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Assessing Forecast Accuracy Measures

Zhuo Chen

Department of Economics

Heady Hall 260

Iowa State University

Ames, Iowa, 50011

Phone: 515-294-5607

Email: zchen@iastate.edu

Yuhong Yang

Department of Statistics

Snedecor Hall

Iowa State University

Ames, IA 50011-1210

Phone: 515-294-2089

Fax: 515-294-4040

Email: yyang@iastate.edu

March 14, 2004

1

Abstract

This paper looks into the issue of evaluating forecast accuracy measures. In the theoretical

direction, for comparing two forecasters, only when the errors are stochastically ordered, the ranking

of the forecasts is basically independent of the form of the chosen measure. We propose well-

motivated Kullback-Leibler Divergence based accuracy measures. In the empirical direction, we

study the performance of several familiar accuracy measures and some new ones in two important

aspects: in terms of selecting the known-to-be-better forecaster and the robustness when subject to

random disturbance. In addition, our study suggests that, for cross-series comparison of forecasts,

individually tailored measures may improve the performance of diﬀerentiating between good and

poor forecasters.

Keywords: Accuracy Measure, forecasting competition

Biographies:

Zhuo Chen is Ph.D. candidate in the Department of Economics at Iowa State University. He received

his BS and MS degrees in Management Science from the University of Science and Technology of China

in 1996 and 1999 respectively. He graduated from the Department of Statistics at Iowa State University

with MS degree in May, 2002.

Yuhong Yang (Corresponding author) received his Ph.D. in Statistics from Yale University in 1996.

Then he joined the Department of Statistics at Iowa State University as assistant professor and be-

came associate professor in 2001. His research interests include nonparametric curve estimation, pattern

recognition, and combining procedures. He has published papers in statistics and related journals in-

cluding Annals of Statistics,Journal of the American Statistical Association,Bernoulli,Statistica Sinica,

Journal of Multivariate Analysis, IEEE Transaction on Information Theory, International Journal of

Forecasti ng and Econometric Theory.

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1Introduction

Needless to say, forecasting is an important task in modern life. With many diﬀerent methods in

forecasting, understanding their relative performance is critical for more accurate prediction of the

quantities of interest. Various accuracy measures have been used in the literature and their properties

have been discussed to some extent. A fundamental question is: are some accuracy measures better

than others? If so, in which sense? Addressing such questions is not only intellectually interesting,

but also highly relevant to the application of forecasting. Not surprisingly, it is a commonly accepted

wisdom that there cannot be any single best forecasting method or any single best accuracy measure,

and that assessing the forecasts and the accuracy measures is necessarily subjective. However, can there

be certain degree of objectivity? Obviously, it is one thing that no accuracy measure dominates the

others and it is another that all reasonable accuracy measures are equally ﬁne.

Adiﬃculty in assessing forecast accuracy is that when diﬀerent forecasts and diﬀerent forecast

accuracy measures are involved, the comparison of forecasts and the comparison of accuracy measures

are very much entangled. Is it possible to separate these two issues?

In this work, having the above questions in mind, we intend to go one-step further both theoretically

and empirically on assessing forecast accuracy measures. In the theoretical direction, when two fore-

casts have error distributions stochastically ordered, then the two forecasts can be compared basically

regardless of the choice of the accuracy measure; on the other hand, when the forecast errors are not

stochastically ordered (as is much more often the case in application), which forecast is better depends

on the choice of the accuracy measure and then in general the comparison of diﬀerent forecasts cannot be

totally objective. As will be seen, the ﬁrst part of this fact can be used to objectively compare diﬀerent

accuracy measures from a certain appropriate angle. If one has a good understanding of the distribution

of the future uncertainty, we advocate the use of the Kullback-Leibler divergence based measures. For

cross-series comparison, we argue that there can be advantage using diﬀerent accuracy measures for

diﬀerent series. We demonstrate this advantage with several examples. In the empirical direction, we

compare the popular accuracy measures and some new ones in terms of their ability to select the better

forecast as well as in terms of the stability of the measures with slight perturbation of the original series.

As will be seen, such forecast comparisons provide us very useful information about the behaviors of the

diﬀerent measures.

In the rest of the introduction we brieﬂy review some previous related works in the literature. More

details of the existing accuracy measures will be given in Sections 3 and 4.

Econometricians and Statisticians have constructed various accuracy measures to evaluate and rank

forecasting methods. Diebold & Mariano (1995) proposed tests of the null hypothesis that there is no

diﬀerence in accuracy between two competing forecasts. Christoﬀersen & Diebold (1998) suggested a

forecast accuracy measure that can value the maintenance of cointegration relationships among variables.

It is generally agreed that the mean squared error (Henceforth MSE) or MSE based accuracy measures

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are not good choices for cross-series comparison since they are typically not invariant to scale changes.

Armstrong & Fildes (1995) suggested no single accuracy measure would be the best in the sense of

capturing necessary complexity of real data. This, of course, does not mean that one can arbitrarily

choose a performance measure that meets a basic requirement (e.g., scale invariance). It is desirable

to compare diﬀerent accuracy measures to ﬁnd out which measures perform better in what situations

and which ones have very serious ﬂaws and thus should be avoided in practice. We notice that only

a handful of studies compared multiple forecast accuracy measures (e.g., Tashman, 2000; Makridakis

1993, Yokum and Armstrong 1995). Tashman (2000) and Koehler (2001) discussed the results of the

latest M-Competition (Makridakis & Hibon, 2000) focusing on forecast accuracy measures.

The comparison of diﬀerent performance measures is a very challenging task since there is no obvious

way to do it objectively. To our knowledge, there has not been any systematically empirical investigation

in this direction in the literature. In this work, we approach the problem from two angles: the ability of

a measure to distinguish between good and bad forecasts and the stability of the measure when there is

a small perturbation of the data.

Section 2 of this paper studies the theoretical comparability of diﬀerent forecasts for one series and

provides the theoretical motivation for the new accuracy measures. Section 3 reviews accuracy measures

for cross-series comparison and we show an advantage of the use of individually tailored accuracy mea-

sures. In Section 4 we give details of the accuracy measures investigated in our empirical study. The

comparison results are given in Section 5. Conclusions are in Section 6.

2 Theoretical comparability of diﬀerent forecasts for a single

series

Suppose that we have a time series {Yi}to be forecasted and there are two forecasters (or two methods)

with forecasts ˆ

Yi,1and ˆ

Yi,2of Yimade at time i−1based on the series itself up to Yi−1and possibly

with outside information available to the forecasters (such as exogenous variables). The forecast errors

are ei,1=ˆ

Yi,1−Yiand ei,2=ˆ

Yi,2−Yifor the two forecasters respectively.

A fundamental question is how should the two forecasters be compared? Can we have any objective

statement on which forecaster is doing a better job?

There are two types of comparisons of diﬀerent forecasts. One is theoretical and the other is empirical.

For a theoretical comparison, assumptions on the nature of the data (i.e., data generating process) must

be made. But such assumptions are not needed for empirical comparisons, which draw conclusions based

on data.

In this section, we consider the issue of whether two forecasters can be compared fairly. We realize

the complexity of this issue and will focus our attention on a very simple setting where some theoretical

understanding is possible. Basically, under a simplifying assumption on the forecast errors, we show that

sometimes the two forecasts can be ordered consistently in terms of prediction risk under any reasonable

loss function; for other cases, the conclusion regarding which forecaster is better depends subjectively

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on the loss function chosen (i.e., it can happen that forecaster one is better under one loss function but

forecaster two is better under another loss function). For the latter case, clearly, unless one can justify

a particular loss function (or certain type of losses) as the only appropriate one for the problem, there

is no completely objective ordering of the two forecasters.

Let the cumulative distribution functions of |ˆ

Yi,1−Yi|and |ˆ

Yi,2−Yi|be F1and F2respectively.

Obviously, the supports of F1and F2are contained in [0,∞).

Following the statistical decision theory framework, we usually use a loss function for comparing

estimators or predictions. Let L(Y, ˆ

Y)be a chosen loss function. Here we only consider loss functions

of the type L(Y, ˆ

Y)=g(|Y−ˆ

Y|)for a nonnegative function gdeﬁned on [0,∞).This class contains the

familiar losses such as absolute error loss and squared error loss.

Given a loss function g(|Y−ˆ

Y|),we say that forecaster 1 is (theoretically) better (equal or worse)

than forecaster 2 if Eg(|ei1|)<Eg(|ei2|)(Eg(|ei1|)=Eg(|ei2|)or Eg(|ei1|)>Eg(|ei2|)), where the

expectation is with respect to the true data generating process (assumed for the theoretical investigation).

Note that, given a loss function, two forecasts ˆ

Yi,1and ˆ

Yi,2can always be compared by the above deﬁnition

at each time i.

Clearly, when multiple periods are involved, to compare two forecasters in an overall sense, assump-

tions on the errors are necessary. One simple assumption is that for each forecaster, the errors at diﬀerent

times are independent and identical distributed. Then the theoretical comparison of the forecasters is

simpliﬁed to the comparison at any given time i.

In reality, however, the forecast errors are typically not iid and the comparison between the forecasters

becomes theoretically intractable. Indeed, it is quite possible that forecaster 1 is better than forecaster

2 for some sample sizes but worse for other sample sizes. Even though the results in this section do not

address such cases, we hope that the insight gained under the simple assumption can be helpful more

generally.

2.1 When the forecasting error distributions are stochastically ordered

Can two forecasters be theoretically compared independently of the loss function chosen? We give a

result more or less in that direction.

Here we assume that F1is stochastically smaller than F2,i.e., for any x≥0,F

1(x)≥F2(x).This

means that the absolute errors of the forecasters are ordered in a probabilistic sense. It is then not

surprising that the loss function does not play any important role in the theoretical comparison of the

two forecasters.

Deﬁnition:AlossfunctionL(Y, ˆ

Y)=g(|Y−ˆ

Y|)is said to be monotone if gis a non-decreasing

function.

Proposition 1: If the error distributions satisfy that F1is stochastically smaller than F2,then for

any monotone loss function L(Y, ˆ

Y)=g(|Y−ˆ

Y|),forecaster 1 is (theoretically) no worse than forecaster

2.

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The proof of Proposition 1 is not diﬃcult and thus omitted.

From the proposition, when the error distributions are stochastically ordered, regardless of the loss

function (as long as being monotone), the forecasters are consistently ordered. Therefore there is an

objective ordering of the two forecasters.

Let us comment brieﬂy on the stochastic ordering assumption. For example, if the forecast errors

of forecaster 1 and 2 are both normally distributed with mean zero but diﬀerent variances. Then the

assumption is met. More generally, if the distributions of |ˆ

Yi,1−Yi|and |ˆ

Yi,2−Yi|both fall in a scale

family, then they are stochastically ordered, and thus the forecasters are comparable naturally without

the need of specifying a loss function.

However, the situation is quite diﬀerent when the forecasting error distributions are not stochastically

ordered, as we will see next.

2.2 When the forecasting error distributions are not stochastically ordered

Suppose that F1and F2are not stochastically ordered, i.e., there exists 0<x

1<x

2such that F1(x1)>

F2(x1)and F1(x2)<F

2(x2).

Proposition 2:WhenF1and F2are not stochastically ordered, we can ﬁnd two monotone loss

functions L1(Y, ˆ

Y)=g1(|Y−ˆ

Y|)and L2(Y, ˆ

Y)=g2(|Y−ˆ

Y|)such that forecaster 1 is better than

forecaster 2 under loss function g1and forecaster 1 is worse than forecaster 2 under loss function g2.

Thus, from the Proposition, in general, there is no hope to order the forecasters objectively. The rel-

ative performance of the forecasts depends heavily on the loss function chosen. The proof of Proposition

2 is left to the reader.

2.3 Comparing forecast accuracy measures based on stochastically ordered

errors

An important implication of Proposition 1 is that it can be used to objectively compare two accuracy

measures from an appropriate angle. The idea is that when the errors from two forecasts are stochastically

ordered, then one forecast is better than another, independently of the loss function. Consequently, we

can compare the accuracy measures through their ability to pick the better forecast. This is a basis for

the empirical comparison in Section 5.1.

2.4 How should the loss function be chosen for comparing forecasts for one

series?

From the section 2.2, we know that generally, in theory, we cannot avoid the use of a loss function

to compare forecasts. In this subsection, we brieﬂy discuss the issue of choosing a loss function for

comparing forecasts for one series. The issue of cross-series comparison will be addressed in Section 3.

There are diﬀerent approaches. One is to use a familiar and/or mathematically convenient loss

function such as squared error loss and absolute error loss. Squared error loss seems to be the most

popular in statistics for mathematical convenience. Another approach is to use an intrinsic measure

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which does not depend on transformations of the data. For this approach, one must make assumptions

on the data generating process so that transformation-invariant measures can be derived, as will be seen

soon. The third approach is to choose a loss function that seems most natural for the problem at hand

based on non-statistical considerations (e.g., how the accuracy of the forecast may be related to the

ultimate good of interest). Perhaps except few cases, there may be diﬀerent views regarding the most

natural loss functions for a particular problem.

2.5 Some intrinsic measures

Here we derive some intrinsic and new measures to compare diﬀerent forecasts. They are obtained under

strong assumptions on the data generating process. In a certain sense, these measures can pay a heavy

price when the assumed data generating process does not describe the data well but they do have the

advantage of a substantial gain of diﬀerentiating diﬀerent forecasts when the assumed data generating

process reasonably capture the nature of the data. In addition, even if the assumption on the data

generating process is wrong, these measures are still sensible and better than MSE and absolute error

because they are invariant under location-scale transformations.

2.5.1 The K-L based measure is optimal in certain sense

We assume that conditional on the previous observations of Yprior to time iand the outside information

available, Yihas conditional probability density of the form 1

σif(y−mi

σi),where fis a probability density

function (pdf) with mean zero and variance 1. Let ˆ

Yibe a forecast of Yi.We will consider an intrinsic

distance to measure performance of ˆ

Yi.

Kullback-Leibler divergence (information, distance) is a fundamentally important quantity in sta-

tistics and information theory. Let pand qbe two probability densities with respect to a dominating

measure µ. Then the K-L divergence between pand qis deﬁned as D(pkq)=Rplog p

qdµ. Let Xbe

a random variable with pdf pwith respect to µ. Then D(pkq)=Elog p(X)

q(X). It is well-known that

D(pkq)≥0(though it does not satisfy the triangle inequality and is asymmetric). Let X0=h(X),

where his a one-to-one transformation. Let p0denote the pdf of X0and let q0denote the pdf of h(e

X)

where e

Xhas pdf q. An important property of the K-L divergence is its invariance under a one-to-one

transformation. That is, D(pkq)=D(p0kq0).K-L divergence plays crucial roles in statistics, for ex-

amples, in hypothesis testing (Cover and Thomas 1991), minimax function estimation in deriving upper

and lower bounds (e.g., Yang and Barron (1999) and earlier references thereof), and adaptive estimation

(Barron (1987) and Yang (2000)).

We ﬁrst assume that σiis known. Then with the forecast ˆ

Yireplacing mi,we have an estimated

conditional pdf of Yi:1

σif(y−ˆ

Yi

σi). The K-L divergence between p(y)= 1

σif(y−mi

σi)and q(y)= 1

σif(y−ˆ

Yi

σi)

is D(pkq)=Rf(x)log f(x)

f³x−(mi−ˆ

Yi)

σi´dµ. Let J(a)=Rf(x)log f(x)

f(x−a)dµ. Note that the function Jis well

deﬁnedoncewespecifythepdff. Now, D(pkq)=J(mi−ˆ

Yi

σi).From the invariance property of the K-L

divergence, for linear transformations of the series, as long as the forecasting methods are equi-variant

7

under linear transformations, the K-L divergence stays unchanged.

From the above points, it makes sense that if computable, J(mi−ˆ

Yi

σi)would be a good measure of

performance. Due to that miand σare unknown, one can naturally replace σiby an estimate ˆσibased

on earlier data and replace miby the observed value Yi.This motivates the loss function

L(Yi,ˆ

Yi)=JÃ(Yi−ˆ

Yi)

ˆσi!.

Note that if ˆσiis location-scale invariant and the forecasting method is location-scale invariant, then

L(Yi,ˆ

Yi)is location-scale invariant.

Now let’s consider two special cases. One is Gaussian and the other is double exponential (Laplace).

For no r mal, f(y)= 1

√2πexp ¡−1

2y2¢and consequently, J(a)= a2

2σ2.Then

L(Yi,ˆ

Yi)=JÃYi−ˆ

Yi

ˆσi!=(Yi−ˆ

Yi)2

2ˆσ2

i

.

It perhaps is worth pointing out that Ei(Yi−ˆ

Yi)2=σ2

i+(mi−ˆ

Yi)2,where Eidenotes expectation

conditional on the information prior to observing Yi.Since σ2

idoes not depend on any forecasting method,

for this case, J³Yi−ˆ

Yi

ˆσi´is essentially equivalent to J³mi−ˆ

Yi

σi´.

For double exponential, f(y)=1

2exp(−|y|)and J(a)=exp(−|a|

σ)−1−|a|

σ.Then

L(Yi,ˆ

Yi)=JÃ(Yi−ˆ

Yi)

ˆσi!=exp(−|Yi−ˆ

Yi|

ˆσi

)−1−|Yi−ˆ

Yi|

ˆσi

.

For our approach, the main diﬃculty is the estimation of σi, especially when dealing with nonsta-

tionary series.

2.5.2 The best choice of loss depends on the nature of the data

Here we show that the MSE is the optimal choice of performance measure in a certain appropriate sense

when the errors have normal distributions and absolute error (ABE) is the optimal choice when the

errors have double exponential distributions.

Consider two forecasts ˆ

Yi,1and ˆ

Yi,2of Yiwith the forecast errors ei,1=ˆ

Yi,1−Yiand ei,2=ˆ

Yi,2−Yi

respectively. We assume that the errors are iid with mean zero for both the forecasters.

MSE or ABE? Firstweassumethatei,1∼N(0,σ2

1)and ei,2∼N(0,σ2

2).From Proposition 1, we

know that forecaster 1 is theoretically better than forecaster 2 if σ2

1<σ2

2for any monotone loss function.

In reality, of course, one does not know the variances and need to compare the forecasters empirically by

looking at the history of their forecasting errors. In this last task, the choice of a loss function becomes

important.

Under our assumptions, (e1,1, ..., en,1)has joint pdf

Ã1

p2πσ2

1!n

exp Ã−Pn

i=1 e2

i,1

2σ2

1!

8

and (e1,2, ..., en,2)has joint pdf

Ã1

p2πσ2

2!n

exp Ã−Pn

i=1 e2

i,2

2σ2

2!.

Thus for each of the two forecasters, Pn

i=1 e2

i,j is a suﬃcient statistic for j=1,2.In contrast, Pn

i=1 |ei,j |

is not suﬃcient. This suggests that for each forecaster, when the errors are normally distributed, the

use of MSE (Pn

i=1 e2

i,j ) better captures the information in the errors than other choices including ABE

(Pn

i=1 |ei,j |). On the other hand, when the errors have double exponential distribution, Pn

i=1 |ei,j |is

suﬃcient but Pn

i=1 e2

i,j is not, and thus the choice of ABS is better than MSE. Note also that when the

errors of the two forecasts are all independent and normally distributed, for testing H0:σ2

1=σ2

2versus

H1:σ2

16=σ2

2,there is a uniformly most powerful unbiased test based on Pn

i=1 e2

i,1/Pn

i=1 e2

i,2,which

again is in the form of MSE.

A simulation Here we study the two types of errors mentioned above.

Case 1 (normal). ei,1∼N(0,1) and ei,2∼N(0,1.5) for i=1,··· ,100.Replicate 1000 times and

record the numbers of times Pn

i=1 e2

i,1>Pn

i=1 e2

i,2and Pn

i=1 |ei,1|>Pn

i=1 |ei,2|, respectively.

Case 2 (double exponential). ei,1∼DE(0,1) and ei,2∼DE(0,1.5) for i=1,··· ,100.Replicate 1000

times and record the number of times Pn

i=1 e2

i,1>Pn

i=1 e2

i,2and Pn

i=1 |ei,1|>Pn

i=1 |ei,2|, respectively.

The numbers of times that the above inequalities hold are presented in Table 1.

Squared Error Absolute Error

Normal 0.311 0.327

DE 0.270 0.247

Table 1: Comparing MSE and ABE

From the above simulations, we clearly see that for diﬀerentiating the competing forecasters, the

choice of loss function does matter. When the errors are normally distributed, MSE is better and

when the errors are double exponentially distributed, ABE is better. A sensible recommendation for

application is that when the errors look like normally distributed (e.g., by examining the Q-Q plot),

MSE is a good choice; and when the errors seem to have a distribution with heavier tail, ABE is a better

choice.

3 Accuracy measures for cross-series comparison

Forecast accuracy measures have been used in empirical evaluation of forecasting methods, e.g., in the

M-Competitions (Makridakis, Hibon & Moser 1979; Makridakis & Hibon 2000). Measures used in M1

Competition are: MSE (Mean square error), MAPE (Mean average percentage error), and Theil’s U2-

statistic. More measures are used in M3 Competition, i.e., symmetric mean absolute percentage error

(sMAPE), average ranking, percentage better, median symmetric absolute percentage error (msAPE),

and median relative absolute error (mRAE).

9

Here we classify the forecast accuracy measures into two types. The ﬁrst category is stand-alone

measures, i.e., measures can be determined by the forecast under evaluation alone. The second type is

the relative measures that compare the forecasts to a baseline/naive forecast, i.e., random walk, or a

(weighted) average of available forecasts.

3.1 Stand-Alone Accuracy Measures

Stand-alone accuracy measures are those that can be obtained without additional reference forecasts.

They are usually associated with a certain loss function though there are a few exceptions (e.g., Granger

& Jeon (2003a,b) proposed a time-distance criterion for evaluating forecasting models). In our study,

we include several accuracy measures that are based on quadratic and absolute loss functions.

Accuracy measures based on mean squared error criterion, especially MSE itself, have been used

widely for a long time in evaluating forecasts for a single series. Indeed, Carbone and Armstrong (1982)

foundthatRootMeanSquaredError(RMSE)hadbeenthe most preferred measure of forecast accuracy.

However, for cross-series comparison, it is well known that MSE and the like are not appropriate since

they are not unit-free. Newbold (1983) criticized the use of MSE in the ﬁrst M-Competition (Makridakis

et al., 1982). Clements & Hendry (1993) proposed the Generalized Forecast Error Second Moment

(GFESM) as an improvement to the MSE. Armstrong & Fildes (1995) again suggested that the empirical

evidence showed that the mean square error is inappropriate to serve as a basis for comparison.

Ahlburg (1992) found that out of seventeen population research papers he surveyed, ten used Mean

Absolute Percentage Error (MAPE). However, MAPE was criticized for the problem of asymmetry and

instability when the original value is small (Koehler, 2001; Goodwin & Lawton, 1999).

In addition, Makridakis (1993) pointed out that MAPE may not be appropriate in certain situations,

such as budgeting, where the average percentage errors may not properly summarize accounting results

and proﬁts. MAPE as accuracy measure is aﬀected by four problems: (1) Equal errors above the

actual value result in a greater APE; (2) Large percentage errors occur when the value of the original

series is small; (3) Outliers may distort the comparisons in forecasting competitions or empirical studies;

(4) MAPEs cannot be compared directly with naïve models such as random work (Makridakis 1993).

Makridakis (2000) proposed modiﬁed MAPE measure (Symmetric Median Absolute Percent Error) and

used it in the M2 and M3 competitions. However, Koehler (2001) found sMAPE penalizes low forecasts

more than high forecasts and thus favors large predictions than smaller ones.

3.2 Relative Measures

The idea of relative measures is to evaluate the performance of a forecast relative to that of a benchmark

(sometimes just a “naive”) forecast. Measures may produce very big numbers due to outliers and/or

inappropriate modeling, which in turn make the comparison of diﬀerent forecasts not feasible or not

reliable. A shock may make all forecasts perform very poorly, and stand-alone measures may put

excessive weight on this period and choose a measure that is less eﬀective in most other periods. Relative

10

measures may eliminate the bias introduced by potential trends, seasonal components and outliers,

provided that the benchmark forecast handles these issues appropriately. However, we need to note that

choosing the benchmark forecast is subjective and not necessarily easy. The earliest relative forecast

accuracy measure seems to be Theil’s U2-statistic, of which the benchmark forecast is the value of the

last observation.

Collopy and Armstrong (1992a) suggested that Theil’s U2 had not gained more popularity because it

was less easy to communicate. Collopy and Armstrong (1992b, p.71) proposed a similar measure (RAE).

Thompson (1990) proposed an MSE based statistic— log mean squared error ratio— as an improvement

of MSE to evaluate the forecasting performances across diﬀerent series.

3.3 The same measure across series or individually tailored measures?

As far as we know, in cross-series comparison of diﬀerent forecasters, for each measure under investiga-

tion, it is applied to all the series. A disadvantage of this approach is that a ﬁxed measure may be well

suited for some series but may be inappropriate for others (e.g., due to a lack of power to distinguish

diﬀerent forecasts or too strong inﬂuence by a few points). For such cases, using individually tailored

measures may improve the comparison of the forecasters.

Example 1: Suppose that the data set has 100 series. The sample size for each series is 50. The

ﬁrst 75 series are generated as y=α0+α1x1+α2x2+α3x3+e,whereα0,α1,α2,α3are generated as

random draws from uniform distribution unif(−1,1),x1,x2,x3are exogenous variables independently

distributed as N(0,1) and eis independent and normal distributed as N(0,5). The remaining 25 series

are generated as y=α0+α1x1+α2x2+α3x3+eas above except that eis distributed as double

exponential DE(0,√10/2).

We compare the two forecasts y1and y2, which are generated by:

y1

t=α0+α1x1t+α2x2t+α3x3t,

y2

t=bα0t+bα1tx1t+bα2tx2t+bα3tx3t,

where bα0t,bα1t,bα2t,bα3tare estimated adaptively by regressing yon x1,x2and x3(with a constant term)

on previous data, i.e., x1,1,x

1,2,···,x1,t−1;x2,1,x

2,2,···,x2,t−1;x3,1,x

3,2,···,x3,t−1.Notethaty1

tis

the “ideal” forecast with the parameters known.

We consider three measures to compare the two forecasts. One is the KL-N , another is the KL-DE21,

and the third is an adaptive measure that uses KL-N for the ﬁrst 75 series and KL-DE2 for the remaining

25 series. The two forecasts are evaluated based on their forecasts of the last 10 periods. We make 2000

replications and record the percentage of choosing the better forecast2(i.e., y1

t) by the three measures.

We report the means and their corresponding standard errors of the diﬀerence between the percentage

of choosing the better forecast by the individually tailored measure and the other two measures in Table

1Please refer to the next section for the details of KL-N and KL-DE2.

2We understand that there might be concerns over whether the conditional mean is ideal or not, but it is deﬁnitely free

of estimation error. Furthermore, since we are varying the coeﬃcients for each series and average the percentage over the

100 series, we pretty much eliminate the possibility that y2

t“happens” to be superior to the conditional mean.

11

2. The table shows that the individually tailored measure improves the ability to distinguish between

forecasts with diﬀerent accuracy. The improvement of the percentage of choosing the better forecast

is about 0.19% to the KL-DE2 and 0.58% to KL-N. Besides being statistically signiﬁcant, even though

these numbers seem to be small, they are not practically insigniﬁcant (note that Makridakis & Hibon

(2000) showed that the percentage better of sixteen forecasting procedures with respect to a baseline

method was from -1.7% to 0.8%).

KL −NKL−DE2Adaptive Measures

Example 1

Percent 71.60% 71.99% 72.18%

Diﬀerence with the Adaptive measure 0.58% 0.19%

Standard error of the diﬀerence 0.03% 0.05%

Example 2

Percent 65.00% 72.90% 73.00%

Diﬀerence with the Adaptive measure 1.30% 0.14%

Standard error of the diﬀerence 0.04% 0.04%

Example 3

Percent 81.11% 81.18% 81.68%

Diﬀerence with the Adaptive measure 0.57% 0.50%

Standard error of the diﬀerence 0.04% 0.03%

Table 2: Percentage of Choosing the Better Forecast

Example 2: Example 2 has the same setting as in Example 1 except that we change the ratio of

series with normal error and double exponential error to 1:1. The new measure is still better than that

of the two original measures but the extent varies, which gives another evidence that the performance

of accuracy measures may be inﬂuenced by the error structure.

Example 3: To address the concern that the conditional mean may not necessarily be better than

the other forecast, we generate yas a series random drawn from a uniform distribution is unif(0,1)

and the two forecasts are: y1=y+e1,y

2=y+e2,wheree1tis distributed as iid N(0,1) and e2t

is distributed as iid N(0,2) for the ﬁrst 50 series and e1tis distributed as iid DE(0,√2/2) and e2tis

distributed as iid DE(0,2) for the remaining 50 series. Replicate it for 2000 times and we report the

quantities in the lower part of Table 2. In this case, it is obvious that y2is stochastically dominated by

y1in forecast accuracy, and thus we know for sure that y1is the better forecast. The result is similar to

those in Examples 1 and 2.

The examples show that it is potentially better to use adaptive measures (as opposed to a ﬁxed

measure) when comparing forecasts. The adaptive measure (or individually tailored measures) can

better distinguish the candidate forecasts using the individual characteristics of the series. It should

be mentioned that in these examples, KL-N and KL-DE2 are applied with the knowledge of the nature

of the series. In a real application, of course, one is not typically told whether the forecast errors are

normally distributed or double-exponentially distributed. One then needs to analyze this aspect using,

e.g., Q-Q plots or formal tests. We leave this for future work.

12

4 Measures in Use in our Empirical Study

In the empirical study of this paper, we try to assess eighteen accuracy measures, including a few new

ones motivated from K-L divergence.

4.1 Stand-Alone Accuracy measures

We consider eleven stand-alone accuracy measures. MAPE, sMAPE and RMSE are familiar in the

literature. We propose several new measures based on Kullback-Leibler divergence, i.e., KL-N, KL-N1,

KL-N2, KL-DE1, and KL-DE2. We also suggest several variations of MSE and APE based measures,

i.e., msMAPE, NMSE. IQR is a new measure based on MSE and adjusted by inter quartile range. Let

mbe the number of observations we use in the evaluation of forecasts. Below we give the details of the

aforementioned measures.

The commonly used MAPE (mean absolute percentage error) has the form:

1

m

m

X

i=1

|byi−yi|

|yi|.

Makridakis & Hibon (2000) used sMAPE (symmetric mean absolute percentage error):

1

m

m

X

i=1

|byi−yi|

(|yi|+|byi|)/2.

The measure reaches the maximum value of two when either |yi|or |byi|equals to zero (undeﬁned when

both are zero).

To avoid the possibility of an inﬂation of sMAPE caused by zero values in the series, we add a

component in the denominator of the symmetric MAPE and denote it msMAPE (modiﬁed sMAPE),

which is formulated as:

1

m

m

X

i=1

|byi−yi|

(|yi|+|byi|)/2+Si

,

where Si=1

i−1Pi−1

k=1 |yk−yi−1|, yi−1=1

i−1Pi−1

k=1 yk.

RMSE is the usual root mean square error measure:

v

u

u

t1

m

m

X

i=1

(byi−yi)2.

NMSE (normalized MSE) is formulated as:

sPi(byi−yi)2

Pi(yi−y)2,

where y=1

nPn

k=1 yk.

KL-N is proposed based on the Kullback-Leibler (KL) divergence. The measure corresponds to the

quadratic loss function (normal error) scaled with (adaptively moving) variance estimate. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

S2

i

,

13

where S2

i=1

i−1Pi−1

k=1(yk−yi−1)2, yi−1=1

i−1Pi−1

k=1 yk. We discussed the theoretical motivation of K-L

divergence based measures in Section 2.5.1.

KL-N1 is a modiﬁed version of KL-N. We use a diﬀerent variance estimate that only considers the

last 5 periods. The reason for considering only a few recent periods is to allow the variance estimator

to perform well when S2

idoes not converge properly due to e.g., un-removed trends. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

S2

i,5

,

where S2

i,5=1

5Pi−1

k=i−6(yk−yi−1,5)2, yi−1,5=1

5Pi−1

k=i−6yk.

KL-N2 uses a variance estimate that considers the last 10 period. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

S2

i,10

,

where S2

i,10 =1

10 Pi−1

k=i−10(yk−yi−1,10)2, yi−1,10 =1

10 Pi−1

k=i−10 yk.

KL-DE1 is an accuracy measure we proposed based on the K-L divergence and the assumption of

double exponential error. Its formula is:

1

m

m

X

i=1

(e−|byi−yi|

bσi+|byi−yi|

bσi−1),

where bσ2

i=1

i−1Pi−1

j=1(yj−yi−1)2.

KL-DE2 is an accuracy measure similar to KL-DE1 but with a diﬀerent estimator of the scale

parameter from the one used in KL-DE1. Its formula is same with KL-DE1 but bσi=1

i−1Pi−1

j=1 |yj−yi−1|.

IQR is an accuracy measure based on inter quartile range. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

Iqr2,

where Iqr is the inter quartile range of Y1, ..., Ymdeﬁned as the diﬀerence between the third quartile

and the ﬁrst quartile of the data. Note that this measure is local-scale transformation invariant and

normalizes the absolute error in terms of Iqr.

4.2 Relative Accuracy Measures

We will use seven relative forecast accuracy measures.

RSE (Relative Squared Error) is the square root of the mean of the ratio of MSE relative to that of

random walk forecast at the evaluated time periods. It is motivated by RAE (relative absolute error)

proposed by Collopy and Armstrong (1992b). It is formulated as:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

(yi−yi,rw)2,

where yi,rw =yi−1.

14

We propose mRSE (modiﬁed RSE) to improve RSE in the case when the series remains unchanged for

one or more time periods. To achieve this, we add a variance estimates component to the denominator,

thus its formula can be written as:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

(yi−yi,rw )2+S2

i

,

where yi,rw =yi−1,S

2

i=1

i−1Pi−1

k=1(yk−yi−1)2, yi−1=1

i−1Pi−1

k=1 yk(an alternative is to replace S2

i−1

by the average of (yi−yi,rw)2).

Theil’s U2 is: sP(byi−yi)2

P(yi−yi,rw)2,

RAE (Collopy and Armstrong, 1992b) is:

sP|byi−yi|

P|yi−yi,rw |,

It should be pointed out that the relative measures are not without any problem. For example, if for

one series, a forecasting method is much worse than random walk, then the measure can be arbitrarily

large, which can be overly inﬂuential when multiple series are compared. Another weakness is that when

the random walk forecast is very poor, then the measures take very small values and consequently these

series play a less important role compared to series where random walk forecast is comparable to the

other forecasts.

MSEr1 (MSE relative 1) is the square root of the mean of the ratio of MSE relative to the variance

of the available forecasts at the current time. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

1

kPk

j=1(byji −yi)2,

where byji is the jth forecast of ith observation.

MSEr2 (MSE relative 2) is the square root of the mean of the ratio of MSE relative to the sample

variance of the diﬀerence between Yand the mean of the competing forecasts. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

1

i−1Pi−1

l=1(yl−byl)2,

where byl=1

kPk

j=1 byjl .

MSEr3 (MSE relative 3) is the square root of the mean of the ratio of MSE relative to the average

mean squared errors of the candidate forecasts. Its formula is:

v

u

u

t1

m

m

X

i=1

(byi−yi)2

1

kPk

l=1 1

i−1Pi−1

j=1(yj−bylj )2.

15

5 Evaluating the Accuracy Measures

Armstrong & Fildes (1995) pointed out that the purpose of an error measure is to provide an informative

and clear summary of the error distribution. They suggested that error measure should use a well-

speciﬁed loss function, be reliable, resistant to outliers, comprehensible to decision makers and should

also provide a summary of the forecast error distribution for diﬀerent lead times. Clements and Hendry

(1993) emphasized that the robustness of an error measure to the linear transformation of the original

series is an important factor to consider.

In this section we evaluate the performance of the forecast accuracy measures from two angles.

We investigate the ability of the measures in picking up the “better” forecast; study the stability of

the forecasts to small disturbances on the original series and the stability of the measures to linear

transformations of the series.

5.1 Ability to select the better forecast

Naturally, we hope that a forecast accuracy measure can diﬀerentiate between good and poor forecasts.

For real data sets, we cannot decide which forecast is really the best if diﬀerent measures disagree and

there is no dominant forecast. Part of the reason is that we have no deﬁnite information on the real

data generating process (DGP).

When selecting the “better” (or “best”) forecast is the criterion, of course, deﬁning “better” (or

“best”) appropriately is crucial. However this becomes somewhat circular because an accuracy measure

is typically needed to quantify the comparison between the forecasts. To overcome the diﬃculty, our

strategies are as follows.

Suppose a forecaster is given the information of the DGP with known structural parameters. Then the

conditional mean can be naturally used as a good forecast. For a forecaster who is given the form of the

DGP but with unknown structural parameters, he/she needs to estimate the parameters for forecasting,

which clearly introduces additional variability in the forecast. Since the ﬁrst one should be advantageous

compared to the second one, we can evaluate an accuracy measure in terms of its tendency to prefer the

ﬁrst one. Moving further in this direction, we can work with two forecasters that have stochastically

ordered error distributions and assess the goodness of an accuracy measure using the frequency that it

yields a better value for the better forecaster.

We agree with Armstrong & Fildes (1995) that simulated data series might not be a good represen-

tation of real data. Given a forecast accuracy measure, data sets can be used to evaluate the competing

forecasts objectively. For assessing an accuracy measure, however, due to the fact that the eﬀects of the

forecasts and the accuracy measure are entangled, maintaining objective and informative is much more

challenging. The use of simulated data then becomes important for a rigorous comparison of accuracy

measures.

We consider nine cases in this subsection.

16

5.1.1 Cases 1-7

The seven cases in this subsection represent various scenarios we may encounter in real applications (but

obviously by no means they give a complete representation) and they can give us some useful information

regarding the performance of the accuracy measures. We replicate all the simulation 20000 times. The

numbers reported in Table 3 are the percentages that each measure chooses the better forecast over all

the replications.

1. Data generating process is AR(1) with auto-regressive coeﬃcient 0.75, and the series length

is 50. Random disturbance is distributed as N(0,1). Using the eighteen measures, we compare the

forecasts generated by the true model, in which we know the true model structure but not the structural

parameters, to the better forecast available, which is the conditional mean of the series (i.e., when the

auto-regression coeﬃcient is known).

Figure 1 presents the boxplot of the values measured for the forecasts produced by the conditional

mean and when m=20. The values greater than 20 are clipped. From the ﬁgure, clearly for some of

themeasures,thedistribution are highly asymmetric.

05101520

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18

Accuracy Measures (difference)

Measure

Value

Figure 1: Boxplot of the Values of the Accuracy Measues

2. Data generating process is white noise distributed as N(0,1), and the series length is 50. We

compare forecasts generated by a white noise, which is also distributed as N(0,1)(true model) to the

better forecast, which is the conditional mean of the series, zero.

3. Data generating process is AR(1) with auto-regressive coeﬃcient 0.75, and the series length is

50. Random disturbance is distributed as N(0,1). We compare the forecast generated by white noise

process distributed as N(0,1),to the better forecast available, i.e., the conditional mean of the series.

4. Data generating process is:

17

y=α0+α1x1+α2x2+α3x3+e,whereα0is generated as a random draw from uniform distribution

unif(0,1),α1,α2,α3are generated as random draws from uniform distribution unif(−1,1).Thesample

size n=50,x1t,x2t,x3tare exogenous variables independently generated from N(0,1) and etis the

random disturbance distributed as iid N(0,1),t=1,··· ,n. We compare the two forecasts y1and y2,

where y1is generated by assuming we know the true parameters and y2is generated using coeﬃcients

estimated based on available data:

y1

t=α0+α1x1t+α2x2t+α3x3t

y2

t=bα0t+bα1tx1t+bα2tx2t+bα3tx3t

for t=n−m+1,··· ,n, where bα0t,bα1t,bα2t,bα3tare estimated by regressing yon x1,x2,andx3with

a constant term using previous data, i.e., x1,1,x

1,2,···,x1,t−1;x2,1,x

2,2,···,x2,t−1;x3,1,x

3,2,···,

x3,t−1.

5. The setting of Case 5 is the same as Case 4 except that etis distributed as double exponential

DE(0,1) for t=1,··· ,n.

6. Data generating process is: y=x1,wherex1is exogenous variables independently distributed as

unif(0,1). The sample size n=50.Wecomparethetwoforecastsy1and y2, which are generated by:

y1

t=x1t+e1t

y2

t=x1t+e2t

for t=n−m+1,··· ,n, where e1tdistributed as iid N(0,1) and e2tdistributed as iid N(0,2).Note

that here y1

tdominates y2

tindependently of the loss function.

7. The setting of Case 7 is the same as Case 6 except that e1tdistributed as iid DE(0,1) and e2t

distributed as iid DE(0,2).AsinCase6,y1

tbeats y2

t.

The results in Table 3 reveal the following. First, sMAPE performs very poorly when the true value

is close to zero. A forecast of zero will be deemed as the worst (maximum in value) of the measured

performance, no matter what values the other forecasts take. If the true value is zero, the measure

will also give out the maximum error measure of 2 for any forecast not equal to zero. After adding a

non-negative component to the denominator, the msMAPE is superior to sMAPE and MAPE (except

Case 2, when compared to MAPE). Second, measures with diﬀerent error structure motivation seem

to perform better when they correspond to the true error structure. Third, Theil’s U2, RSE, IQR and

KL-divergence based measures perform relatively well. Lastly, the table shows that the measures choose

the better forecaster more often when using more observations to evaluate the forecasts.

5.1.2 Case 8

We consider another case in which the original DGP is white noise, series length is 30.

We compare two forecasts both generated by independent white noise with the same noise level. Our

interest is to see whether the measures wrongly claim one forecast is better than the other though they

18

are actually the same. In each replication we generate 40 series and evaluate the two forecasts with the

eighteen measures. Thus for each replication we produce two series of values of measured performance.

We test the null hypothesis of that the two forecasts perform equally well (poor) by a paired t-test

with signiﬁcance level of 0.05. The empirical size is recorded as the number of rejection of the null

based on the accuracy measures. We make 10000 replications and present the mean of the empirical

sizes of the test for the 16 measures in Table 4 with diﬀerent number of periods (m=2,5,10).Note

that Armstrong and Fildes (1995) suggested that geometric mean might be better than arithmetic mean

when evaluating forecasts with multi-series. We introduce the geometric mean of NMSE, Theil’s U2 and

RAE as GmNMSE, GmTheil’sU2 and GmRAE. We have not observed consistent improvements over the

arithmetic mean in our simulation.

From the table, clearly, in general, large myields size closer to 0.05 for the measures. MAPE, RSE,

and MSEr3 are too conservative. The other measures are satisfactory for this aspect.

5.1.3 Case 9

We construct another setting to study the performance of the accuracy measures dealing with series of

diﬀerent natures.

For each replication, we have kseries with series length n=50. The data generating process is:

y=α0+α1x1+α2x2+α3x3+e, where for 50 percent of the replications, α0,α1,α2,α3are generated

as random draw from uniform distribution unif(−1,1) and from unif(−10,10) for the other half. Here

x1,x2,x3are exogenous variables independently distributed as N(0,1) and eis independent normal

distributed as N(0,σ2)(or double exponential DE(0,√2σ/2))3with σ=1for 10% of the series and

σ=0.05 for the remaining 90%. This way, the diﬀerent series are not homogenous. We compare the

two forecasts y1and y2, which are generated by:

y1

t=α0+α1x1t+α2x2t+α3x3t

y2

t=bα0t+bα1tx1t+bα2tx2t+bα3tx3t

for t=n−m+1,··· ,n, where bα0t,bα1t,bα2t,bα3tare estimated by regressing yon x1,x2,x3,anda

constant term using previous data, i.e., y1,y

2,···,yt−1;x1,1,x

1,2,···,x1,t−1;x2,1,x

2,2,···,x2,t−1;

x3,1,x

3,2,···,x3,t−1.

For each replication, we sum the numbers produced by the accuracy measures across the 100 series.

We declare that a measure chooses the right forecast if the sum of the measured value of y1

tis less than

that of y2

t.

We repeat the replication for 10000 times and record the percentages of choosing the better forecast

by the accuracy measures over the replications in Table 5. We also evaluate the three geometric mean

methods along with others.

3We multiply √2/2to make the variance of the exponential component equal to that of the normal error component.

This makes the simulation “fair”.

19

The table suggests that: ﬁrst, it is better when the number of series used in each replication is

larger, which supports the idea of M3-competition that including more series can reduce the inﬂuence of

dominating series; second, evaluating with ﬁveperiodsisbetterthanevaluatingwithjusttwoperiods;

third, geometric means slightly improves for the case of NMSE but not exactly so for Theil’s U2 and

RAE. MAPE, sMAPE, RSE, and MSEr3 perform poorly relative to others.

5.2 Stability of the Accuracy Measures

Stability of accuracy measures is another issue worthy of serious consideration. Since the observation

are typically subject to errors, measures that are robust to minor contaminations have an advantage of

reliably capturing the performance of the forecasts. With a minor contamination at a sensible level, the

more a measure changes, the less it is credible. Obviously, being stable does not qualify an accuracy

measure to be a good one, but being unstable with a minor contamination at a level typically seen in

an application is deﬁnitely a serious weakness.

5.2.1 Stability to Linear Transformation

As Clements and Hendry (1993) suggested, stability of accuracy measures with respect to the linear

transformation of the original series is an important factor. Here we use a series of monthly Austria/U.S.

foreign exchange rate from January 1998 to December 2001. The original series is measured as how many

Austrian Schillings are equivalent to one U.S. Dollar. The data was obtained from the web page of Federal

Reserve Bank of St. Louis. It is calculated as the average of daily noon buying rates in New York City

for cable transfers payable in foreign currencies. We round it to the ﬁrst digit after the decimal point

and perform a linear transformation of the original series by minus the mean of the series and multiply

10, i.e.,

ynew =10·yoriginal −10 ·mean(yorig inal)

We have four forecasts generated by random walk, ARIMA(1,1,0), ARIMA(0,1,1), and a forecast

generated by a model selected based on BIC criterion from ARIMA models with AR, MA and diﬀerence

orders from zero to one. Table 6 presents the change of the values produced by the accuracy measures

using the last 20 points. We note that the ﬁrst ﬁve accuracy measures produced very diﬀerent values

after the transformation since they are not location-scale transformation invariant. Note also that the

last three accuracy measures had some minor changes. This suggested that the ﬁrst ﬁve measures are

generally not good for cross-series comparison of forecasting procedures since a linear transformation of

the original series may change the ranking of the forecast.

5.2.2 Stability to Perturbation

In addition to robustness to linear transformation, a good accuracy measure should be robust to mea-

surement error. It is common that available quantities are subject to some disturbances, e.g., due to

20

rounding, truncation or measurement errors. When the original series (F) is added with a disturbance

term simulating the rounded digit, the accuracy measures may produce a diﬀerent ranking of the fore-

casts. The change of the best ranked forecast indicates the instability of the accuracy measures with

respect to such a disturbance. In addition, we can add a small normally distributed disturbance on the

original series.

The data set used is Earnings Yield of All Common Stocks on the New York Stock Exchange from

1871 to 1918. The series is obtained from NBER (National Bureau of Economic Research) website. The

unitispercentandthenumbersareroundedtotwodecimals.Wehavetwoforecasts: onegeneratedby

random walk and the other from ARIMA with AR, MA and diﬀerence orders selected by BIC over the

choices of zero and one. The forecasts are ranked using the accuracy measures. We perturb the data by

adding a small disturbance.

(1) Rounding: F0=F+u,whereuis generated from a uniform distribution Unif(−0.005,0.005).

This addition is used to simulate the actual numbers which were rounded up to two decimals (as given

in the data).

(2) Truncation: F0=F+u,whereuis generated from a uniform distribution Unif(0,0.01).Thisis

used to simulate the actual numbers assuming that the numbers in the data were truncated up to two

decimals.

(3) Normal 1: F0=F+e,whereeis random draw from a normal distribution of N(0,(0.1σF)2),

where σFis the sample standard deviation of the original series.

(4) Normal 2: F0=F+e,whereeis random draw from N(0,(0.12σF)2).

The perturbation is replicated 5000 times. Then we can make forecasts based on the perturbed

dataset and obtain the new ranking of the two diﬀerent forecast methods. Table 7 shows the percentage

change for the earnings yield data set. Note that KL −N1,KL−N2,MSEr1,MSEr2,MSEr3are

relatively unstable when subject to rounding, truncation, or normal perturbation. The poor performance

of these measures is probably due to the poor variance estimation in the denominator of the measures.

It is rather surprising that RSE performs so well in this example, but we suspect that this does not hold

generally. Note that RSE faces a problem when the denominator happens to be close to zero, which

is reﬂected in its poor performance shown in the earlier tables. Its modiﬁcation mRSE addresses this

diﬃculty and has a good overall performance. Not surprisingly, the measures are less stable when the

variance of the normal perturbation is greater. Even though MAPE performs well under rounding and

normal perturbations, it is highly unstable when truncation is involved.

5.3 Evaluating at one point vs. evaluating at multiple points

As to how many points we should use to compare diﬀerent forecasts under MSE based on a single

series, Ashley (2003) presented an analysis from statistical signiﬁcance point of view. For cross-series

comparison, our earlier experiments suggest that the ability of choosing the better forecast improves

signiﬁcantly when using more points for the evaluation as found in Tables 3, 4 and 5. Another observation

21

is that when mis small, accuracy measures of diﬀerent error structure motivation perform more similarly

than when mis large. An extreme example is that linear loss function and absolute value loss function

are equivalent when m=1.

6ConcludingRemarks

In this paper, we studied various forecast accuracy measures. Theoretically speaking, for comparing

two forecasters, only when the errors are stochastically ordered, the ranking of the forecasts is basi-

cally independent of the form of the chosen accuracy measure. Otherwise, the ranking depends on the

speciﬁcation of the accuracy measure. Under some conditions on the conditional distribution of Y,K-L

divergence based accuracy measures are well-motivated and have certain nice invariance properties.

In the empirical direction, we studied the performance of the familiar accuracy measures and some

new ones. They were compared in two important aspects: in selecting the known-to-be-better forecaster

and the robustness when subject to random disturbance, e.g., measurement error.

The results suggest the following:

(1) For cross-series comparison of forecasts, individually tailored measures may improve the perfor-

mance of diﬀerentiating between good and poor forecasters. More work needs to be done on how to

select a measure based on the characteristics of each individual series. For example, we may use a QQ

plot and/or other means to have a good sense on the shape of the error distribution and then apply the

corresponding accuracy measures.

(2) Stochastically ordered forecast errors provide a tool for objectively comparing diﬀerent forecast

accuracy measures by assessing their ability to choose the better or best forecast.

(3) In addition to the known facts that MAPE and sMAPE are not location invariant, and they have

amajorﬂaw when the true value of the forecast is close to zero, we obtained new information on MAPE

and related measures: their ability to pick out the better forecast is substantially worse than the other

accuracy measures. The proposed msMAPE showed a signiﬁcant improvement over MAPE and sMAPE

in this aspect. The MSE based relative measures are generally better than MAPE and sMAPE, but not

as good as K-L divergence based measures.

(4) We proposed the well motivated KL-divergence and IQR based measures, which were shown to

have relatively good performance in the simulations.

7 Acknowledgments

The work of the second author was supported by the United States National Science Foundation CA-

REER Award Grant DMS-00-94323.

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23

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

m20 220 220 220 220 220 220 2

MAPE 0.596 0.546 0.999 0.764 0.830 0.750 0.639 0.568 0.700 0.594 0.786 0.648 0.736 0.617

sMAP E 0.623 0.506 0.000 0.000 0.963 0.729 0.675 0.560 0.760 0.588 0.751 0.578 0.726 0.572

msM AP E 0.688 0.530 0.575 0.537 0.986 0.771 0.739 0.574 0.823 0.600 0.815 0.614 0.780 0.599

RMS E 0.757 0.544 0.979 0.721 0.996 0.791 0.825 0.576 0.824 0.592 0.938 0.662 0.841 0.627

NMSE 0.757 0.544 0.979 0.721 0.996 0.791 0.825 0.576 0.824 0.592 0.938 0.662 0.841 0.627

KL −N0.760 0.544 0.977 0.720 0.996 0.791 0.824 0.576 0.821 0.592 0.937 0.662 0.840 0.627

KL −N10.709 0.546 0.946 0.711 0.979 0.777 0.778 0.576 0.770 0.593 0.910 0.661 0.823 0.626

KL −N20.752 0.547 0.966 0.717 0.992 0.788 0.807 0.577 0.799 0.591 0.931 0.662 0.836 0.625

KL −DE10.758 0.543 0.976 0.721 0.996 0.790 0.819 0.580 0.840 0.596 0.929 0.661 0.860 0.626

KL −DE20.757 0.543 0.975 0.720 0.996 0.789 0.817 0.579 0.844 0.596 0.928 0.661 0.860 0.625

IQR 0.758 0.544 0.977 0.720 0.996 0.791 0.822 0.577 0.820 0.593 0.934 0.663 0.841 0.627

RSE 0.633 0.541 0.836 0.701 0.992 0.851 0.600 0.564 0.642 0.581 0.699 0.639 0.671 0.613

mRSE 0.781 0.549 0.986 0.721 1.000 0.824 0.763 0.574 0.792 0.589 0.911 0.656 0.817 0.623

Theil

0sU20.757 0.544 0.979 0.721 0.996 0.791 0.825 0.576 0.824 0.592 0.938 0.662 0.841 0.627

RAE 0.724 0.544 0.970 0.716 0.995 0.788 0.784 0.577 0.854 0.602 0.925 0.659 0.861 0.624

MSEr10.610 0.521 0.916 0.674 0.975 0.747 0.658 0.557 0.776 0.591 0.864 0.636 0.810 0.615

MSEr20.691 0.529 0.953 0.647 0.984 0.708 0.759 0.550 0.740 0.571 0.902 0.610 0.796 0.589

MSEr30.686 0.529 0.926 0.647 0.961 0.708 0.749 0.550 0.730 0.571 0.865 0.610 0.778 0.589

Table 3: Percentage of Choosing the best model

m=2 m=5 m=10

MAPE 0.022 0.020 0.021

sM AP E 0.057 0.045 0.051

msM AP E 0.057 0.047 0.052

RMSE 0.053 0.052 0.050

NMSE 0.044 0.048 0.049

KL −N0.051 0.051 0.051

KL −N10.051 0.051 0.048

KL −N20.053 0.050 0.049

KL −DE10.050 0.049 0.051

KL −DE20.051 0.049 0.050

IQR 0.052 0.051 0.051

RSE 0.018 0.021 0.018

mRSE 0.051 0.050 0.054

Theil0sU20.039 0.050 0.050

RAE 0.040 0.047 0.048

GmNMSE 0.055 0.050 0.049

GmT heil0sU 20.055 0.050 0.049

GmRAE 0.055 0.051 0.050

MSEr10.053 0.050 0.050

MSEr20.045 0.041 0.041

MSEr30.024 0.020 0.018

Table 4: Empirical Size of the Paired ttest

24

#ofseries 60 20

Error Normal Error Double Exp. Normal Error Double Exp.

m5 2 5 2 5 2 5 2

MAPE 0.750 0.736 0.837 0.792 0.735 0.663 0.792 0.794

sM AP E 0.776 0.749 0.863 0.814 0.751 0.678 0.802 0.793

msM AP E 0.997 0.939 0.999 0.978 0.925 0.818 0.973 0.902

RMSE 1.000 0.981 1.000 0.993 0.979 0.871 0.990 0.928

NMSE 0.997 0.892 0.998 0.932 0.950 0.786 0.966 0.853

KL −N1.000 0.964 0.999 0.986 0.970 0.863 0.980 0.906

KL −N10.998 0.954 0.998 0.987 0.961 0.847 0.972 0.890

KL −N21.000 0.956 0.998 0.982 0.972 0.862 0.981 0.904

KL −DE10.973 0.906 0.975 0.908 0.941 0.838 0.939 0.842

KL −DE20.973 0.909 0.971 0.909 0.939 0.836 0.941 0.847

IQR 1.000 0.963 0.999 0.986 0.967 0.849 0.980 0.903

RSE 0.701 0.707 0.772 0.771 0.687 0.661 0.721 0.730

mRSE 0.997 0.948 0.997 0.980 0.954 0.850 0.968 0.887

Theil0sU20.999 0.913 0.998 0.941 0.961 0.815 0.973 0.860

RAE 0.993 0.891 0.998 0.952 0.937 0.797 0.974 0.879

GmNMSE 0.999 0.907 1.000 0.979 0.965 0.791 0.977 0.899

GmT heil0sU20.999 0.907 1.000 0.979 0.965 0.791 0.977 0.899

GmRAE 0.998 0.898 1.000 0.986 0.950 0.788 0.987 0.909

MSEr10.972 0.862 0.999 0.987 0.884 0.762 0.968 0.899

MSEr20.936 0.827 0.949 0.879 0.858 0.715 0.875 0.791

MSEr30.782 0.720 0.823 0.757 0.762 0.665 0.796 0.710

Table 5: Percentage of Choosing the Better Forecaster

forecast Random Walk ARIMA(1,1,0) ARIMA(0,1,1) ARIMA (BIC)

series original new original new original new original new

MAPE 0.024 0.302 0.023 0.306 0.022 0.296 0.030 0.381

sM AP E 0.024 0.280 0.023 0.264 0.022 0.256 0.030 0.371

msM AP E 0.023 0.207 0.022 0.196 0.021 0.190 0.029 0.263

RMSE 0.428 4.278 0.431 4.305 0.421 4.207 0.569 5.489

NMSE 0.426 0.135 0.428 0.135 0.423 0.134 0.492 0.153

KL −N0.410 0.410 0.415 0.415 0.409 0.409 0.590 0.574

KL −N10.869 0.869 0.822 0.822 0.805 0.805 1.147 1.089

KL −N20.677 0.677 0.666 0.666 0.649 0.649 0.853 0.830

KL −DE10.071 0.071 0.071 0.071 0.069 0.069 0.132 0.122

KL −DE20.109 0.109 0.109 0.109 0.106 0.106 0.202 0.188

IQR 0.398 0.398 0.419 0.419 0.414 0.414 0.626 0.611

RSE 0.975 0.975 1.158 1.158 1.113 1.113 1.612 1.470

mRSE 0.348 0.348 0.347 0.347 0.341 0.341 0.501 0.478

Theil0sU21.000 1.000 1.006 1.006 0.983 0.983 1.331 1.283

RAE 1.000 1.000 0.965 0.965 0.934 0.934 1.247 1.156

MSEr11.078 1.100 0.929 0.934 0.870 0.878 1.104 1.071

MSEr20.703 0.710 0.726 0.733 0.705 0.711 0.880 0.864

MSEr30.729 0.739 0.752 0.762 0.731 0.740 0.912 0.899

Table 6: Stability of Accuracy Measure to Linear Transformation

25

Rounding Truncation Normal 1 Normal 2

MAPE 0.068 0.959 0.213 0.520

sM AP E 0.092 0.024 0.362 0.704

msM AP E 0.089 0.025 0.431 0.728

RMSE 0.056 0.043 0.620 0.877

NMSE 0.056 0.043 0.619 0.877

KL −N0.089 0.070 0.612 0.867

KL −N10.278 0.896 0.368 0.134

KL −N20.148 0.062 0.598 0.861

KL −DE10.061 0.030 0.605 0.831

KL −DE20.047 0.024 0.602 0.828

IQR 0.071 0.053 0.611 0.874

RSE 0.004 0.001 0.017 0.067

mRSE 0.018 0.010 0.249 0.356

Theil0sU20.056 0.043 0.620 0.877

RAE 0.044 0.031 0.541 0.873

MSEr10.169 0.859 0.604 0.170

MSEr20.201 0.077 0.571 0.634

MSEr30.243 0.086 0.562 0.627

Table 7: Rate of ranking change

26