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.     

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.     

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.     

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.     

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.     

Linear constraints and the efficiency of combined forecasts

This note extends some recent results, achieved by Clemen, on constraining the weights of a combined forecast. There is a great potential for improving the ordinary least squares forecast by imposing linear restrictions, and it will be shown how this potential can be exhausted by using an F-test. The corresponding decision procedure leads to a pre-test forecast with good statistical properties.     

Linear constraints and the efficiency of combined forecasts

Studies of combined forecasts have typically constrained the combining weights to sum to one and have not included a constant term in the combination. In a recent paper, Granger and Ramanathan (1984) have argued in favour of an unrestricted linear combination, including a constant term. This paper shows that for the purpose of prediction it may make sense to impose restrictions on the combining model because of potential increases in forecasting efficiency. Empirical results show that small gains in forecasting efficiency can be obtained by restricting the linear combination of GNP forecasts from four econometric models.     

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.     

Subjective versus objective combining of forecasts: An experiment

The paper reports results of an experiment conducted to evaluate subjective versus objective combination of forecasts. The subjects were undergraduate students at Texas A&M. The students forecasted two different types of time series. The results found show that the subjective combination of forecasts improves their accuracy as compared with individual efforts. Four ex-ante weighting methods were also used to combine the forecasts. They all improve the accuracy of the forecasts. The best results, though, were from the subjective combination of forecasts.     

The combining of forecasts using recursive techniques with non-stationary weights

This paper evaluates six optimal and four ad hoc recursive combination methods on five actual data sets. The performance of all methods is compared to the mean and recursive least squares. A modification to one method is proposed and evaluated. The recursive methods were found to be very effective from start-up on two of the data sets. Where the optimal methods worked well so did the ad hoc ones, suggesting that often combination methods allowing ‘local bias’ adjustment may be preferable to the mean forecast and comparable to the optimal methods.     

An empirical investigation of combinations of economic forecasts

This paper examines the effects of combining three econometric and three times-series forecasts of growth and inflation in the U.K. If forecasts are unbiased then a combination exploiting this fact will be more efficient than an unrestricted combination. Ex post econometric forecasts may be biased but ex ante they are unbiased. The results of the study are that a restricted linear combination of the econometric forecasts is superior to an unrestricted combination and also to the unweighted mean of the forecasts. However, it is not preferred to the best of the individual forecasts.     

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.     

Combining exchange rate forecasts: What is the optimal consensus measure?

In this paper a high-quality disaggregate database is utilized to examine whether individual forecasters produce efficient exchange rate predictions and also if the properties of the forecasts change when they are combined. The paper links a number of themes in the exchange rate literature and examines various methods of forecast combination. It is demonstrated, inter alia, that some forecasters are better than others, but that most are not as good as a naive no-change prediction. Combining forecasts adds to the accuracy of the predictions, but the gains mainly reflect the removal of systematic and unstable bias.     

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.     

Combination of forecasts in electricity demand prediction

Accurate demand prediction is of great importance in the electricity supply industry. Electricity cannot be stored, and generating plant must be scheduled well in advance to meet future demand. Up to now, where online information about external conditions is unavailable, time series methods on the historical demand series have been used for short-term demand prediction. These have drawbacks, both in their sensitivity to changing weather conditions and in their poor modelling of the daily/weekly business cycles. To overcome these problems a framework has been constructed whereby forecasts from different prediction methods and different forecasting origins can be selected and combined, solely on the basis of recent forecasting performance, with no a priori assumptions of demand behaviour. This added flexibility in univariate forecasting provides a significant improvement in prediction accuracy.     

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.     

The uses and abuses of ‘consensus’ forecasts

Clemen's (1989) review of the forecast-combining literature amply illustrates both the interest in and the importance of this subject. This article stresses the tautological properties of various consensus measures that assure their success relative to most individual forecasts. It confirms the finding of earlier studies that for each specific macroeconomic variable roughly one-third of individual forecasters are more accurate than a consensus. However, each individual does relatively poorly for some variable while the consensus, in contrast, necessarily never fails relative to most individuals. These results, like most previous studies, describe consensus measures that are synthetic constructs derived from a pre-existing set of individual forecasts. Strictly speaking, this contemporaneous consensus is not available to individual forecasters when their forecasts are made. A prior consensus measure, which is in their information sets, was relatively much less accurate than the contemporaneous measure. Nevertheless, a small subset of individual forecasters were generally inferior to the known, prior consensus forecast.     

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.     

N-step combinations of forecasts

While there is general agreement that a linear combination of forecasts can outperform the individual forecasts, there is controversy about the appropriateness of the combination method to be used in a given situation. Hence, in any given application it may be more beneficial to combine different sets of combined forecasts rather than picking one of them. This paper introduces the concept of N-step combinations of forecasts which involves combining the combined forecasts obtained from different combination procedures used at the preceding step. Using quarterly GNP data, evidence supporting the increase in the accuracy of the one-period-ahead ex-ante forecasts as the combination step increases is provided. The MSE, MAE, MAPE and their corresponding standard deviations are used to evaluate the accuracy of the forecasts obtained.