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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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http://www.tandfonline.com/page/terms-and-conditionshttp://dx.doi.org/10.1080/13518470500146082http://www.tandfonline.com/loi/rejf207/30/2019 Volatiliy of Stock Prices Index.
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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: robert.faff@buseco.monash.edu.au
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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174 E. Balaban et al.
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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176 E. Balaban et al.
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
Actual Relative Rank Actual Relative Rank Actual Relative Rank Actual Relative
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
Actual Relative Rank Actual Relative Rank Actual Relative Rank Actual Relative
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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Forecasting Stock Market Volatility 179
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
MME(U) MME(O) % % MME(U) MMEUnder- Over-
Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela
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
MME(U) MME(O) % % MME(U) MMEUnder- Over-
Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela
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
MME(U) MME(O) % % MME(U) MMEUnder- Over-
Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela
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
MME(U) MME(O) % % MME(U) MMEUnder- Over-
Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela
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
MME(U) MME(O) % % MME(U) MMEUnder- Over-
Actual Relative Rank Actual Relative Rank estimation estimation Actual Relative Rank Actual Rela
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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184 E. Balaban et al.
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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Forecasting Stock Market Volatility 185
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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186 E. Balaban et al.
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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