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.     

Bootstrapping forecast intervals: An application to AR(p) models

Forecast intervals typically depend upon an assumption of normal forecast errors due to lack of information concerning the distribution of the forecast. This article applies the bootstrap to the problem of estimating forecast intervals for an AR(p) model. Box-Jenkins intervals are compared to intervals produced from a naive bootstrap and a bias-correction bootstrap. Substantial differences between the three methods are found. Bootstrapping an AR(p) model requires use of the backward residuals which typically are not i.i.d. and hence inappropriate for bootstrap resampling. A recently developed method of obtaining i.i.d. backward residuals is employed and was found to affect the bootstrap prediction intervals.     

Bias‐corrected bootstrap prediction intervals for autoregressive model: new alternatives with applications to tourism forecasting

This paper proposes the use of the bias‐corrected bootstrap for interval forecasting of an autoregressive time series with an arbitrary number of deterministic components. We use the bias‐corrected bootstrap based on two alternative bias‐correction methods: the bootstrap and an analytic formula based on asymptotic expansion. We also propose a new stationarity‐correction method, based on stable spectral factorization, as an alternative to Kilian's method exclusively used in past studies. A Monte Carlo experiment is conducted to compare small‐sample properties of prediction intervals. The results show that the bias‐corrected bootstrap prediction intervals proposed in this paper exhibit desirable small‐sample properties. It is also found that the bootstrap bias‐corrected prediction intervals based on stable spectral factorization are tighter and more stable than those based on Kilian's stationarity‐correction. The proposed methods are applied to interval forecasting for the number of tourist arrivals in Hong Kong. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparison of methods for bootstrapping in the local level model

Bootstrap in time series models is not straightforward to implement, as in this case the observations are not independent. One of the alternatives is to bootstrap the residuals in order to obtain the bootstrap series and thus use these series for inference purposes. This work deals with the problem of assessing the accuracy of hyperparameters in structural models. We study the simplest case, the local level model, where the hyperparameters are given by the variances of the disturbance terms. As their distribution is not known, we employ the bootstrap to approximate the true distribution, using parametric and non‐parametric approaches. Bootstrap standard deviations are computed and their performances compared to the asymptotic and empirical standard errors, calculated using a Monte Carlo simulation. We also build confidence intervals to the hyperparameters, using four bootstrap methods and the results are compared by means of the length, shape and coverage probabilities of the intervals. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

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.     

Expert judgments of political riskiness

Professional analysts' judgments of the political riskiness of 49 focal countries for the period 1983-1985 were studied. Data were collected on nine predictor variables; each was significantly correlated at the 0.01 level with ratings of political riskiness. The highest correlation was with infant mortality and life expectancy; either accounted for roughly 50% of the variance in ratings. Different variables were better predictors of political risk within different geographic regions. A factor analysis suggested the presence of three underlying factors. The predictor variable with the highest loading was chosen to represent each of the three factors. These were: exchange rate differential; estimated inflation rate; and infant mortality rate. Approximately 75% of the variance in ratings could be accounted for on the basis of a linear combination of the three predictor variables. These three variables were capable of good prediction even for various subsets of countries based on geographic region or other criteria. Using all nine variables as predictors resulted in only marginal improvement. A cluster analysis revealed little difference among clusters of judges. Ratings by undergraduate students closely paralled those of professional analysts. As in previous studies of expert predictions and forecasts, claims of expertise in political risk analysis were better supported by command of factual knowledge than by differentially superior predictive ability.     

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.     

Multiple prediction intervals for time series: Comparison of simultaneous and marginal intervals

Simultaneous prediction intervals for forecasts from time series models that contain L (L ≤ 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.     

Model uncertainty and forecast accuracy

In time-series analysis, a model is rarely pre-specified but rather is typically formulated in an iterative, interactive way using the given time-series data. Unfortunately the properties of the fitted model, and the forecasts from it, are generally calculated as if the model were known in the first place. This is theoretically incorrect, as least squares theory, for example, does not apply when the same data are used to formulates and fit a model. Ignoring prior model selection leads to biases, not only in estimates of model parameters but also in the subsequent construction of prediction intervals. The latter are typically too narrow, partly because they do not allow for model uncertainty. Empirical results also suggest that more complicated models tend to give a better fit but poorer ex-ante forecasts. The reasons behind these phenomena are reviewed. When comparing different forecasting models, the BIC is preferred to the AIC for identifying a model on the basis of within-sample fit, but out-of-sample forecasting accuracy provides the real test. Alternative approaches to forecasting, which avoid conditioning on a single model, include Bayesian model averaging and using a forecasting method which is not model-based but which is designed to be adaptable and robust.     

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 with an econometric model: The ‘ragged edge’ problem

In practical econometric forecasting exercises, incomplete data on current and immediate past values of endogenous variables are available. This paper considers various approaches to this ‘ragged edge’ problem, including the common device of treating as ‘temporarily exogenous’ an endogenous variable whose value is known, by deleting it from the set of endogenous variables for whose forecast values the model is solved and suppressing the corresponding structural equation. It is seen that this forecast can be adjusted to coincide with the optimal forecast. The initial discussion concerns the textbook linear simultaneous equation model; extensions to non-linear dynamic models are described.     

Short-term forecasting of the US unemployment rate

This paper aims to assess whether Google search data are useful when predicting the US unemployment rate among other more traditional predictor variables. A weekly Google index is derived from the keyword “unemployment” and is used in diffusion index variants along with the weekly number of initial claims and monthly estimated latent factors. The unemployment rate forecasts are generated using MIDAS regression models that take into account the actual frequencies of the predictor variables. The forecasts are made in real time, and the forecasts of the best forecasting models exceed, for the most part, the root mean squared forecast error of two benchmarks. However, as the forecasting horizon increases, the forecasting performance of the best diffusion index variants decreases over time, which suggests that the forecasting methods proposed in this paper are most useful in the short term.     

Prediction of final data with use of preliminary and/or revised data

In the case of US national accounts the data are revised for the first few years and every decade, which implies that we do not really have the final data. In this paper we aim to predict the final data, using the preliminary data and/or the revised data. The following predictors are introduced and derived from a context of the non-linear filtering or smoothing problem, which are: (1) prediction of the final data of time t given the preliminary data up to time t- 1, (2) prediction of the final data of time t given the preliminary data up to time t, (3) prediction of the final data of time t given the preliminary data up to time T, (4) prediction of the final data of time t given the revised data up to time t -1 and the preliminary data up to time t- 1, and (5) prediction of the final data of time t given the revised data up to time t-1 and the preliminary data up to time t. It is shown that (5) is the best predictor but not too different from (3). The prediction problem is illustrated using US per capita consumption data.     

The Volatility and Density Prediction Performance of Alternative GARCH Models

This study compares the volatility and density prediction performance of alternative GARCH models with different conditional distribution specifications. The conditional residuals are specified as normal, skewedHyphen;t or compound Poisson (jump) distribution based upon a nonlinear and asymmetric GARCH (NGARCH) model framework. The empirical results for the S&P 500 and FTSE 100 index returns suggest that the jump model outperforms all other models in terms of both volatility forecasting and density prediction. Nevertheless, the superiority of the nonHyphen;normal models is not always significant and diminished during the sample period on those occasions when volatility experiences an obvious structural change. Copyright © 2011 John Wiley & Sons, Ltd.     

Predicting nominal variable relationships with multiple response

For forecasting purposes, it is useful to predict the most likely response of an individual to a nominally-scaled variable using the response to a predictor variable which is also nominally scaled. Traditional statistical approaches are not suitable when respondents provide multiple responses. For practical applications it is desirable to provide a simple measure of prediction that is easy to calculate and understand. Two situations are described where predictions of multiple response are implemented and two indices of predictive association are developed for the situations. These indices provide predictive explanations where none were possible using traditional methods of predictive association. The need to complement these indices with conditional probabilities and log-linear models is suggested. The evaluation and implications of these indices are discussed.     

Predicting the Distribution of Stock Returns: Model Formulation,Statistical Evaluation,VaR Analysis and Economic Significance

A large literature has investigated predictability of the conditional mean of low‐frequency stock returns by macroeconomic and financial variables; however, little is known about predictability of the conditional distribution. We look at one‐step‐ahead out‐of‐sample predictability of the conditional distribution of monthly US stock returns in relation to the macroeconomic and financial environment. Our methodological approach is innovative: we consider several specifications for the conditional density and combinations schemes. Our results are as follows: the entire density is predicted under combination schemes as applied to univariate GARCH models with Gaussian innovations; the Bayesian winner in relation to GARCH‐skewed‐t models is informative about the 5% value at risk; the average realised utility of a mean–variance investor is maximised under the Bayesian winner as applied to GARCH models with symmetric Student t innovations. Our results have two implications: the best prediction model depends on the evaluation criterion; and combination schemes outperform individual models. Copyright © 2015 John Wiley & Sons, Ltd.     

Financial distress prediction model: The effects of corporate governance indicators

This paper constructs a financial distress prediction model that includes not only traditional financial variables, but also several important corporate governance variables. Using data from Taiwan, the empirical results show that the best in-sample and out-of-sample prediction models should combine the financial variables with the corporate governance variables. Moreover, the prediction accuracy is higher for the models using dynamic distress threshold values than those with tradition threshold values. Most financial ratios, except for the debt ratio, are higher in financially sound companies than in financial distressed ones. With regard to the corporate governance variables, we find that the CEO/Chairman duality may not result in the outbreak of financial distress, but higher equity pledge ratios of managers (shareholding ratios by board members and insiders) positively (negatively) correlate with financial distress.     

Bootstrap Replacement to Validate the Influence of the Economic Cycle on the Structure and the Accuracy Level of Business Failure Prediction Models

The aim of this study was to answer the question of how the economic cycle affects the stability and efficiency of business failure prediction models, using bootstrap replacement method for validation. We analyse 2228 Spanish small and medium‐sized enterprises for the period 2001–2009, and divide it into three different phases of the economic cycle (growth, crisis, recession). We find that the structure and the ability of business failure prediction models are different according to the economic cycle. These findings are relevant for the debate on the most suitable financial ratios when developing business failure prediction models and to pose their accuracy level in these prediction models. Copyright © 2015 John Wiley & Sons, Ltd.     

Prediction intervals for growth curve forecasts

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.