Basic Notation

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.