Long Memory Time Series Models Research Articles

Confidence Regions for Parameters in Stationary Time Series Models With Gaussian Noise

This article develops two new empirical likelihood methods for long-memory time series models based on adjusted empirical likelihood and mean empirical likelihood. By application of Whittle likelihood, one obtains a score function that can be viewed as the estimating equation of the parameters of the long-memory time series model. An empirical likelihood ratio is obtained which is shown to be asymptotically chi-square distributed. It can be used to construct confidence regions. By adding pseudo samples, we simultaneously eliminate the non-definition of the original empirical likelihood and enhance the coverage probability. Finite sample properties of the empirical likelihood confidence regions are explored through Monte Carlo simulation, and some real data applications are carried out.

Inference for Long-memory Time Series Models Based on Modified Empirical Likelihood

Empirical likelihood method has been applied to short-memory time series models by Monti (1997) through the Whittle's estimation method. Yau (2012) extended this idea to long-memory time series models. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, it experiences the undercoverage issue which causes the coverage probabilities of parameters lower than the given nominal levels, especially for small sample sizes. In this paper, we propose a modified empirical likelihood which combines the advantages of the adjusted empirical likelihood and the transformed empirical likelihood to modify the one proposed by Yau (2012) for autoregressive fractionally integrated moving average (ARFIMA) model for the purpose of improving coverage probabilities.Asymptotic null distribution of the test statistic has been established as the standard chi-square distribution with the degree of freedom one. Simulations have been conducted to investigate the performance of the proposed method as well as the comparisons of other existing methods to illustrate that the proposed method can provide better coverage probabilities especially for small sample sizes.

Forecasting of Price of Rice in India Using Long-Memory Time-Series Model

Price forecasting of agricultural commodities plays an important role in efficient planning and formulation of executive decisions. In this study, an attempt has been made to apply autoregressive fractionally integrated moving average (ARFIMA) model to the daily all India maximum, minimum and modal wholesale price data of rice in order to capture the observed long-run persistency. The price series under consideration are stationary, but there is a significant presence of long memory in the price data. Accordingly, ARFIMA model is applied to obtain the forecasts and window-based evaluation of forecasting is carried out with the help of relative mean absolute percentage error, root mean square error and mean absolute error. To this end, a comparative study has also been made between the best fitted ARFIMA model and the best fitted ARIMA model and it is observed that ARFIMA model outperforms the usual ARIMA model.

S&P500 volatility analysis using high-frequency multipower variation volatility proxies

The availability of ultra-high-frequency data has sparked enormous parametric and nonparametric volatility estimators in financial time series analysis. However, some high-frequency volatility estimators are suffering from biasness issues due to the abrupt jumps and microstructure effect that often observed in nowadays global financial markets. Hence, we motivate our studies with two long-memory time series models using various high-frequency multipower variation volatility proxies. The forecast evaluations are illustrated using the S&P500 data over the period from year 2008 to 2013. Our empirical studies found that higher-power variation volatility proxies provide better in-sample and out-of-sample performances as compared to the widely used realized volatility and fractionally integrated ARCH models. Finally, these empirical findings are used to estimate the one-day-ahead value-at-risk of S&P500.

Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations

This paper deals with the estimation of seasonal long-memory time series models in the presence of ‘outliers’. It is long known that the presence of outliers can lead to undesirable effects on the statistical estimation methods, for example, substantially impacting the sample autocorrelations. Thus, the aim of this work is to propose a semiparametric robust estimator for the fractional parameters in the seasonal autoregressive fractionally integrated moving average (SARFIMA) model, through the use of a robust periodogram at both very low and seasonal frequencies. The model and some theories related to the estimation method are discussed. It is shown by simulations that the robust methodology behaves like the classical one to estimate the long-memory parameters if there are no outliers (no contamination). On the other hand, in the contaminated scenario (presence of outliers), the standard methodology leads to misleading results while the proposed method is unaffected. The methodology is applied to model and forecast sulfur dioxide (SO2) pollutant concentrations which have seasonal long-memory features and occasional large peak pollutant concentrations.

Adjusted empirical likelihood for long-memory time-series models

The empirical likelihood method has been applied to short-memory time-series models by Monti through the Whittle’s estimation method. Yau extended this idea to long-memory time series models. Nordman and Lahiri showed a new formulation of empirical likelihood for inference of dependent data. Asymptotic distributions of the empirical likelihood ratio statistic for short and long-memory time series have been derived to construct confidence regions for the corresponding model parameters. However, computing profile empirical likelihood function involving constrained maximization does not always have a solution, which leads to several drawbacks. In this article, we propose an adjusted empirical likelihood procedure to modify the one proposed by Yau for the autoregressive fractionally integrated moving average (ARFIMA) model. It guarantees the existence of a solution to the required maximization problem and maintains an asymptotic chi-squared distribution. Simulations have been carried out to illustrate that the adjusted empirical likelihood method for different long-time series models provides competitive confidence regions and coverage probabilities compared to the unadjusted ones, especially for small sample sizes.

Long-Memory Effects in Linear Response Models of Earth’s Temperature and Implications for Future Global Warming

Abstract A linearized energy-balance model for global temperature is formulated, featuring a scale-invariant long-range memory (LRM) response and stochastic forcing representing the influence on the ocean heat reservoir from atmospheric weather systems. The model is parameterized by an effective response strength, the stochastic forcing strength, and the memory exponent. The instrumental global surface temperature record and the deterministic component of the forcing are used to estimate these parameters by means of the maximum-likelihood method. The residual obtained by subtracting the deterministic solution from the observed record is analyzed as a noise process and shown to be consistent with a long-memory time series model and inconsistent with a short-memory model. By decomposing the forcing record in contributions from solar, volcanic, and anthropogenic activity one can estimate the contribution of each to twentieth-century global warming. The LRM model is applied with a reconstruction of the forcing for the last millennium to predict the large-scale features of Northern Hemisphere temperature reconstructions, and the analysis of the residual also clearly favors the LRM model on millennium time scale. The decomposition of the forcing shows that volcanic aerosols give a considerably greater contribution to the cooling during the Little Ice Age than the reduction in solar irradiance associated with the Maunder Minimum in solar activity. The LRM model implies a transient climate response in agreement with IPCC projections, but the stronger response on longer time scales suggests replacing the notion of equilibrium climate sensitivity by a time scale–dependent sensitivity.

A long-memory integer-valued time series model, INARFIMA, for financial application

A model to account for the long-memory property in a count data framework is proposed and applied to high-frequency stock transactions data. By combining features of the INARMA and ARFIMA models, an Integer-valued Auto Regressive Fractionally Integrated Moving Average (INARFIMA) model is proposed. The unconditional and conditional first- and second-order moments are given. The CLS, FGLS and GMM estimators are discussed. In its empirical application to two stock series for AstraZeneca and Ericsson B, we find that both series have a fractional integration property.

Long memory or shifting means in geophysical time series?

In the literature many papers state that long-memory time series models such as Fractional Gaussian Noises (FGN) or Fractionally Integrated series (FI(d)) are empirically indistinguishable from models with a non-stationary mean, but which are mean reverting. We present an analysis of the statistical cost of model mis-specification when simulated long memory series are analysed by Atheoretical Regression Trees (ART), a structural break location method. We also analysed three real data sets, one of which is regarded as a standard example of the long memory type. We find that FGN and FI(d) processes do not account for many features of the real data. In particular, we find that the data sets are not H-self-similar. We believe the data sets are better characterized by non-stationary mean models.

Rank-based Inference for Multivariate Nonlinear and Long-memory Time Series Models

The portfolio of the Japanese Government Pension Investment Fund (GPIF) consists of a linear combination of five benchmarks of financial assets. Some of these exhibit long-memory and nonlinear behavior. Their analysis therefore requires multivariate nonlinear and long-memory time series models. Moreover, the assumption that the innovation densities underlying those models are known seems quite unrealistic. If those densities remain unspecified, the model becomes a semiparametric one, and rank-based inference methods naturally come into the picture. Rank-based inference methods under very general conditions are known to achieve the semiparametric efficiency bounds. Defining ranks in the context of multivariate time series models, however, is not obvious. We propose two distinct definitions. The first one relies on the assumption that the innovation density is some unspecified elliptical density. The second one relies on the assumption that the innovation process is described by some unspecified independent component analysis model. Applications to portfolio management problems are discussed.

A class of nearly long-memory time series models

We consider an autoregressive regime-switching model for the dynamic mean structure of a univariate time series. The model allows for a variety of stationary and nonstationary alternatives, and includes the possibility of approximate long-memory behavior. The proposed model includes as special cases white noise, first-order autoregression, and random walk models as well as regime-switching models and the random level-shift model proposed by Chen and Tiao, Journal of Business and Economic Statistics, 8 (1990) p. 83. We describe properties of the model, focusing on its resemblance to long-memory under a certain asymptotic parameterization. We develop a reversible-jump Markov chain Monte Carlo method for Bayesian inference on unknown model parameters and apply the methodology to the Nile River data.

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.

Unemployment hysteresis in Canada: an approach based on long-memory time series models

Long-memory univariate time series models, known as ARFIMA, are applied to measure shock persistence for Canada and the USA. Both output and unemployment exhibit higher persistence in Canada than in the USA. The finding that the rate of unemployment in Canada exhibits high shock persistence supports the notion of hysteresis in unemployment. The analysis of disaggregated Canadian unemployment rates indicates that hysteresis is most pronounced for adult male workers. The same class of models is used to obtain a generalization of the Beveridge-Nelson decomposition of time series into a permanent and a cyclical component. The permanent component of the Canadian unemployment rate exhibits a kink in the early 1970s.

BIAS IN AN ESTIMATOR OF THE FRACTIONAL DIFFERENCE PARAMETER

Abstract. An estimator of the difference parameter in a class of long‐memory time series models is examined. It is shown that, in particular circumstances, the estimator can be badly biased, and tests based on it consequently seriously misleading. The source of this bias is identified, and it is shown that its magnitude can readily be predicted through straightforward analytical arguments.

Long memory relationships and the aggregation of dynamic models

By aggregating simple, possibly dependent, dynamic micro-relationships, it is shown that the aggregate series may have univariate long-memory models and obey integrated, or infinite length transfer function relationships. A long-memory time series model is one having spectrum or order ω -2 d for small frequencies ω, d>0. These models have infinite variance for d≧ 1 2 but finite variance for d< 1 2 . For d=1 the series that need to be differenced to achieve stationarity occur, but this case is not found to occur from aggregation. It is suggested that if series obeying such models occur in practice, from aggregation, then present techniques being used for analysis are not appropriate.