Semiparametric forecast intervals

Consider forecasting the economic variable Y_t+h with predictors X _t, where h is the forecast horizon. This paper introduces a semiparametric method that generates forecast intervals of Y_t+h| X _t from point forecast models. First, the point forecast model is estimated, thereby taking advantage of its predictive power. Then, nonparametric estimation of the conditional distribution function (CDF) of the forecast error conditional on X _t builds the rest of the forecast distribution around the point forecast, from which symmetric and minimum‐length forecast intervals for Y_t+h| X _t can be constructed. Under mild regularity conditions, asymptotic analysis shows that (1) regardless of the quality of the point forecast model (i.e., it may be misspecified), forecast quantiles are consistent and asymptotically normal; (2) minimum length forecast intervals are consistent. Proposals for bandwidth selection and dimension reduction are made. Three sets of simulations show that for reasonable point forecast models the method has significant advantages over two existing approaches to interval forecasting: one that requires the point forecast model to be correctly specified, and one that is based on fully nonparametric CDF estimate of Y_t+h| X _t. An application to exchange rate forecasting is presented. Copyright © 2010 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.     

Forecasting the duration of short‐term deflation episodes

The paper proposes a simulation‐based approach to multistep probabilistic forecasting, applied for predicting the probability and duration of negative inflation. The essence of this approach is in counting runs simulated from a multivariate distribution representing the probabilistic forecasts, which enters the negative inflation regime. The marginal distributions of forecasts are estimated using the series of past forecast errors, and the joint distribution is obtained by a multivariate copula approach. This technique is applied for estimating the probability of negative inflation in China and its expected duration, with the marginal distributions computed by fitting weighted skew‐normal and two‐piece normal distributions to autoregressive moving average ex post forecast errors and using the multivariate Student t copula.     

Forecast Combination and Bayesian Model Averaging: A Prior Sensitivity Analysis

In this study we evaluate the forecast performance of model‐averaged forecasts based on the predictive likelihood carrying out a prior sensitivity analysis regarding Zellner's g prior. The main results are fourfold. First, the predictive likelihood does always better than the traditionally employed ‘marginal’ likelihood in settings where the true model is not part of the model space. Secondly, forecast accuracy as measured by the root mean square error (RMSE) is maximized for the median probability model. On the other hand, model averaging excels in predicting direction of changes. Lastly, g should be set according to Laud and Ibrahim (1995: Predictive model selection. Journal of the Royal Statistical Society B 57 : 247–262) with a hold‐out sample size of 25% to minimize the RMSE (median model) and 75% to optimize direction of change forecasts (model averaging). We finally apply the aforementioned recommendations to forecast the monthly industrial production output of six countries, beating for almost all countries the AR(1) benchmark model. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi‐step forecasting for long‐memory processes

In this paper we present results of a simulation study to assess and compare the accuracy of forecasting techniques for long‐memory processes in small sample sizes. We analyse differences between adaptive ARMA(1,1) L‐step forecasts, where the parameters are estimated by minimizing the sum of squares of L‐step forecast errors, and forecasts obtained by using long‐memory models. We compare widths of the forecast intervals for both methods, and discuss some computational issues associated with the ARMA(1,1) method. Our results illustrate the importance and usefulness of long‐memory models for multi‐step forecasting. Copyright © 1999 John Wiley & Sons, Ltd.     

ARARMA models for time series analysis and forecasting

Methods of time series forecasting are proposed which can be applied automatically. However, they are not rote formulae, since they are based on a flexible philosophy which can provide several models for consideration. In addition it provides diverse diagnostics for qualitatively and quantitatively estimating how well one can forecast a series. The models considered are called ARARMA models (or ARAR models) because the model fitted to a long memory time series (t) is based on sophisticated time series analysis of AR (or ARMA) schemes (short memory models) fitted to residuals Y(t) obtained by parsimonious‘best lag’non-stationary autoregression. Both long range and short range forecasts are provided by an ARARMA model Section 1 explains the philosophy of our approach to time series model identification. Sections 2 and 3 attempt to relate our approach to some standard approaches to forecasting; exponential smoothing methods are developed from the point of view of prediction theory (section 2) and extended (section 3). ARARMA models are introduced (section 4). Methods of ARARMA model fitting are outlined (sections 5,6). Since‘the proof of the pudding is in the eating’, the methods proposed are illustrated (section 7) using the classic example of international airline passengers.     

A Time‐Simultaneous Prediction Box for a Multivariate Time Series

A sample‐based method in Kolsrud (Journal of Forecasting 2007; 26 (3): 171–188) for the construction of a time‐simultaneous prediction band for a univariate time series is extended to produce a variable‐ and time‐simultaneous prediction box for a multivariate time series. A measure of distance based on the L_∞ ‐norm is applied to a learning sample of multivariate time trajectories, which can be mean‐ and/or variance‐nonstationary. Based on the ranking of distances to the centre of the sample, a subsample of the most central multivariate trajectories is selected. A prediction box is constructed by circumscribing the subsample with a hyperrectangle. The fraction of central trajectories selected into the subsample can be calibrated by bootstrap such that the expected coverage of the box equals a prescribed nominal level. The method is related to the concept of data depth, and thence modified to increase coverage. Applications to simulated and empirical data illustrate the method, which is also compared to several other methods in the literature adapted to the multivariate setting. Copyright © 2015 John Wiley & Sons, Ltd.     

A new parsimonious recurrent forecasting model in singular spectrum analysis

Singular spectrum analysis (SSA) is a powerful nonparametric method in the area of time series analysis that has shown its capability in different applications areas. SSA depends on two main choices: the window length L and the number of eigentriples used for grouping r. One of the most important issues when analyzing time series is the forecast of new observations. When using SSA for time series forecasting there are several alternative algorithms, the most widely used being the recurrent forecasting model, which assumes that a given observation can be written as a linear combination of the L?1 previous observations. However, when the window length L is large, the forecasting model is unlikely to be parsimonious. In this paper we propose a new parsimonious recurrent forecasting model that uses an optimal m(<L?1) coefficients in the linear combination of the recurrent SSA. Our results support the idea of using this new parsimonious recurrent forecasting model instead of the standard recurrent SSA forecasting model.     

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.     

Optimal prediction with nonstationary ARFIMA model

We propose two methods to predict nonstationary long‐memory time series. In the first one we estimate the long‐range dependent parameter d by using tapered data; we then take the nonstationary fractional filter to obtain stationary and short‐memory time series. In the second method, we take successive differences to obtain a stationary but possibly long‐memory time series. For the two methods the forecasts are based on those obtained from the stationary components. Copyright © 2007 John Wiley & Sons, Ltd.     

Guaranteed‐content prediction intervals for non‐linear autoregressions

In this paper we present guaranteed‐content prediction intervals for time series data. These intervals are such that their content (or coverage) is guaranteed with a given high probability. They are thus more relevant for the observed time series at hand than classical prediction intervals, whose content is guaranteed merely on average over hypothetical repetitions of the prediction process. This type of prediction inference has, however, been ignored in the time series context because of a lack of results. This gap is filled by deriving asymptotic results for a general family of autoregressive models, thereby extending existing results in non‐linear regression. The actual construction of guaranteed‐content prediction intervals directly follows from this theory. Simulated and real data are used to illustrate the practical difference between classical and guaranteed‐content prediction intervals for ARCH models. Copyright © 2001 John Wiley & Sons, Ltd.     

Estimating and forecasting the long‐memory parameter in the presence of periodicity

We consider one parametric and five semiparametric approaches to estimate D in SARFIMA (0, D, 0)_s processes, that is, when the process is a fractionally integrated ARMA model with seasonality s. We also consider h‐step‐ahead forecasting for these processes. We present the proof of some features of this model and also a study based on a Monte Carlo simulation for different sample sizes and different seasonal periods. We compare the different estimation procedures analyzing the bias, the mean squared error values, and the confidence intervals for the estimators. We also consider three different methods to choose the total number of regressors in the regression analysis for the semiparametric class of estimation procedures. We apply the methodology to the Nile River flow monthly data, and also to a simulated seasonal fractionally integrated time series. Copyright © 2007 John Wiley & Sons, Ltd.     

Identification of asymmetric prediction intervals through causal forces

When causal forces are specified, the expected direction of the trend can be compared with the trend based on extrapolation. Series in which the expected trend conflicts with the extrapolated trend are called contrary series. We hypothesized that contrary series would have asymmetric forecast errors, with larger errors in the direction of the expected trend. Using annual series that contained minimal information about causality, we examined 671 contrary forecasts. As expected, most (81%) of the errors were in the direction of the causal forces. Also as expected, the asymmetries were more likely for longer forecast horizons; for six‐year‐ahead forecasts, 89% of the forecasts were in the expected direction. The asymmetries were often substantial. Contrary series should be flagged and treated separately when prediction intervals are estimated, perhaps by shifting the interval in the direction of the causal forces. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling European industrial production with multivariate singular spectrum analysis: A cross‐industry analysis

In this paper, an optimized multivariate singular spectrum analysis (MSSA) approach is proposed to find leading indicators of cross‐industry relations between 24 monthly, seasonally unadjusted industrial production (IP) series for German, French, and UK economies. Both recurrent and vector forecasting algorithms of horizontal MSSA (HMSSA) are considered. The results from the proposed multivariate approach are compared with those obtained via the optimized univariate singular spectrum analysis (SSA) forecasting algorithm to determine the statistical significance of each outcome. The data are rigorously tested for normality, seasonal unit root hypothesis, and structural breaks. The results are presented such that users can not only identify the most appropriate model based on the aim of the analysis, but also easily identify the leading indicators for each IP variable in each country. Our findings show that, for all three countries, forecasts from the proposed MSSA algorithm outperform the optimized SSA algorithm in over 70% of cases. Accordingly, this new approach succeeds in identifying leading indicators and is a viable option for selecting the SSA choices L and r, which minimizes a loss function.     

Consistent forecast intervals when the forecast-period exogenous variables are stochastic

Derivation of prediction intervals in the k-variable regression model is problematic when future-period values of exogenous variables are not known with certainty. Even in the most favourable case when the forecasts of the exogenous variables are jointly normal, the distribution of the forecast error is non-normal, and thus traditional asymptotic normal theory does not apply. This paper presents an alternative bootstrap method. In contrast to the traditional predictor of the future value of the endogenous variable, which is known to be inconsistent, the bootstrap predictor converges weakly to the true value. Monte Carlo results show that the bootstrap prediction intervals can achieve approximately nominal coverage.     

Enhancing survey‐based investment forecasts

We investigate the accuracy of capital investment predictors from a national business survey of South African manufacturing. Based on data available to correspondents at the time of survey completion, we propose variables that might inform the confidence that can be attached to their predictions. Having calibrated the survey predictors' directional accuracy, we model the probability of a correct directional prediction using logistic regression with the proposed variables. For point forecasting, we compare the accuracy of rescaled survey forecasts with time series benchmarks and some survey/time series hybrid models. In addition, using the same set of variables, we model the magnitude of survey prediction errors. Directional forecast tests showed that three out of four survey predictors have value but are biased and inefficient. For shorter horizons we found that survey forecasts, enhanced by time series data, significantly improved point forecasting accuracy. For longer horizons the survey predictors were at least as accurate as alternatives. The usefulness of the more accurate of the predictors examined is enhanced by auxiliary information, namely the probability of directional accuracy and the estimated error magnitude.     

System-based weights versus series-specific weights in the combination of forecasts

System-based combination weights for series r/step-length h incorporate relative accuracy information from other forecast step-lengths for r and from other series for step-length h. Such weights are examined utilizing the West and Fullerton (1996) data set-4275 ex ante employment forecasts from structural simultaneous equation econometric models for 19 metropolitan areas at 10 quarterly step-lengths and a parallel set of 4275 ARIMA forecasts. The system-based weights yielded combined forecasts of higher average accuracy and lower risk of large inaccuracy than seven alternative strategies: (1) averaging; (2) relative MSE weights; (3) outperformance (per cent best) weights; (4) Bates and Granger (1969) optimal weights with a convexity constraint imposed; (5) unconstrained optimal weights; (6) select a ‘best’ method (ex ante) by series and; (7) experiment in the Bischoff (1989) sense and select either method (2) or (6) based on the outcome of e experiment. Accuracy gains of the system-based combination were concentrated at step-lengths two to five. Although alternative (5) was generally outperformed, none of the six other alternatives was systematically most accurate when evaluated relative to each other. This contrasts with Bischoff's (1989) results that held promise for an empirically applicable guideline to determine whether or not to combine.     

Step-ahead spot price densities using daily synchronously reported prices and wind forecasts

This paper uses non-linear methodologies to follow the synchronously reported relationship between the Nordic/Baltic electric daily spot auction prices and geographical relevant wind forecasts in MWh from early 2013 to 2020. It is a well-known market (auctions) microstructure fact that the daily wind forecasts are information available to the market before the daily auction bid deadline at 11 a.m. The main objective is therefore to establish conditional and marginal step ahead spot price density forecast using a stochastic representation of the lagged, synchronously reported and stationary spot price and wind forecast movements. Using an upward expansion path applying the Schwarz (Bayesian information criterion [BIC]) criterion and a battery of residual test statistics, an optimal maximum likelihood process density is suggested. The optimal specification reports a significant negative covariance between the daily price and wind forecast movements. Conditional on bivariate lags from the SNP information and using the known market information for wind forecast movements at t₁, the paper establishes one-step-ahead bivariate and marginal day-ahead spot price movement densities. The result shows that wind forecasts significantly influence the synchronously reported spot price densities (means and volatilities). The paper reports day-ahead bivariate and marginal densities for spot price movements conditional on several very plausible price and wind forecast movements. The paper suggests day-ahead spot price predictions from conditional and synchronously reported wind forecasts movements. The information should increase market participants spot market insight and consequently make spot price predictions more accurate and the confidence interval considerably narrower.     

Robust modelling of ARCH models

The autoregressive conditional heteroscedastic (ARCH) model and its extensions have been widely used in modelling changing variances in financial time series. Since the asset return distributions frequently display tails heavier than normal distributions, it is worth while studying robust ARCH modelling without a specific distribution assumption. In this paper, rather than modelling the conditional variance, we study ARCH modelling for the conditional scale. We examine the L₁‐estimation of ARCH models and derive the limiting distributions of the estimators. A robust standardized absolute residual autocorrelation based on least absolute deviation estimation is proposed. Then a robust portmanteau statistic is constructed to test the adequacy of the model, especially the specification of the conditional scale. We obtain their asymptotic distributions under mild conditions. Examples show that the suggested L₁‐norm estimators and the goodness‐of‐fit test are robust against error distributions and are accurate for moderate sample sizes. This paper provides a useful tool in modelling conditional heteroscedastic time series data. Copyright © 2001 John Wiley & Sons, Ltd.     

Forecasting relative quality

This paper develops a model to forecast the likely quality rating of new automobiles. A scheme is devised and implemented for 1982 whereby the probability that a specific model will have one of five quality ratings is computed. The quality ratings are based on the trouble index computed by Consumer Reports based on a survey of its subscribers. A multinomial logit specification is used whereby the relative probability that a given level of quality is realized is a function of previous quality ratings, the location of the manufacturer of the automobile, the size of the automobile and the list price (or port-of-entry prices, in the case of imports). The forecast results when compared in a qualitative way to actual 1982 quality ratings prove to be acceptable.