Garch Vola Measurepdf

8/3/2019 Garch Vola Measurepdf

1/26Electronic copy available at: http://ssrn.com/abstract=1941699

Swiss Finance InstituteResearch

Paper Series N11 4

5

Forecasting Financial TimeSeries: Normal GARCH with

Outliers or Heavy TailedDistribution Assumptions?Christoph HARTZUniversity of MunichMarc S. PAOLELLAUniversity of Zurich and Swiss Finance Institute



Established at the initiative of the Swiss Bankers' Association, the Swiss

Finance Institute is a private foundation funded by the Swiss banks and

Swiss Stock Exchange. It merges 3 existing foundations: the International

Center FAME, the Swiss Banking School and the Stiftung "Banking and

Finance" in Zurich. With its university partners, the Swiss Finance Institute

pursues the objective of forming a competence center in banking and

finance commensurate to the importance of the Swiss financial center. It will

be active in research, doctoral training and executive education while also

proposing activities fostering interactions between academia and the

industry. The Swiss Finance Institute supports and promotes promising

research projects in selected subject areas. It develops its activity in

complete symbiosis with the NCCR FinRisk.

The National Centre of Competence in Research Financial Valuation and

Risk Management (FinRisk) was launched in 2001 by the Swiss National

Science Foundation (SNSF). FinRisk constitutes an academic forum that

fosters cutting-edge finance research, education of highly qualified finance

specialists at the doctoral level and knowledge transfer between finance

academics and practitioners. It is managed from the University of Zurich and

includes various academic institutions from Geneva, Lausanne, Lugano,

St.Gallen and Zurich. For more information see www.nccr-finrisk.ch .

This paper can be downloaded without charge from the Swiss Finance

Institute Research Paper Series hosted on the Social Science Research

Network electronic library at:

http://ssrn.com/abstract=1658362



Forecasting Financial Time Series: Normal GARCH with Outliers

or Heavy Tailed Distribution Assumptions?

Christoph Hartza Marc S. Paolellaa,b,

aDepartment of Statistics, University of Munich, Munich, GermanybDepartment of Banking and Finance, University of Zurich, Switzerland

cSwiss Finance Institute

September 2011

Abstract

The use of GARCH models is widely used as an effective method for capturing

the volatility clustering inherent in financial returns series. The residuals from such

models are however often non-Gaussian, and two methods suggest themselves for

dealing with this; outlier removal, or use of non-Gaussian innovation distributions.

While there are benefits to both, we show that the latter method is better if interest

centers on volatility and value-at-risk prediction. New volatility measures based on

OHLC (open high low close) data are derived and used. Use of OHLC measures are

shown to be superior to use of the naive estimator used in other GARCH outlierstudies.

KeywordsOutliers, fat-tailed distributions, GARCH, volatility.

Corresponding author. E-mail address: [email protected]. Part of this research has been carriedout within the National Centre of Competence in Research Financial Valuation and Risk Management (NCCRFINRISK), which is a research program supported by the Swiss National Science Foundation.


4/26

1 INTRODUCTION 2

1 Introduction

The growth of interest in modeling and forecasting financial time series continues unabated, both

in academic and financial institutions, with a steady proliferation of models being proposed andtested for use in the prediction of volatility and Value-at-Risk, hereafter VaR. It is virtually

undisputed that returns of financial time series sampled at weekly or higher frequencies deviate

markedly from being independently and identically distributed (iid), most notably in the form

of (i) blatant volatility clustering, which is also reflected by high autocorrelation of absolute and

squared returns; (ii) substantial kurtosis, i.e., the density of the unconditional return distribu-

tion is more peaked around the center and possesses much fatter tails than the normal density;

and (iii) mild skewness. With respect to the first of these, use of (generalized) autoregressive

conditional heteroscedasticity (GARCH) models and their numerous extensions constitute byfar the most popular modeling method. While they are indeed quite capable of accounting for

the volatility clustering and, thus, a significant amount of the excess kurtosis (compared to the

normal distribution) andfor asymmetric GARCH generalizationssome of the skewness, the

residuals of fitted GARCH models are most often still leptokurtic and skewed. Because of this, it

is commonplace to allow the innovations process in the GARCH model to be an iid sequence of

random variables from a distributional class which still contains normality as a special case, but

is otherwise fat-tailed and asymmetric.

A different way which is, in some respects, more in line with the traditional approach in

statistics to handle aberrant observations not in accord with the normal distribution, is to

classify them as outliers and remove them from the analysis. Several authors have contributed

to this idea. Frances and Ghijsels (1999) proposed an Additive Outlier GARCH model, in short

AO-GARCH, driven by normal innovations. Their method is based on that of Chen and Liu

(1993) for detecting different types of outliers in time series and is described in more detail in

Section 2 below. Their model was extended to support innovative outliers, also based on the Chen

and Liu (1993) approach, by Charles and Darne (2005a); see also Charles and Darne (2005b) for

further analysis of this model. Alternative approaches, involving outlier removal either before or

after applying a GARCH filter, are discussed in Carnero et al. (2001).

We concentrate herein on the method in Frances and Ghijsels (1999). They fit their model

to the returns on four weekly European stock indices and demonstrated that volatility forecasts

are more accurate with the AO-GARCH model than when using the normal-GARCH ormore

importantlya GARCH model driven by Students t innovations with (jointly) estimated degrees

of freedom parameter, denoted t-GARCH. This finding would have important consequences for

risk management, given the predominance of t-GARCH and other variants for deriving VaR


5/26

1 INTRODUCTION 3

predictions.

If the true data generating process (DGP) is (or, realistically, is closer to) a normal-GARCH

process with occasional additive outliers, then it is certainly reasonable to expect that an AO-

GARCH model with estimated parameters will result in better out-of-sample forecasts than al-

ternative models such as t-GARCH. However, the validity of concluding the converse, i.e., that

improved forecasting ability of model A over model B implies that model A is a better descrip-

tion of the DGP than model B, strongly depends on the method of forecast measurement. In

the case of Frances and Ghijsels (1999), they compare the predicted volatility of the competing

models with the absolute return (demeaned by a long time average) as a proxy for the realized

volatility. This type of realized volatility has been, unfortunately, shown to be highly inaccu-

rate by Andersen and Bollerslev (1998). This simplistic, faulty measure was also used by Park

(2002) to demonstrate superiority of his outlier robust GARCH model for exchange rate returns.

Ideally, intra-day transactions would be used to construct volatility measures, because they have

been shown to be an excellent method of eliciting the true, unobservable, daily volatility (An-

dersen et al., 2003). For the data used in our study below, intra-day prices are not available,

so we instead follow Parkinson (1980) and Wiggins (1991) and make use of the OHLC-prices

(open-high-low-close) to obtain better predictions of the true volatility.

Another point of contention with Frances and Ghijsels (1999) is that their simulation studies

consist only of estimating and detecting outliers in the GARCH(1,1) model from simulated,

outlier-infected GARCH data and subsequently verifying that the procedure is indeed able to

successfully label and remove most of the actual generated outliers. While this is important

and useful, they do not study the arguably more important issue of the behavior of their model

and detection procedure, and the quality of the ensuing forecasts, assuming a data generating

process which closely mimics the stylized facts of financial time series. In particular, one could

consider data generated from a GARCH-type model with fat-tailed innovations such as from

the generalized asymmetric Students t, or GAt (Mittnik and Paolella, 2000), the asymmetric

stable Paretian (see Doganoglu et al., 2007; Broda et al., 2011; and the references therein), or the

generalized hyperbolic (see Broda and Paolella, 2009; Yang, 2011; Paolella and Polak, 2011; and

the references therein). In such a case, one would expect the outlier-based method to perform

poorly because it will ultimately remove the extreme data pointswhich are precisely those

which become relevant for understanding and modeling the tail behavior of the distribution and,

consequently, for prediction of risk.

The aim of this paper is to show via the use of more appropriate measures of volatility and

the use of several real data sets that the AO-GARCH method is inferior to the use of fat-tailed


6/26

2 MODELS UNDER INVESTIGATION 4

GARCH models with respect to volatility forecasting and VaR prediction. An added plus to

this conclusion is that the AO-GARCH model is, relatively to other GARCH models, far more

time-intensive to estimate and also involves the use of a user-decided tuning parameter for the

threshold of outlier determination which is decisive for its performance.

The remainder of this paper is as follows. In Section 2, the models under comparison are

briefly introduced, while Section 3 presents the time series used for comparison and discusses

the in-sample results for both models. In Section 4, the out-of-sample performance of both

models with respect to volatility forecasting and VaR prediction is discussed in detail. Section 5

concludes.

2 Models under Investigation

Let the observed time series be denoted by rt, t = 1, . . . , T , where rt = 100(ln Pt ln Pt1) isthe usual definition of the one-period return of the asset with price Pt at time t. The normal-

GARCH(p,q) model assumes

rt = + t, t = ztt, (1)

2t = c0 +

pi=1

ci2ti +

qj=1

dj2tj , (2)

where ztiid

N(0, 1). For the vast majority of applications, p = q = 1 is deemed adequate. For thedata sets considered in Section 3, this choice was also found to be appropriate.

As discussed in the Introduction, a popular method of augmenting the normal-GARCH model

to accommodate the typically observed non-normality of the residuals is simply to replace the

normal distribution with one which can exhibit asymmetry and tails fatter than those of the

normal. We use the GAt distribution, which generalizes Students t distribution by adding an

additional shape and and asymmetry parameter. In particular, its density is given by

fGAt(z; d,,) = K

1 + (z)d

(+

1

d)

, if z < 0,1 + (z/)

d

(+ 1d)

, if z 0,(3)

with K1 = ( + 1)d11/d B(d1, ). The density is asymmetric for = 1 and coincides with

the usual Students t density with m degrees of freedom with m = /2, d = 2 and = 1. The

raw integer moments are given by

E[Zr] =(1)r(r+1) + r+1

1 +

B(

r+1d , rd

)B(1d , ) r/d (4)

for r < d; see Paolella (2007) for derivation and further details of the GAt.


7/26

2 MODELS UNDER INVESTIGATION 5

We now briefly describe the AO-GARCH model proposed by Frances and Ghijsels (1999),

which is based on the method of Chen and Liu (1993) for detecting additive outliers in ARMA

time series. Let time series yt be given by the ARMA(r, s) model r(L)yt = s(L)t, where t is

a white noise process and the roots of r(L) and s(L) lie outside the unit circle. This series is

not observed, but rather yt, where

yt = yt + It(), (5)

with It() = 1 for t = and zero otherwise. That is, the observed series is an ARMA process

plus an additive outlier. By modeling the observed series, yt, with an ARMA(r, s) model, the

estimated residuals t = (L)yt, = t + (L)It() are obtained, with (L) = (L)/(L), from

which the estimated impact, (), can be elicited from the regression t = xt + t, with xt = 0

for t < , xt = 1 for t = , and xt+k = k for t and k = 1, 2, . . . .Outlier detection involves testing the significance of the (). Chen and Liu (1993) propose

three ways for estimating the residual standard deviation, a, from which Frances and Ghijsels

(1999) use the omit-one method, i.e., use the sample variance and omit the observation at t = .

The test statistic they obtain is given by

= [()/a]

/[T

t=

x2t

]1/2, (6)

where the impact of an AO is deemed significant if exceeds the critical value C, which is setequal to 4 by Frances and Ghijsels (1999). If the impact is significant, then the observation yt

can be adjusted to obtain the AO-corrected yt = yt It().Frances and Ghijsels (1999) apply the aforementioned AO correction for ARMA models to

the normal GARCH(1,1) model by defining t = 2t 2t and rewriting the GARCH equation to

obtain

2t = c0 + (c1 + d1)2t1 + t d1t1. (7)

This corresponds to an ARMA(1,1) model for

2

t , to which the AO correction is applied. Afterestimating a GARCH(1,1) model for the original series t, they calculate t =

2t 2t and

(L) = (1 (c1 + d1)L)/(1 d1L) and perform the regression for the estimated residuals t forevery t = to obtain () and the test statistic . Then, t is replaced by

t for that t = with

the largest value of > C = 4. Next, the series 2t = t +

2t is constructed to calculate the

AO-corrected series, where

t =

t, for t = ,

sign(t)(2t )

1/2, for t = .(8)


8/26

3 DATA ANALYSIS AND IN-SAMPLE RESULTS 6

For the corrected series, t , the procedure of estimating the normal GARCH(1,1) model and the

correcting for AO is repeated until no test statistic exceeds the critical value C = 4. The

estimated GARCH(1,1) model for the final AO corrected series t is then used for forecasting

purposes.

In the subsequent empirical analysis, we follow Frances and Ghijsels (1999) and use C = 4.

It is, however, important to realize that estimation and forecasting results for the AO-GARCH

model are highly dependent on the choice of C and no objective criteria for choosing a correct

value of C has been proposed.

3 Data Analysis and In-Sample Results

The data used in our study are based on the daily prices of five major European stock indices:

the German DAX, the Dutch AEX, the French CAC40, the Italian MIBTEL and the British

FTSE100. While the ending periods of each coincide (last trading day in 2001), the starting

dates differ because of data availability and are, respectively, Jan. 1992, Jan. 1993, Jan. 1991,

Jan. 1994 and Jan. 1992.

Figure 1 plots the returns for each of the five series, from which the volatility clustering is quite

blatant. The usual measures of sample skewness and kurtosis were also computed; each series

exhibits negative skewness, i.e., has a longer left tail, and a sample kurtosis which is considerably

larger than the appropriate value of three for the normal distribution.1

With respect to computing time, estimation of the GAt-GARCH model takes about five

times longer than normal-GARCH. Time for estimation of the AO-GARCH model with normal

innovations depends mainly on the number of outliers detected, as each detected outlier invokes a

normal-GARCH estimation and T regressions for detecting subsequent outliers. This is, in turn,

dictated by the choice of tuning parameter C. Roughly speaking, with C = 4 we found about 5%

of the returns under investigation to be outliers. With that one can say that the AO-GARCH

model takes more than 0.05T times longer than estimation of the normal-GARCH model. (All

programs were written in Matlab and compiled to increase execution speed.)

Because of the nature of the AO-GARCH method of fitting a model to the data, it does

not lend itself to likelihood-based comparisons. In particular, after the outliers are chosen and

removed, the resulting fit isessentially by constructionexcellent. As such, for in-sample com-

parisons, we will study the choice of outliers removed by the AO-GARCH method, and argue

1 Standard errors and classical inference methods to assess the significance of the deviation of these statisticsfrom their null values corresponding to normality are not reliable in this context because the data are clearlynot iid, and the existence of low moments is not at all certain. As such, the statistics are not presented.


9/26


1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

DAX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

AEX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

CAC40

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

MIBTEL

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

FTSE100

Figure 1: Return plots

that it is inappropriate for financial return data.

Table 1 presents the estimated parameters and several test statistics for the AO-GARCH

model applied to each of the five time series under study. For every index there are two columns,

labeled 0 and AO, with 0 referring to no outlier detection and use of just the usual normal-GARCH

model, and AO, which refers to the final iteration after full AO detection and correction. For

all five series, the mean of the standardized residuals as well as the mean parameter is not


10/26


Table 1: Estimation results for the normal-GARCH and the AO-GARCH modelsa

DAX(0/AO)

AEX(0/AO)

CAC40(0/AO)

MIBTEL(0/AO)

FTSE100(0/AO)

0.067(0.020)

/ 0.064(0.019)

0.070(0.019)

/ 0.078(0.018)

0.041(0.022)

/ 0.059(0.019)

0.053(0.027)

/ 0.039(0.025)

0.043(0.016)

/ 0.043(0.015)

c0 0.023(0.006)

/ 0.009(0.004)

0.016(0.005)

/ 0.010(0.004)

0.054(0.015)

/ 0.008(0.005)

0.092(0.023)

/ 0.048(0.017)

0.010(0.004)

/ 0.004(0.002)

c1 0.091(0.012)

/ 0.059(0.009)

0.101(0.013)

/ 0.065(0.013)

0.066(0.012)

/ 0.028(0.007)

0.145(0.020)

/ 0.076(0.015)

0.065(0.010)

/ 0.039(0.009)

d1 0.897(0.012)

/ 0.934(0.011)

0.891(0.013)

/ 0.924(0.016)

0.900(0.019)

/ 0.965(0.009)

0.814(0.024)

/ 0.890(0.024)

0.925(0.012)

/ 0.955(0.011)

mean 0.022/0.013 0.015/0.011 0.010/0.008 0.020/0.006 0.020/0.015var 0.998/ 0.997 0.997/ 0.997 1.003/ 1.002 1.000/ 1.001 0.997/ 0.998

skew 0.242/0.158 0.399/0.130 0.353/0.032 0.249/0.050 0.074/0.050kurt 4.230/ 2.721 3.992/ 2.577 5.057/ 2.480 5.279/ 2.603 4.072/ 2.402

LB10 15.02(0.13)

/ 12.64(0.24)

14.00(0.17)

/ 14.19(0.16)

19.29(0.04)

/ 18.23(0.05)

25.61(0.00)

/ 15.86(0.10)

25.29(0.00)

/ 25.35(0.00)

QLB10 6.25(0.79)

/ 8.53(0.58)

15.12(0.13)

/ 8.76(0.56)

5.66(0.84)

/ 10.14(0.43)

6.65(0.76)

/ 16.94(0.08)

6.41(0.78)

/ 9.90(0.45)

# AO 99(3.9%)

118(5.2%)

128(4.6%)

74(3.7%)

152(6.0%)

# AO 51(2.0%)

62(2.7%)

71(2.6%)

35(1.7%)

77(3.1%)

# AO+ 48(1.9%)

56(2.5%)

57(2.1%)

39(1.9%)

75(3.0%)

R(AO) 0.06 0.00 0.00 0.00 0.00

R(+) 0.92 0.00 0.00 0.00 0.00a For every index there are two columns, labeled 0 for normal-GARCH model without any outlier detectionand AO for final iteration after full outlier detection and correction.Rows 1 to 4 contain the parameter estimates (standard errors given in parenthesis); rows 5 to 10 givestatistics for the standardized residuals, where LB10 (QLB10) is the Ljung-Box test statistic of order 10 forthe (squared) standardized residuals (p-values are given in parentheses); rows 11 to 13 give the number ofoutliers (# AO), the number of negative outliers (# AO-), and the number of positive outliers (# AO+)detected and corrected by the AO-GARCH model; and rows 14 and 15 give the p-values for the runs-teststatistic for the sequence of outliers, R(AO), and the sequence of positive and negative outliers, R(+).

significantly affected by the outlier correction. This and the number of outliers of either sign

(#AO+ and #AO-) indicate that the method is equally likely to detect a negative or a positive

outlier.

For the fitted AO-GARCH model, the constant term of the GARCH equation, c0, and the

ARCH term, c1, are smaller for all five indices in comparison to the normal-GARCH model,

whereas the GARCH parameter d1 is greater after the outlier correction. This finding is in line

with Figure 2, where the corrected return series are plotted. It is obvious that the AO correction


11/26


removes all of the peaks of the original series. By cutting these peaks, any information about

the DGP contained in these extreme returns is lost. If these observations are indeed genuine

outliers resulting from, say, incorrect data entry, or (far more likely) correspond to information

arrivals via political or economic events which are truly exceptional and will not have any relevance

for future modeling, then the procedure apparently succeeds admirably in their removal.

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

DAX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

AEX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

CAC40

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

MIBTEL

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

5

0

5

FTSE100

Figure 2: AO corrected return plots

The standardized residuals of the normal-GARCH model still exhibit some of the properties as


12/26


the return series themselves. For all five indices, they have negative skewness and higher kurtosis

than imposed by the normal distribution. After applying the AO-correction, the negative skewness

is reduced and the kurtosis drops down below the anticipated value of three for the normal

distribution, which leads to the conjecture that the critical value C = 4 may be too small and

too many returns are corrected by the procedure. Hence, the AO-correction works as expected:

All observations not in line with the assumed model and error distribution are corrected.

As indicated by the Ljung-Box test statistic for the squared standardized residuals for order

10 (QLB10) and corresponding p-values, there is no heteroscedasticity left in the standardized

residuals of the normal-GARCH model for all five time series. In comparison to the normal-

GARCH model, the QLB10 test statistic for the AO-GARCH model rises for all indices except

for the AEX index. For the MIBTEL index, QLB10 = 16.94, with p-values 0.08, indicating that

there is still an amount of heteroscedasticity remaining in the standardized AO-GARCH model

residuals.

The occurrence of the detected and removed outliers from the AO-GARCH model is illustrated

in Figure 3. From the plot, it is strikingly evident that their positions are not randomly distributed

throughout the sample, but rather occur in clusters. This undermines the presumption that

outliers are sprinkled throughout the data in a random, unpatterned fashion, but rather that

the AO-GARCH model is removing observations occurring in groups of returns with larger than

average heteroscedasticity. It is hardly tenable that these observations are outliers in the sense

that their consideration would only corrupt the underlying signal in the data, but rather that

they are an intimate part of the time-varying volatility phenomena containing information both

about the GARCH parameters and, separate from any particular parametric model, about the

actual probability of extreme events.

To statistically test whether the observations deemed to be outliers are occurring at random

or are clustered, define a new time series {Bt} with Boolean elements, depending on whether anoutlier is detected or not, i.e., Bt = 1 if an outlier is detected, and zero otherwise. The runs test

statistic

R(AO) =s 2T p0p1

2

np0p1

asy N(0, 1) (9)

can then be applied, where p0 and p1 are the proportions of zeros and ones, respectively and s is

the number of runs.2

We also test for the random occurrence of positive and negative outliers. Let {St} be thetime series such that St = sign(rt)Bt. With p1 and p1 the proportion of negative and positive

2 See Paolella (2006, Sec. 6.3) for the exact distribution of this and related runs distributions, original references,further details and applications.


13/26


1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1

0

1

DAX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1

0

1

AEX

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1

0

1

CAC40

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1

0

1

MIBTEL

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1

0

1

FTSE100

Figure 3: Outlier occurrence: a cross (x) marks the occurrence of an outlier of eithersign, whereas a plus (+) marks the same outliers, but differenced by sign. So,the cross series sums the plus series.

outliers, respectively, and p0 and s are defined as before, the test statistic is given by

R(+) =s n

1

1i=1

p2i

n

1

i=1p2i 2

1

i=1p3i +

1

i=1p2i

2 asy N(0, 1). (10)


14/26


With the single exception of the DAX index, the statistical significance of the resulting test

statistics provide evidence to suggest that the occurrence of the outliers is non-random and that

useful information about the process is lost by removing the outliers.

Table 2: Estimation results for the GAt-GARCH modela

DAX AEX CAC40 MIBTEL FTSE100

0.178(0.035)

0.203(0.032)

0.179(0.045)

0.097(0.049)

0.133(0.030)

c0 0.022(0.009)

0.022(0.008)

0.055(0.020)

0.133(0.039)

0.014(0.006)

c1 0.133(0.021)

0.154(0.024)

0.101(0.019)

0.202(0.035)

0.109(0.025)

d1 0.913(0.013)

0.901(0.015)

0.919(0.016)

0.840(0.026)

0.930(0.011)

5.879(0.000)

5.991(0.187)

5.968(0.299)

5.979(0.338)

5.981(3.068)

d 2.019(0.061)

2.060(0.090)

2.029(0.084)

2.016(0.094)

2.100(0.455)

0.922(0.020)

0.903(0.021)

0.924(0.023)

0.966(0.025)

0.926(0.022)

mean(theo.)

0.097(0.098)

0.117(0.122)

0.098(0.094)

0.042(0.041)

0.094(0.091)

var(theo.)

0.610(0.601)

0.591(0.587)

0.607(0.595)

0.606(0.595)

0.574(0.570)

skewness(theo.)

0.235(0.180)

0.398(0.217)

0.387(0.172)

0.249(0.077)

0.074(0.159)

kurtosis(theo.)

4.350(3.762)

4.133(3.671)

5.499(3.720)

5.355(3.725)

4.086(3.576)

LB10 14.96(0.13)

13.21(0.21)

19.21(0.04)

26.46(0.00)

25.61(0.00)

QLB10 4.53(0.92)

13.47(0.20)

5.16(0.88)

6.96(0.73)

6.76(0.75)

a Rows 1 to 7 contain the parameter estimates (standard errors given in parenthesis); rows 8to 14 give statistics for the standardized residuals, where LB10 (QLB10) is the Ljung-Box teststatistic of order 10 for the (squared) standardized residuals (p-values are given parenthesis);

We now turn to the analysis of the fitted GAt-GARCH models. The estimation results are

given in Table 2. Denote by zt the residuals of the estimated model. If the GARCH model is

appropriate, then the zt should be approximately iid with zero location, unit scale, and density

given by the GAt density with the estimated values of the shape and skewness parameters (these

having been jointly estimated with the GARCH parameters). As for the normal-GARCH model

and the AO-GARCH model, the values for the mean and the variance are close to the theoretical

values computed from (4) with the estimated parameter values. But, in addition, the GAt

distribution is able to capture parts of the skewness and the excess kurtosis. Figure 4 shows the


15/26


kernel and fitted probability density functions of the zt. It is visually clear that the model fits

the data quite well.

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

DAX

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

AEX

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

CAC40

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

MIBTEL

4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

FTSE100

Figure 4: Fitted GAt density (solid) vs. empirical kernel density (dashed) for thestandardized residuals.


16/26

4 OUT-OF-SAMPLE FORECASTING RESULTS 14

4 Out-of-Sample Forecasting Results

4.1 Volatility Forecasts

Judging the quality of a forecast of the volatility obviously necessitates a benchmark value against

which it can be compared. As discussed in detail by Andersen and Bollerslev (1998), the com-

monly used proxy for the true unobservable daily volatility,

vst = |rt r|, r =1

T

rt, (11)

is a poor measure for the true daily volatility owing to its enormous variance (although it is un-

biased). This is, unfortunately, the measure employed by Frances and Ghijsels (1999). Andersen

and Bollerslev investigate and recommend the use of intra-day returns for measuring daily volatil-

ity. However, because of the limited availability of such data, we propose four different volatility

measures, each based on the widely available OHLC (open-high-low-close) data for stock prices

to approximate the realized volatility.

pot

pht

plt

pct

Figure 5: Diagrammatic occurrence of OHLC-Prices: pot (pht , p

lt and p

ct ) denote the

logarithmic open (high, low and close) price at time t.

The first approach is to use the four available prices per day to construct six possible intra-day

returns, i.e. the returns that may occur by intra-day trading and using the limited information

set of the OHLC-prices. These artificial returns are then used in conjunction with the usual

method for calculating a sample standard deviation, i.e.,

vat =

16

(x,y)A

(r(x,y)t E[rt])2, A = {(o, c), (o, h), (o, l), (h, l), (h, c), (l, c)}, (12)

where r(x,y)t = 100 |pyt pxt | for (x, y) A, are the absolute values of the six artificial returns

between two realized prices of one day.


17/26


The second approach also uses the four available prices of one day, as well as the closing price

of the previous day, to derive four artificial returns in terms of an inter-day trading with limited

information, i.e., the returns that may occur when trading takes place between the closing price

the day before, and one of four possible prices today. As before, the standard deviation of these

returns is used as a proxy for todays realized volatility:

vet =

14

(x,y)E

(r(x,y)t E[rt])2, E = {(c, c), (c, o), (c, l), (c, h)}, (13)

where r(x,y)t = 100 |pyt pxt1| with (x, y) E, are the absolute values of the four possible

inter-day returns.

A third way to calculate the daily volatility is a mixture of the two previously mentioned

approaches, which we refer to as the mixed volatility measure :

vmt =

110

(x,y)M

(r(x,y)t E[rt])2, M = A + E, (14)

where r(x,y)t are the ten possible intra- and inter-day returns as defined before.

The three aforementioned approaches are straightforward and obviously simplistic in construc-

tion. The fourth approach we consider was proposed in Garman and Klass (1980) for estimating

the volatility constant of a diffusion process for security prices by the use of OHLC-prices. They

derive six different estimators for , one of which we use, given by

vxt =

a

((pot pct1) 100)2f

+ (1 a) ((pht plt) 100)2

(1 f)4log2 , (15)

with f is the portion of day where no trading takes place; that is, the time between closing at day

t 1 and opening of day t. For the empirical applications below, f = 0.6 is used, which turnedout to be a good approximation for the different values of f for the different indices. Finally,

a = 0.17 is a constant which minimizes vxt , independent of f (this is the 23 estimator of Garman

and Klass, 1980).

The five volatility approximations described above are used to judge the forecasting accuracy

of the two competing GARCH models. For both models, 1000 one-step-ahead volatility forecasts

are constructed for all five indices, with an estimation period of 250 days (about one year of

trading data) for both models. To make the comparison more realistic, the model parameters

were updated for each t = T 1000, . . . , T 1. Note that such an exercise will extremely time-consuming when using the AO-correction method.


18/26


The three popular error criteria, mean squared error (MSE), median squared error (MedSE)

and the mean absolute percentage error (MAPE), were used to judge the performance. In addi-

tion, we ran a regression analogous to that of Andersen and Bollerslev (1998) of the form

vf = 0 + 1 vf, f = 1, . . . , 1000, (16)

with vf being the realized volatility measured by one of the volatility proxies and vf is the

one-step-ahead prediction of volatility of one of the models. Note that vf equals f for the

AO-GARCH model (with normal innovations) and vf = f

E[(Z E[z])2] for the GAt dis-tribution. For a model which provides correct one-step volatility predictions, the parameters 0

and 1 are expected to be 0 and 1, respectively.

The results for the standard volatility proxy (11), as well as for the mixed and the extreme

value approximations are given in Tables 3 to 5. (The results for the intra- and inter-day approx-

imations are similar to the results obtained by the mixed volatility approximation, so we do not

present them here to save space.) Beginning with the standard volatility proxy vst in (11), we see

that, comparatively speaking, it is by far the worst: The low R2 value is in agreement with the

results reported in Andersen and Bollerslev (1998) and is the lowest R2 of all competing proxies

across both models and all indices. Similarly, vst exhibits the largest error criteria values; in the

case of the MAPE, it is on average 10 times higher than for the vmt in (14) and about 15 times

higher as the MAPE for the vxt in (15).

Table 3: Forecasting results for standard volatility approximation (11)a

DAX(AO/GAt)

AEX(AO/GAt)

CAC40(AO/GAt)

MIBTEL(AO/GAt)

FTSE100(AO/GAt)

0 0.341(0.145)

/ 0.083(0.104)

0.210(0.121)

/ 0.015(0.081)

0.283(0.234)

/ 0.113(0.130)

0.331(0.141)

/ 0.074(0.082)

0.073(0.142)

/ 0.005(0.107)

1 1.144(0.102)

/ 0.745(0.063)

1.119(0.099)

/ 0.776(0.053)

1.147(0.185)

/ 0.719(0.087)

1.196(0.112)

/ 0.739(0.053)

0.834(0.131)

/ 0.790(0.085)

R2 0.112/ 0.121 0.113/ 0.174 0.037/ 0.064 0.102/ 0.163 0.039/ 0.080

MSE 1.026/ 1.110 0.896/ 0.934 0.917/ 0.979 1.054/ 1.086 0.624/ 0.659MedSE 0.488/ 0.557 0.324/ 0.409 0.448/ 0.556 0.424/ 0.472 0.343/ 0.399

MAPE 550.1/ 583.0 417.7/ 468.6 458.1/ 530.8 411.3/ 451.9 485.0/ 538.9a For every index there are two columns, labeled AO for the AO-GARCH model and GAt for the GAt-GARCHmodel. Values 0 and 1 are the estimated regression coefficients (standard errors in parentheses).

The most accurate proxy for realized volatility seems to be the extreme value estimator from

Garman and Klass (1980), vxt , in (15). It results in the highest R2, the lowest MAPE and, with

two exceptions, the lowest MedSE of all proxies. For the MSE, it took second place, with a slight

preference being given to the volatility approximations given in equations (12) to (14). Thus, the


19/26


Table 4: Forecasting results for mixed volatility approximation (14)a

DAX(AO/GAt)

AEX(AO/GAt)

CAC40(AO/GAt)

MIBTEL(AO/GAt)

FTSE100(AO/GAt)

0 0.363(0.095)

/ 0.090(0.067)

0.260(0.076)

/ 0.008(0.048)

0.610(0.146)

/ 0.123(0.077)

0.341(0.089)

/ 0.135(0.050)

0.024(0.094)

/ 0.045(0.069)

1 1.249(0.067)

/ 0.820(0.041)

1.209(0.063)

/ 0.820(0.032)

1.494(0.116)

/ 0.955(0.051)

1.241(0.071)

/ 0.727(0.032)

0.985(0.087)

/ 0.840(0.055)

R2 0.260/ 0.286 0.272/ 0.401 0.143/ 0.258 0.235/ 0.337 0.114/ 0.190

MSE 0.436/ 0.459 0.358/ 0.361 0.362/ 0.344 0.423/ 0.452 0.269/ 0.271

MedSE 0.151/ 0.210 0.099/ 0.172 0.119/ 0.180 0.141/ 0.186 0.106/ 0.142

MAPE 42.2/ 49.3 42.5/ 53.1 38.6/ 46.4 49.3/ 57.2 43.0/ 50.7a For every index there are two columns, labeled AO for the AO-GARCH model and GAt for the GAt-GARCHmodel. Values 0 and 1 are the regression coefficients (standard errors in parentheses).

Table 5: Forecasting results for extreme value volatility approximation (15)a

DAX(AO/GAt)

AEX(AO/GAt)

CAC40(AO/GAt)

MIBTEL(AO/GAt)

FTSE100(AO/GAt)

0 0.450(0.116)

/ 0.150(0.082)

0.336(0.093)

/ 0.013(0.058)

0.854(0.181)

/ 0.177(0.094)

0.450(0.104)

/ 0.193(0.058)

0.006(0.112)

/ 0.101(0.082)

1 1.650(0.081)

/ 1.081(0.050)

1.587(0.077)

/ 1.078(0.038)

2.026(0.143)

/ 1.284(0.063)

1.622(0.083)

/ 0.936(0.037)

1.355(0.103)

/ 1.102(0.065)

R2 0.292/ 0.321 0.300/ 0.444 0.168/ 0.297 0.279/ 0.388 0.149/ 0.225

MSE 0.882/ 0.688 0.688/ 0.439 0.759/ 0.523 0.696/ 0.492 0.529/ 0.394

MedSE 0.152/ 0.146 0.091/ 0.095 0.131/ 0.109 0.112/ 0.110 0.125/ 0.106

MAPE 30.3/ 30.5 29.8/ 30.3 28.4/ 27.3 32.1/ 32.5 31.3/ 29.9a For every index there are two columns, labeled AO for the AO-GARCH model and GAt for the GAt-GARCHmodel. Values 0 and 1 are the regression coefficients (standard errors in parentheses).

simple estimators vat , vet and v

mt proposed herein are seen to be competitive with v

xt from Garman

and Klass (1980). (We elected not to report the results based on the most efficient extreme value

estimator proposed by Garman and Klass (1980) because the one used here has a straightforward

interpretation, is related to the volatility estimator of Parkinson (1980), and leads to qualitatively

similar results.)

While interesting in themselves, the different volatility approximations are not our main goal,

but rather to compare the volatility forecasts of the competing GARCH models. The plots for

volatility forecasts for both models as well as the realized volatility, measured by the extreme

value estimator, vxt , are given in Figure 6. We see that the AO-GARCH model is not able

to predict higher fluctuations of the realized volatility. Its volatility forecasts are very smooth

and slowly varying. While this might be appealing from an applied point of view, in the sense


20/26


0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

DAX

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

AEX

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

CAC40

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

MIBTEL

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

FTSE100

Underlying grey line: realized volatiliy, measured by the extreme value approximation, vx

Dashed (black) line: volatility forecast of the AO-GARCH modelSolid (blue) line: volatility forecast of the GAt-GARCH model

Figure 6: Realized Volatility vs. Volatility Forecasts

that the frequency of portfolio weight adjustment (and the associated costs) are lessened, the

predictions are simply too poor and would not pass the tests required by the Basle Accord.

The behavior of the forecasts is what one would expect based on the results of the runs tests

in Section 3. In particular, as the extreme values are not randomly dispersed throughout the


21/26


data, they are informative for risk prediction, but the method removes all such extraordinary

peaks of the original series and labels them as outliers. In contrast, the volatility forecasts of the

GAt-GARCH model are able to capture the higher fluctuations of the volatility process and are,

thus, considerably more accurate.

With regard to the error criteria, the conclusion as to which model is best in predicting future

volatility depends on the chosen volatility approximation. For the standard approximation in

(11) as well as for the intra-day measure in (12), the inter-day measure in (13) and the mixed

volatility measure in (14), the AO-GARCH model yields smaller error criteria values for all

indices (with two exceptions for the intra and one exception for the mixed approximation). For

the extreme value approximation, vxt in (15), we draw opposite conclusions. The GAt-GARCH

model is preferred for all indices by the MSE, for 4 out of 5 indices by the MedSE, and for

2 out of 5 indices by the MAPE. The conclusion is, however, unequivocal when based on the

regression results: Irrespective of the chosen volatility approximation, the R2 value is highest for

the GAt-GARCH model for all indices.

Furthermore, the GAt-GARCH model not only leads to higher R2 values but also the number

of regression coefficients which are insignificantly different from their expected values, are higher

for all the volatility measures, with the exception of the standard approximation, vst . Based on the

most accurate volatility approximation, vxt , there are 4 (3) of the 0 (1) estimates insignificantly

different for the GAt-GARCH model, whereas for the AO-GARCH model, all parameter estimates

are found to be significantly in violation of the null hypothesis.

4.2 VaR-Forecasts

Besides predicting future volatility, GARCH models can be, and are, used to measure the down-

side risk inherent in a given financial position, i.e., the Value-at-Risk (VaR) of a financial position.

The VaR is defined to be the maximum amount of loss of a financial position that may occur

within a given number of days for a given probability . For our purposes, the VaR is given by

the cut-off point for which the next days predicted return will not fall with probability 1 .It is obtained by inverting the equation

Pr (rt+1 t+1()) = . (17)

When using a GARCH model for predicting the VaR, the calculation is done by using the condi-

tional density ft+1|t(rt+1|; rt, rt1, . . . ), where refers to the vector of estimated model param-eters, and f() is given by the assumed error distribution.

As for the volatility predictions above, 1000 one-step-ahead forecasts of the conditional re-

turn densities for the five indices are evaluated for both models under investigation. Again, an


22/26

5 CONCLUSIONS 20

estimation period of 250 days for both models is used and the model parameters are updated for

every forecast. To compare the accuracy of the VaR predictions, the empirical tail probabilities

=

1

1000

T1t=T1000

I

[rt+1 < t+1()] (18)are calculated for three different probability levels, = 0.01, = 0.025 and = 0.05, where I

is the indicator function. When based on the true distribution at time t + 1, the empirical tail

probability will coincide, in terms of the long-run average, with the given probability level. If the

observed probability is higher (less) than , then the model tends to underestimate (overestimate)

the risk for the index returns, i.e., the implied absolute -values tend to be too small (large). The

results for the different indices and probabilities are given in Table 6.

As expected from the results of forecasting volatility, the AO-GARCH model is not able

to measure the VaR correctly. By detecting and correcting the time series for anomalous

observations and, thus, removing the information contained in them, the AO-GARCH model

suffers when attempting to predict the tails of the distribution. For all three probability levels,

the AO-GARCH model underestimates the VaR of all time series by a factor of 1 .7 for the 5%

level and a factor of 3.3 for the 1% level. On the other hand, the GAt-GARCH model works quite

well, in agreement with the findings in Mittnik and Paolella (2000). For all probability levels, the

empirical tail probabilities are close to the given probability levels, i.e., the conditional modeling

of volatility, in conjunction with a more flexible return distribution, enhances the ability to draw

conclusions about the future tail probabilities of returns.

Table 6: Empirical tail probabilitiesa

5%(AO/GAt)

2.5%(AO/GAt)

1%(AO/GAt)

DAX 9.0 / 5.4 5.3 / 2.5 2.5 / 0.7

AEX 8.4 / 4.5 4.7 / 2.3 3.0 / 1.0

CAC40 8.3 / 5.2 5.5 / 2.7 3.2 / 0.7

MIBTEL 9.4 / 6.4 5.7 / 3.3 2.7 / 1.2FTSE100 8.8 / 5.2 6.1 / 2.4 3.3 / 1.0

a Observed tail probabilities from (17) multiplied by 100. For acorrectly specified model, we expect .

5 Conclusions

We compare two possible ways to accommodate the well-documented stylized fact of non-normal,

heteroskedastic financial returns for the purpose of risk prediction. One way involves systemati-


23/26

REFERENCES 21

cally removing aberrant values; the other is to use distributions which allow for explicit modeling

of the observed features of the data. The differences between both ways can be summarized

as follows. The AO-GARCH model is successful in identifying and correcting the original time

series for additive outliers. For all the time series investigated in this paper, the occurrence of the

outliers reveals strong systematic patterns, against the assumption that outliers occur at random.

Instead of labeling them as outliers, such aberrant values appear to be chronic in financial return

series and form an important part of the analysis and prediction of risk.

The volatility forecasts obtained from the two competing models are very different, although

it is hard to judge which model leads to better forecasts when judging is based on the standard

volatility approximation and standard error criteria. The results are far clearer if more efficient

volatility approximations are used, as proposed herein.

References

Andersen, T. G. and Bollerslev, T. (1998). Answering the Skeptics: Yes, Standard Volatility

Models Do Provide Accurate Forecasts. International Economic Review, 39(4), 885905.

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (2003). Modeling and Forecasting

Realized Volatility. Econometrica, 71(2), 579625.

Broda, S. A., Haas, M., Krause, J., Paolella, M. S., and Steude, S.-C. (2011). Stable Mixture

GARCH Models. Swiss Finance Institute Occasional Paper Series No. 11 39.

Broda, S. A. and Paolella, M. S. (2009). CHICAGO: A Fast and Accurate Method for Portfolio

Risk Calculation. Journal of Financial Econometrics, 1, 125.

Carnero, M. A., Pena, D., and Ruiz, E. (2001). Outliers and Conditional Autoregressive Het-

eroscedasticity in Time Series. Revista Estadistica, 53, 143213.

Charles, A. and Darne, O. (2005a). Outliers and GARCH Models in Daily Financial Data.Economics Letters, 86, 347352.

Charles, A. and Darne, O. (2005b). Relevance of Detecting Outliers in GARCH Models for

Modelling and Forecasting Financial Data. Finance: Revue de Lassociation Francaise de

Finance, 26(1), 3371.

Chen, C. and Liu, L.-M. (1993). Joint Estimation of Model Parameters and Outlier Effects in

Time Series. Journal of American Statistical Association, 88(421), 284297.


24/26

REFERENCES 22

Doganoglu, T., Hartz, C., and Mittnik, S. (2007). Portfolio Optimization when Risk Factors are

Conditionally Varying and Heavy Tailed. Computational Economics, 29(3-4), 333354.

Frances, P. H. and Ghijsels, H. (1999). Additive Outliers, GARCH and Forecasting Volatility.

International Journal of Forecasting, 15, 19.

Garman, M. B. and Klass, M. J. (1980). On the Estimation of Security Price Volatilities from

Historical Data. Journal of Business, 53(1), 6778.

Mittnik, S. and Paolella, M. S. (2000). Conditional Density and Value-at-Risk Prediction of Asian

Currency Exchange Rates. Journal of Forecasting, 19, 313333.

Paolella, M. S. (2006). Fundamental Probability: A Computational Approach. Chichester.

Paolella, M. S. (2007). Intermediate Probability: A Computational Approach. Chichester.

Paolella, M. S. and Polak, P. (2011). MARC-MARS: Modeling Asset Returns via Conditional

Multivariate Asymmetric Regime-Switching. Submitted.

Park, B.-J. (2002). An Outlier Robust GARCH Model and Forecasting Volatility of Exchange

Rate Returns. Journal of Forecasting, 21, 381393.

Parkinson, M. (1980). The Extreme Value Method for Estimating the Variance of the Rate of

Return. Journal of Business, 53(1), 6165.

Wiggins, J. B. (1991). Empirical Tests of the Bias and Efficiency of the Extreme-Value Variance

Estimator for Common Stocks. Journal of Business, 64(3), 417432.

Yang, M. (2011). Volatility feedback and risk premium in garch models with generalized hyper-

bolic distributions. Studies in Nonlinear Dynamics & Econometrics, 15(3).


25/26

c/o University of Geneva40 bd du Pont d'Arve

1211 Geneva 4Switzerland

T +41 22 3798471F +41 22 379 82 [email protected]

www.SwissFinanceInstitute.ch


26/26

c/o University of Geneva40 bd du Pont d'Arve

1211 Geneva 4Switzerland

T +41 22 379 84 71

Garch Vola Measurepdf

Documents

Transcript of Garch Vola Measurepdf