Linear combination of forecasts with an intercept: A bayesian approach

The standard approach to combining n expert forecasts involves taking a weighted average. Granger and Ramanathan proposed introducing an intercept term and unnormalized weights. This paper deduces their proposal from Bayesian principles. We find that their formula is equivalent to taking a weighted average of the n expert forecasts plus the decision-maker's prior forecast.     

A non-linear combination of experts' forecasts: A Bayesian approach

A general Bayesian approach to combining n expert forecasts is developed. Under some moderate assumptions on the distributions of the expert errors, it leads to a consistent, monotonic, quasi-linear average formula. This generalizes Bordley's results.     

Bayesian consensus forecasts of macroeconomic variables

Economists, like other forecasters, share knowledge, data and theories in common. Consequently, their forecast errors are likely to be highly dependent. This paper reports on an empirical study of 16 macroeconomic forecasters. Composite forecasts are computed using a sequential weighting scheme that takes dependence into account; these are compared to a simple average and median forecasts. A within-sample composite is also calculated. Both these methods perform significantly better than the average or median of the forecasts. This improvement in accuracy is apparently because the dependence between the forecasters' errors is so high that the optimal composite forecasts sometimes lie outside the range of the individual forecasts.     

Linear combination of forecasts: A general Bayesian model

We provide a general Bayesian model for combining forecasts from experts (or forecasting models) who might be biased and correlated with each other. The combination procedure involves debiasing and then combining unbiased forecasts. We also provide a sequential method for learning about the forecasters' biases in the process of combining information from them.     

Combining forecasts using optimal combination weight and generalized autoregression

In this paper, we consider a combined forecast using an optimal combination weight in a generalized autoregression framework. The generalized autoregression provides not only a combined forecast but also an optimal combination weight for combining forecasts. By simulation, we find that short‐ and medium‐horizon (as well as partly long‐horizon) forecasts from the generalized autoregression using the optimal combination weight are more efficient than those from the usual autoregression in terms of the mean‐squared forecast error. An empirical application with US gross domestic product confirms the simulation result. Copyright © 2008 John Wiley & Sons, Ltd.     

Unbiasedness,efficiency and the combination of economic forecasts

This paper considers the problem of determining whether forecasts are unbiased and examines the implications this has for combining different forecasts. The practical issues of how economic forecasts might be combined are discussed. There is an empirical illustration of the procedures in which the properties of UK forecasts from the London Business School, the National Institute, the Henley Centre for Forecasting, Phillips and Drew and the OECD are examined.     

To combine or not to combine? issues of combining forecasts

This paper addresses issues such as: Does it always pay to combine individual forecasts of a variable? Should one combine an unbiased forecast with one that is heavily biased? Should one use optimal weights as suggested by Bates and Granger over twenty years ago? A simple model which accounts for the main features of individual forecasts is put forward. Bayesian analysis of the model using noninformative and informative prior probability densities is provided which extends and generalizes results obtained by Winkler (1981) and compared with non-Bayesian methods of combining forecasts relying explicitly on a statistical model for the individual forecasts. It is shown that in some instances it is sensible to use a simple average of individual forecasts instead of using Bates and Granger type weights. Finally, model uncertainty is considered and the issue of combining different models for individual forecasts is addressed.     

Combination of forecasts using self‐organizing algorithms

Based on the theories and methods of self‐organizing data mining, a new forecasting method, called self‐organizing combining forecasting method, is proposed. Compared with optimal linear combining forecasting methods and neural networks combining forecasting methods, the new method can improve the forecasting capability of the model. The superiority of the new method is justified and demonstrated by real applications. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparative evaluation of objective and subjective weather forecasts in the united states

This paper is concerned primarily with the evaluation and comparison of objective and subjective weather forecasts. Operational forecasts of three weather elements are considered: (1) probability forecasts of precipitation occurrence, (2) categorical (i.e. non-probabilistic) forecasts of maximum and minimum temperatures and (3) categorical forecasts of cloud amount. The objective forecasts are prepared by numerical-statistical procedures, whereas the subjective forecasts are based on the judgements of individual forecasters. In formulating the latter, the forecasters consult information from a variety of sources, including the objective forecasts themselves. The precipitation probability forecasts are found to be both reliable and skilful, and evaluation of the temperature/cloud amount forecasts reveals that they are quite accurate/skilful. Comparison of the objective and subjective forecasts of precipitation occurrence indicates that the latter are generally more skilful than the former for shorter lead times (e.g. 12–24 hours), whereas the two types of forecasts are of approximately equal skill for longer lead times (e.g. 36–48 hours). Similar results are obtained for the maximum and minimum temperature forecasts. Objective cloud amount forecasts are more skilful than subjective cloud amount forecasts for all lead times. Examination of trends in performance over the last decade reveals that both types of forecasts for all three elements increased in skill (or accuracy) over the period, with improvements in objective forecasts equalling or exceeding improvements in subjective forecasts. The role and impact of the objective forecasts in the subjective weather forecasting process are discussed in some detail. The need to conduct controlled experiments and other studies of this process, with particular reference to the assimilation of information from different sources, is emphasized. Important characteristics of the forecasting system in meteorology are identified, and they are used to describe similarities and differences between weather forecasting and forecasting in other fields. Acquisition of some of these characteristics may be beneficial to other forecasting systems.     

Forecasting with more than one model

This paper comprises an editorial review for the Special Issue on Combining Forecasts. It gives a background to the current growth of interest in this topic and speculates upon some of the reasons for this popularity. Some of the main methodological issues in practice are also described.     

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.     

Extracting information shocks from the Bank of England inflation density forecasts

This paper shows how to extract the density of information shocks from revisions of the Bank of England's inflation density forecasts. An information shock is defined in this paper as a random variable that contains the set of information made available between two consecutive forecasting exercises and that has been incorporated into a revised forecast for a fixed point event. Studying the moments of these information shocks can be useful in understanding how the Bank has changed its assessment of risks surrounding inflation in the light of new information, and how it has modified its forecasts accordingly. The variance of the information shock is interpreted in this paper as a new measure of ex ante inflation uncertainty that measures the uncertainty that the Bank anticipates information perceived in a particular quarter will pose on inflation. A measure of information absorption that indicates the approximate proportion of the information content in a revised forecast that is attributable to information made available since the last forecast release is also proposed.     

The combination of macroeconomic forecasts

The paper examines combined forecasts based on two components: forecasts produced by Chase Econometrics and those produced using the Box-Jenkins ARIMA technique. Six series of quarterly ex ante and simulated ex ante forecasts are used over 37 time periods and ten horizons. The forecasts are combined using seven different methods. The best combined forecasts, judged by average relative root-mean-square error, are superior to the Chase forecasts for three variables and inferior for two, though averaged over all six variables the Chase forecasts are slightly better. A two-step procedure produces forecasts for the last half of the sample which, on average, are slightly better than the Chase forecasts.     

Optimal selection of forecasts

Many studies have shown that, in general, a combination of forecasts often outperforms the forecasts of a single model or expert. In this paper we postulate that obtaining forecasts is costly, and provide models for optimally selecting them. Based on normality assumptions, we derive a dynamic programming procedure for maximizing precision net of cost. We examine the solution for cases where the forecasters are independent, correlated and biased. We provide illustrative examples for each case.     

Linear combination of restrictions and forecasts in time series analysis

An important tool in time series analysis is that of combining information in an optimal way. Here we establish a basic combining rule of linear predictors and show that such problems as forecast updating, missing value estimation, restricted forecasting with binding constraints, analysis of outliers and temporal disaggregation can be viewed as problems of optimal linear combination of restrictions and forecasts. A compatibility test statistic is also provided as a companion tool to check that the linear restrictions are compatible with the forecasts generated from the historical data. Copyright © 2000 John Wiley & Sons, Ltd.     

Measuring the market risk of freight rates: A forecast combination approach

This paper addresses the issue of freight rate risk measurement via value at risk (VaR) and forecast combination methodologies while focusing on detailed performance evaluation. We contribute to the literature in three ways: First, we reevaluate the performance of popular VaR estimation methods on freight rates amid the adverse economic consequences of the recent financial and sovereign debt crisis. Second, we provide a detailed and extensive backtesting and evaluation methodology. Last, we propose a forecast combination approach for estimating VaR. Our findings suggest that our combination methods produce more accurate estimates for all the sectors under scrutiny, while in some cases they may be viewed as conservative since they tend to overestimate nominal VaR.     

An exploratory analysis of portfolio managers' probabilistic forecasts of stock prices

This study reports the results of an experiment that examines (1) the effects of forecast horizon on the performance of probability forecasters, and (2) the alleged existence of an inverse expertise effect, i.e., an inverse relationship between expertise and probabilistic forecasting performance. Portfolio managers are used as forecasters with substantive expertise. Performance of this ‘expert’ group is compared to the performance of a ‘semi-expert’ group composed of other banking professionals trained in portfolio management. It is found that while both groups attain their best discrimination performances in the four-week forecast horizon, they show their worst calibration and skill performances in the 12-week forecast horizon. Also, while experts perform better in all performance measures for the one-week horizon, semi-experts achieve better calibration for the four-week horizon. It is concluded that these results may signal the existence of an inverse expertise effect that is contingent on the selected forecast horizon.     

The performance of non‐linear exchange rate models: a forecasting comparison

In recent years there has been a considerable development in modelling non‐linearities and asymmetries in economic and financial variables. The aim of the current paper is to compare the forecasting performance of different models for the returns of three of the most traded exchange rates in terms of the US dollar, namely the French franc (FF/$), the German mark (DM/$) and the Japanese yen (Y/$). The relative performance of non‐linear models of the SETAR, STAR and GARCH types is contrasted with their linear counterparts. The results show that if attention is restricted to mean square forecast errors, the performance of the models, when distinguishable, tends to favour the linear models. The forecast performance of the models is evaluated also conditional on the regime at the forecast origin and on density forecasts. This analysis produces more evidence of forecasting gains from non‐linear models. Copyright © 2002 John Wiley & Sons, Ltd.     

Collinearity and the use of latent root regression for combining GNP forecasts

In combining economic forecasts a problem often faced is that the individual forecasts display some degree of dependence. We discuss latent root regression for combining collinear GNP forecasts. Our results indicate that latent root regression produces more efficient combining weight estimates (regression parameter estimates) than ordinary least squares estimation (OLS), although out-of-sample forecasting performance is comparable to OLS.