Forecast combination for outlier protection and forecast combination under heavy tailed errors

Abstract

Forecast combination has been proven to be a very important technique to obtain accurate predictions. Numerous forecast combination schemes with distinct properties have been proposed. However, to our knowledge, little has been discussed in the literature on combining forecasts with minimizing the occurrence of forecast outliers in mind. An unnoticed phenomenon is that robust combining, which often improves predictive accuracy (under square or absolute error loss) when innovation errors have a tail heavier than a normal distribution, may have a higher frequency of prediction outliers. Given the importance of reducing outlier forecasts, it is desirable to seek new loss functions to achieve both the usual accuracy and outlier-protection simultaneously.In the second part of this dissertation, we propose a synthetic loss function and apply it on a general adaptive theoretical and numeric results support the advantages of the new method in terms of providing combined forecasts with relatively fewer large forecast errors and comparable overall performances. For various reasons, in many applications, forecast errors exhibit heavy tail behaviors. Unfortunately, to our knowledge, little has been done to deal with forecast combination for such situations. The familiar forecast combination methods such as simple average, least squares regression, or those based on variance-covariance of the forecasts, may perform very poorly in such situations. In the third part of this dissertation, we propose two forecast combination methods to address the problem. One is specially proposed for the situations that 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 shortage of data and/or evolving data generating process. adaptive risk bounds of both methods are developed. Simulations and a real example show the excellent performance of the new methods.