Abstract
Forecast combination has been proven to be a very important technique to obtain accurate predictions for various applications in economics, finance, marketing and many other areas. In many applications, forecast errors exhibit heavy-tailed behaviors for various reasons. Unfortunately, to our knowledge, little has been done to obtain reliable forecast combinations for such situations. The familiar forecast combination methods, such as simple average, least squares regression or those based on the variance-covariance of the forecasts, may perform very poorly due to the fact that outliers tend to occur, and they make these methods have unstable weights, leading to un-robust forecasts. To address this problem, in this paper, we propose two nonparametric forecast combination methods. One is specially proposed for the situations in which the forecast errors are strongly believed to have heavy tails that can be modeled by a scaled Student’s t-distribution; the other is designed for relatively more general situations when there is a lack of strong or consistent evidence on the tail behaviors of the forecast errors due to a shortage of data and/or an evolving data-generating process. Adaptive risk bounds of both methods are developed. They show that the resulting combined forecasts yield near optimal mean forecast errors relative to the candidate forecasts. Simulations and a real example demonstrate their superior performance in that they indeed tend to have significantly smaller prediction errors than the previous combination methods in the presence of forecast outliers.
Highlights
When multiple forecasts are available for a target variable, well-designed forecast combination methods can often outperform the best individual forecaster, as demonstrated in the literature of the applications of forecast combinations in areas, such as economics, finance, tourism and wind power generation in the last fifty years.Many combination methods have been proposed from different perspectives since the seminal work of forecast combination by Bates & Granger [1]
Remarks: 1. When only Condition 2 is satisfied, Theorem 1 shows that the cumulative distance between the true densities and their estimators from the t-adaptive forecasting through exponential re-weighting (AFTER) is upper bounded by the cumulative forecast errors of the best candidate forecaster plus a penalty that has two parts: the squared relative estimation errors of the scale parameters and the logarithm of the initial weights
It is expected that the t-AFTER and g-AFTER will outperform the L1 -AFTER and L2 -AFTER when forecasting data generating processes (DGPs) with heavy-tailed distributions in the errors
Summary
When multiple forecasts are available for a target variable, well-designed forecast combination methods can often outperform the best individual forecaster, as demonstrated in the literature of the applications of forecast combinations in areas, such as economics, finance, tourism and wind power generation in the last fifty years. If the forecast errors are assumed to follow a normal or a double-exponential distribution with zero mean, the conditional probability density functions used in the combining process of the AFTER scheme can be estimated relatively for all of the candidate forecasters, because the estimation of the conditional scale parameters is straightforward Cheng & Yang [18] advocate the incorporation of a smooth surrogate of the L0 -loss in the performance measure for weighting to reduce the occurrence of outlier forecasts None of these methods can deal with forecast errors with tails as heavy as that of Student’s t-distributions.
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