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13.4.3 How to Report Density Forecasts

A problem that frequently arises in density forecasting is how to report the predictive density. Point estimation typically requires reporting only a single number, for example the predicted number of jobs created next month may be 100,000, and is a number easily communicated to the user of the forecast. Density forecasting, in contrast, requires the entire distribution of the forecast to be communicated. If the density forecast is parametric, the functional form of the density along with the estimated parameters can be reported. For less parametric or Bayesian approaches, this becomes more difficult.

Often only partial information is transmitted at the end of the forecasting process. For example, the Bank of England’s fan charts report selected percentiles. Alternatively, a graph of the prediction distribution could be presented. A difficulty with these methods is that they cannot be directly used by any forecaster without a loss function to generate their optimal forecast. If we take seriously the motivation of density forecasting—that it allows us to evaluate the expected loss (13.1) for a broad array of loss functions—then the full density must be communicated to the user. For methods that result in a simulated set of pairs ${y_{i} ∣ z, p_{y i} (y ∣ z)}$ for a range of y-values, one approach is to provide d such pairs to the decision maker. These can be used to approximate (13.1) by the sum

$d^{- 1} i = 1 \sum d L (f, y_{i})_{l P y i} (y_{i} ∣ z),$

which in turn can be minimized to find the optimal forecast, $f,$ given $L$ . The difficulties with this approach are twofold. First, we need a large number of pairs d in order to obtain a satisfactory approximation to the expected loss. This means choosing a large value for d and so requires a computational approach, $e.g.,$ making draws available electronically. When y is multivariate, this becomes more of an issue. Second, the pairs ${y_{i} ∣ z, p_{y i} (y ∣ z)}, i = 1, \dots, d$ must be computed for all useful sets of $z,$ on which they are conditioned. For an agency that provides density estimates in real time this is not too difficult since the relevant z will be known.

练习题

What is the primary challenge in reporting predictive density in density forecasting compared to point estimation?

A. Point estimation requires reporting the entire distribution.

B. Density forecasting requires reporting only a single number.

C. Density forecasting requires communicating the entire distribution of the forecast.

D. Point estimation requires communicating the expected loss.

How is a parametric density forecast typically reported?

A. By providing a graph of the prediction distribution.

B. By reporting selected percentiles.

C. By reporting the functional form of the density along with the estimated parameters.

D. By providing a simulated set of pairs

{y_{i} ∣ z, p_{y i} (y ∣ z)}

What are the difficulties associated with non-parametric or Bayesian approaches in density forecasting?

A. They require reporting only a single number.

B. They make it difficult to report the functional form of the density.

C. They require communicating the entire distribution of the forecast.

D. They involve complex computational methods.

Partial information transmission in forecasting, such as reporting selected percentiles, allows forecasters to directly use the information without a loss function to generate their optimal forecast.

The full density must be communicated to the user to evaluate the expected loss for a broad array of loss functions in density forecasting.

The sum used to approximate the expected loss with simulated pairs is given by $d^{- 1} \sum_{i = 1}^{d} L (f, y_{i}) p_{y_{i}} (y_{i} ∣ z)$ , where $d$ is the number of ___.

To minimize the sum $d^{- 1} \sum_{i = 1}^{d} L (f, y_{i}) p_{y_{i}} (y_{i} ∣ z)$ and find the optimal forecast $f$ , given $L$ , a ___ number of pairs $d$ is required.

Explain why the pairs ${y_{i} ∣ z, p_{y i} (y ∣ z)}$ must be computed for all useful sets of $z$ in the simulated pairs approach.

What is the main advantage of using the two-piece normal distribution in density forecasting, as applied by the Bank of England?

Which of the following are challenges associated with the simulated pairs approach in density forecasting?

A. Requires a small number of pairs

d

B. Requires a computational approach to handle a large number of pairs.

C. Pairs must be computed for all useful sets of

z

D. Only applicable to univariate distributions.

When reporting a parametric density forecast, which of the following must be communicated to the user?

A. Only the estimated parameters

B. Only the functional form of the density

C. Both the functional form of the density and the estimated parameters

D. A graph of the prediction distribution

Which of the following are challenges associated with reporting non-parametric density forecasts? Select all that apply.

A. Requires reporting only the functional form

B. More difficult to communicate the full distribution

C. Often relies on partial information transmission

D. Simulated pairs approach requires large computational resources

The full density must be communicated to the user in density forecasting to evaluate the expected loss for a broad array of loss functions.

To approximate the expected loss using simulated pairs ${y_{i} ∣ z, p_{y i} (y ∣ z)}$ , the sum $d^{- 1} i = 1 \sum d L (f, y_{i})_{l P y i} (y_{i} ∣ z)$ is used, where $d$ represents the ___.

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13.4.3 How to Report Density Forecasts