1 Estimation with Asymmetric Loss

Suppose you are interested in estimating a value Q , which is well

- defined and can be measured
at some later point. Your estimat or is . Your loss is a function of your estimation error, x = Q

– ; denote the loss function as L ( x ). The estimation problem is to find the estimator that minimizes the expected loss, E [ L ( x )], where the expectation is over the available information.

By far the most common loss function used is quadratic, L ( x ) = x 2 , and minimizing the quadratic
loss yields the estimator, = E ( Q ) , the mean of Q . In some circumstances, however, loss
is asymmetric. Suppose, for example, that a greater loss is incurred by incorrectly choosing an

estimate which is bigger than Q , than by choos ing an estimate which is less than Q . Because

x = Q

– , this means that the loss function is asymmetric, and steeper on the left than the right

of zero.

Two convenien t asymmetric loss functions are the Linex and Lin

- Lin loss functions. The Linex loss function is,

Linex: L ( x ) = b [ e ax

– ax

– 1] ( 1 )

wh ere a and b are constants. The Linex loss function is approximately exponential on one side and approximately linear on the other; when a < 0, the loss is greater when x is negative. Near zero, it is locally quadratic. The Lin

- Lin loss function is,

Lin

- Lin: L (

x ) =

, ( 2 )

c 1 and c 2 are positive constants. The Lin

- Lin loss function is linear on both sides, but with different slopes. When c 1 > c 2 , the loss is greater when x is negative.

Figure 1 shows the Linex, Lin

- Lin, and quadratic loss functions, for the parameterization in the
numerical exercise below.2 Figure 1. The Linex, Lin

- Lin, and quadratic loss f unctions

. All three loss functions have a slope of

- 2 at x =

- 200;
the Linex and Lin

- Lin loss functions have a slope of 0.4 at x = 200.

The Linex and Lin

- Lin loss functions are convenient because they have closed

- form solutions for the optimal estimator . For the Linex loss function, if Q has a normal distribution, then this closed form is,

= E ( Q ) + 1⁄2 a  2

, ( 3 )

where 

2 is the variance of the distribution of q

. For example, if a =

- 1, then the Linex estimator

is the mean of Q (which is the quadratic

- loss estimator), minus 1⁄2  2

.

For the Lin

- Lin loss function, the optimal estimator is the c 2 /( c 1 + c 2 )

- th percentile of the
distribution of Q :

= , ( 4 )

where F is the cumulative distributi on function of Q given currently available information. For
example, if c 1 = 3 and c 2 = 1 , then is the 25 th percentile of the distribution of Q . Note that

if c 1 = c 2 , the Lin

- Lin estimator is the median. The Lin

- Lin loss function is the loss function that underlies quantile regression.3

For both the Linex and Lin

- Lin loss functions, if the loss is greater when exceeds Q than

when is less than Q , then the optimal is less than a measure of the center of the
distribution, in the case of Linex, it is less than the mean, in the case of Lin

- Lin, it is less than the
median.

Numerical example Implementation of these formulas requires calibration of th e loss function parameters, plus knowledge of 

2 (for Linex) and of F (for Lin

- Lin). Here we work through an example.

To calibrate the loss function, let  ( x 0 ) = | L  (

- x 0 )/ L  (

x 0 ) | , which is the ratio of the absolute slopes

of the loss functions evaluated at  x 0 for some x 0 > 0. This is a measure of the marginal loss from overpredicting to underpredicting, as a size of the overprediction error. For a symmetric loss function, 

( x 0 ) = 1. For the Linex and Lin

- Lin loss functions,

Linex : 

( x 0 ) = ( 5 )

Lin

- Lin : 

( x 0 ) = ( 6 )

For Lin

- Lin, the ratio is just the ratio of slope coefficients. For Linex, the ratio depends on the point of evaluation, and given x 0 , a must be obtained by solving ( 5 ) nonlinearly.

We consider three values of  : 5, 10, and 20. For Linex, we evaluate the derivative at x 0 = 200, which yields the following parameters:

 c 1 c 2 a
5 .8 3 3 .167

- .00805

10 .909 .091

- .0115

20 .952 .048

- .015 0

The Linex and Lin

- Lin loss functions in Figure 1 are for  = 5, with coefficients scaled so that the slope at x 0 =

- 200 is 2 (the increase in the loss when going from x = 200 to x = 201 is 2 units).4

The implied optimal additive terms (under the assumption of normality) are given in the following table:

 (std. dev. of Q ) 100 100 100 200 200 200

 5 10 20 5 10 20 Linex bias term

- 40

- 58

- 75

- 161

- 230

- 300 Lin

- Lin bias term

- 97

- 134

- 167

- 193

- 267

- 334 Lin

- Lin percentile .167 .091 .048 .167 .091 .048

For example, for  = 5 and  = 200, the optimal Linex bias is

- 161, so that the optimal Linex estimator would be the mean, minus 161. For Lin

- Lin, the optimal bias term is

- 193, so that the optimal estimator would be the median (which for normality is the mean), minus 193.

The fina l row gives the percentile of the distribution of Q which would be the optimal Lin

- Lin estimator. This row is included because it is valid under a general distribution, not just normality (the previous two bias rows were derived under normality). For exa mple, for  = 5, the optimal

Lin

- Lin estimator would be the 16.7% percentile of the distribution of Q

.

Reference

Christoffersen, P.F. and F.X. Diebold, 1997. “Optimal Prediction under Asymmetric Loss,” Econometric Theory 13, 808

- 817.