Prediction intervals in the ARFIMA model using bootstrap G (original) (raw)
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2014
Background:Frequently, an estimated mean squared error is the only indicator or yardstick of measuring error in a prediction. However, the statement that the future values falls in an interval with a specified probability is more informative. Prediction intervals have this probabilistic interpretation, which is similar to that of tolerance intervals. Two resampling methods yield prediction intervals that obtain some types of asymptotic invariance to the sampling distribution. The resampling procedure proposed here utilizes the bootstrap method. The bootstrap interval derives from an empirical distribution generated using bootstrap resampling. The bootstrap is a resampling technique whose aim is to gain information on the distribution of an estimator. Objective: The bootstrap method for measures of Statistical accuracy such as standard error, bias, prediction error and to complicated data structures such as autoregressive models are considered. We estimated the parameters and the bootstrap t confidence interval with an autoregressive model fitted to the real data. Results:Bootstrap prediction intervals provide a non parametric measure of the probable error of forecast from a standard linear autoregressive model. Empirical measure prediction error rate motivate the choice of these intervals, which are calculated by an application of the bootstrap methods, to a time series data. Conclution: Bootstrap prediction intervals represent a useful addition to the traditional set of measures to assess the accuracy of forecast. The asymptotic properties of the intervals do not depend upon the sampling distribution, and the bootstrap results suggest that the invariance approximately holds for relatively all sample sizes.
Prediction Intervals for ARIMA Models
Journal of Business & Economic Statistics, 2001
The problem of constructing prediction intervals for linear time series (ARIMA) models is examined. The aim is to find prediction intervals which incorporate an allowance for sampling error associated with parameter estimates. The effect of constraints on parameters arising from stationarity and invertibility conditions is also incorporated. Two new methods, based to varying degrees on first-order Taylor approximations, are proposed. These are compared in a simulation study to two existing methods: a heuristic approach and the `plug-in' method whereby parameter values are set equal to their maximum likelihood estimates
Another look at the forecast performance of ARFIMA models
International Review of Financial Analysis, 2004
This paper investigates the out-of-sample forecast performance of the autoregressive fractionally integrated moving average [ARFIMA (0,d,0)] specification, both when the underlying value of the fractional differencing parameter (d) is known a priori and when it is unknown. Forecast performance is measured relative to simple deterministic models and a random walk model, for forecast horizons up to 100 periods ahead. Overall, the linear models tend to outperform the ARFIMA specification for both the positive and negative values of d for the simulated series, and for positive d values from the real time-series data. The results of the study question the use of the ARFIMA specification as a forecast tool.
Journal of Applied Statistics, 2020
The bootstrap procedure has emerged as a general framework to construct prediction intervals for future observations in autoregressive time series models. Such models with outlying data points are standard in real data applications, especially in the field of econometrics. These outlying data points tend to produce high forecast errors, which reduce the forecasting performances of the existing bootstrap prediction intervals calculated based on non-robust estimators. In the univariate and multivariate autoregressive time series, we propose a robust bootstrap algorithm for constructing prediction intervals and forecast regions. The proposed procedure is based on the weighted likelihood estimates and weighted residuals. Its finite sample properties are examined via a series of Monte Carlo studies and two empirical data examples.
Bootstrap predictive inference for ARIMA processes
Journal of Time Series Analysis, 2004
In this study, we propose a new bootstrap strategy to obtain prediction intervals for autoregressive integrated moving average processes. Its main advantage over other bootstrap methods previously proposed for autoregressive integrated processes is that variability due to parameter estimation can be incorporated into prediction intervals without requiring the backward representation of the process. Consequently, the procedure is very flexible and can be extended to processes even if their backward representation is not available. Furthermore, its implementation is very simple. The asymptotic properties of the bootstrap prediction densities are obtained. Extensive finite sample Monte Carlo experiments are carried out to compare the performance of the proposed strategy vs. alternative procedures. The behaviour of our proposal equals or outperforms the alternatives in most of the cases. Furthermore, our bootstrap strategy is also applied for the first time to obtain the prediction density of processes with moving average components.
Prediction intervals for fractionally integrated time series and volatility models
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The two of the main formulations for modeling long range dependence in volatilities associated with financial time series are fractionally integrated generalized autoregressive conditional heteroscedastic (FIGARCH) and hyperbolic generalized autoregressive conditional heteroscedastic (HYGARCH) models. The traditional methods of constructing prediction intervals for volatility models, either employ a Gaussian error assumption or are based on asymptotic theory. However, many empirical studies show that the distribution of errors exhibit leptokurtic behavior. Therefore, the traditional prediction intervals developed for conditional volatility models yield poor coverage. An alternative is to employ residual bootstrap-based prediction intervals. One goal of this dissertation research is to develop methods for constructing such prediction intervals for both returns and volatilities under FIGARCH and HYGARCH model formulations. In addition, this methodology is extended to obtain prediction...
Computational Statistics & Data Analysis, 1998
Multiple forecasts for autoregressive-integrated moving-average (ARIMA) models are useful in many areas such as economics and business forecasting. In recent years, approximation methods to construct simultaneous prediction intervals for multiple forecasts arc developed. These methods were based on highex-order Bonfcrroni and product-type inequalities. In this article, we compare the 'exact' method which requires the evaluation of multivariate normal probabilities to the approximation methods. It is found that the exact method is computationally far more efficient. Furthermore, the exact method can be applied to all ARIMA models while the approximation methods are limited to only a subset of ARIMA models. Illustrative examples are given to compare the performance of various procedures. (~) 1998 Elsevier Science B.V. All rights reserved.
Bootstrap prediction intervals for power-transformed time series
International Journal of Forecasting, 2005
In this paper we propose a bootstrap resampling scheme to construct prediction intervals for future values of a variable after a linear ARIMA model has been fitted to a power transformation of it. The advantages over existing methods for computing prediction intervals of power transformed time series are that the proposed bootstrap intervals incorporate the variability due to parameter estimation, and do not rely on distributional assumptions neither on the original variable nor on the transformed one. We show the good behavior of the bootstrap approach versus alternative procedures by means of Monte Carlo experiments. Finally, the procedure is illustrated by analysing three real time series data sets. ______________________________________________________________________ ___ Abstract In this paper we propose a bootstrap resampling scheme to construct prediction intervals for future values of a variable after a linear ARIMA model has been …tted to a power transformation of it. The advantages over existing methods for computing prediction intervals of power transformed time series are that the proposed bootstrap intervals incorporate the variability due to parameter estimation, and do not rely on distributional assumptions neither on the original variable nor on the transformed one. We show the good behavior of the bootstrap approach versus alternative procedures by means of Monte Carlo experiments. Finally, the procedure is illustrated by analysing three real time series data sets.
Effects of parameter estimation on prediction densities: a bootstrap approach
International Journal of Forecasting, 2001
We use a bootstrap procedure to study the impact of parameter estimation on prediction densities, focusing on seasonal ARIMA processes with possibly non normal innovations. We compare prediction densities obtained using the Box and Jenkins approach with bootstrap densities which may be constructed either taking into account parameter estimation variability or using parameter estimates as if they were known parameters. By means of Monte Carlo experiments, we show that the average coverage of the intervals is closer to the nominal value when intervals are constructed incorporating parameter uncertainty. The effects of parameter estimation are particularly important for small sample sizes and when the error distribution is not Gaussian. We also analyze the effect of the estimation method on the shape of prediction densities comparing prediction densities constructed when the parameters are estimated by Ordinary Least Squares (OLS) and by Least Absolute Deviations (LAD). We show how, when the error distribution is not Gaussian, the average coverage and length of intervals based on LAD estimates are closer to nominal values than those based on OLS estimates. Finally, the performance of the bootstrap intervals is illustrated with two empirical examples.
On the test and estimation of fractional parameter in ARFIMA model: bootstrap approach
Applied Mathematical Sciences, 2014
One of the most important problems concerning Autoregressive Fractional Integrated Moving Average (AFRIMA) time series model is the estimation of the fractional parameter. This research work was aimed to show efficiency of the different methods used to test and estimate fractional parameter in the fractionally integrated autoregressive moving-average (AFRIMA) model. In this study, estimates were obtained by smoothed spectral regression method and truncated geometric bootstrap method which aid in the test and estimation of fractional parameter by obtaining estimates through regression estimation method. The results indicate that the semi-parametric methods outperformed the parametric method when elements of AR or MA components are involved. Performance of one of the semi-parametric method (Robinson estimator) usually is not as good as the other semiparametric method: it has large bias, standard deviation and mean square error. The use of smoothed periodogram in this method improves the estimate; however, they are still not as good as the usual semi-parametric methods. In the long run, having compared the result of the two estimates, it was discovered that the bootstrap approach that is, the stimulation obtained using the truncated geometric bootstrap method produced a set of 4784 T. O. Olatayo and A. F. Adedotun fractional parameter of the estimates of the parameters of ARFIMA models that behave better and has reasonably good power.