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3.1.3 Optimal Forecasts Conditional on Future Variables

Often we are interested in conditional forecasts, i.e., forecasts of Y conditional on a specific future path taken by another random variable, W. In the above analysis and in what follows, the results can be extended to this case by replacing $ρ_{Y} (y ∣ z, θ)$ and $p_{Z} (z ∣ θ)$ with the distributions conditional on setting the outcome of W to $w .$ i.e., $P Y (y ∣ z, w, θ)$ and $p_{Z} (z ∣ w, θ)$ . For example, the Federal Reserve might want to forecast inflation conditional on a path of future Federal funds rates. While this extension may seem trivial conceptually, it can be difficult to implement in practice. The difficulty arises in constructing $p_{Y} (y ∣ z, w, θ)$ . Typically a structural model of the data is required to understand the distribution of the outcome given future values of the variables we are conditioning on. In the case of inflation forecasting conditional on the Federal funds rate, we require a model for the structural relationship to understand how inflation reacts to the Federal funds rate without feedback effects from inflation.

3.2 CLASSICAL APPROACH

Applying any of the results of the previous section requires estimating the form of the model for the appropriate feature of the conditional distribution of Y given $z$ along with the unknown parameters that arise in this estimation. Suppose we have a parametric model for the optimal forecast, $f (z, β)$ , where $β$ are parameters of the model, which in turn are a function of θ . Our (infeasible) optimal forecast model $f (z, \hat{β})$ must be estimated and now becomes $f (z, \hat{β})$ , where $\hat{β}$ will be a function of $z,$ the data available to construct estimates of the forecast model. Because the parameter estimates are a function of the available data, $z,$ we can write $f (z, \hat{β}) = f (z)$ . In the classical approach, a good estimator and hence a good forecasting model will be one that results in low average loss, where we average over $Z$ as well as Y . This is an estimator (or forecast model) that has low risk, where risk is defined as

$R (θ, f) = E_{Y, Z} [L (f (Z), Y)] = \int_{z} \int_{y} L (f (z), y)_{P Y} (y ∣ z, θ)_{P Z} (z ∣ θ) d y d z . (3.15)$

This is the risk that classical forecasting methods—methods for choosing $f (z) -$ attempt to control and minimize.

练习题

What is the main focus of conditional forecasts?

A. Forecasts of

W

conditional on

Y

B. Forecasts of

Y

conditional on a specific future path of

W

C. Forecasts of

Z

conditional on

Y

D. Forecasts of

Y

without considering

W

In extending results to conditional forecasts, which distributions are replaced?

P_{Y} (y ∣ z, θ)

and

p_{Z} (z ∣ θ)

are replaced with

P_{Y} (y ∣ z, θ)

and

p_{Z} (z ∣ θ)

P_{Y} (y ∣ z, θ)

and

p_{Z} (z ∣ θ)

are replaced with

P_{Y} (y ∣ z, w, θ)

and

p_{Z} (z ∣ w, θ)

P_{Y} (y ∣ z, w, θ)

and

p_{Z} (z ∣ w, θ)

are replaced with

P_{Y} (y ∣ z, θ)

and

p_{Z} (z ∣ θ)

D. Only

P_{Y} (y ∣ z, θ)

is replaced with

P_{Y} (y ∣ z, w, θ)

What are the difficulties in implementing conditional forecasts? (Select all that apply)

A. Constructing

p_{Y} (y ∣ z, w, θ)

B. Understanding the distribution of the outcome given future values of the variables we are conditioning on

C. Calculating the average of

Y

and

Z

D. Estimating the parameters of the model easily

What is an example of conditional forecasting?

A. Forecasting

Z

conditional on

Y

B. Forecasting inflation conditional on a path of future Federal funds rates

C. Forecasting

W

without considering

Y

D. Forecasting

Y

based on past values of

Y

only

The classical approach to forecasting aims to minimize the average loss over $Y$ only.

Risk in classical forecasting is defined as the expected loss when integrating over both $Y$ and $Z$ .

In the parametric model for the optimal forecast $f (z, β)$ , $β$ are the __________ of the model.

The estimated optimal forecast model $f (z, \hat{β})$ can be written as $f (z)$ because $\hat{β}$ is a function of __________.

Explain the goal of classical forecasting methods.

How does the classical approach define a good forecasting model?

When constructing a conditional forecast of $Y$ given a future path of $W$ , which of the following distributions must be replaced with their conditional counterparts?

p_{Y} (y ∣ θ)

and

p_{Z} (z ∣ θ)

ρ_{Y} (y ∣ z, θ)

and

p_{Z} (z ∣ θ)

p_{Y} (y ∣ θ)

and

p_{W} (w ∣ θ)

ρ_{Y} (y ∣ z, θ)

and

p_{W} (w ∣ θ)

Which of the following statements are true regarding the classical approach to forecasting?

A. A good forecasting model results in low average loss over

Y

and

Z

B. Risk is defined as the expected loss over

Y

only.

C. The goal of classical forecasting methods is to minimize risk.

D. The optimal forecast model

f (z, \hat{β})

is always feasible to estimate.

The difficulty in implementing conditional forecasts arises primarily from the need to construct $p_{Y} (y ∣ z, w, θ)$ , which typically requires a structural model of the data.

In the classical approach, risk is defined as $R (θ, f) = E_{Y, Z} [L (f (Z), Y)] = \int_{z} \int_{y} L (f (z), y) p_{Y} (y ∣ z, θ) p_{Z} (z ∣ θ) d y d z$ . This measure is used to evaluate the performance of a ___.

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3.1.3 Optimal Forecasts Conditional on Future Variables