Fractional ARIMA Model Research Articles

A Fractional ARIMA (ARFIMA) Model in the Analysis of Historical Crude Oil Prices

We investigate historical data for crude oil prices using autoregressive fractionally integrated moving average (ARFIMA) models to determine whether shocks in the series have transitory or permanent effects. Our best specification is an ARFIMA(2,d,2) with an estimated value of d around 0.4, but its confidence interval is wide and does not allow us to either reject the I(0) or the I(1) hypotheses. This high level of uncertainty may be due to the presence of breaks or non-linear trends in the data.

Moment estimator for an AR(1) model driven by a long memory Gaussian noise

In this paper, we consider an inference problem for the first order autoregressive process driven by a given long memory stationary Gaussian noise. Suppose that the covariance function of the noise is positive and can be expressed as |k|2H−2 times a positive function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA(0, d, 0) model with d∈(0,12) and some others Gaussian noise are special examples that satisfy this assumption. We propose a moment estimator and prove the strong consistency, the asymptotic normality and the almost sure central limit theorem. Moreover, we give the upper Berry–Esséen bound by means of Fourth moment theorem.

Time series forecasting methods for the Baltic dry index

AbstractThis paper forecasts the daily Baltic Dry Index (BDI) using time series and machine learning methods. Significant business cycles and freight rate volatility present in the ocean‐going shipping industry make the ability to forecast freight rates and cycles extremely important for business decisions. Data‐driven model selection based on data characteristics is performed through ARIMA, fractional ARIMA (FARIMA), and ARIMA and FARIMA models with GARCH and EGARCH errors. The corresponding machine learning techniques utilized are feed‐forward fully connected artificial neural networks (ANNs), support vector regression (SVR), and multivariate adaptive regression splines (MARS). Among others, FARIMA and MARS models are used for the first time in forecasting the BDI. Diebold–Mariano tests reveal that time series and machine learning approaches have comparable performance. However, combinations of forecasts of the selected models lead to better forecasting accuracy overall. Bai and Perron tests are utilized to check the robustness of the results over different cycles through the detection of breakpoints in the series.

Improving quality of adaptive video by traffic prediction with (F)ARIMA models

During the past years, adaptive video based on hypertext transfer protocol HTTP has become very popular. Streaming of the adaptive video relies heavily on an estimation of end-to-end network throughput, which can be challenging especially in mobile networks, where the capacity highly fluctuates. In this work, we propose to predict the network throughput using its past measurements. As the analysis shows, the network throughput forms a long range-dependent process, thus, for the throughput prediction we apply fractional ARIMA (FARIMA) model. Our approach does not require any modifications to the network infrastructure or the TCP stack. The predictions are performed for data traces obtained from measurements of throughput of a real mobile network. As the experiment shows, the obtained traffic model is able to enhance the performance of an adaptive streaming algorithm. Compared to the throughput predictors employed in contemporary systems dedicated to adaptive video streaming, the proposed technique obtains better results when taking into account effectiveness of network capacity utilisation and stability of video play-out.

An Application of the Seasonal Fractional ARIMA Model to the Semiconductor Manufacturing

In semiconductor manufacturing industry, run to run (R2R) model based process control (MBPC), which is based on the exponentially weighted moving-average (EWMA) filter, has been widely applied to chemical-mechanical planarization (CMP), chemical vapor deposition (CVD) processes in order to provide the ability to track and compensate for process drifts without a priori assumptions on magnitude or consistency. However, once human interactions and periodic components are added to the R2R process, overall system becomes complex. Therefore a fractional autoregressive integrated moving average (FARIMA) model is proposed to maintain an accurate estimation of how the etching rate changes from R2R or even chamber to chamber (C2C).

Nonparametric regression with nearly integrated regressors under long-run dependence

We study nonparametric estimation of regression function with nonstationary (integrated or nearly integrated) covariates and the error series of the regressor process following a fractional ARIMA model. A local linear estimation method is developed to estimate the unknown regression function. The asymptotic results of the resulting estimator at both interior points and boundaries are obtained. The asymptotic distribution is mixed normal, associated with the local time of an Ornstein-Uhlenbeck (O-U) fractional Brownian motion. Furthermore, we study the Nadaraya-Watson estimator and examine its asymptotic results. As a result, it shares exactly the same asymptotic results as those for the local linear estimator for the zero energy situation. But for the non-zero energy case, the local linear estimator is superior over the Nadaraya-Watson estimator in terms of optimal convergence rate. Moreover, a comparison of our results with the conventional results for stationary covariates is presented. Finally, a Monte Carlo simulation is conducted to illustrate the finite sample performance of the proposed estimator. This article is protected by copyright. All rights reserved

Estimation for seasonal fractional ARIMA with stable innovations via the empirical characteristic function method

Seasonal fractional ARIMA (ARFISMA) model with infinite variance innovations is used in the analysis of seasonal long-memory time series with large fluctuations (heavy-tailed distributions). Two methods, which are the empirical characteristic function (ECF) procedure developed by Knight and Yu [The empirical characteristic function in time series estimation. Econometric Theory. 2002;18:691–721] and the Two-Step method (TSM) are proposed to estimate the parameters of stable ARFISMA model. The ECF method estimates simultaneously all the parameters, while the TSM considers in the first step the Markov Chains Monte Carlo–Whittle approach introduced by Ndongo et al. [Estimation of long-memory parameters for seasonal fractional ARIMA with stable innovations. Stat Methodol. 2010;7:141–151], combined with the maximum likelihood estimation method developed by Alvarez and Olivares [Méthodes d'estimation pour des lois stables avec des applications en finance. Journal de la Société Française de Statistique. 2005;1(4):23–54] in the second step. Monte Carlo simulations are also used to evaluate the finite sample performance of these estimation techniques.

Analyzing South Africa’s inflation persistence using an ARFIMA model with Markov-switching fractional differencing parameter

The successful conduct of monetary policy relies on accurately characterising inflation’s data generating properties. Monetary policy errors that allow inflation to transition to a high inflation regime that is very persistent might have costly economic implications as the central bank attempts to bring inflation to a lower regime, say at some target level. This paper studies the duration of inflation persistence over time and across various policy regimes. We test the inertial properties of South African inflation in a Markov-Switching autoregressive fractionally integrated moving average model. We isolate period of high inflation and low inflation and analyse how persistent it is. This is an unique application to South Africa. The use of a fractional differencing ARIMA model allows for the possibility that inflation is close to a unit root, however, still mean reverting. This implies that shocks to inflation is very persistent and take long to dissipate. The inflation persistence is measured using a test by Ng and Perron (2001). We show that inflation is more persistent during high inflation episodes relative to low inflation episodes and more volatile during low inflation periods compared to high inflation periods. We estimate that it takes approximately 70 months for 50 percent of the shocks to dissipate in a high inflation regime compared to 10 months in a low inflation regime. The model identifies three structural breaks - a low inflation regime from 1920 until 1960, a high inflation regime from 1961 until 2003, and another low inflation regime over part of the inflation targeting period, 2003-2014. We also show that inflation persistence in the high inflation regime transitioned to a low inflation regime only much later than the implementation of inflation targeting - hinting that agents take time to adjust expectations. This has an important consequence for monetary policy - monetary policy errors that allow inflation to transition to a high inflation regime may take many months for any corrective policy to become effective.

장기기억 특성과 이분산성을 고려한 인터넷 트래픽 예측을 위한 시계열 모형 연구

본 논문은, 장기기억 특성과 이분산성을 고려한 인터넷 트래픽 예측 모형을 제안하고자 한다. 트래픽 과부하를 대비하기 위해서, 트래픽 용량은 트래픽의 예측치와 트래픽의 변동 크기에 따라 트래픽의 최대용량을 설정하여야 한다. 이를 위하여 교내 트래픽 자료 중 교내로 들어오는 트래픽과 교외로 나가는 트래픽에 이분산성과 장기기억 모형의 유용성을 확인하였다. 이에 대하여 AR-GARCH 모형, ARMA-GARCH 모형과 장기기억모형인 Fractional ARIMA와 장기기억과 이분산성을 고려한 Fractional ARMA-GARCH 모형을 적용하여 모형의 예측성능을 비교하였다. In this paper, we propose the time series forecasting models for internet traffic with long memory and heteroscedasticity. To control and forecast traffic volume, we first introduce the traffic forecasting models which are determined by the volatility and heteroscedasticity of the traffic. We then analyze and predict the heteroscedasticity and the long memory properties for forecasting traffic volume. Depending on the characteristics of the traffic, Fractional ARIMA model, Fractional ARIMA-GARCH model are applied and compared with the MAPE(Mean Absolute Percentage Error) Criterion.

Assessment of Risk of Rainfall Events with a Hybrid of ARFIMA-GARCH

Hazardous situations related to rainfall events can be due to very intense rainfall, or to the persistence of rainfall over a long period of time. Such events may result to an exceedence of the capacity of drainage systems resulting in the heap of basements which may lead to landslides and flooding. This study assesses the persistence dependence of rainfall time series of Chui Chak, a station in Peninsular Malaysia that observed the highest rainfall event for the period 01/01/1975-31/12/2008. The persistence dependence of the rainfall time series was modelled via fractional ARIMA model augmented with the GARCH model. The Ljung-Box test for testing autocorrelation proves that the combined ARFIMA-GARCH model captures the temporal persistence behaviour in the Chui Chak rainfall time series data with persistence measure 0.839. This measure represents a relatively lasting persistence, that is, the process variability should return to the historical average after a relatively long period of time which may have a risk of extreme event.

A semiparametric approach to estimate two seasonal fractional parameters in the SARFIMA model

This paper explores seasonal and long-memory time series properties by using the fractional ARIMA model when the data have one and two seasonal periods and short-memory components. The stationarity and invertibility parameter conditions are established for the model studied. To estimate the seasonal fractional long-memory parameters, a semiparametric estimation method is proposed. The asymptotic properties of the estimator are established and the accuracy of the method is investigated through Monte Carlo experiments. The good performance of the estimator indicates that it can be an alternative procedure to estimate long-memory time series data with two seasonal periods. Series of PM10 concentrations and electricity hourly demand are considered as examples of applications of the proposed estimation method.

Performance Evaluation of Time Series Models using Short-Term Air Passenger Data

We perform a comparison of time series models that include seasonal ARIMA, Fractional ARIMA, and Holt-Winters models; in addition, we also consider hourly and daily air passenger data. The results of the performance evaluation of the models show that the Holt-Winters methods outperforms other models in terms of MAPE.

Unified asymptotic theory for nearly unstable AR([formula omitted]) processes

A unified asymptotic theory for nearly unstable higher order autoregressive processes and their least squares estimates is established. A novel version of Jordan’s canonical decomposition with perturbations together with a suitable plug-in principle is proposed to develop the underlying theories. Assumptions are stated in terms of the domain of attraction of partial Fourier transforms. The machinery is applied to recapture some of the classical results with the driving noise being martingale differences. Further, we show how to extend the results to higher order fractional ARIMA models in nearly unstable settings, thereby offering a comprehensive theory to analyse nearly unstable time series.

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of ( g , F ) -processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those ( g , F ) -processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.

A general asymptotic theory for time-series models

This paper develops a general asymptotic theory for the estimation of strictly stationary and ergodic time–series models. Under simple conditions that are straightforward to check, we establish the strong consistency, the rate of strong convergence and the asymptotic normality of a general class of estimators that includes LSE, MLE and some M-type estimators. As an application, we verify the assumptions for the long-memory fractional ARIMA model. Other examples include the GARCH(1,1) model, random coefficient AR(1) model and the threshold MA(1) model.

A General Asymptotic Theory for Time Series Models

This paper develops a general asymptotic theory for the estimation of strictly stationary and ergodic time series models. Under simple conditions that are straightforward to check, we establish the strong consistency, the rate of strong convergence and the asymptotic normality of a general class of estimators that includes LSE, MLE, and some M-type estimators. As an application, we verify the assumptions for the long-memory fractional ARIMA model. Other examples include the GARCH(1,1) model, random coefficient AR(1) model and the threshold MA(1) model.

Testing for change points in time series models and limiting theorems for NED sequences

This paper first establishes a strong law of large numbers and a strong invariance principle for forward and backward sums of near-epoch dependent sequences. Using these limiting theorems, we develop a general asymptotic theory on the Wald test for change points in a general class of time series models under the no change-point hypothesis. As an application, we verify our assumptions for the long-memory fractional ARIMA model.

Do long swings in the business cycle lead to strong persistence in output?

This paper investigates how the occasional long swing in the business cycle can produce long-memory behavior in US output. To prove this theoretical relationship, we extend the Hamilton Markov chain regime switching model of real aggregate output to include the occasional long regime. We do this by modeling the duration length of the expansion and recession regimes as draws from a fat-tailed distribution with realized durations that are high in variability and occasionally extreme in value. Empirically, we find that the tail indices for the length of US economic booms and busts correspond with the long-memory parameter estimates of Diebold and Rudebusch [1989. Long memory and persistence in aggregate output. Journal of Monetary Economics 24, 189–209] and Sowell [1992a. Modeling long-run behavior with the fractional ARIMA model. Journal of Monetary Economics 29, 277–302] for real US output. Estimates of our extended regime switching model produce better short- and long-run forecasts of output in comparison to forecasts with a fractionally integrated model. Furthermore, our estimated regime-switching model finds US expansions to be fragile during their infancy, but become more and more likely to continue after surviving the first seven quarters.

Nelson and Plosser Revisited: Evidence from Fractional ARIMA Models

In this study fractionally integrated ARIMA (ARFIMA) models are estimated using an extended version of Nelson and Plosser[1]’s data set. The analysis employs[11]’s maximum likelihood procedure. Such a parametric approach requires the model to be correctly specified in order for the estimates to be consistent. A model-selection procedure based on diagnostic tests on the residuals, together with several likelihood criteria, is adopted to determine the correct specification for each series. The results suggest that all series, except unemployment and bond yields, are integrated of order greater than one. Thus, the standard approach of taking first differences may result in stationary series with long memory behavior.

Trend assessment: applications for hydrology and climate research

Abstract. The assessment of trends in climatology and hydrology still is a matter of debate. Capturing typical properties of time series, like trends, is highly relevant for the discussion of potential impacts of global warming or flood occurrences. It provides indicators for the separation of anthropogenic signals and natural forcing factors by distinguishing between deterministic trends and stochastic variability. In this contribution river run-off data from gauges in Southern Germany are analysed regarding their trend behaviour by combining a deterministic trend component and a stochastic model part in a semi-parametric approach. In this way the trade-off between trend and autocorrelation structure can be considered explicitly. A test for a significant trend is introduced via three steps: First, a stochastic fractional ARIMA model, which is able to reproduce short-term as well as long-term correlations, is fitted to the empirical data. In a second step, wavelet analysis is used to separate the variability of small and large time-scales assuming that the trend component is part of the latter. Finally, a comparison of the overall variability to that restricted to small scales results in a test for a trend. The extraction of the large-scale behaviour by wavelet analysis provides a clue concerning the shape of the trend.