Bootstrapping prediction intervals for autoregressive models

Methods of improving the coverage of Box–Jenkins prediction intervals for linear autoregressive models are explored. These methods use bootstrap techniques to allow for parameter estimation uncertainty and to reduce the small-sample bias in the estimator of the models’ parameters. In addition, we also consider a method of bias-correcting the non-linear functions of the parameter estimates that are used to generate conditional multi-step predictions.

Bootstrap prediction intervals for autoregressive time series

The calculation of interval forecasts for highly persistent autoregressive (AR) time series based on the bootstrap is considered. Three methods are considered for countering the small-sample bias of least-squares estimation for processes which have roots close to the unit circle: a bootstrap bias-corrected OLS estimator; the use of the Roy–Fuller estimator in place of OLS; and the use of the Andrews–Chen estimator in place of OLS. All three methods of bias correction yield superior results to the bootstrap in the absence of bias correction. Of the three correction methods, the bootstrap prediction intervals based on the Roy–Fuller estimator are generally superior to the other two. The small-sample performance of bootstrap prediction intervals based on the Roy–Fuller estimator are investigated when the order of the AR model is unknown, and has to be determined using an information criterion.

An historical perspective on forecast errors

Using annual observations on industrial production over the last three centuries, and on GDP over a 100-year period, we seek an historical perspective on the forecastability of these UK output measures. The series are dominated by strong upward trends, so we consider various specifications of this, including the local linear trend structural time-series model, which allows the level and slope of the trend to vary. Our results are not unduly sensitive to how the trend in the series is modelled: the average sizes of the forecast errors of all models, and the wide span of prediction intervals, attests to a great deal of uncertainty in the economic environment. It appears that, from an historical perspective, the postwar period has been relatively more forecastable.

Assessing macro uncertainty in real-time when data are subject to revision

Model-based estimates of future uncertainty are generally based on the in-sample fit of the model, as when Box-Jenkins prediction intervals are calculated. However, this approach will generate biased uncertainty estimates in real time when there are data revisions. A simple remedy is suggested, and used to generate more accurate prediction intervals for 25 macroeconomic variables, in line with the theory. A simulation study based on an empirically-estimated model of data revisions for US output growth is used to investigate small-sample properties.

Multiprobabilistic prediction in early medical diagnoses

This paper describes the methodology of providing multiprobability predictions for proteomic mass spectrometry data. The methodology is based on a newly developed machine learning framework called Venn machines. Is allows to output a valid probability interval. The methodology is designed for mass spectrometry data. For demonstrative purposes, we applied this methodology to MALDI-TOF data sets in order to predict the diagnosis of heart disease and early diagnoses of ovarian cancer and breast cancer. The experiments showed that probability intervals are narrow, that is, the output of the multiprobability predictor is similar to a single probability distribution. In addition, probability intervals produced for heart disease and ovarian cancer data were more accurate than the output of corresponding probability predictor. When Venn machines were forced to make point predictions, the accuracy of such predictions is for the most data better than the accuracy of the underlying algorithm that outputs single probability distribution of a label. Application of this methodology to MALDI-TOF data sets empirically demonstrates the validity. The accuracy of the proposed method on ovarian cancer data rises from 66.7 % 11 months in advance of the moment of diagnosis to up to 90.2 % at the moment of diagnosis. The same approach has been applied to heart disease data without time dependency, although the achieved accuracy was not as high (up to 69.9 %). The methodology allowed us to confirm mass spectrometry peaks previously identified as carrying statistically significant information for discrimination between controls and cases.