Prediction Intervals for ARIMA Models (original) (raw)
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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.
Prediction intervals in the ARFIMA model using bootstrap G
Financial statistical journal, 2018
This paper presents a bootstrap resampling scheme to build prediction intervals for future values in fractionally autoregressive moving average (ARFIMA) models. Standard techniques to calculate forecast intervals rely on the assumption of normality of the data and do not take into account the uncertainty associated with parameter estimation. Bootstrap procedures, as nonparametric methods, can overcome these diculties. In this paper, we test two bootstrap prediction intervals based on the nonparametric bootstrap in the residuals of the ARFIMA model. In this paper, two bootstrap prediction intervals are proposed based on the nonparametric bootstrap in the residuals of the ARFIMA model. The rst one is the well known percentile bootstrap, (Thombs and Schucany, 1990; Pascual et al., 2004), never used for ARFIMA models to the knowledge of the authors. For the second approach, the intervals are calculated using the quantiles of the empirical distribution of the bootstrap prediction errors (Masarotto, 1990; Bisaglia e Grigoletto, 2001). The intervals are compared, through a Monte Carlo experiment, to the asymptotic interval, under Gaussian and non-Gaussian error distributions. The results show that the bootstrap intervals present coverage rates closer to the nominal level assumed, when compared to the asymptotic standard method. An application to real data of temperature in New York city is also presented to illustrate the procedures.
Multiple prediction intervals for time series: Comparison of simultaneous and marginal intervals
Journal of Forecasting, 1991
Simultaneous prediction intervals for forecasts from time series models that contain L ( L 2 1) unknown future observations with a specified probability are derived. Our simultaneous intervals are based on two types of probability inequalities, i.e. the Bonferroni-and product-types. These differ from the marginal intervals in that they take into account the correlation structure between the forecast errors. For the forecasting methods commonly used with seasonal time series data, we show how to construct forecast error correlations and evaluate, using an example, the simultaneous and marginal prediction intervals. For all the methods, the simultaneous intervals are accurate with the accuracy increasing with the use of higher-order probability inequalities, whereas the marginal intervals are far too short in every case. Also, when L is greater than the seasonal period, the simultaneous intervals based on improved probability inequalities will be most accurate.
Prediction intervals for growth curve forecasts
Journal of Forecasting, 1995
Since growth curves are often used to produce medium- to long-term forecasts for planning purposes, it is obviously of value to be able to associate an interval with the forecast trend. The problems in producing prediction intervals are well described by Chatfield. The additional problems in this context are the intrinsic non-linearity of the estimation procedure and the requirement for a prediction region rather than a single interval. The approaches considered are a Taylor expansion of the variance of the forecast values, an examination of the joint density of the parameter estimates, and bootstrapping. The performance of the resultant intervals is examined using simulated data sets. Prediction intervals for real data are produced to demonstrate their practical value.
On Boostrap Prediction Intervals for Autoregressive Model
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.
Model uncertainty and the forecast accuracy of ARMA models: A survey
2015
The objective of this paper is to analyze the effects of uncertainty on density forecasts of linear univariate ARMA models. We consider three specific sources of uncertainty: parameter estimation, error distribution and lag order. For moderate sample sizes, as those usually encountered in practice, the most important source of uncertainty is the error distribution. We consider alternative procedures proposed to deal with each of these sources of uncertainty and compare their finite properties by Monte Carlo experiments. In particular, we analyze asymptotic, Bayesian and bootstrap procedures, including some very recent procedures which have not been previously compared in the literature.
A comparison of the forecasting ability of ARIMA models
Journal of Property Investment & Finance, 2007
This study compares the relative performance of ARIMA models in the forecasting of UK rents across the office, retail and industrial sectors. The performance of each model is assessed both in the estimation phase and out-of-sample. The ranked performance of each of the models is then compared to examine whether the best fitting model also tends to provide the most accurate forecast of future rental movements. The results show that while there is little evidence of a strong positive relationship between estimation and forecast performance, there is also a lack of evidence of a consistent negative relationship. In addition, in the majority of cases all ARIMA variations correctly predict general market movements.
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.
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.
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.