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5.1.1 Empirical Bayes Methods

Forecasts or forecast distributions generally depend on so-called hyperparameters that must be chosen by the forecaster. This can readily be seen for the point forecasts in the above examples. For example, for the normal regression forecast in, we must choose the hyperparameters $β_{0}$ and $Σ_{0}$ to construct the forecast $\tilde{β}^{'} x_{T}$ In empirical Bayes methods, the hyperparameters are estimated from the data in a model consistent way, rather than being imposed by the forecaster.

The typical empirical Bayes approach is to use the marginal density of the (observed) data that arises from integrating out the prior density, i.e., $m (z) =$ $\int p z (z ∣ θ) π (θ) d θ$ . Since this is a function of the hyperparameters in $π (θ)$ , this relates the distribution that generates the observed data, $z_{t}$ , to the hyperparameters. For many problems the hyperparameters can therefore be estimated from the observed data in a simple and intuitive way.

Empirical applications in economics or finance that use so-called g -priors often involve estimating hyperparameters and priors based on the data and then determining the “strength” of the prior as some fraction of the information available in the data sample at hand. For a discussion of both parametric and nonparametric empirical Bayes methods in the context of forecast models with many predictors, see Stock and Watson (2006).

5.1.2 Bayesian Modeling of the Predictive Distribution

Construction of the predictive distribution, $p_{Y} (y_{T + 1} ∣ Z_{T})$ , i.e., the posterior distribution of the variable we wish to forecast conditional only on the current data, $Z_{t} = {Y_{t}, X_{t}}, t = 1, ..., T .$ , is the main object of interest in Bayesian forecasting. Having constructed a posterior density for $θ, π (θ ∣ Z_{T})$ , the predictive density is the average density for y after integrating out the unknown parameters. Formally, we have

$p_{y}^{p} (y_{T + 1} ∣ Z_{T}) = \int p_{y} (y_{T + 1} ∣ θ, Z_{T}) π (θ ∣ Z_{T}) d θ .$

An illustration of this point is the result in Example 5.1.1, where for the i.i.d. normal case the predictive density is a normal distribution with mean equal to the posterior mean and variance equal to the sum of the variance of $γ_{t}$ plus the variance from the posterior for the mean.

Just as in the classical case, uncertainty over the parameters of the model results in greater risk and less certainty over the location of the forecast. In the previous example this shows up in a strictly larger variance than if the mean were known.

How the forecast is constructed differs depending on the difficulties involved in the computation of the integral in (5.7) and also in the construction of the posterior which itself requires computing an integral. Closed-form solutions are rarely available in interesting forecasting problems and so instead the predictive distribution is computed using numerical methods that can be tailored to the specific problem at hand.

练习题

In forecasting, what are hyperparameters?

A. Parameters that are estimated from the data using numerical methods

B. Parameters that are directly observed in the data

C. Parameters that must be chosen by the forecaster

D. Parameters that are always known with certainty

What is the key characteristic of empirical Bayes methods for hyperparameters?

A. Hyperparameters are imposed by the forecaster

B. Hyperparameters are estimated from the data in a model-consistent way

C. Hyperparameters are ignored in the forecasting process

D. Hyperparameters are set to zero

Which of the following are true about the typical empirical Bayes approach? (Select all that apply)

A. It uses the marginal density of the observed data

B. It integrates out the prior density

C. It is unrelated to the hyperparameters in

π (θ)

D. It provides a simple and intuitive way to estimate hyperparameters

Empirical applications in economics and finance that use g -priors involve determining the “strength” of the prior as some fraction of the information available in the data sample at hand.

The typical empirical Bayes approach uses the marginal density of the observed data $m (z) = \int p z (z ∣ θ) π (θ) d θ$ , which is a function of the hyperparameters in ___.

Explain how empirical Bayes methods differ from traditional methods in choosing hyperparameters.

What is the main object of interest in Bayesian forecasting?

A. The prior distribution of the parameters

B. The joint distribution of the data

C. The posterior distribution of the variable to be forecast conditional on the current data

D. The likelihood function of the data

What is the predictive density formula in Bayesian forecasting?

p_{y}^{p} (y_{T + 1} ∣ Z_{T}) = \int p_{y} (y_{T + 1} ∣ θ) d θ

p_{y}^{p} (y_{T + 1} ∣ Z_{T}) = \int p_{y} (y_{T + 1} ∣ θ, Z_{T}) π (θ) d θ

p_{y}^{p} (y_{T + 1} ∣ Z_{T}) = \int p_{y} (y_{T + 1} ∣ θ, Z_{T}) π (θ ∣ Z_{T}) d θ

p_{y}^{p} (y_{T + 1} ∣ Z_{T}) = \int p_{y} (y_{T + 1} ∣ Z_{T}) π (θ ∣ Z_{T}) d θ

Which of the following are true about the predictive density in the i.i.d. normal case? (Select all that apply)

A. It is a normal distribution

B. Its mean is equal to the prior mean

C. Its mean is equal to the posterior mean

D. Its variance is equal to the sum of the variance of

γ_{t}

and the variance from the posterior for the mean

Uncertainty over the parameters of the model in Bayesian forecasting results in less risk and more certainty over the location of the forecast.

In Bayesian forecasting, the predictive density is computed using numerical methods when closed - form solutions are rarely available, and these methods can be tailored to the specific ___.

Explain the relationship between uncertainty over model parameters and forecast risk in Bayesian forecasting.

Which of the following statements are correct regarding the integration of prior knowledge and current section concepts? (Select all that apply)

A. The forecaster’s loss function is involved in the construction of the predictive density in Bayesian forecasting.

B. Hyperparameters in empirical Bayes methods are estimated from the data, similar to how the posterior for parameters is constructed using Bayes' rule where the marginal likelihood is calculated by integrating out the prior.

C. The optimal Bayesian point forecast is the forecast that maximizes expected loss with respect to the posterior predictive density.

D. In the i.i.d. normal case of predictive density, the mean is related to the posterior mean and variance is related to the sum of variances from different sources, just as the posterior for parameters is calculated using the relationship between the likelihood, prior, and marginal likelihood.

How does the concept of minimizing expected loss in Bayesian point forecasting relate to the construction of the predictive density and the use of prior information in empirical Bayes methods?

In empirical Bayes methods, how are hyperparameters typically estimated? Choose the most accurate description.

A. By using a fixed value chosen by the forecaster

B. By integrating out the prior density to obtain the marginal density of the observed data

C. By minimizing the expected loss function directly

D. By assuming a uniform prior distribution

Which of the following statements correctly describes the relationship between the predictive density and the posterior density in Bayesian forecasting?

A. The predictive density is the posterior density of the parameters.

B. The predictive density is the average density for

y

after integrating out the unknown parameters using the posterior density.

C. The predictive density is constructed without considering the posterior density.

D. The predictive density is always a normal distribution.

Select all the statements that are true regarding the computation of the predictive distribution in Bayesian forecasting.

A. Closed-form solutions are commonly available for interesting forecasting problems.

B. The predictive distribution is computed using numerical methods when closed-form solutions are not available.

C. The computation of the predictive distribution involves integrating out the unknown parameters.

D. The predictive distribution is unaffected by uncertainty over the model parameters.

E. The predictive distribution can be tailored to the specific problem at hand using numerical methods.

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5.1.1 Empirical Bayes Methods