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.     

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.     

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.     

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.     

Time series forecasts and extra-model information

Often a forecaster has supplementary information (e.g. field reports or forecasts from another source) that cannot be included directly in a time series model. Especially interesting are cases where this information is given at time intervals that are different from those of the time series model forecasts. Previous authors have considered a numerical and a model-based statistical method for combining extra-model information of this type with ARIMA model forecasts. This paper extends both methods to vector ARMA model forecasts and dynamic regression (transfer function) model forecasts. It is also shown that a Lagrange multiplier numerical procedure arises as a special case of the model-based procedure. An empirical example is given.     

Testing for unreliable estimators and insignificant forecasts in combined forecasts

The reliability and precision of the weights used in combining individual forecasts, irrespective of the method of combination, is important in evaluating a combined forecast. The objective of this study is not to suggest the ‘best’ method of combining individual forecasts, but rather to propose exploratory procedures, that make use of all available sample information contained in the covariance matrix of individual forecast errors, to (1) detect if the weights used in combining forecasts are ‘reliable’ (and ‘stable’ if it is known that the covariance matrix of forecast errors is stationary over time) and (2) test for ‘insignificant’ individual forecasts used in forming a combined forecast. We present empirical applications using two-year sales and individual forecast data provided by a major consumer durables manufacturer to illustrate the feasibility of our proposed procedures.     

A robust Cusum test for SETAR‐type nonlinearity in time series

As a part of an effective self‐exciting threshold autoregressive (SETAR) modeling methodology, it is important to identify processes exhibiting SETAR‐type nonlinearity. A number of tests of nonlinearity have been developed in the literature. However, it has recently been shown that all these tests perform poorly for SETAR‐type nonlinearity detection in the presence of additive outliers. In this paper, we develop an improved test for SETAR‐type nonlinearity in time series. The test is an outlier‐robust test based on the cumulative sums of ordered weighted residuals from generalized maximum likelihood fits. A Monte Carlo study confirms that the proposed test is competitive with existing tests for data from uncontaminated SETAR models and superior to them for SETAR data contaminated with additive outliers. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

Daily volatility forecasts: reassessing the performance of GARCH models

Volatility plays a key role in asset and portfolio management and derivatives pricing. As such, accurate measures and good forecasts of volatility are crucial for the implementation and evaluation of asset and derivative pricing models in addition to trading and hedging strategies. However, whilst GARCH models are able to capture the observed clustering effect in asset price volatility in‐sample, they appear to provide relatively poor out‐of‐sample forecasts. Recent research has suggested that this relative failure of GARCH models arises not from a failure of the model but a failure to specify correctly the ‘true volatility’ measure against which forecasting performance is measured. It is argued that the standard approach of using ex post daily squared returns as the measure of ‘true volatility’ includes a large noisy component. An alternative measure for ‘true volatility’ has therefore been suggested, based upon the cumulative squared returns from intra‐day data. This paper implements that technique and reports that, in a dataset of 17 daily exchange rate series, the GARCH model outperforms smoothing and moving average techniques which have been previously identified as providing superior volatility forecasts. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

On a robust test for SETAR‐type nonlinearity in time series analysis

There has been growing interest in exploiting potential forecast gains from the nonlinear structure of self‐exciting threshold autoregressive (SETAR) models. Statistical tests have been proposed in the literature to help analysts check for the presence of SETAR‐type nonlinearities in observed time series. However, previous studies show that classical nonlinearity tests are not robust to additive outliers. In practice, time series outliers are not uncommonly encountered. It is important to develop a more robust test for SETAR‐type nonlinearity in time series analysis and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null distribution of the test statistic is derived. A Monte Carlo experiment is carried out to compare the power of the proposed test with other existing tests under the influence of time series outliers. The effects of additive outliers on nonlinearity tests with misspecification of the autoregressive order are also studied. The results indicate that the proposed method is preferable to the classical tests when the observations are contaminated with outliers. Finally, we provide illustrative examples by applying the statistical tests to three real datasets. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian analysis of concurrent time series with application to regional IBM revenue data

Business data frequently arise in the form of concurrent time series. We present a general framework for simultaneous modeling and fitting of such series using the class of Box—Jenkins models. This framework is an exchangeable hierarchical Bayesian model incorporating dependence among the series. Our motivating data set consists of regional IBM revenue available monthly for several geographic regions. Stationary seasonal autoregressive models are simultaneously fit to the regional data series using various error covariance specifications for the strong interregional dependence. A modified Gibbs sampling algorithm is used to carry out the fitting and to enable all subsequent inference. Graphical techniques using predictive distributions are employed to assess model adequacy and to select among models. Outlier estimation and prediction under the chosen model are used for planning and to measure the effect of special promotional events.     

Ex-post and ex-ante prediction of unobserved economic time series: a case study

In several countries, some macro-economic variables are not observed frequently (e.g. quarterly) and economic authorities need estimates of these high-frequency figures to make econometric analyses or to follow closely the country's economic growth. Two problems are involved in this context. The first is to make these estimates after observing low-frequency values and some related indicators, and the second is to obtain predictions using just the observed indicators, i.e. before observing a new low-frequency figure. This paper gives a new optimal solution to the first problem, and solves the second using a recursive optimal approach. In the second situation, additionally, statistical tests are developed for detecting structural changes at current periods in the macro-economic variable involved. © 1998 John Wiley & Sons, Ltd.     

Modelling and forecasting time series sampled at different frequencies

This paper discusses how to specify an observable high‐frequency model for a vector of time series sampled at high and low frequencies. To this end we first study how aggregation over time affects both the dynamic components of a time series and their observability, in a multivariate linear framework. We find that the basic dynamic components remain unchanged but some of them, mainly those related to the seasonal structure, become unobservable. Building on these results, we propose a structured specification method built on the idea that the models relating the variables in high and low sampling frequencies should be mutually consistent. After specifying a consistent and observable high‐frequency model, standard state‐space techniques provide an adequate framework for estimation, diagnostic checking, data interpolation and forecasting. An example using national accounting data illustrates the practical application of this method. Copyright © 2008 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.