Encompassing tests for evaluating multi-step system forecasts invariant to linear transformations

The paper suggests two encompassing tests for evaluating multi-step system forecasts invariant to linear transformations. An invariant measure for forecast accuracy is necessary as the conclusions otherwise can depend on how the forecasts are reported (e.g., as in level or growth rates). Therefore, a measure based on the prediction likelihood of the forecast for all variables at all horizons is used. Both tests are based on a generalization of the encompassing test for univariate forecasts where potential heteroscedasticity and autocorrelation in the forecasts are considered. The tests are used in evaluating quarterly multi-step system forecasts made by Statistics Norway