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2.3 MULTIVARIATE LOSS FUNCTIONS

When a decision maker’s objectives depend on multiple variables, the loss function needs to be extended from being defined over scalar outcomes to depend on a vector of outcomes. This situation arises, for example, for a central bank concerned with both inflation and employment prospects.

Conceptually it is easy to generalize univariate loss functions to the multivariate case, although difficulties may arise in determining how costly different combinations of forecast errors are. How individual forecast errors or their cross products are weighted becomes particularly important.

The most common multivariate loss function is multivariate quadratic error loss, also known as multivariate MSE loss; see Clements and Hendry (1993). This loss function maps a vector of forecast errors $e = (e_{1}, \dots, e_{n})^{'}$ to the real number line and so is simply a weighted average of the individual squared forecast errors and their cross products:11

$MSE (A) = e^{'} A e .$

Here the $(n \times n)$ matrix A is required to be nonnegative and positive definite. This is the matrix equivalent of the univariate assumption for MSE loss that $a > 0$ in (2.11).

As noted in the discussion of MAPE loss, the loss function in (2.23) may be difficult to interpret when the predicted variables are measured in different units. This concern is related to obtaining a reasonable specification of the loss function whose role it is to compare and trade off losses of different sizes across different variables. Hence this is not really a limitation of the loss function itself but of applications of the loss function.

The loss function in (2.23) is “bowl shaped” in the sense that the level sets are convex and symmetric around 0. It is easily verified that (2.23) satisfies the basic assumptions for a loss function in (2.3). If the entire vector of forecast errors is 0, then the loss is 0. A positive-definite and nonnegative weighting matrix A ensures that losses rise as forecast errors get larger, so assumption (2.3c) holds.12

A special case arises when $A = I_{n},$ , the $(n \times n)$ identity matrix. In this case covariances can be ignored and the loss function simplifies to MSE $(I_{n}) = E [e^{'} e] =$ tr $E [(e e^{'})]$ , i.e., the sum of the individual mean squared errors. Thus, a loss function based on the trace of the covariance matrix of forecast errors is simply a special case of the general form in (2.23). In general, however, covariances between forecast errors come into play, reflecting the cross products corresponding to the off-diagonal terms in A.

As a second example of a multivariate loss function, Komunjer and Owyang (2012) provides an interesting generalization of the Elliott, Komunjer, and Timmermann (2005) loss function in (2.17) to the case where $e = (e_{1}, \dots, e_{n})^{'}$

Let $∣∣ e ∣ ∣_{p} = (∣ e_{1} ∣^{p} + \cdot \cdot \cdot + ∣ e_{n} ∣^{p})^{1/ p}$ be the $l_{p}$ norm of e and assume that the n-vector of asymmetry parameters, $α,$ , satisfies $∣∣ α ∣ ∣_{q} < 1$ . Further, let $1 \leq P \leq \infty$ and, for a given value of $p_{: l}$ , set q so that $1/ p + 1/ q = 1$ . The multivariate loss function proposed by Komunjer and Owyang takes the form

$L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1} . (2.24)$

As in the univariate case, the extent to which large forecast errors are penalized relative to small ones is determined by the exponent, $P .$ . However, now the full vector $α = (α_{1}, \dots, α_{n})$ characterizes the asymmetry in the loss function, with $α = 0$ representing the symmetric case. Since α is a vector, this loss function offers great flexibility in both the magnitude and direction of asymmetry for multivariate loss functions.

Other multivariate loss functions have been used empirically. Laurent, Rombouts, and Violante (2013) consider a multivariate version of the family of loss functions introduced by Patton (2011), and apply it to volatility forecasting.

练习题

When a central bank is concerned with both inflation and employment prospects, what type of loss function is needed?

A. Univariate loss function

B. Multivariate loss function

C. Binary loss function

D. Level-dependent loss function

What is the form of the most common multivariate loss function?

L (e) = e^{2}

L (e) = ∣ e ∣

MSE (A) = e^{'} A e

L (e) = α e^{2} + (1 - α) e

What property must the matrix $A$ in the multivariate MSE loss function $MSE (A) = e^{'} A e$ have?

A. Negative definite

B. Positive semi-definite

C. Nonnegative and positive definite

D. Diagonal

Which of the following are true about the multivariate MSE loss function when $A = I_{n}$ ?

A. Covariances can be ignored

B. The loss function simplifies to the sum of individual mean squared errors

C. It is no longer a multivariate loss function

D. The cross products corresponding to the off-diagonal terms in

A

are considered

What are the features of the loss function $L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1}$ proposed by Komunjer and Owyang?

A. It is a multivariate loss function

B. It generalizes the Elliott, Komunjer, and Timmermann (2005) loss function

C. It only considers the magnitude of forecast errors

D. The asymmetry in the loss function is characterized by the vector

α

The multivariate MSE loss function $MSE (A) = e^{'} A e$ is difficult to interpret when the predicted variables are measured in different units.

If the entire vector of forecast errors in the multivariate MSE loss function $MSE (A) = e^{'} A e$ is 0, then the loss is greater than 0.

The loss function $L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1}$ takes the form of a generalization of the ___ loss function.

Explain the significance of the matrix $A$ in the multivariate MSE loss function $MSE (A) = e^{'} A e$ .

How does the vector $α$ affect the loss function $L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1}$ ?

Which of the following statements about the multivariate quadratic error loss function is correct?

A. The matrix A in

MSE (A) = e^{'} A e

must be negative definite.

B. The multivariate quadratic error loss function is a weighted average of the individual squared forecast errors and their cross products.

C. When

A = I_{n}

, the loss function cannot be simplified to the sum of individual mean squared errors.

D. The loss function in

MSE (A) = e^{'} A e

is not bowl-shaped.

Which of the following are true about the Komunjer and Owyang's multivariate loss function $L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1}$ ?

A. The exponent

p

determines the extent to which large forecast errors are penalized relative to small ones.

B. The vector

α

characterizes the asymmetry in the loss function.

C. When

α = 0

, the loss function is symmetric.

D. The loss function is only applicable when

p = 2

E. The loss function is a generalization of the EKT loss function.

The multivariate quadratic error loss function $MSE (A) = e^{'} A e$ is difficult to interpret when the predicted variables are measured in different units because it is challenging to compare and trade off losses of different sizes across different variables.

In the Komunjer and Owyang's multivariate loss function $L (e) = (∣∣ e ∣ ∣_{p} + α^{'} e) ∣∣ e ∣ ∣_{p}^{p - 1}$ , the condition $∣∣ α ∣ ∣_{q} < 1$ ensures that the loss function is well-defined, where $q$ is set such that $1/ p + 1/ q = 1$ for a given value of $p$ with $1 \leq p \leq \infty$ . The vector $α = (α_{1}, \dots, α_{n})$ characterizes the ___ in the loss function.

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2.3 MULTIVARIATE LOSS FUNCTIONS