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2.2.3 Loss Functions That Depend on Other State Variables

Under some simplifying assumptions we saw earlier that the binary loss function takes a particularly simple form. More generally, if the loss function depends on Z and the loss associated with a perfect forecast depends on the outcome Y, then the loss function for the binary problem becomes

$L (f, y, z) = ⎩ ⎨ ⎧ - u_{1, 1} (z) - u_{1, 0} (z) - u_{0, 1} (z) - u_{0, 0} (z) if f = 1 and y = 1, if f = 1 and y = 0, if f = 0 and y = 1, if f = 0 and y = 0,$

where $u_{i, j} (z)$ are the utilities gained when $f = i, y = j$ , and $Z = z .$

In this general form, the loss function cannot be simplified to depend only on the forecast error. Again restrictions need to be imposed on the losses in (2.21). First, we require that $u_{0, 0} (z) > u_{1, 0} (z)$ and $u_{1, 1} (z) > u_{0, 1} (z)$ so that losses associated with correct forecasts are not higher than those associated with incorrect forecasts. We might also impose that min ${u_{0, 0} (z), u_{1, 1} (z)} > max {u_{1, 0} (z), u_{1, 0} (z)}$ so that correct forecasts result in a lower loss (higher utility) than incorrect forecasts. Finally, it is quite reasonable to assume that correct forecasts are associated with different losses $u_{0, 0} (z) \neq = u_{1, 1} (z)$ , in which case normalizing the loss associated with a perfect forecast to 0 will not be possible for both outcomes. This is an example of level-dependent loss being built directly into the loss function.

2.2.4 Consistent Ranking of Forecasts with Measurement Errors in the Outcome

Hansen and Lunde (2006) and Patton (2011) consider the problem of comparing and consistently ranking volatility forecasts from different models when the observed outcome is measured with noise. This situation is common in volatility forecasting or in macro forecasting where the outcome may subsequently be revised. The volatility of asset returns is never actually observed although a proxy for it can be constructed. Volatility forecast comparisons typically use realized volatility, squared returns, or range-based proxies, $\overset{σ}{^}^{\hat{2}}$ , in place of the true variance, $σ^{2}$

Hansen and Lunde establish sufficient conditions under which noisy proxies can be used in the forecast evaluation without giving rise to rankings that are inconsistent with the (infeasible) ranking based on the true outcome.

Patton defines a loss function as being robust to measurement errors in the outcome if it gives the same expected-loss ranking of two forecasts whether based on the true (but unobserved) outcome or some unbiased proxy thereof. Specifically, a loss function is robust to such measurement errors if, for two forecasts $f_{1}$ and $f_{2}$ the ranking based on the true outcome, $y,$

$E [L (f_{1}, y)] \supseteq E [L (f_{2}, y)]$

is the same as the ranking based on the proxied outcome, $\overset{y}{^}$ :

$E [L (f_{1}, \overset{y}{^})] \supseteq E [L (f_{2}, \overset{y}{^})],$

for unbiased proxies $\overset{y}{^}$ satisfying $E [\overset{y}{^} ∣ Z] = y$ , where Z is again the information set used to generate the forecasts.

Patton (2011, Proposition 1) establishes conditions under which robust loss functions must belong to the following family:

$L (f, \overset{y}{^}) = \tilde{C} (f) + B (\overset{y}{^}) + C (f) (\overset{y}{^} - f),$

where $B$ and $C$ are twice continuously differentiable functions, $C$ is strictly decreasing, and $\tilde{C}$ is the antiderivative of $C_{i}$ , i.e., $\tilde{C}^{'} = C .^{10}$ In Patton’s analysis $f$ is a volatility forecast and $\overset{y}{^}$ is a proxy for the realized volatility. Examples of loss functions in the family (2.22) include MSE and QLIKE loss:

$MSE : L (f, \overset{y}{^}) = (\overset{y}{^} - f)^{2},$

$QLIKE : L (f, \overset{y}{^}) = lo g (f) + \frac{y ^}{f} .$

练习题

Which of the following correctly represents the loss function $L (f, y, z)$ when $f = 1$ and $y = 0$ ?

- u_{1, 1} (z)

- u_{1, 0} (z)

- u_{0, 1} (z)

- u_{0, 0} (z)

What condition ensures that losses associated with correct forecasts are not higher than those associated with incorrect forecasts?

u_{0, 0} (z) > u_{1, 0} (z)

and

u_{1, 1} (z) > u_{0, 1} (z)

u_{0, 0} (z) < u_{1, 0} (z)

and

u_{1, 1} (z) < u_{0, 1} (z)

u_{0, 0} (z) = u_{1, 0} (z)

and

u_{1, 1} (z) = u_{0, 1} (z)

u_{0, 0} (z) > u_{0, 1} (z)

and

u_{1, 1} (z) > u_{1, 0} (z)

Which of the following are valid restrictions on the losses for the binary loss function? (Select all that apply)

u_{0, 0} (z) > u_{1, 0} (z)

u_{1, 1} (z) < u_{0, 1} (z)

min {u_{0, 0} (z), u_{1, 1} (z)} > max {u_{1, 0} (z), u_{0, 1} (z)}

u_{0, 0} (z) = u_{1, 1} (z)

It is reasonable to assume that correct forecasts are associated with different losses $u_{0, 0} (z) \neq = u_{1, 1} (z)$ .

What is the main problem addressed by Hansen and Lunde (2006) and Patton (2011)?

A. Comparing and ranking volatility forecasts when the outcome is perfectly observed

B. Comparing and ranking volatility forecasts when the outcome is measured with noise

C. Developing new loss functions for binary outcomes

D. Normalizing loss functions for perfect forecasts

Which of the following are sufficient conditions for using noisy proxies in forecast evaluation? (Select all that apply)

A. Noisy proxies can be used without affecting the rankings

B. Noisy proxies must be biased

C. Noisy proxies must be unbiased

D. Noisy proxies must satisfy

E [\overset{y}{^} ∣ Z] = y

A loss function is robust to measurement errors if it gives different expected-loss rankings of two forecasts based on the true outcome and the proxied outcome.

Patton (2011) establishes that robust loss functions must belong to the family: $L (f, \overset{y}{^}) = \tilde{C} (f) + B (\overset{y}{^}) + C (f) (\overset{y}{^} - f)$ , where $B$ and $C$ are twice continuously differentiable functions, and $C$ is strictly ___.

Explain the significance of the condition $E [\overset{y}{^} ∣ Z] = y$ in the context of using noisy proxies for forecast evaluation.

Which of the following is an example of a loss function in the family established by Patton (2011)?

L (f, \overset{y}{^}) = (\overset{y}{^} - f)^{3}

L (f, \overset{y}{^}) = lo g (f) + \frac{y ^}{f}

L (f, \overset{y}{^}) = \tilde{C} (f) + B (\overset{y}{^})

L (f, \overset{y}{^}) = C (f) (\overset{y}{^} - f)^{2}

Why is it important that $u_{0, 0} (z) \neq = u_{1, 1} (z)$ in the context of binary loss functions?

The MSE loss function is an example of a robust loss function as defined by Patton (2011).

Consider a binary forecasting problem where the loss function depends on a state variable $Z$ and the outcome $Y$ . The general form of the loss function is given by $L (f, y, z)$ . If we impose the restriction that $u_{0, 0} (z) > u_{1, 0} (z)$ and $u_{1, 1} (z) > u_{0, 1} (z)$ , which of the following statements is true?

A. Correct forecasts will always have higher losses than incorrect forecasts.

B. Correct forecasts will always have lower losses than incorrect forecasts.

C. The losses for correct forecasts will be equal to the losses for incorrect forecasts.

D. The losses for correct forecasts will depend only on the forecast error.

Which of the following are examples of loss functions that can be used in the family of robust loss functions as defined by Patton (2011)? Select all that apply.

A. Mean Squared Error (MSE):

L (f, \overset{y}{^}) = (\overset{y}{^} - f)^{2}

B. Mean Absolute Error (MAE):

L (f, \overset{y}{^}) = ∣ \overset{y}{^} - f ∣

C. QLIKE loss:

L (f, \overset{y}{^}) = lo g (f) + \frac{y ^}{f}

D. Linex loss:

L (e) = a_{1} (exp (a_{2} e) - a_{2} e - 1)

A loss function is considered robust to measurement errors if it provides the same expected-loss ranking of two forecasts based on the true outcome $y$ and the proxied outcome $\overset{y}{^}$ , where $\overset{y}{^}$ is an unbiased proxy satisfying $E [\overset{y}{^} ∣ Z] = y$ .

In the context of comparing volatility forecasts with measurement errors, Hansen and Lunde establish sufficient conditions under which noisy proxies can be used in forecast evaluation without giving rise to rankings that are inconsistent with the ranking based on the true outcome. This ensures that the ranking based on the true outcome $y$ is the same as the ranking based on the proxied outcome $\overset{y}{^}$ , for unbiased proxies satisfying $E [\overset{y}{^} ∣ Z] =$ ___$.

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2.2.3 Loss Functions That Depend on Other State Variables