3.4.1 Density Forecasts

3.4.1 Density Forecasts

The search for a solution to minimize $E_Y[L(f(z),Y)|Z]$ across different decision rules requires choosing a forecast, i.e., a function of the forecast density that minimizes the risk

$$f^{\ast }(z)=\arg \underset{f(z)}{\min }{\int }_{y}L(f(z),y{)}_{PY}(y|z)dy.\text{\hspace{1em}(3.29)}$$

This expression is suggestive of two ways in which density rather than point forecasts are useful. First, forecasters with different loss functions will generally construct different optimal forecasts even though the true density for the data is the same for all forecasters. Under some loss functions (e.g., piecewise linear loss), the optimal forecast will be a quantile of the outcome for which the quantile depends on the degree of loss asymmetry; see Example 3.1.3. Two forecasters with different loss functions in this family will want different quantiles of the predictive distribution, so an agency that merely reports a single number could never provide both with an optimal forecast.

Second, point forecasts are often criticized on the grounds that they fail to communicate the degree of uncertainty surrounding the forecast. For example, a point forecast of changes in future property prices might be positive although the forecast density could place significant weight on a price fall.

An alternative to providing point forecasts is thus to report an estimate of the predictive density $p_Y(y|z)$ . When closed-form solutions are available, point forecasts can be directly computed from summary statistics of the provided distribution. More generally, numerical integration over the density forecast is required to find the optimal point forecast.

Density forecasts solve the first problem since two forecasters requiring different quantiles of the distribution would both be able to construct their optimal forecasts from the densities provided. Predictive densities also work for an MSE forecaster interested in the conditional mean. This forecaster can compute the conditional mean from the density forecast and examine how reasonable the mean is as a summary statistic for the center of the conditional distribution of Y given Z.

It would appear that it is sufficient to provide the entire predictive distribution (density forecast) for all possible outcomes. A number of caveats arise for this solution, however. While there seems to be a clear gain in generality from providing density forecasts rather than point forecasts, this is somewhat illusory since typically the predictive density depends on parameters, θ, that must be estimated. Hence an estimated predictive density will be provided, with different estimators of the density (e.g., different estimators of θ for parametric densities), resulting in different density estimates. Forecasters with different loss functions will generally prefer different estimators. The provision of the predictive density does not avoid the issue that the best estimator ultimately depends on the loss function. From the perspective of forecasters with piecewise linear (lin-lin) loss, a density estimator might well trade off accuracy near the median for accuracy at the tails (or vice versa). Hence a forecaster requiring a quantile near the mean will prefer a different estimate of the predictive density than a forecaster requiring a quantile in the tails.

The Bayesian equivalent to the predictive density is constructed via the posterior distribution for θ. The Bayesian approach chooses f (z) to minimize

$$\begin{array}{rlr}& & r(\pi ,f)={\int }_z({\int }_{\theta }\left\{{\int }_{y}L(f(z),y,z)p_Y(y|z,\theta )dy\right\}{p}_Z(z|\theta )\pi (\theta )d\theta )dz\\ & & ={\int }_z\left({\int }_{\theta }\left\{{\int }_{y}L(f(z),y,z)p_Y(y|z,\theta )dy\right\}\pi (\theta |z)d\theta \right)m(z)dz\\ & & ={\int }_z\left({\int }_y\left\{L(f(z),y,z){\int }_{\theta }{p}_Y(y|z,\theta )\pi (\theta |z)d\theta \right\}dy\right)m(z)dz.\end{array}\text{\hspace{1em}(3.30)}$$

Here we allow the integral over θ to pass by the loss function since the loss function is not explicitly a function of the parameters, θ . Also, $\begin{array}{r}{\int }_{\theta }{p}_Y(y|z,\theta )\pi (\theta |z)d\theta =p_Y^P(y|z)\end{array}$ is the predictive density obtained by integrating over θ , using π (θ |z) as weights.