Forecasting Process Details

Forecast Combination Models

This section discusses the computation of predicted values and confidence limits for forecast combination models. See Chapter 48: Specifying Forecasting Models, for information about how to specify forecast combination models and their combining weights.

Given the response time series ${\{ y_{t} : 1 \le t \le n \} }$ with previously generated forecasts for the m component models, a combined forecast is created from the component forecasts as follows:

 

Predictions:

${\hat{y}_{t}= \sum _{i=1}^{m}{w_{i}\hat{y}_{i,t} }}$

 

Prediction Errors:

${\hat{e}_{t} = y_{t} - \hat{y}_{t}}$

where ${\hat{y}_{i,t}}$ are the forecasts of the component models and ${w_{i}}$ are the combining weights.

The estimate of the root mean square prediction error and forecast confidence limits for the combined forecast are computed by assuming independence of the prediction errors of the component forecasts, as follows:

 

Standard Errors:

${\hat{{\sigma }}_{t}= \sqrt {\sum _{i=1}^{m}{w^{2}_{i} \hat{{\sigma }}^{2}_{i,t}}}}$

 

Confidence Limits:

${{\pm } \hat{{\sigma }}_{t}\mr{Z}_{{\alpha }/2}}$

where ${\hat{{\sigma }}_{i,t}}$ are the estimated root mean square prediction errors for the component models, ${{\alpha }}$ is the confidence limit width, $1-\alpha $ is the confidence level, and ${\mr{Z}_{{\alpha }/2}}$ is the ${\frac{{\alpha }}{2}}$ quantile of the standard normal distribution.

Since, in practice, there might be positive correlation between the prediction errors of the component forecasts, these confidence limits may be too narrow.