Notation

The prediction errors are defined with a reference $i$ to the information set available at the time the forecast was made:

where \(\mathcal{F}_{i}\) need not only include lags of \(y_{t}\). In practice, the information that will be actually used may be a small subset of \(\mathcal{F}_{i}\).

The properties of these forecast errors can be assessed in isolation or relative to a benchmark, which we will define as \(\breve{e}_{t|i}\). The benchmark may be a naive forecast, e.g. random walk, in which case \(\breve{y}_{t|\mathcal{F}_{i}}\) would be equal to \(\breve{y}_{t|y_{t-1}}=y_{t-1}\). However, the benchmark could also be a prediction regularly published by a forecasting institute or market analysts, i.e. Bloomberg, which is not necessarily model-based. In that case, \(\breve{y}_{t|\mathcal{F}_{i}}\) would be given by methods and a subset of \(\mathcal{F}_{i}\) which is unknown to us.

For model-based forecasts, we use the following notation:

\(\hat{y}_{t|\mathcal{F}_{i}}=E_{\theta}[y_{t}|\mathcal{F}_{i}]\) to highlight the fact that they are based on model-consistent expectations given by the parameter vector \(\theta\) .

In forecasting comparisons involving competing forecasts resulting from the same information set, the subindex $i$ will be removed because it does not play a role. One could test the following hypothesis involving forecast errors:

The subsequent pages describe the implementation details of the various tests within JDemetra+ and examples of how to construct them:

• Diebold-Mariano
• Encompassing
• Bias