Bias‐corrected bootstrap prediction regions for vector autoregression

This paper examines small sample properties of alternative bias‐corrected bootstrap prediction regions for the vector autoregressive (VAR) model. Bias‐corrected bootstrap prediction regions are constructed by combining bias‐correction of VAR parameter estimators with the bootstrap procedure. The backward VAR model is used to bootstrap VAR forecasts conditionally on past observations. Bootstrap prediction regions based on asymptotic bias‐correction are compared with those based on bootstrap bias‐correction. Monte Carlo simulation results indicate that bootstrap prediction regions based on asymptotic bias‐correction show better small sample properties than those based on bootstrap bias‐correction for nearly all cases considered. The former provide accurate coverage properties in most cases, while the latter over‐estimate the future uncertainty. Overall, the percentile‐t bootstrap prediction region based on asymptotic bias‐correction is found to provide highly desirable small sample properties, outperforming its alternatives in nearly all cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Bootstrap prediction intervals for autoregressive models of unknown or infinite lag order

Recent studies on bootstrap prediction intervals for autoregressive (AR) model provide simulation findings when the lag order is known. In practical applications, however, the AR lag order is unknown or can even be infinite. This paper is concerned with prediction intervals for AR models of unknown or infinite lag order. Akaike's information criterion is used to estimate (approximate) the unknown (infinite) AR lag order. Small‐sample properties of bootstrap and asymptotic prediction intervals are compared under both normal and non‐normal innovations. Bootstrap prediction intervals are constructed based on the percentile and percentile‐t methods, using the standard bootstrap as well as the bootstrap‐after‐bootstrap. It is found that bootstrap‐after‐bootstrap prediction intervals show small‐sample properties substantially better than other alternatives, especially when the sample size is small and the model has a unit root or near‐unit root. Copyright © 2002 John Wiley & Sons, Ltd.     

A Bayesian nonlinear support vector machine error correction model

The use of linear error correction models based on stationarity and cointegration analysis, typically estimated with least squares regression, is a common technique for financial time series prediction. In this paper, the same formulation is extended to a nonlinear error correction model using the idea of a kernel‐based implicit nonlinear mapping to a high‐dimensional feature space in which linear model formulations are specified. Practical expressions for the nonlinear regression are obtained in terms of the positive definite kernel function by solving a linear system. The nonlinear least squares support vector machine model is designed within the Bayesian evidence framework that allows us to find appropriate trade‐offs between model complexity and in‐sample model accuracy. From straightforward primal–dual reasoning, the Bayesian framework allows us to derive error bars on the prediction in a similar way as for linear models and to perform hyperparameter and input selection. Starting from the results of the linear modelling analysis, the Bayesian kernel‐based prediction is successfully applied to out‐of‐sample prediction of an aggregated equity price index for the European chemical sector. Copyright © 2006 John Wiley & Sons, Ltd.     

Predicting Stock Return Volatility: Can We Benefit from Regression Models for Return Intervals?

We study the performance of recently developed linear regression models for interval data when it comes to forecasting the uncertainty surrounding future stock returns. These interval data models use easy‐to‐compute daily return intervals during the modeling, estimation and forecasting stage. They have to stand up to comparable point‐data models of the well‐known capital asset pricing model type—which employ single daily returns based on successive closing prices and might allow for GARCH effects—in a comprehensive out‐of‐sample forecasting competition. The latter comprises roughly 1000 daily observations on all 30 stocks that constitute the DAX, Germany's main stock index, for a period covering both the calm market phase before and the more turbulent times during the recent financial crisis. The interval data models clearly outperform simple random walk benchmarks as well as the point‐data competitors in the great majority of cases. This result does not only hold when one‐day‐ahead forecasts of the conditional variance are considered, but is even more evident when the focus is on forecasting the width or the exact location of the next day's return interval. Regression models based on interval arithmetic thus prove to be a promising alternative to established point‐data volatility forecasting tools. Copyright ©2015 John Wiley & Sons, Ltd.     

Simultaneous prediction intervals for ARMA processes with stable innovations

We describe a method for calculating simultaneous prediction intervals for ARMA times series with heavy‐tailed stable innovations. The spectral measure of the vector of prediction errors is shown to be discrete. Direct computation of high‐dimensional stable probabilities is not feasible, but we show that Monte Carlo estimates of the interval width is practical. Copyright © 2008 John Wiley & Sons, Ltd.     

Beating the VAR: Improving Swedish GDP Forecasts Using Error and Intercept Corrections

This paper examines the forecast accuracy of an unrestricted vector autoregressive (VAR) model for GDP, relative to a comparable vector error correction model (VECM) that recognizes that the data are characterized by co‐integration. In addition, an alternative forecast method, intercept correction, is considered for further comparison. Recursive out‐of‐sample forecasts are generated for both models and forecast techniques. The generated forecasts for each model are objectively evaluated by a selection of evaluation measures and equal accuracy tests. The result shows that the VECM consistently outperforms the VAR models. Further, intercept correction enhances the forecast accuracy when applied to the VECM, whereas there is no such indication when applied to the VAR model. For certain forecast horizons there is a significant difference in forecast ability between the intercept corrected VECM compared to the VAR model. Copyright © 2015 John Wiley & Sons, Ltd.     

Functional methods for time series prediction: a nonparametric approach

The problem of prediction in time series using nonparametric functional techniques is considered. An extension of the local linear method to regression with functional explanatory variable is proposed. This forecasting method is compared with the functional Nadaraya–Watson method and with finite‐dimensional nonparametric predictors for several real‐time series. Prediction intervals based on the bootstrap and conditional distribution estimation for those nonparametric methods are also compared. Copyright © 2010 John Wiley & Sons, Ltd.     

The Role of High‐Frequency Intra‐daily Data,Daily Range and Implied Volatility in Multi‐period Value‐at‐Risk Forecasting

This paper assesses the informational content of alternative realized volatility estimators, daily range and implied volatility in multi‐period out‐of‐sample Value‐at‐Risk (VaR) predictions. We use the recently proposed Realized GARCH model combined with the skewed Student's t distribution for the innovations process and a Monte Carlo simulation approach in order to produce the multi‐period VaR estimates. Our empirical findings, based on the S&P 500 stock index, indicate that almost all realized and implied volatility measures can produce statistically and regulatory precise VaR forecasts across forecasting horizons, with the implied volatility being especially accurate in monthly VaR forecasts. The daily range produces inferior forecasting results in terms of regulatory accuracy and Basel II compliance. However, robust realized volatility measures, which are immune against microstructure noise bias or price jumps, generate superior VaR estimates in terms of capital efficiency, as they minimize the opportunity cost of capital and the Basel II regulatory capital. Copyright © 2013 John Wiley & Sons, Ltd.     

Forecasting in the presence of level shifts

This article addresses the problem of forecasting time series that are subject to level shifts. Processes with level shifts possess a nonlinear dependence structure. Using the stochastic permanent breaks (STOPBREAK) model, I model this nonlinearity in a direct and flexible way that avoids imposing a discrete regime structure. I apply this model to the rate of price inflation in the United States, which I show is subject to level shifts. These shifts significantly affect the accuracy of out‐of‐sample forecasts, causing models that assume covariance stationarity to be substantially biased. Models that do not assume covariance stationarity, such as the random walk, are unbiased but lack precision in periods without shifts. I show that the STOPBREAK model outperforms several alternative models in an out‐of‐sample inflation forecasting experiment. Copyright © 2005 John Wiley & Sons, Ltd.     

Short‐term salmon price forecasting

This study establishes a benchmark for short‐term salmon price forecasting. The weekly spot price of Norwegian farmed Atlantic salmon is predicted 1–5 weeks ahead using data from 2007 to 2014. Sixteen alternative forecasting methods are considered, ranging from classical time series models to customized machine learning techniques to salmon futures prices. The best predictions are delivered by k‐nearest neighbors method for 1 week ahead; vector error correction model estimated using elastic net regularization for 2 and 3 weeks ahead; and futures prices for 4 and 5 weeks ahead. While the nominal gains in forecast accuracy over a naïve benchmark are small, the economic value of the forecasts is considerable. Using a simple trading strategy for timing the sales based on price forecasts could increase the net profit of a salmon farmer by around 7%.     

Forecasting nonlinear time series with feed‐forward neural networks: a case study of Canadian lynx data

The forecasting capabilities of feed‐forward neural network (FFNN) models are compared to those of other competing time series models by carrying out forecasting experiments. As demonstrated by the detailed forecasting results for the Canadian lynx data set, FFNN models perform very well, especially when the series contains nonlinear and non‐Gaussian characteristics. To compare the forecasting accuracy of a FFNN model with an alternative model, Pitman's test is employed to ascertain if one model forecasts significantly better than another when generating one‐step‐ahead forecasts. Moreover, the residual‐fit spread plot is utilized in a novel fashion in this paper to compare visually out‐of‐sample forecasts of two alternative forecasting models. Finally, forecasting findings on the lynx data are used to explain under what conditions one would expect FFNN models to furnish reliable and accurate forecasts. Copyright © 2005 John Wiley & Sons, Ltd.     

Hybrid Forecasting with Estimated Temporally Aggregated Linear Processes

We introduce a new strategy for the prediction of linear temporal aggregates; we call it ‘hybrid’ and study its performance using asymptotic theory. This scheme consists of carrying out model parameter estimation with data sampled at the highest available frequency and the subsequent prediction with data and models aggregated according to the forecasting horizon of interest. We develop explicit expressions that approximately quantify the mean square forecasting errors associated with the different prediction schemes and that take into account the estimation error component. These approximate estimates indicate that the hybrid forecasting scheme tends to outperform the so‐called ‘all‐aggregated’ approach and, in some instances, the ‘all‐disaggregated’ strategy that is known to be optimal when model selection and estimation errors are neglected. Unlike other related approximate formulas existing in the literature, those proposed in this paper are totally explicit and require neither assumptions on the second‐order stationarity of the sample nor Monte Carlo simulations for their evaluation. Copyright © 2014 John Wiley & Sons, Ltd.     

Prediction of α‐stable GARCH and ARMA‐GARCH‐M models

The best prediction of generalized autoregressive conditional heteroskedasticity (GARCH) models with α‐stable innovations, α‐stable power‐GARCH models and autoregressive moving average (ARMA) models with GARCH in mean effects (ARMA‐GARCH‐M) are proposed. We present a sufficient condition for stationarity of α‐stable GARCH models. The prediction methods are easy to implement in practice. The proposed prediction methods are applied for predicting future values of the daily SP500 stock market and wind speed data.     

Forecasting the recent behavior of US business fixed investment spending: an analysis of competing models

We evaluate forecasting models of US business fixed investment spending growth over the recent 1995:1–2004:2 out‐of‐sample period. The forecasting models are based on the conventional Accelerator, Neoclassical, Average Q, and Cash‐Flow models of investment spending, as well as real stock prices and excess stock return predictors. The real stock price model typically generates the most accurate forecasts, and forecast‐encompassing tests indicate that this model contains most of the information useful for forecasting investment spending growth relative to the other models at longer horizons. In a robustness check, we also evaluate the forecasting performance of the models over two alternative out‐of‐sample periods: 1975:1–1984:4 and 1985:1–1994:4. A number of different models produce the most accurate forecasts over these alternative out‐of‐sample periods, indicating that while the real stock price model appears particularly useful for forecasting the recent behavior of investment spending growth, it may not continue to perform well in future periods. Copyright © 2007 John Wiley & Sons, Ltd.     

Short‐term forecasts of economic activity: Are fortnightly factors useful?

A short‐term mixed‐frequency model is proposed to estimate and forecast Italian economic activity fortnightly. We introduce a dynamic one‐factor model with three frequencies (quarterly, monthly, and fortnightly) by selecting indicators that show significant coincident and leading properties and are representative of both demand and supply. We conduct an out‐of‐sample forecasting exercise and compare the prediction errors of our model with those of alternative models that do not include fortnightly indicators. We find that high‐frequency indicators significantly improve the real‐time forecasts of Italian gross domestic product (GDP); this result suggests that models exploiting the information available at different lags and frequencies provide forecasting gains beyond those based on monthly variables alone. Moreover, the model provides a new fortnightly indicator of GDP, consistent with the official quarterly series.     

Bootstrap prediction bands for forecast paths from vector autoregressive models

The problem of forecasting from vector autoregressive models has attracted considerable attention in the literature. The most popular non‐Bayesian approaches use either asymptotic approximations or bootstrapping to evaluate the uncertainty associated with the forecast. The practice in the empirical literature has been to assess the uncertainty of multi‐step forecasts by connecting the intervals constructed for individual forecast periods. This paper proposes a bootstrap method of constructing prediction bands for forecast paths. The bands are constructed from forecast paths obtained in bootstrap replications using an optimization procedure to find the envelope of the most concentrated paths. From extensive Monte Carlo study, it is found that the proposed method provides more accurate assessment of predictive uncertainty from the vector autoregressive model than its competitors. Copyright © 2010 John Wiley & Sons, Ltd.     

Forecasting business failure in China using case‐based reasoning with hybrid case respresentation

Case‐based reasoning (CBR) is a very effective and easily understandable method for solving real‐world problems. Business failure prediction (BFP) is a forecasting tool that helps people make more precise decisions. CBR‐based BFP is a hot topic in today's global financial crisis. Case representation is critical when forecasting business failure with CBR. This research describes a pioneer investigation on hybrid case representation by employing principal component analysis (PCA), a feature extraction method, along with stepwise multivariate discriminant analysis (MDA), a feature selection approach. In this process, sample cases are represented with all available financial ratios, i.e., features. Next, the stepwise MDA is used to select optimal features to produce a reduced‐case representation. Finally, PCA is employed to extract the final information representing the sample cases. All data signified by hybrid case representation are recorded in a case library, and the k‐nearest‐neighbor algorithm is used to make the forecasting. Thus we constructed a hybrid CBR (HCBR) by integrating hybrid case representation into the forecasting tool. We empirically tested the performance of HCBR with data collected for short‐term BFP of Chinese listed companies. Empirical results indicated that HCBR can produce more promising prediction performance than MDA, logistic regression, classical CBR, and support vector machine. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid forecasting approach for piece‐wise stationary time series

We consider the problem of forecasting a stationary time series when there is an unknown mean break close to the forecast origin. Based on the intercept‐correction methods suggested by Clements and Hendry (1998) and Bewley (2003), a hybrid approach is introduced, where the break and break point are treated in a Bayesian fashion. The hyperparameters of the priors are determined by maximizing the marginal density of the data. The distributions of the proposed forecasts are derived. Different intercept‐correction methods are compared using simulation experiments. Our hybrid approach compares favorably with both the uncorrected and the intercept‐corrected forecasts. Copyright © 2006 John Wiley & Sons, Ltd.     

Forecasting the Nikkei spot index with fractional cointegration

We investigate the forecast performance of the fractionally integrated error correction model against several competing models for the prediction of the Nikkei stock average index. The competing models include the martingale model, the vector autoregressive model and the conventional error correction model. We consider models with and without conditional heteroscedasticity. For forecast horizons of over twenty days, the best forecasting performance is obtained for the model when fractional cointegration is combined with conditional heteroscedasticity. Our results reinforce the notion that cointegration and fractional cointegration are important for long‐horizon prediction. Copyright © 1999 John Wiley & Sons, Ltd.     

Forecasting trend output in the Euro area

This paper is an applied study about forecasting trend output and the output gap in the Euro area. The need for trend output forecasts is justified by an analysis of the monetary strategy of the European Central Bank. Trend output serves as a direct inflation indicator and helps to determine the reference value for money. For both purposes, trend output has to be forecasted. A permanent–transitory decomposition based on cointegration restrictions gives an estimate of trend output in the Euro area. Ex‐ante point forecasts of trend output are computed and bootstrap simulation is employed to construct prediction intervals that take estimation uncertainty into consideration. The uncertainty of trend output and the output gap is quite large and raises questions about their usefulness as indicators for monetary policy. Copyright © 2002 John Wiley & Sons, Ltd.