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17.9 RELATING THE METHODS

Differences between forecast encompassing tests and tests for equal predictive accuracy are perhaps best understood by considering the relation between these tests and forecast combination under MSE loss for a pair of forecasts. As we saw in chapter 14, if two forecasts have the same forecast error variance and we fail to reject the null of equal forecast accuracy, then it is optimal to use equal weights in a combined forecast, rather than use only one or the other forecast. In this case the combined forecast will reduce the forecast error variance, at least asymptotically. Conversely, forecast encompassing tests inspect whether it is optimal to assign a weight of 1 on one forecast and 0 on the other. These tests therefore examine very different hypotheses regarding the relative usefulness of the forecasts.

This reasoning does not extend to the situation with more than two forecasts in the comparison. With multiple forecasts, a combined forecast need not assign nonzero weights to all forecasts even if these have the same accuracy and it is possible for a forecast to produce the same forecast error variance as the remaining forecasts, yet still be encompassed by these forecasts.

Tests for forecast encompassing are equivalent to pre-testing a model. The forecast encompassing test addresses whether there is useful information in a forecast, just as pre-testing asks whether there is useful information in a candidate regressor. The general approach is to subsequently ignore models that are encompassed, just as regressors are ignored if they do not contribute enough to be marginally helpful in explaining the outcome.

Tests for equal predictive accuracy are more difficult to consider since neither a rejection nor a failure to reject imply a specific course of action. If the proposal is to replace one model with another, equal predictive accuracy might be regarded as a failure of the competing model to improve on the benchmark forecast. However, equal predictive accuracy does not mean that a new model is not useful, since it is possible that a combination of forecasts could lead to improvements. Moreover, even if we reject that the forecasts are equally good, it still may be that the forecasts can be usefully combined to provide a better forecast.

练习题

If two forecasts have the same forecast error variance and we fail to reject the null of equal forecast accuracy, what is the optimal weighting strategy for a combined forecast?

A. Assign a weight of 1 to one forecast and 0 to the other

B. Use equal weights for both forecasts

C. Assign weights based on the inverse of the forecast error variance

D. Use weights proportional to the forecast accuracy

What does a forecast encompassing test examine regarding the weights of two forecasts?

A. Whether equal weights are optimal

B. Whether weights should be proportional to forecast accuracy

C. Whether it is optimal to assign a weight of 1 to one forecast and 0 to the other

D. Whether weights should be assigned based on forecast error variance

Which of the following statements are true about multiple forecasts and weight assignment?

A. A combined forecast must assign nonzero weights to all forecasts if they have the same accuracy

B. A forecast can produce the same forecast error variance as others and still be encompassed by them

C. The reasoning for two forecasts extends directly to multiple forecasts without modification

D. With multiple forecasts, a combined forecast need not assign nonzero weights to all forecasts even if they have the same accuracy

Forecast encompassing tests are equivalent to pre-testing a model to determine if it contains useful information.

Equal predictive accuracy tests always imply a specific course of action, such as replacing one model with another.

If two forecasts have the same forecast error variance and equal forecast accuracy, the combined forecast will reduce the forecast error variance ___.

Forecast encompassing tests inspect whether it is optimal to assign a weight of _ to one forecast and _ to the other.

Explain the relationship between forecast encompassing tests and pre-testing a model.

Why might equal predictive accuracy not imply that a new model is not useful?

Which of the following statements are true about the implications of equal predictive accuracy tests?

A. Equal predictive accuracy means a new model is not useful

B. Equal predictive accuracy does not imply a specific course of action

C. Rejecting equal predictive accuracy means forecasts cannot be combined usefully

D. Combining forecasts might still lead to improvements even if predictive accuracy is equal

When comparing two forecasts with equal forecast error variance, what is the optimal weight assignment for a combined forecast if we fail to reject the null of equal forecast accuracy?

A. Assign a weight of 1 to one forecast and 0 to the other

B. Assign a weight of 0.5 to each forecast

C. Assign a weight of 0.7 to one forecast and 0.3 to the other

D. Assign a weight of 1 to the forecast with the smaller error variance

Which of the following statements are true regarding forecast encompassing tests and tests for equal predictive accuracy?

A. Forecast encompassing tests inspect whether it is optimal to assign a weight of 1 to one forecast and 0 to the other

B. Tests for equal predictive accuracy always imply a specific course of action

C. Forecast encompassing tests are equivalent to pre-testing a model

D. Equal predictive accuracy means that a new model is not useful

E. Even if forecasts are not equally good, they can still be usefully combined

With multiple forecasts, a combined forecast must assign nonzero weights to all forecasts if they have the same accuracy.

Tests for forecast encompassing are equivalent to ___, which addresses whether there is useful information in a candidate regressor.

Explain why equal predictive accuracy does not necessarily mean that a new model is not useful.

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17.9 RELATING THE METHODS