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Forecasting stock market volatility: Furtherinternational evidenceErcan Balaban

a, Asli Bayar

b& Robert W. Faff

c

aManagement School and Economics, The University of Edinburgh, UK

bDepartment of Management, ankaya University, Ankara, Turkey

cDepartment of Accounting and Finance, Monash University, Victoria,

AustraliaVersion of record first published: 17 Feb 2007.

To cite this article: Ercan Balaban , Asli Bayar & Robert W. Faff (2006): Forecasting stock market volatility:Further international evidence, The European Journal of Finance, 12:2, 171-188

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The European Journal of Finance

Vol. 12, No. 2, 171188, February 2006

Forecasting Stock Market Volatility: FurtherInternational Evidence

ERCAN BALABAN, ASLI BAYAR & ROBERT W. FAFFManagement School and Economics, The University of Edinburgh, UK, Department of Management, ankayaUniversity, Ankara, Turkey, Department of Accounting and Finance, Monash University, Victoria, Australia

ABSTRACT This paper evaluates the out-of-sample forecasting accuracy of eleven models for monthly

volatility in fifteen stock markets. Volatility is defined as within-month standard deviation of continuouslycompounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997.The first half of the sample is retained for the estimation of parameters while the second half is for the forecastperiod. The following models are employed: a random walk model, a historical mean model, moving averagemodels, weighted moving average models, exponentially weighted moving average models, an exponentialsmoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model, andan EGARCH model. First, standard (symmetric) loss functions are used to evaluate the performance ofthe competing models: mean absolute error, root mean squared error, and mean absolute percentage error.

According to all of these standardloss functions, the exponentialsmoothing model provides superior forecastsof volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models.

Asymmetric loss functions are employed to penalize under-/over-prediction. When under-predictions arepenalized more heavily, ARCH-type models provide the best forecasts while the random walk is worst.However, when over-predictions of volatility are penalized more heavily, the exponential smoothing model

performs best while the ARCH-type models are now universally found to be inferior forecasters.

KEY WORDS: Stock market volatility, forecasting, forecast evaluation

1. Introduction

In a recent review article, Poon and Granger (2003) present a persuasive case for why the forecast-

ing of volatility is a critical activity in financial markets it has a very wide sphere of influence

including investment, security valuation, risk management, and monetary policy making(p. 478).

Specifically, they emphasize the importance of volatility forecasting in: (a) pricing of options,

underlined by the massive growth in the trading of derivative securities in recent times; (b) finan-

cial risk management in the global banking sector, stimulated by the Basle Accords; and (c) USmonetary policy, especially in the wake of major world events such as September 11.

This begs the obvious question(s): how can we effectively forecast volatility and is it pos-

sible to clearly identify a preferred technique? Various methods by which such forecasts can

be achieved have been developed in the literature and applied in practice. Such techniques range

from the extremely simplistic models that use nave assumptions through to the relatively complex

Correspondence Address: Robert W. Faff, Department of Accounting and Finance, Monash University, Victoria, 3800,

Australia. Email: [email protected]

1351-847X Print/1466-4364 Online/06/02017118 2006 Taylor & FrancisDOI: 10.1080/13518470500146082


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172 E. Balaban et al.

conditional heteroscedastic models of the GARCH family (see Bollerslev et al., 1992; Bera and

Higgins, 1993; and Bollerslev et al., 1994 for general reviews of this literature).

However, despite the appeal of complexity and despite their popularity, it is by no means agreed

that complex models such as GARCH provide superior forecasts of return volatility. Dimson

and Marsh (1990) is a notable example in which simple models have prevailed although it shouldbe pointed out that ARCH models were not included in their analysis. Specifically, Dimson and

Marsh apply five different types of forecasting model to a set of UK equity data, namely, (a) a

random walk model; (b) a long-term mean model; (c) a moving average model; (d) an exponential

smoothing model; and (e) regression models. They recommend the final two of these models and,

in so doing, sound an early warning in this literature that the best forecasting models may well be

the simple ones.

Other papers in this literature however spell out a mixed set of findings on this issue. For exam-

ple, Akgiray (1989) found in favour of a GARCH (1,1) model (over more traditional counterparts)

when applied to monthly US data. Brailsford and Faff (1996) investigate the out-of-sample predic-

tive ability of several models of monthly stock market volatility in Australia. In the measurement

of the performance of the models, in addition to symmetric loss functions, they use asymmetric

loss functions to penalize under-/over-prediction. They conclude that the ARCH class of models

and a simple regression model provide superior forecast of volatility. However, the various model

rankings are shown to be sensitive to the error statistics used to assess the accuracy of the forecasts.

In contrast, Tse (1991) and Tse and Tung (1992) investigated Japanese and Singaporean data and

found that an exponentially weighted moving average (EWMA) model produced better volatility

forecasts than ARCH models.1

In the finance literature, generally the existing evidence concerning the relative quality of

volatility forecasts is related to an individual countrys stock market: Australia (Brailsford and Faff,

1996; Walsh and Tsou, 1998), Germany (Bluhm and Yu, 2000), Japan (Tse, 1991), New Zealand

(Yu, 2002), Netherlands, Germany, Spain and Italy (Franses and Ghijsels, 1999), Singapore (Tse

and Tung, 1992), Sweden (Frennberg and Hannsson, 1996), Switzerland (Adjaoute et al., 1998),

Turkey (Balaban, 1999), UK (Dimson and Marsh, 1990; Loudon et al., 2000; McMillan et al.,

2000), USA (Akgiray, 1989; Pagan and Schwert, 1990; Hamilton and Lin, 1996; Brooks, 1998).2

Notably, in any one of these papers the range of forecasting models is often restricted to a relatively

narrow set of models.

The current paper seeks to extend and supplement this existing evidence by, in a single unifying

framework, analysing a wide range of volatility forecasting approaches across broad cross-section

of countries that reflect both developed and emerging markets. Specifically, in the context of

volatility forecasting we consider more countries than ever before evaluated in a single paper

namely, fifteen countries comprising Belgium; Canada; Denmark; Finland; Germany; Hong Kong;

Italy; Japan; Malaysia; Netherlands; Philippines; Singapore; Thailand; the UK and the USA.

Moreover, a considerable range of forecasting models is used a random walk model, a historical

mean model, moving average models, weighted moving average models, exponentially weightedmoving average models, an exponential smoothing model, a regression model, an ARCH model,

a GARCH model, a GJR-GARCH model, and an EGARCH model.

Furthermore, following Brailsford and Faff (1996), we compare the forecasting techniques

based on both symmetric (the mean error, the mean absolute error, the root mean squared error and

the mean absolute percentage error) and asymmetric error statistics. As established by Brailsford

and Faff (1996), the consideration of asymmetric error statistics is of considerable practical interest

for short and long positions in asset markets, and is especially motivated in the context of option

pricing and risk management.3 Specifically, it can be argued that option buyers versus sellers will


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Forecasting Stock Market Volatility 173

not place equal importance on over and under predictions of volatility. For example, consider the

case of call options under-predictions (over-predictions) of volatility will induce a downward

(an upward) biased estimate of the call option price. The resulting under-estimate (over-estimate)

of price is of major concern to the option seller (buyer) since they stand to lose money on any

transaction based on these inferior volatility forecasts. Under-/over-prediction of volatility hasalso significant implications for market risk management models such as value-at-risk (VaR). An

overestimated VaR will only cause the firm to tie up capital in an unprofitable form whereas an

underestimate could lead either to pressure from the regulator or even bankruptcy.4

From a general standpoint, we choose to conduct this experiment in a multi-country context

for two main reasons. First, a major motivation for this choice (as opposed to examining yet

another US or other single country sample) is to accommodate the general argument of Leamer

(1983) and extended by Lo and MacKinlay (1990) regarding the concern about data snooping in

finance research. Specifically, it is argued that investigating alternative data samples (either across

time or across markets) is a credible out-of-sample robustness check. We have the added advantage

of conducting such multi-country analysis in a single unified framework that will eliminate the

vagaries of alternate research designs impacting on the comparability of results across markets.

The second motivation for our multi-country focus is that it permits us to choose countries that

involve a representative set of both developed and emerging markets. It is well known and widely

accepted that emerging markets display risk (and return) characteristics that somewhat contrast

that observed in developed market settings (see for example; Harvey, 1995; and Bekaert and

Harvey, 1997). Accordingly, the inclusion of Malaysia, the Philippines and Thailand provide the

emerging market focus, thereby enabling us to explore whether developed and emerging markets

exhibit similar or different forecasting ability.

The main results of our study can be summarized as follows. First, based on the conventional

symmetric loss functions, we find that the exponential smoothing model provides superior fore-

casts of volatility. Second, the ARCH-based models generally prove to be the worst forecasting

models in the context of these symmetric measures. Third, when under-predictions are penalized

more heavily ARCH-type models provide the best forecasts while the random walk is worst.

Finally, when over-predictions of volatility are penalized more heavily, the exponential smooth-

ing model performs best while the ARCH-type models are now universally found to be inferior

forecasters.

The remainder of this paper is organized as follows: in the second section, the data and method-

ology are described, in the third section the empirical results are presented, and finally in the fourth

section the paper is concluded.

2. Empirical Methodology

2.1 Data and Sample Description

We employ daily observations of stock market indices of fifteen countries covering the period

December 1987 to December 1997. The data are sourced from Datastream. The investigated

countries (indices) are Belgium (Brussels All Shares Price Index); Canada (Toronto SE 300

Composite Price Index); Denmark (Copenhagen SE General Price Index); Finland (Hex General

Price Index); Germany (Faz General Price Index); Hong Kong (Hang Seng Price Index); Italy

(Milan Comit General Price Index); Japan (Nikkei 500 Price Index); Malaysia (Kuala Lumpur

Composite); the Netherlands (CBS All Share General Price Index); the Philippines(the Philippines


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SE Composite Price Index); Singapore (Singapore All Share Price Index); Thailand (Bangkok

S.E.T. Price Index); the UK (FTSE All Share Index) and the US (NYSE Composite Index).

Our analysis involves monthly volatility forecasts. Continuously compounded daily returns are

calculated as follows:

Rm,t = ln(Im,t/Im,t1) (1)

where Im,t and Rm,t denote the value of stock market index and continuously compounded return

on trading day t in month m, respectively. We define monthly realised volatility as the within-month

standard deviation of continuously compounded daily returns as follows:

a,m =

1

(n 1)n

t=1(Rm,t m)2

0.5(2)

m

=(1/n)

n

t=1

Rm,t (3)

The number of trading days in a month is n. In the dataset for each country, there are 120

monthly volatility observations. Of these observations, the first 60 (from December 1987 to

November 1992) are used for estimation, while the second 60 observations (from December 1992

to December 1997) are used for forecasting purposes.

2.2 Forecasting Techniques

There exist a wide range of potentially useful models for forecasting volatility. It is clearly impos-

sible to employ all models in a single paper and challenging to convincingly justify the choice of a

narrow set purely on objective grounds (see for example; West and Chou, 1995). No doubt whileour choice is influenced by personal taste and practical implementation issues, it also impor-

tantly reflects our assessment of the models most widely used by practitioners. Generally we

employ time series models, but there will inevitably be versions of these models omitted that

others would have liked included.5 With this background, the competing models employed are

displayed in Table 1.

3. Forecast Evaluation and Empirical Results

3.1 Symmetric Error Statistics

Four commonly used loss functions or error statistics: the mean error (ME), the mean absolute error

(MAE), the root mean squared error (RMSE), and the mean absolute percentage error (MAPE)

are employed to measure the performance of the forecasting models.

ME= 160

12061

(f,m a,m) (4)

MAE= 160

12061

|f,m a,m| (5)


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Table 1. Volatility forecasting models

Model Equationa Paramet

Random walk (RW) f,m(RW) = a,m1 (6)

Historical mean (HM) f,m(HM) =1

(m 1)m1j=1

a,j (7)

Moving average (MA) f,m(MA()) =1

m1j=m

a,j (8) = 3, 6, 12, 24

Weighted moving average(WMA)b

f,m(WMA()) =m1

j=mja,j (9) = 3, 6, 12, 24

Exponential smoothing (ES) f,m(ES) = f,m1(ES) + (1 )a,m1 (10) The optimal vaerror statistic

Exponentially weighted moving

average (EWMA)

f,m(EWMA

)

=f,m

1(EWMA

)

+(1 )f,m(MA ) (11)

=3, 6, 12, 24

, are obtaine

Regression (REG) f,m(REG) =

c + b a,m1 (12) One-step ahead60 obs.

ARCH(1)

GARCH(1, 1)

GJRGARCH(1, 1)

EGARCH (1, 1)

ht = 0 + 1 2t1 (13)

ht = 0 + 12t1 + 1ht1 (14)

ht = 0 + 12t1 + 22t1Dt1 + 1ht1 (15)

ln(ht) = 0 +

t1ht1

+

|t1|ht1

2

0.5

+ ln(ht

1) (16)

In all of the ARmonthly vola1,250 daily o(on the averagmade to consdeviations foARCH type mother modelschallenging f

a In each case, monthly volatility is forecast for the latter 60 months in our sample, i.e. m = 61, . . . , 120.bThe weight of each observation, j, is chosen to decline by 10%, giving the highest (lowest) weight to the newest (oldest) information.cNote that if = 0, the ES model collapses to a RW model. The initial forecasts in the ES and EWMA models are MA3 forecasts.dNote that if = 0, the EWMA model collapses to a MA model.eAll ARCH-type models fulfil the standard requirements for nonnegativity of conditional variance and parameter restrictions.


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RMSE=

1

60

12061

(f,m a,m)20.5

(17)

MAPE=1

60

12061

f,m a,ma,m (18)where f,m and a,m denote the volatility forecast and the realized volatility in month m,

respectively.

First, it should be recognized that the mean error (ME) metric suffers from the fact that large

errors of positive and negative sign may offset each other and, hence, may lead to an unreliable

ranking device across the various forecasting models. Accordingly, we provide only a brief dis-

cussion of the mean error measures across the various volatility forecasting models and markets.

Perhaps, the most useful guide that the ME provides is the degree of average under- or over-

prediction of volatility. On this score we generally found that across all countries, the ARCH-type

models tend to over-predict volatility, while the non-ARCH-type models under-predict volatility.

Moreover, the degree of over-prediction of the former is considerably more pronounced than the

under-prediction of the latter.

Table 2 provides results of actual and relative forecast error statistics for each model according

to the MAE error metric. The relative forecast error is obtained by taking the ratio of the actual

error statistic of a given model divided by the actual error statistic of the worst-performing model

for that country. The table also shows the ranking of forecasting models, for each country, from

1 (best forecast) to 11 (worst forecast). We can identify a number of key features from Table 2.

First (and most notably), the exponential smoothing method clearly produces the most accurate

volatility forecasts for 13 out of the 15 countries, ES is ranked number one. Moreover, for the

two cases (Finland and Malaysia) in which ES is not ranked first it ranks fourth but on both

occasions only marginally behind the most preferred models. For example according to MAE,

we find that in the case of Finland the relative difference in forecasting performance between ES

and the top-ranked model (MA-3) is about 3%. Second according to the MAE criterion, the worst

model is the standard ARCH model which ranks last for nine countries. Interestingly however,

the random walk model also does quite poorly. Third, Hong Kong produces the largest relative

difference between the best model (ES) and the worst-ranked model (ARCH) a difference of

53%. In contrast, based on the MAE the narrowest relative gap between the best and worst models

is found for Canada and Malaysia a difference of only 22% between ES (GJR-GARCH) and

RW (ARCH). Fourth, in terms of MAE, the moving average group of models provide the second

best volatility forecast behind ES.

For the RMSE metric,in unreported results, findings mostly reinforce the results gained from the

MAE analysis.6 As wasthe case above with MAEthe exponential smoothing approach dominates

gaining a number one ranking for 12 out of the 15 countries. Also, based on RMSE there is aconsistently poor showing of the ARCH model it ranks last seven times and second-last on a

further three occasions. Indeed, the non-ARCH based models again are generally superior to the

ARCH based models. In further unreported results, the outcome of the MAPE metric reveals that

themajor patterns identified for MAE and RMSE are predominantly intact and so only brief further

comment will be made. Once again it is found that the ES model is clearly most favoured it

provides superior forecasts in all but one instance (Finland). Based on the MAPE the REG model

is superior in forecasting volatility for Finland. Furthermore, MA models continue to perform

quite well while ARCH-based models remain as the poorest volatility forecasters.7


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Table 2. Mean absolute error (MAE) of forecasting monthly volatility

Belgium Canada Denmark Finland

Actual Relative Rank Actual Relative Rank Actual Relative Rank Actual Relative

RW 0.1420 0.677 6 0.1827 1.000 11 0.1776 0.838 7 0.3998 1.000 HM 0.1718 0.819 7 0.1596 0.874 10 0.1757 0.829 6 0.3359 0.840 MA-3 0.1369 0.653 3(4) 0.1555 0.851 8 0.1514 0.714 2(3) 0.2968 0.742 WMA-3 0.1361 0.649 2 0.1547 0.847 6 0.1515 0.715 4 0.2995 0.749 EWMA-3 0.1369 0.653 3(4) 0.1478 0.809 2 0.1514 0.714 2(3) 0.3086 0.772 ES 0.1295 0.617 1 0.1425 0.780 1 0.1478 0.697 1 0.3086 0.772 REG 0.1386 0.661 5 0.1496 0.819 3 0.1605 0.757 5 0.3079 0.770 ARCH(1) 0.2098 1.000 11 0.1564 0.856 9 0.1879 0.886 8 0.3189 0.798 GARCH(1, 1) 0.1860 0.887 9 0.1548 0.847 7 0.2120 1.000 11 0.3124 0.781 GJRGARCH(1, 1) 0.1805 0.860 8 0.1519 0.831 4 0.2108 0.994 10 0.3213 0.804

EGARCH 0.1924 0.917 10 0.1532 0.839 5 0.1934 0.912 9 0.3097 0.775 Hong Kong Italy Japan Malaysia


RW 0.4784 0.513 7 0.3290 1.000 11 0.2573 0.544 3 0.4391 0.906 HM 0.5382 0.577 10 0.2727 0.829 7 0.3241 0.685 8 0.4749 0.980 MA-3 0.4563 0.489 5 0.2627 0.798 5 0.2859 0.604 5(6) 0.3866 0.798 WMA-3 0.4534 0.486 3 0.2626 0.798 4 0.2816 0.595 4 0.3847 0.794 EWMA-3 0.4545 0.487 4 0.2502 0.760 2 0.2859 0.604 5(6) 0.3866 0.798 ES 0.4366 0.468 1 0.2474 0.752 1 0.2543 0.538 1 0.3852 0.795 REG 0.4521 0.484 2 0.2612 0.794 3 0.2571 0.544 2 0.4340 0.896 ARCH(1) 0.9332 1.000 11 0.2818 0.857 9 0.4730 1.000 11 0.4844 1.000 GARCH(1, 1) 0.5142 0.551 9 0.2817 0.856 8 0.3618 0.765 10 0.3843 0.793 GJRGARCH(1, 1) 0.5034 0.539 8 0.2833 0.861 10 0.3275 0.692 9 0.3779 0.780

EGARCH 0.4571 0.490 6 0.2661 0.809 6 0.2975 0.629 7 0.3917 0.809


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Table 2. Continued

Philippines Singapore Thailand UK


RW 0.4026 0.722 5 0.2454 0.787 6 0.4218 0.740 6 0.1135 0.570 HM 0.5122 0.918 10 0.2524 0.809 7 0.4687 0.823 10 0.1790 0.899 MA-3 0.3494 0.627 3(4) 0.2267 0.727 3(4) 0.3881 0.681 5 0.1179 0.592

WMA-3 0.3467 0.622 2 0.2247 0.720 2 0.3869 0.679 4 0.1163 0.584 EWMA-3 0.3494 0.627 3(4) 0.2267 0.727 3(4) 0.3858 0.677 3 0.1162 0.584 ES 0.3371 0.604 1 0.2157 0.692 1 0.3755 0.659 1 0.1068 0.537 REG 0.4125 0.740 6 0.2375 0.761 5 0.3773 0.662 2 0.1455 0.731 ARCH(1) 0.5577 1.000 11 0.3119 1.000 11 0.5697 1.000 11 0.1990 1.000 GARCH(1, 1) 0.4329 0.776 8 0.2705 0.867 9 0.4370 0.767 8 0.1730 0.869 GJRGARCH(1, 1) 0.4523 0.811 9 0.2717 0.871 10 0.4280 0.751 7 0.1603 0.806 EGARCH 0.4237 0.760 7 0.2670 0.856 8 0.4379 0.769 9 0.1487 0.747

The mean absolute error, (MAE), actual figures must be multiplied by 10 2. Actual is the calculated error statistic. Relative is the ratio bmodel and that of the worst performing model. The best performing model has a rank 1.


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3.2 Asymmetric Error Statistics

The conventional error statistics used in the previous subsection, ME, MAE, RMSE, and MAPE,

are symmetric; i.e. they give an equal weight to under- and over-predictions of volatility of similar

magnitude. However, many investors do not give equal importance to under- and over-prediction of

volatility, for example, in the pricing of options, while under-prediction of volatility is undesirablefor a seller, over-prediction is undesirable for a buyer. Following Brailsford and Faff (1996), to

penalize under-/over-predictions more heavily, the following mean mixed error statistics, MME,

are constructed:8

MME(U ) = 160

O

t=1|f,m a,m| +

Ut=1

|f,m a,m|

(19)

MME(O) = 160

U

t=1|f,m a,m| +

O

t=1 |f,m a,m|

(20)

where O is the number of over-predictions, and U is the number of under-predictions. MME(U)

and MME(O) penalize the under-predictions and over-predictions more heavily, respectively.

In Table 3 we report the results of the eleven volatility forecasting methods across the fif-

teen countries when assessed by the MME(U) and the MME(O) error metrics. The key features

of this analysis in the context of MME(U) which penalizes under-prediction more heavily can

be summarized as follows. First, given that we have already established that the ARCH-based

models are more heavily prone to (average) over-prediction, it is no surprise that these models

do particularly well according to MME(U). Indeed, in stark contrast to the previous analysis

(based on the symmetric measures) ARCH-based models are ranked number 1 in twelve coun-

tries. Second, it is interesting to note that of these models the EGARCH variation fairs best by

providing the supreme volatility forecast for five of the fifteen countries. Third, we find that the

difference between the forecasting ability of theARCH-based models is typically quite small. For

example, in the case of Hong Kong there is just 6.4% difference between the relative MME(U)

measures of all four ARCH-based models.

Fourth,we observe that the previously preferred ES method, according to MAE, mostly achieves

middle rankings. Clearly, the smaller over-prediction tendency of ES (relative to the ARCH-based

models) drives this result. Fifth, similar to the now poor showing of the ES, the MA models also

rate quite badly often ranked at 7 or worse according to MME(U). Sixth, according to the

MME(U) metric, the worst performing model at forecasting monthly volatility is the random

walk model it achieves the worst ranking in seven of the fifteen countries. This is not surprising

when it is recognized that RW typically produces the highest incidence of under-estimation of

monthly volatility.Turning now to the MME(O) results reported in Table 3, we tend to see a return to the same

sort of pattern discussed earlier with regard to the symmetric error measures. Specifically, the

key features of this analysis (which penalizes over-prediction more heavily) can be summarized

as follows. First, similar to all the symmetric measures, ES dominates it achieved supreme

ranking for eight out of the 15 countries and a second ranking on another two occasions. Second,

again (this time based on MME(O)) we see a relatively small gap between the accuracy of the

non-ARCH based methods. Third, the non-ARCH based models again are consistently superior

to the ARCH based models.


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Table 3. Mean mixed error (MME) statistics from forecasting monthly volatility

Belgium Cana

MME(U) MME(O) % % MME(U) MMEUnder- Over-

Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela

RW 1.8239 1.000 11 1.7146 0.551 2 0.500 0.500 2.0160 1.000 11 1.9569 0.9

HM 1.3590 0.745 4 2.6474 0.851 7 0.317 0.680 1.7387 0.862 7 1.9953 0.9MA-3 1.7819 0.977 9(10) 1.7293 0.556 3(4) 0.467 0.533 1.6450 0.816 2 2.0053 1.0WMA-3 1.7810 0.976 8 1.7315 0.556 5 0.467 0.533 1.6645 0.826 3 1.9951 0.9EWMA-3 1.7819 0.977 9(10) 1.7293 0.556 3(4) 0.467 0.533 1.6234 0.805 1 1.8918 0.9ES 1.7676 0.969 7 1.6392 0.527 1 0.467 0.533 1.6692 0.828 4 1.7509 0.8

REG 1.4224 0.780 5 2.0960 0.673 6 0.383 0.617 1.7930 0.889 10 1.8155 0.9ARCH(1) 1.4231 0.780 6 3.1107 0.999 10 0.717 0.283 1.7922 0.889 9 1.8827 0.9GARCH(1, 1) 1.2880 0.706 2 3.0303 0.974 9 0.734 0.266 1.6975 0.842 5 1.9226 0.9GJRGARCH(1, 1) 1.3531 0.742 3 2.8719 0.923 8 0.717 0.283 1.7201 0.853 6 1.8937 0.9EGARCH 1.2672 0.695 1 3.1122 1.000 11 0.734 0.266 1.7753 0.881 8 1.8930 0.9

Denmark Finla



RW 1.7814 0.997 9 2.0905 0.667 5 0.467 0.533 3.0987 0.748 9 3.1082 0.9HM 1.4910 0.835 6 2.5418 0.811 7 0.367 0.633 4.1430 1.000 11 1.3014 0.3MA-3 1.7865 1.000 10(11) 1.8807 0.600 1(2) 0.500 0.500 2.6057 0.629 6 2.5952 0.7WMA-3 1.7715 0.992 8 1.8874 0.602 3 0.500 0.500 2.6216 0.633 7 2.6245 0.7

EWMA-3 1.7865 1.000 10(11) 1.8807 0.600 1(2) 0.500 0.500 2.2049 0.532 3(4) 3.1922 0.9ES 1.6633 0.931 7 1.9065 0.608 4 0.450 0.550 2.2049 0.532 3(4) 3.1922 0.9REG 1.4044 0.786 5 2.4229 0.773 6 0.367 0.633 3.1841 0.769 10 2.1471 0.6ARCH(1) 1.2825 0.718 4 2.9346 0.936 8 0.717 0.283 2.6767 0.646 8 2.7941 0.8GARCH(1, 1) 1.1925 0.668 3 3.1113 0.992 10 0.700 0.300 2.2612 0.546 5 3.0638 0.9

GJRGARCH(1, 1) 1.1310 0.633 2 3.1354 1.000 11 0.717 0.283 2.1361 0.516 2 3.3710 1.0EGARCH 1.0490 0.587 1 3.0692 0.979 9 0.734 0.266 2.0751 0.501 1 3.2514 0.9


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Germany Hong K



RW 2.4129 1.000 11 2.3466 0.496 3 0.517 0.483 3.250 0.910 10 3.1315 0.47HM 1.6339 0.677 6 4.0257 0.852 7 0.267 0.733 3.570 1.000 11 3.4678 0.52MA-3 2.2181 0.919 8 2.3840 0.504 5 0.417 0.583 3.174 0.889 7 3.1800 0.47

WMA-3 2.2101 0.916 7 2.3689 0.501 4 0.417 0.583 3.191 0.894 8 3.1613 0.47EWMA-3 2.2819 0.946 10 2.2304 0.472 2 0.467 0.533 3.164 0.886 6 3.1695 0.47ES 2.2344 0.926 9 2.0977 0.444 1 0.467 0.533 3.001 0.841 5 3.1633 0.47REG 1.5408 0.639 4 3.5973 0.761 6 0.283 0.717 3.206 0.898 9 3.1973 0.4

ARCH(1) 1.5870 0.658 5 4.2072 0.890 9 0.767 0.233 2.915 0.817 4 6.6425 1.00GARCH(1, 1) 1.3077 0.542 1 4.7271 1.000 11 0.800 0.200 2.688 0.753 1 4.0484 0.60GJRGARCH(1, 1) 1.4074 0.583 2 4.4821 0.948 10 0.783 0.217 2.707 0.758 2 3.9225 0.59EGARCH 1.4748 0.611 3 4.0719 0.861 8 0.750 0.250 2.713 0.760 3 3.7850 0.57

Italy Japan

MME(U) MME(O) % % MME(U) MME

Under- Over- Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela

RW 2.7336 0.871 9 2.8764 1.000 11 0.500 0.500 2.4734 0.975 8 2.3647 0.4

HM 3.1402 1.000 11 1.8894 0.657 1 0.617 0.383 1.9291 0.760 5 3.7118 0.64MA-3 2.2841 0.727 5 2.6255 0.913 6 0.450 0.550 2.5364 0.999 9(10) 2.5756 0.44WMA-3 2.2831 0.727 4 2.6084 0.907 4 0.467 0.533 2.5372 1.000 11 2.5344 0.4EWMA-3 2.2026 0.701 3 2.5382 0.882 3 0.467 0.533 2.5364 0.999 9(10) 2.5756 0.44ES 2.1142 0.673 2 2.6124 0.908 5 0.433 0.567 2.4685 0.973 7 2.3251 0.40

REG 2.7567 0.878 10 2.1989 0.764 2 0.517 0.483 2.0316 0.801 6 2.9029 0.50ARCH(1) 2.3810 0.758 8 2.7984 0.973 9 0.550 0.450 1.2401 0.489 1 5.7738 1.00GARCH(1, 1) 2.3801 0.758 7 2.6347 0.916 7 0.500 0.500 1.4687 0.579 2 4.5315 0.7GJRGARCH(1, 1) 2.3459 0.747 6 2.6635 0.926 8 0.550 0.450 1.6083 0.634 3 4.0092 0.69EGARCH 2.0618 0.657 1 2.8298 0.984 10 0.583 0.417 1.7132 0.675 4 3.7359 0.64


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Table 3. Continued

Malaysia Netherla



RW 3.1778 0.953 9 3.1115 0.760 6 0.450 0.550 2.2855 0.979 10 2.1665 0.82

HM 3.3340 1.000 11 3.2046 0.783 7 0.467 0.533 2.0189 0.865 5 2.6375 1.00MA-3 3.0356 0.910 7(8) 2.7469 0.671 1(2) 0.500 0.500 2.2359 0.958 9 1.8378 0.69WMA-3 3.0038 0.901 6 2.7754 0.678 3 0.483 0.517 2.1995 0.942 8 1.8514 0.70EWMA-3 3.0356 0.910 7(8) 2.7469 0.671 1(2) 0.500 0.500 2.3351 1.000 11 1.6608 0.6

ES 2.9669 0.890 5 2.8355 0.693 4 0.467 0.533 2.1960 0.940 7 1.7056 0.64

REG 3.2130 0.964 10 3.0404 0.743 5 0.467 0.533 1.9682 0.843 3 2.1423 0.8ARCH(1) 2.5657 0.770 1 4.0917 1.000 11 0.667 0.333 2.1816 0.934 6 2.3459 0.8GARCH(1, 1) 2.7225 0.817 2 3.3425 0.817 9 0.617 0.383 1.8266 0.782 2 2.4547 0.9GRJGARCH(1, 1) 2.7414 0.822 3 3.2753 0.800 8 0.600 0.400 1.9849 0.850 4 2.2977 0.87

EGARCH 2.7585 0.827 4 3.3603 0.821 10 0.600 0.400 1.8069 0.774 1 2.4305 0.92

Philippines Singapo



RW 3.1985 1.000 11 3.0967 0.542 5 0.500 0.500 2.3663 1.000 11 2.3192 0.59

HM 2.0710 0.647 2 5.2125 0.913 10 0.283 0.717 1.6810 0.710 4 3.0127 0.76MA-3 3.0325 0.948 8(9) 2.6569 0 .465 3(4) 0.533 0.467 2.2029 0.931 9(10) 2.2389 0.57WMA-3 3.0441 0.952 10 2.6280 0.460 1 0.567 0.433 2.1829 0.922 8 2.2387 0.57

EWMA-3 3.0325 0.948 8(9) 2.6569 0.465 3(4) 0.533 0.467 2.2029 0.931 9(10) 2.2389 0.57ES 2.8762 0.899 7 2.6546 0.465 2 0.517 0.483 2.1584 0.912 7 2.1191 0.54

REG 2.1268 0.665 4 4.3861 0.768 6 0.300 0.700 1.7074 0.722 5 2.8521 0.72ARCH(1) 1.9493 0.609 1 5.7084 1.000 11 0.767 0.233 1.4862 0.628 3 3.9188 1.00GARCH(1, 1) 2.1669 0.677 6 4.4727 0.784 8 0.683 0.317 1.4659 0.619 2 3.5067 0.89GRJGARCH(1, 1) 2.1274 0.665 5 4.6318 0.811 9 0.683 0.317 1.4573 0.616 1 3.4995 0.89

EGARCH 2.0975 0.656 3 4.4460 0.779 7 0.700 0.300 1.7183 0.726 6 3.2306 0.82


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Thailand UK



RW 3.0649 1.000 11 3.1456 0.605 5 0.450 0.550 1.6947 1.000 11 1.4682 0.4HM 2.9870 0.975 7 3.7965 0.730 7 0.417 0.583 0.8624 0.509 1 3.2718 0.9MA-3 2.9941 0.977 8 3.0188 0.580 4 0.467 0.533 1.6212 0.957 9 1.6449 0.4WMA-3 3.0096 0.982 9 3.0138 0.579 3 0.483 0.517 1.6115 0.951 8 1.6275 0.4EWMA-3 2.9824 0.973 6 3.0000 0.577 2 0.467 0.533 1.5459 0.912 7 1.6425 0.4

ES 3.0289 0.988 10 2.9650 0.570 1 0.483 0.517 1.6525 0.975 10 1.4493 0.4REG 2.5726 0.839 5 3.3052 0.635 6 0.383 0.617 0.9541 0.563 4 2.8041 0.7ARCH(1) 2.4625 0.803 4 5.2030 1.000 11 0.733 0.266 0.9068 0.535 3 3.5110 1.0GRJGARCH(1, 1) 2.3341 0.762 3 4.3094 0.828 8 0.683 0.317 1.0012 0.591 6 3.1228 0.8TARCH 2.2250 0.726 1 4.3345 0.833 9 0.700 0.300 0.8843 0.522 2 3.0486 0.8

EGARCH 2.3122 0.754 2 4.3747 0.841 10 0.700 0.300 0.9893 0.584 5 2.7669 0.7

USA

MME(U) MME(O) % %Under- Over-

Actual Relative Rank Actual Relative Rank estimation estimation

RW 1.8228 0.915 6 1.8596 0.570 5 0.433 0.567HM 1.3985 0.702 1 3.2614 1.000 11 0.283 0.717MA-3 1.9712 0.990 10 1.8001 0.552 3 0.500 0.500

WMA-3 1.9450 0.977 9 1.8043 0.553 4 0.483 0.517EWMA-3 1.9913 1.000 11 1.7688 0.542 2 0.483 0.517ES 1.8497 0.929 8 1.6407 0.503 1 0.467 0.533REG 1.3991 0.703 2 2.6199 0.803 8 0.333 0.667ARCH(1) 1.8229 0.915 7 3.0547 0.937 10 0.667 0.333

GARCH(1, 1) 1.6125 0.810 3 2.6069 0.799 7 0.633 0.367GRJGARCH(1, 1) 1.7373 0.872 5 2.7464 0.842 9 0.617 0.383EGARCH 1.7101 0.859 4 2.5757 0.790 6 0.600 0.400

MME(U) and MME(O) are the mean mixed error statistics that penalise the under-predictions and over-predictions more heavily, respectstatistics. MME(U) and MME(O) actual figures must be multiplied by 10 2. Relative is the ratio between the actual error statistic of a modmodel. The best performing model has a rank 1.


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3.3 Discussion

Here we briefly discuss three dimensions of our analysis: (1) symmetric error metric results;

(2) asymmetric error metric results; and (3) developed versus emerging market comparisons. Our

initial set of findings relate to the outcome of the analysis when employing the standard symmetric

error metrics to assess volatility forecasting performance. First, we consistently found that theExponential Smoothing approach dominates in providing superior forecasts of monthly volatility.

Generally in comparison with the existing literature, these results have cases of similarity and

others of divergence. This finding largely confirms that of Dimson and Marsh (1990) for the UK

FT All shares index (though it should be noted that they did not include ARCH-type models in

their analysis). Other studies that similarly suggest a degree of superiority for the Exponential

Smoothing (ES) model include Figlewski and Green (1999) for the S&P 500; McMillan, Speight

and Gwilym (2000) where ES, MA and RW tended to perform well for low frequency UK data that

included the crash; and Ferreira (1999) for the French interbank one-month mid rate. However,

the superiority of the ES model contrasts that of Brailsford and Faff (1996) who found that for an

Australian market index this model performed rather poorly.

The second general aspect of note to our initial findings (based on symmetric errors) is thatnon-ARCH based models are found to be superior to the ARCH based models, consistent with

the findings of Tse (1991) Japan; Tse and Tung (1992) Singapore; and McMillan, Speight

and Gwilym (2000) UK. However, Poon and Granger (2003) report that of those studies they

surveyed that involve historical volatility models andARCH-type models, a slight majority (56%)

found the historical models to generally outperform their ARCH counterparts. A third and final

finding of note in our initial analysis was that within the sub-group of ARCH-type models there

is a tendency for more complex models to be superior. The ranking within ARCH-type models

(in terms of their complexity) tends to be rather mixed throughout the literature and thus no clear

point of comparison can be drawn.

Our second set of findings relate to the outcome of our analysis when employing the non-

standard asymmetric error metrics to assess volatility forecasting performance. Interestingly, ourresults change when the asymmetric loss functions, that penalize under-/over-prediction, are

employed. Specifically, when under-predictions are penalized more heavily, ARCH-type models

provide the best forecasts while the random walk is worst. However, when over-predictions of

volatility are penalized more heavily the exponential smoothing model performs best while the

ARCH-type models are now universally found to be inferior forecasters. The observation that the

rankings are dramatically affected by which asymmetric metric is used reinforces the findings of

Brailsford and Faff (1996). Moreover, the current paper confirms the Brailsford and Faff (1996)

finding of GARCH(1, 1) tending to be preferred when under-predictions are penalized more

heavily. However, with regard to the case when over-predictions are penalized more heavily,

while the current paper shows ES to rank very highly, Brailsford and Faff (1996) find it ranks

worst.Our third point of discussion involves assessing whether any clear trends or patterns are evident

in a comparison of forecasting volatility across developed versus emerging markets. In our sample,

there are three countries commonly classified as emerging: Malaysia, Philippines and Thailand.

In general terms no strong patterns are observed these emerging markets do not display clear

common features that contrast their developed counterparts. It does seem however that Malaysia

is a little different. In terms of the MAE symmetric error analysis Malaysia is one of the few

countries which does not favour the Exponential Smoothing forecasts the GJR-GARCH model

is superior. However, it should be noted in this case that (a) the ES model while ranked 4th,


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provides an MAE extremely close to GJR and (b) the only other country which does not favour

ES is Finland which cannot be viewed as emerging. With regard to the asymmetric analysis again

Malaysia shows up as being a little different to the others but only in the MME(O) case where

it again avoids choosing ES as its superior model.

What then are the major implications of these results? Based on our findings assessed usingstandard symmetric error measures, it seems that in many market situations, in different market

contexts (including both developed and emerging markets) the Exponential Smoothing model is

preferred. Thus, in situations in which the direction of forecast error is equally important (e.g.

portfolio selection), our study suggests that in the absence of other information leading to an

alternative preference, the use of the Exponential Smoothing model is recommended. However,

our results with regard to asymmetric error measures suggest that ARCH-type models (ES model)

are preferred for those cases in which under-predictions (over-predictions) are penalized more

heavily. These results are of greatest relevance to situations applications like VaR analysis wherein

a dramatic underestimate could have dire consequences for investors. Based on our results they

would be best served by using ARCH-type volatility forecasts as these models do best when

under-predictions of volatility are penalized most.

4. Conclusions

Volatility forecasting is a widely researched area in the finance literature. The performance of

forecasting models of varying complexity has been investigated according to a range of measures

and generally mixed results have been recorded. On the one hand some argue that relatively

simple forecasting techniques are superior, while others suggest that the relative complexity of

ARCH-type models is worthwhile. In this paper we seek to extend and supplement this existing

evidence by, in a single unifying framework, analysing a wide range of volatility forecasting

approaches across fifteen countries. Specifically, our analysis encompasses the ten-year period

1988 to 1997 for the market returns of Belgium; Canada; Denmark; Finland; Germany; Hong

Kong; Italy; Japan; Malaysia; Netherlands; Philippines; Singapore; Thailand; the UK; and the

USA.

In our analysis, a considerable range of forecasting models are used a random walk model, a

historical mean model, moving average models, weighted moving average models, exponentially

weighted moving average models, an exponential smoothing model, a regression model, an ARCH

model, a GARCH model, a GJR-GARCH model, and an EGARCH model. Furthermore, we

compare the forecasting techniques based on both symmetric (mean error, the mean absolute

error, the root mean squared error and the mean absolute percentage error) and asymmetric error

statistics.

The key features of our results can be summarized into two parts. First, in the context of standard

symmetric error metrics the exponential smoothing approach dominates. Thus, in situations inwhich the direction of forecast error is equally important (e.g. portfolio selection), our study

suggests that in the absence of other information leading to an alternative preference, the use

of the exponential smoothing model is recommended. Second, in the context of asymmetric

error metrics, when under(over)-predictions are penalized more heavily, ARCH-type (exponential

smoothing) models provide the best forecasts. These results are of greatest relevance to option

pricing, and market risk management applications. For example, in value-at-risk settings, under-

estimation of risk can be disastrous. Based on our results,VaR users would be best served by using

ARCH-type volatility forecasts as these models do best when under-predictions of volatility are


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penalized most. Finally, no major discernible patterns could be isolated regarding forecasting

volatility in emerging versus developed markets.

Acknowledgements

For helpful comments, we thank Chris Adcock, the editor; three anonymous referees; the partic-

ipants at the 13th Annual Meeting of the European Financial Management Association held in

Helsinki, Finland in June 2003; the 11th Conference on the Theories and Practices of Security

and Financial Markets held in Taiwan in December 2002; and seminar participants at the Bank

of England in February 2003 and at the University of Stirling in November 2003. The usual

disclaimers apply.

Notes

1 Evidence withrespect to foreign exchangemarkets includes Taylor(1987), Lee (1991), West and Cho (1995),Andersen

and Bollerslev (1998), Brooks and Burke (1998), Andersen et al. (1999), McKenzie (1999), Andersen et al. (2002),

Klaassen (2002), Vilasuso (2002), Balaban (2004) for example, West and Cho (1995) cannot show superiority ofany forecasting models.

2 The interested reader is referred to Poon and Granger (2003) for an excellent review of 93 papers that have looked at

forecasting volatility in financial markets.3 Interestingly Poon and Granger (2003, p. 490) state that given . . . most investors would treat gains and losses

differently, use of such [asymmetric] functions may be advisable, but their use is not common in the literature.

Variation from the conventional loss functions is not new, however for example, see Pagan and Schwert (1990).4 We thank an anonymous referee for suggesting this VaR example.5 One type of model that we exclude is the regime-switching variety. While a regime-switching model is a good one

for in-sample modelling, it is not readily amenable to an out-of-sample volatility forecasting exercise. Our desire to

have a practical focus underlying our work helps rule this model out of consideration.6 While these results and the MAPE results discussed shortly are not reported in the interests of brevity, details are

available from the authors upon request.7

As stated earlier, the ARCH models are updated monthly using daily data and n-day ahead forecasts are constructed.This process will make it more challenging for them to compete, thereby representing a potential explanation of their

poor predictive performance. We thank an anonymous referee for bringing this issue to our attention.8 Since the absolute values of all forecast errors are less than one, taking their square root increases the penalty for

under/over-prediction.

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