monthly returns of US mutual funds during the period January 2006 to February 2010 which covers the subprime crises.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

Bias correction for time series factor models

In this paper we work with multivariate time series that follow a Factor Model. In particular, we consider the setting where factors are dominated by highly persistent AutoRegressive (AR) processes and samples that are rather small. Therefore, the factors' AR models are estimated using small sample bias correction techniques. A Monte Carlo study reveals that bias-correcting the AR coefficients of the factors allows to obtain better results in terms of prediction interval coverage. As expected, the simulation shows that bias-correction is more successful for smaller samples. We present the results assuming the AR order and number of factors are known as well as unknown. We also study the advantages of this technique for a set of Industrial Production Indexes of several European countries.     

Structural changes in autoregressive models for binary time series

We study autoregressive models for binary time series with possible changes in their parameters. A procedure for detection and testing of a single change is suggested. The limiting behavior of the test statistic is derived. The performance of the test is analyzed under the null hypothesis as well as under different alternatives via a simulation study. Application of the method to a real data set on US recession is provided as an illustration.     

Some simulations and applications of forecasting long-memory time-series models

In this paper, we show some results of forecasting based on the ARFIMA(p,d,q) and ARIMA(p,d,q) models. We show, by simulation, that the technique of forecasting of the ARIMA(p,d,q) model can also be used when d is fractional, i.e., for the ARFIMA(p,d,q) model. We also conduct a simulation study to compare the two estimators of d obtained through regression methods. They are used in the hypothesis test to decide whether or not the series has long memory property and are compared on the basis of their k-step ahead forecast errors. The properties of long-memory models are also investigated using an actual set of data.     

Estimation of cyclic long-memory parameters

This paper studies cyclic long-memory processes with Gegenbauer-type spectral densities. For a semiparametric statistical model, new simultaneous estimates for singularity location and long-memory parameters are proposed. This generalized filtered method-of-moments approach is based on general filter transforms that include wavelet transformations as a particular case. It is proved that the estimates are almost surely convergent to the true values of parameters. Solutions of the estimation equations are studied, and adjusted statistics are proposed. Monte-Carlo study results are presented to confirm the theoretical findings.