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A Better Asymmetric Model of ChangingVolatility in Stock and Exchange RateReturns: Trend-GARCHChristian Bauer aa University of Bayreuth , Bayreuth, GermanyPublished online: 17 Feb 2007.

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The European Journal of FinanceVol. 13, No. 1, 65–87, January 2007

A Better Asymmetric Model of ChangingVolatility in Stock and ExchangeRate Returns: Trend-GARCH

CHRISTIAN BAUERUniversity of Bayreuth, Bayreuth, Germany

ABSTRACT The impact of short run price trending on the conditional volatility is tested empirically. A newfamily of conditionally heteroscedastic models with a trend-dependent conditional variance equation: TheTrend-GARCH model is described. Modern microeconomic theory often suggests the connection betweenthe past behaviour of time series, the subsequent reaction of market individuals, and thereon changes inthe future characteristics of the time series. Results reveal important properties of these models, which areconsistent with stylized facts found in financial data sets. They can also be employed for model identification,estimation, and testing. The empirical analysis supports the existence of trend effects. The Trend-GARCHmodel proves to be superior to alternative models such as EGARCH, AGARCH, TGARCH OR GARCH-in-Mean in replicating the leverage effect in the conditional variance, in fitting the news impact curve andin fitting the volatility estimates from high frequency data. In addition, we show that the leverage effect isdependent on the current trend, i.e. it differentiates between bullish and bearish markets. Furthermore, trendeffects can account for a significant part of the long memory property of asset price volatilities.

KEY WORDS: GARCH, trend, volatility, news impact curve, leverage effect, persistence

1. Introduction

Innovations in economic time series analysis have mainly been driven by the wish to explainstylized facts that puzzled the older models. This paper aims to connect two well-known stylizedfacts, the asymmetry of returns and the importance of technical trading, in order to empiricallyanalyse the impact of short-run price trending on the conditional volatility. We develop a newclass of GARCH models, the Trend-GARCH model. The empirical results are consistent withstylized facts in financial time series.

The volatility feedback effect (Campell and Hetschel, 1992) has been used to explain thepresence of conditional leftskewedness observed in stock returns through an increase in futurevolatility following all kinds of news. However, markets amplify the impact of bad news butdampen the impact of good news on returns. This typically results in the conditional left-skewedness of returns. The news impact curve (NIC) (Engle and Ng, 1993) of such an asset priceseries is thus asymmetric. Several extensions of the GARCH model – e.g. EGARCH; AGARCH

Correspondence Address: Christian Bauer, University of Bayreuth, Chair of Economic Policy (VWL I), 95440 Bayreuth,Germany. Email: christian.bauer@uni-bayreuth.de

1351-847X Print/1466-4364 Online/07/010065–23 © 2007 Taylor & FrancisDOI: 10.1080/13518470600763752

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or TGARCH – catch this specific stylized fact of financial time series. The Trend-GARCH modelproves superior to these alternatives for two reasons. First, it fits better to the empirical data (seeSection 4), e.g. with respect to the asymmetric NIC and the significance of trend effects on futurevolatility. The main difference between the Trend-GARCH model and its alternatives is the vari-ability of the asymmetric effect. In the Trend-GARCH model the effect of an innovation in thetime series depends on its relation to the current trend, e.g. the same negative innovation dampensfuture volatility if it breaks a positive trend but increases volatility if it amplifies a negative trend.Secondly, the structure of the model corresponds to a microeconomic investment decision model.

Current strands of economic literature also include heterogeneous traders in microstructuremodels of asset markets. Typical types of traders are

• fundamentalists who react to fundamental analysis and chartists who base their decisions ontechnical analysis (Lux, 1995),

• noise traders (De Long et al., 1990) who react to market fads and disturbed information,• dealers (market makers) who coordinate the initial buying and selling orders, clear the market

and extract information from order flow and traders who make the initial orders.

These models account for well-known stylized facts of asset prices such as fat tails, bubbles,herd behaviour, etc. The overwhelming majority of these models use at least one type of traderwho follows a positive feedback trading rule. These traders may be approximated by a simpletrend-following trading rule. Bauer and Herz (2004) show in the context of an exchange ratemodel that the presence of such traders in the market increases the conditional volatility. Thesame argument is valid for prices of arbitrary assets.1

Trend-following behaviour only provides a simple and intuitive reason but no rigorous prooffor the relevance of pricing trends in the conditional volatility process. Alternatively, the trendterm may as well represent other aspects of pricing behaviour like the market microstructure orinstitutional settings. The Trend-GARCH model is superior to other GARCH alternatives and thetrend term somehow captures in the data properties that other models do not. This finding mayserve as a starting point for further discussion and research of this topic.

The remainder of the paper introduces the Trend-GARCH model and compares it to alternativeextensions of the GARCH model (Section 2.1). The news impact curve of this model class isthen compared with other GARCH models (Section 2.2). Section 3 presents the first empiricalevidence. Section 4 compares the capability of the different models to capture the asymmetry ofvolatility in empirical data, while Section 5 deals with the long memory properties of real data.Section 6 concludes.

2. GARCH Extensions and the Trend-GARCH Model

2.1 The Models

Following Engle (1982) we use the standard notation for the innovations of a discrete time real-valued conditional heteroscedastic stochastic process {εt },

εt = zt

√ht (1)

where Et−1(zt ) = 0 and VARt−1(zt ) = 1, and the conditional variance ht is a positive time-varyingand measurable function with respect to the information set It−1 available at time t − 1. Et−1(·)andVARt−1(·) denote the expectation and variance operator conditional on the information set It−1.

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A Model of Changing Volatility in Stock and Exchange Rate Returns 67

In the original ARCH(p) model, Engle (1982) defines the conditional variance ht as a linearfunction of the lagged squared innovations. To allow for more flexibility in modelling the variancestructure than the simple Markovian dependency up to lag p of the squared innovations in theARCH model, Bollerslev (1986) proposes the more general GARCH(p, q) model. The conditionalvariance ht in the GARCH model is a linear function of the lagged squared innovations and thelagged conditional volatilities

ht = ω + α(L)ε2t + β(L)ht (2)

L denotes the backshift operator and α(L) ≡ ∑p

i=1 αiLi and β(L) ≡ ∑q

i=1 βiLi . Given the usual

constraints, like finiteness of the fourth moment E(ε4t ) < ∞, the conditional variance process can

be rewritten as ARMA(max(p, q), q) in the squared innovations

[1 − α(L) − β(L)]ε2t = ω + [1 − β(L)](ε2

t − ht ) (3)

The process is stable and covariance stationary if all roots of 1 − α(L) − β(L) and β(L) lieoutside the unit circle.

This type of time series processes has been expanded in many ways. Typically economictime series can be characterized by a long memory property of the volatility process, i.e. thevolatility clusters are more compact than one would expect by standard GARCH processes.This observation leads typically to GARCH parameter estimates which imply a non-stationaryvolatility process or are very close to the stationarity border. To account for these problems Engleand Bollerslev (1986) proposed the IGARCH and Bollerslev et al. (1996) the FIGARCH models.In integrated GARCH(p, i, q) processes the volatility process is integrated of order i indicated by(1 − L)i in the corresponding equation. In fractionally integrated GARCH(p, d, q) processes thevolatility process is fractionally integrated of order d, i.e. the process of the conditional volatilityis defined by

ht = ω[1 − β(1)]−1 + {1 − [1 − β(L)]−1(1 − L)dφ(L)

}ε2t (4)

The power series representation of (1 − L)d gives anARCH(∞) characterization of (4) (Bollerslevet al., 1996).

A second type of generalization of the original model incorporates the asymmetry of economictime series. A stylized characteristic of financial markets is their asymmetric reaction to good andbad news. The negative correlation between past returns and future volatility is called leverageeffect. Engle and Ng (1993) investigate the leverage effect in stock markets quantitatively. Zakoian(1994) and Glosten et al. (1993) use a dummy variable for days with negative innovations tomodel the conditional volatility. This model is known as the TGARCH (threshold GARCH) orGJR model:

ht = ω + αε2t−1 + γ ε2

t−1dt−1 + βht−1, dt−1 ={

1 if εt−1 < 0

0 if εt−1 ≥ 0(5)

Alternative models which account for the leverage effect are the EGARCH model ofNelson (1991)

ln ht = ω + β ln(ht−1) + α

∣∣∣∣ εt−1√ht−1

∣∣∣∣ + γεt−1√ht−1

(6)

and the AGARCH model of Engle (1990)

ht = ω + α(εt−1 − γ )2 (7)

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The leverage effect accounts partly for the asymmetric behaviour of the conditional varianceprocess of economic time series. The different effects of good and bad news depend on theireffects on the trends in the markets.2

Another family of extensions of the GARCH model account for the impact of the actualvolatility on the expected future mean. The ARCH-M models were introduced by Domowitz andHakkio (1985) and Engle et al. (1987). The generalized models are denoted GARCH-in-mean orGARCH-M models. While leaving the equation for the conditional variance unchanged, the meanequation is supplemented by a linear function of the conditional variance or standard deviation.For a typical GARCH-M model with ARMA mean equation this yields the set of equations

ht = ω + α(L)ε2t + β(L)ht (8a)

xt = ωm + αm(L)ε2t + βm(L)xt + γf (h2

t ) (8b)

where f (x) is x or√

x. The in-mean concept may be mixed with other GARCH extensions (seethe GRJ-GARCH-M in Lanne and Saikkonen, 2005). The empirical literature on the influenceof the conditional variance on the mean process is voluminous, but not unanimous. Lanne andSaikkonen (2005) find a positive relation between risk and return within a GARCH-M setup notuntil applying a z-distribution for the conditional distribution of the innovations. The quantitativeeffects are small compared to the trend effects presented here.3

Asset prices in general, such as stock prices, exchange rates or derivatives, are the result of supplyand demand, i.e. of the behaviour of traders and investors. A number of studies (e.g. Menkhoff,1998) affirm the influence of technical analysis on the decisions of traders. Technical analysisusually relies on the interpretation of some sort of trend measure.4 Theoretical models like De Longet al. (1990) or Bauer and Herz (2004) show that technical analysis is rational in an heterogeneousenvironment and may influence the variance of the return process.

The trend-following character of technical trading results in more technical-induced activityand volatility if trends are large.5 Thus the conditional volatility is not only affected by the signof the innovation, but also by its effect on a trend. If an innovation εt amplifies the currentlyobserved trend, market activity and volatility increase. If an innovation dampens the currentlyobserved trend, market activity and volatility decrease. Since trends can be positive or negative,the influence of the sign of the innovation depends on the current trend.

Bauer and Herz (2004) suggest analysing a new variant of GARCH models which accountfor the effects of trends to volatility. The Trend-GARCH(s, p, q) model is characterized by thevolatility process6

ht = ω + β(L)ht + α(L)ε2t + γ

(1

s

s∑i=1

εt−i

)2

(9)

The trend component is simply the average increment of the last s periods. This trend measureis typically used to proxy trend estimates in technical trading models (compare Lux andMarchesi, 2000).7 Alternatively, an exponential trend can be used. The exponential-Trend-GARCH(s, p, q) with parameter λ model is characterized by the volatility process

ht = ω + β(L)ht + α(L)ε2t + γ

(1 − λs

1 − λ

s∑i=1

λi−1εt−i

)2

(10)

In analogy to the TGARCH model asymmetric effects for positive and negative trendsmay be incorporated, too. Equations (11) formulate the volatility equation of an asymmetric

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A Model of Changing Volatility in Stock and Exchange Rate Returns 69

Trend-GARCH model. The current trend is termed et and the indicator variable for a negativetrend is dt . Thus the volatility impact of a positive trend is measured by γ , while the impact of anegative trend is given by γ + γ .

ht = ω + β(L)ht + α(L)ε2t + γ e2

t + γ dt e2t (11)

et = 1

s

s∑i=1

εt−i , dt ={

1 if et < 0

0 if et ≥ 0

We want to stress that the simple Trend-GARCH model already implies asymmetric effects ofnews on volatility because the impact is dependent on the trend. In the asymmetric Trend-GARCHmodel, the impact of news is not only dependent on its influence on the trend but also on the signof the trend, i.e. whether the market is bullish or bearish. We explain this interrelation in greaterdetail in Section 2.2.

The Trend-GARCH model supplements the simple GARCH model by just one additionalparameter and thus has the same number of parameters as EGARCH, AGARCH, TGARCH,and FIGARCH models. Since it accounts for both asymmetry and persistence in volatility, thisparametrization may be viewed as parsimonious.

The formulation of the Trend-GARCH volatility process given in Equation (9) shows that thetrend is more than an external variable determining the conditional variance. As the trend dependson the innovations just as the variance does and both influence the size of future innovations, theconditions determining the stationarity properties of the time series differ from the conditions forordinary GARCH models. The task of a rigorous analysis of these stationarity conditions is leftto future research. Simulations based on coefficients estimated from the economic time seriesanalysed in this article behave stable and may be assumed to have a stationary volatility process.

The ARCH part of the volatility Equation (9) cannot represent the information in the trend.Comparing the ARCH term α(L)ε2

t = ∑p

i=1 αiε2t−i and the trend term γ

(1/s

∑si=1 εt−i

)2 =γ

1/2s

∑si=1 ε2

t−i + 2γ1/2s

∑si<j εt−iεt−j , we see that the cross-products of the past innovations

appear to bring the ‘new’ input. While in the ARCH part is increased by any innovation (ε2t � 0),

innovations with alternative signs may cancel out in the trend term due to the cross-products. Forfour reasons, we prefer not to separate the cross-products from the trend part. First, the ARCH partof the GARCH model is able to account for trends in the volatility process, but not for volatilityeffects of trends in the time series itself. Secondly, the trend has a simple and intuitive meaning.Thirdly, in analogy of mean and variance of a stochastic sample, the squared sum and the sum ofsquares may be viewed as containing quite independent information. For example, the sample 1,0, 1 has a squared sum of 4 and a sum of squares of 2, while the sample 1, 0, −1 has the samesum of squares but a squared sum of 0. Finally, the orders p and s may be different.

Furthermore, the trend component cannot be replaced by switching to a larger time scale. Forexample a Trend-GARCH(20, p, q) model (20 working days approximate one month) is notequivalent to a GARCH(p, q) model on monthly data. The Trend-GARCH(20, p, q) model ondaily data measures the influence of the monthly trend on daily conditional volatility while theGARCH(p, q) model on monthly data measures the monthly conditional volatility.

Finally, the differences of the GARCH-M models to the Trend-GARCH models will be summa-rized. Trend-GARCH models differ significantly from GARCH-M models in two respects. First,while both classes of models constitute a relation between the first and second moment of theinnovations, they are exact opposites with respect to the direction of this relation. Trend-GARCHmodels are based on the influence of the empirical (observed) past mean of the innovations on the

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future volatility, while GARCH-M models represent an impact of the actual volatility on theexpected future mean. Secondly, the relation of trend and conditional variance/standard devia-tion in GARCH-M models is linear. However, the empirical data suggests a U-shaped relation,i.e. a strong trend of either sign causes a rise in future volatility (see Section 3).8 The leverageeffect merely accounts for the asymmetry of this dependence. This behaviour cannot be modelledwithin a GARCH-M setup, since the trend linearly depends on a function of the variance.

As we will show in the remaining empirical sections, the Trend-GARCH model is superiorin explaining and representing the empirical data, i.e. the U-shaped relation of current trend andfuture volatility and the fit of the NIC. It also partially accounts for the long memory property ofthe volatility process in stock returns in a (Trend-) FIGARCH setup.

2.2 The News Impact Curve

The news impact curve (NIC) relates today’s returns to tomorrows volatility. Following Engle andNg (1993):

The news impact curve is the functional relationship between conditional variance at timet and the shock term (error term) at time t − 1, holding constant the information dated

Figure 1. NICs of alternative GARCH(1, 1) models: GARCH(1, 1) and GARCH-M (solid black line),AGARCH (circles), EGARCH (dashed grey line), TGARCH (squares)

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A Model of Changing Volatility in Stock and Exchange Rate Returns 71

t − 2 and earlier, and with all lagged conditional variance evaluated at the level of theunconditional variance.

It is thus the appropriate measure for the model comparison in this paper. Engle and Ng (1993)defined it as the expected conditional variance of the next period conditional on the current shock εt

E(ht+1 | εt ) (12)

For the classicalARCH and GARCH models the NIC is a parabola with minimum at εt = 0. TheGARCH-M model shows the same NIC as the corresponding basic not-in-mean model, since thisextension only influences the mean equation. The NICs of the asymmetric model of Engle (1990)(see Equation (5)) and the EGARCH model of Nelson (1991) have their minimum at εt = 0,too. However, in these cases the NIC is not symmetric around 0, but skewed. Typically, negativenews drive volatility up more than good news. In these models, any news today drive up volatilitytomorrow.

The NIC of the AGARCH model of Engle (1990) is not symmetric around 0, either. In thismodel the NIC is a right-shifted parabola. This potentially suggests slightly positive news as arequirement for the markets to remain as calm as possible. ‘No news’ in this model implies ahigher volatility than in tranquil markets.

In contrast to the former models, where the location of the NIC is uniquely determined throughthe model parameters, the Trend-GARCH models yield a NIC which depends on the current trend.

Figure 2. NIC of the Trend-Garch(1, 1) model for different measured trends: no trend (coincides with simpleGARCH model) (solid curve), negative trend (squares), positive trend (circles)

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Based on Equation (18) a simple calculation shows that for Trend-GARCH models

E(ht+1 |εt ) = ω + β(L)ht+1 + α(L)ε2t + γ

(1

s

s∑i=1

εt+1−i

)2

= ω + β(L)ht+1 + α1ε2t + α∗(L)ε2

t + γ1

s2

(εt +

s−1∑i=1

εt−i

)2

(13)

Table 1. Parameters and formulas of the news impact curve of the alternative GARCH models

Model NIC Plotted formula

GARCH(1, 1) andGARCH-M

E(ht+1|εt ) = αε2t + ω + βht y = x2 + 1

AGARCH E(ht+1|εt ) = α(εt − a)2 + ω + βht , b > 0 y =(

x − 1

2

)2

+ 1

TGARCH E(ht+1|εt ) = αε2t + aε2

t Iεt<0 + ω + βht , b > 0 y = x2 + 1

2x2Ix<0 + 1

EGARCH E(ht+1|εt ) = exp(ω) exp

(α

εt

ht+ a

|εt |ht

)h

βt y = exp

(x + 1

2|x|

)

Trend-GARCH:no trend

E(ht+1|εt ) = αε2t + ω + βht + γ

1

s2

(εt +

s−1∑i=1

εt−i

)2

y = x2 + 1

Trend-GARCH:positive trend

E(ht+1|εt ) = αε2t + ω + βht + γ

1

s2

(εt +

s−1∑i=1

εt−i

)2

y = x2 + x + 1

Trend-GARCH:negative trend

E(ht+1|εt ) = αε2t + ω + βht + γ

1

s2

(εt +

s−1∑i=1

εt−i

)2

y = x2 − x + 1

Model Parameter values

GARCH(1, 1) and GARCH-M β = 0.9, ht = 1, ω = 0.1, α = 1AGARCH β = 0.9, ht = 1, ω = 0.1, α = 1, a = 0.5TGARCH β = 0.9, ht = 1, ω = 0.1, α = 1, a = 0.5EGARCH β = 0.9, ht = 1, ω = 0.1, α = −0.1, a = 0.5

Trend-GARCH: no trend α = 0.5, β = 0.4, γ1/2s = 0.5,

s−1∑i=1

γ εt−i = 0

Trend-GARCH: positive trend α = 0.5, β = 0.4, γ1/2s = 0.5,

s−1∑i=1

γ εt−i = 1

Trend-GARCH: negative trend α = 0.5, β = 0.4, γ1/2s = 0.5,

s−1∑i=1

γ εt−i = −1

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A Model of Changing Volatility in Stock and Exchange Rate Returns 73

where α∗(L) = α(L) − α1L denote the respective polynomial without the linear term. For aTrend-GARCH(s, 1, 1) model, Equation (13) simplifies to

E(ht+1|εt ) = ω + βht + αε2t + γ

1

s2

(εt +

s−1∑i=1

εt−i

)2

(14)

= ω + βht +(

α + γ1

s2

)ε2t + 2γ

s2εt

s−1∑i=1

εt−i + γ1

s2

(s−1∑i=1

εt−i

)2

(15)

Obviously, we see from Equation (15) the impact of news term εt on the future volatility isasymmetric and depends on the term

∑s−1i=1 εt−i . The interpretation of this finding, which seems to

contradict the symmetric parametrization of the model, is easier using Equation (14). The effect

of news is symmetric given its relation to the trend. If the trend is positive(∑s−1

i=1 εt−i > 0)

then slightly negative news (εt < 0), which slow the trend down, calm the market. If the trend isnegative then slightly positive news generate a relatively tranquil market.

Also, the minimum level of future volatility depends on the size of the current trend. Strongtrends have two implications on future volatility. First, the stronger the trend, the higher the futurevolatility, i.e. the minimum of the NIC increases with the absolute size of the trend. Secondly,larger trends require stronger signals to be weakened. To tranquilize the market an innovation hasto be the larger the stronger the trend.

The economic intuition behind this analysis is based on the existence of a positive feedbacktrading rule of some of the market participants. The positive feedback trading rule implies a trend-following behaviour. The larger the trend, the more traders react. Since traders may take short andlong positions, or as an exchange rate trend always is a positive trend for one and a negative trendfor the other country, the influence of the trend is comparatively symmetric. The trend model ismore flexible in reflecting the actual market’s situation than the pure leverage models.

Figures 1 and 2 give an overview of the NIC of the selection of models presented here.A GARCH(1,1) specification is used for each alternative model. Figure 2 shows theTrend-GARCHNICs while Figure 1 displays the alternative models. The influence of the current volatility ht onthe NIC is like an additive constant (multiplicative for the EGARCH model), i.e. different valuesmay only shift the NIC on the vertical axis.9 Table 1 contains the corresponding parameter valuesand formulas for the NICs and the plots for the alternative models and the Trend-GARCH-modelwith null, positive and negative trend. For the asymmetric Trend-GARCH model (11) the locationof the two parabolas for given trends would not be axis-symmetric.

3. Empirical Evidence

The empirical analysis is separated into four parts. In a first step, the Trend-GARCH model isfitted to the data. The estimates of the trend parameter are highly significant for all but one assetprice series. In the second step, the dependence of trend and variance in the data is revealed by akernel regression. Afterwards, we have a look at estimations of the asymmetric Trend-GARCHmodel.

In the following Section 4, all GARCH models are estimated on various asset prices to analysetheir ability to account for asymmetry. Here the Trend-GARCH model has the highest explanatoryvalue for the conditional volatility. Finally in Section 5, the analyses of a Trend-FIGARCH modelshows that the trend accounts for a part of the long memory property of the volatility process inthe analysed time series.

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The data cover daily opening stock prices of six major US companies in the DJII (HomeDepot,IBM, Coca Cola, Johnson & Johnson, Procter & Gamble, and Exxon), four stock indices (Dax,DJII, Hang Seng, and Nikkei), and two US Dollar exchange rates (to the Hungarian Forint andthe Euro).10 Data run from 01/01/1990 to 15/07/2004.11

For all estimations, we use daily returns, i.e. a trivial mean equation. Based on the literature onmodel comparisons (see Bollerslev et al., 1992 or Carnero et al., 2004), we restricted our modelclasses to GARCH(1, 1) extensions. In an appendix, which is available from the author uponrequest, we provide extensive empirical support for our choice of the order of the GARCH model,the order of the ARMA mean equation, and the length s of the trend component. Based on BICstatistics the GARCH(1,1) with trivial mean equation is suited best on average for our diversesample.

3.1 Trend-GARCH Estimation

As a first step in the empirical analysis, the Trend-GARCH model is fitted to the data. Accordingto the standard assumption of perfect markets, the mean equation of the GARCH model resemblesa random walk for the stock prize series or a trivial equation for the series of returns. For ourTrend-GARCH(s, 1, 1) model, the variance equation is amended by a squared trend term.

ht = ω + βht−1 + αε2t−1 + γ

(1

s

s∑i=1

εt−i

)2

= ω + βht−1 + αε2t−1 + γ (trendt−1)

2 (16)

The trend is estimated as the increase of the asset price in the past five observations, i.e. aTrend-GARCH(5,1,1) is estimated.12 The estimation is performed with the GARCH routineimplemented in the SPLUS program package. The trend is calculated a priori and treated asan exogenous variable.

Table 2 gives the results of the estimates of the squared trend parameter γ in the varianceequation.

The estimates of the trend parameter are positive for all series. They are highly significant forall but the Hungarian Forint series.

Table 2. Estimates of the parameter γ of the squared trend in the variance equation

Time series Estimate of γ Std. error t-value Pr(>|t |)HomeDepot 1.34 0.06 23.03 0.00IBM 1.42 0.08 17.98 0.00Coca Cola 2.01 0.06 33.42 0.00Johnson & Johnson 1.07 0.08 13.02 0.00Procter & Gamble 1.07 0.06 16.89 0.00Exxon 1.27 0.06 19.94 0.00Dax 1.02 0.05 18.57 0.00DJII 0.70 0.06 11.66 0.00Hang Seng 1.33 0.05 28.79 0.00Nikkei 1.85 0.07 26.43 0.00HUF/USD 0.29 0.18 1.57 0.12EUR/USD 1.55 0.10 14.74 0.00

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Table 3. Estimates of the parameter γ of the squared trend in the variance equationfrom simulated data

Simulated data from model Estimate of γ Std. error t-value Pr(>|t |)GARCH(1, 1) 0.015 0.014 1.05 0.15AGARCH 0.015 0.018 0.84 0.20TGARCH 0.0087 0.018 0.49 0.31EGARCH 0.019 0.020 0.94 0.17GARCH-M 0.03 0.028 1.3 0.10

Table 3 gives the results of the estimates of the squared trend parameter in the variance equationaccording to Equation (16) from simulated data. The parameters for the simulation are esti-mated from the original DJII price series. Data is simulated for each of the alternative models:GARCH(1,1), AGARCH(1,1), TGARCH(1,1), EGARCH(1,1), and GARCH-M(1,1). The esti-mates of the trend parameters are not significant for any of the models. Therefore these modelsdo not reveal the volatility structure of the true data, since they don’t catch the trend dependenceof the volatility process.

3.2 Kernel Regressions

Another model free regression shows the relationship between the trend and the future conditionalvariance. Trend and conditional variance are estimated by the mean and the empirical variance ofthe first differences of the log asset prices within windows of length 5, i.e.

trendt = 1

s

s∑i=0

εt−i

ht =s∑

i=0

ε2t−i −

(s∑

i=0

εt−i

)2

with s = 5 and εt = 100 (pt − pt−1). The first difference of the logarithmic asset price is upscaledto obtain percentage returns.

To resemble the time structure of the model, data points (trendt , ht+s) are analysed, i.e. theimpact of the current trend on future volatility. Now a kernel regression with a Gauss kernel isperformed on these data points and simultaneous confidence bands around the kernel regressionsare constructed using a bootstrap type estimator (see figure 3). At the given significance level of0.05 the probability that the estimated relation does not leave the confidence band at any point is0.95. Since the windows overlap, the estimates, which are close in time, are correlated (α-mixing).The kernel regression is justified by the amount of data points, since we can asymptotically neglectthis type of dependency between the data points. Also, for the estimation of the confidence bandswe can asymptotically neglect these dependencies using the bootstrap algorithm of Neumann andPolzehl (1998).

Bauer and Herz (2004) show on several examples that regressions on simulated GARCH andFIGARCH data do not result in U-shaped dependencies between current trend and future volatility,which appears in the regressions on the original data. The same holds true for simulations on theGARCH extensions analysed here.13

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Figure 3. Kernel regressions of past trend on future conditional volatility and simultaneous confidence bands at 5% level

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3.3 The Asymmetric Trend-GARCH Model

In this subsection, we take a short look at the asymmetric Trend-GARCH model as parametrizedin Equation (11). This model differentiates between bullish and bearish markets, i.e. the effect ofnews does not only depend on its trend-breaking or trend-strengthening effects, but also whetherthe trend which it breaks or strengthens is positive or negative.

Table 4 displays the results of the estimations. Except for Exxon and the DJII the trend effectson volatility are significantly different in bullish and bearish markets.

From Table 4, we may derive the following stylized facts.

1. Positive trends (impact γ ) increase conditional volatility significantly for each time series.2. Negative trends (impact γ + γ ) increase conditional volatility significantly more than positive

trends for individual stocks.3. Negative trends (impact γ + γ ) increase conditional volatility significantly less than positive

trends for stock indices. For the DJII there is no significant difference. Negative trends do notsignificantly raise volatility for the DAX and the Nikkei.

3.4 Comparison with Volatility Estimates from Intraday Data

Andersen et al. (2003) view daily data as the low frequency sample of a stochastic process thatruns at much higher frequencies or even continuously. They show that the average of the squaredinnovations may serve as a measure for the volatility of the daily innovations if the underlying highfrequency process has a GARCH structure. Their approach yields volatility predictions highlysuperior to a large variation of other volatility estimation approaches – including RiskMetrics,GARCH-, and FIGARCH-models. This volatility estimation approach therefore has become anatural benchmark in volatility estimation models.

Andersen et al. (2003) provide their estimated volatility time series online, henceforth ABDL-volatility series. We will use these estimates of the USDollar/DeutschMark exchange rate as abenchmark for the models being compared, i.e. we perform a linear regression of the volatilityestimates from our models on the ABDL-volatility series. Since the volatility estimates are strictly

Table 4. Estimates of coefficients for positive and negative trend for theasymmetric Trend-GARCH model

γ p-value γ p-value

HomeDepot 0.62 0 7.31 8.9E − 09IBM 0.80 0 13.43 0Coca Cola 0.07 0 1.27 0Johnson & Johnson 2.15 0 2.70 2.8E − 09Procter & Gamble 0.06 0 1.01 0Exxon 0.20 0 0.02 0.23Dax 1.20 0 −1.20 0DJII 0.78 0 −0.01 0.45Hang Seng 2.74 0 −0.90 8.5E − 06Nikkei 6.93 0 −6.56 0HUF/USD 0.01 1.4E − 06 12.25 0EUR/USD 0.52 2.2E − 05 −0.31 6.2E − 06

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Table 5. Regression of estimated volatilites on ABDL-volatility series from intraday data

Estimate of Estimate ofModel intercept Std. error Pr(>|t |) slope coefficient Std. error Pr(>|t − 1|) r2

GARCH(1,1) −0.41 0.30 0.17 0.93 0.029 0.017 0.30AGARCH 0.15 0.31 0.63 0.98 0.029 0.58 0.31TGARCH 1.5 0.34 0.00 1.10 0.032 0.002 0.32EGARCH −0.34 0.31 0.27 0.93 0.029 0.034 0.30Trend-GARCH 1.20 0.27 0.00 1.04 0.026 0.12 0.42

positive and piled around low values, we use the logs of the volatilities for the regression. SinceAndersen et al. (2003) use foreign exchange data from 1986–1999, we adopt this period for theestimations in this subsection.

Table 5 presents the results. The Trend-GARCH model explains volatility – as measured bythe ABDL-volatility series – significantly better than any of the other models.

The estimated slope coefficient for all but the AGARCH and the Trend-GARCH model issignificantly different from unity. Furthermore only the AGARCH and the EGARCH modelsyield estimates of the constant term which don’t significantly differ from 0. Finally, the Trend-GARCH model yields a significantly higher r2 value than any of the other models. These resultssuggest applying the Trend-GARCH concept to intraday data as a topic of future research (compareGiot, 2005 for a comparison of several GARCH models on intraday data). The stylized fact thatintraday traders typically have chartistic attitudes supports extending the Trend-GARCH approachto intraday data from the theoretical grounding of the model.

4. Asymmetry of Volatility

The news impact curves of the five GARCH models presented in Section 2.2 are estimated on theempirical data. As the AGARCH, EGARCH, and TGARCH models were designed to replicatethe leverage effect, this comparison presents a contest for the ability to catch the asymmetry of thevolatility series. We noted above, that the asymmetric effect in the Trend-GARCH model dependson the size and sign of the current trend.

The NIC is directly connected to the conditional variance of the time series. This leads to thefollowing problem when comparing alternative models. The true conditional variance of the assetprice p is unknown. The estimation of each of the models – as a side effect – yields an estimation ofthe conditional variance, too. These estimated conditional variances differ from model to model.Estimates of the conditional volatility resulting from one of the GARCH-models by definitionfulfil the variance equation of the respective model. Using one of these estimates for the modelcomparison would bias the outcome in favour of the model whose variance estimate was used.In order to circumvent this complication, a model-free estimation of the variance is used: theempirical variance of the past s innovations εt = pt − pt−1 with s = 3.

ht =s∑

i=0

ε2t−i −

(s∑

i=0

εt−i

)2

(17)

To provide a second test, which does not rely on an estimate of the volatility, we also test whichof the models implies residuals most close to normal.14

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Table 6. OLS regression equations for the NIC of all five models

Model NIC

GARCH(1,1) and GARCH-M ε2t+1 ∼ ε2

t + ht

AGARCH ε2t+1 ∼ ε2

t + εt + ht

TGARCH ε2t+1 ∼ ε2

t + Iεt<0ε2t + ht

EGARCH ln ε2t+1 ∼

∣∣∣∣∣∣∣εt√ht

∣∣∣∣∣∣∣ + εt√ht

+ ln ht

Trend-GARCH ε2t+1 ∼ ε2

t + ht + trend2t

When using the proxy (17) for the conditional variance as the dependent variable in the regres-sions for the NIC, another problem emerges. The windows from which the dependent variableht+1 and the independent ht are calculated would overlap. In order to estimate the NIC of the fivemodels the future conditional variance is therefore proxied by the square of the next innovationε2t+1. This is in line with the common interpretation of the NIC that describes the influence of the

actual innovation on the size of the future innovations. Table 6 gives the five regression equations.For the Trend-GARCH model the trend is estimated as the average increase of the asset price

over the past three observations. For the EGARCH model, the estimated log variance has beenrecalculated to assure comparability with the other models.

The regression is performed by OLS. Table 7 displays the r2 of each estimate.Clearly the Trend-GARCH model is superior. Each of the four GARCH modifications has the

same number of estimated parameters (3). The explanatory value of the Trend-GARCH modelis larger than that of any of the other models for each time series, and this better performancein not due to a higher number of parameters. As the conventional GARCH(1,1) model has oneparameter less, the advantage of the Trend-GARCH model over the simple GARCH model needsto be treated separately, as we have done in Section 3.15

For all estimates the signs of the coefficients are either as expected or the estimates are insignif-icant. The trend component in the Trend-GARCH model is positive and highly significant for all12 time series.16

Table 7. r2 values of the OLS regressions for the NIC of all five models

GARCH(1, 1)and GARCH-M AGARCH TGARCH EGARCH Trend-GARCH

HomeDepot 0.53 0.53 0.53 0.43 0.59IBM 0.51 0.51 0.51 0.37 0.55Coca Cola 0.44 0.44 0.44 0.39 0.57Johnson & Johnson 0.51 0.51 0.51 0.33 0.53Procter & Gamble 0.61 0.61 0.61 0.44 0.64Exxon 0.53 0.53 0.53 0.43 0.58Dax 0.58 0.58 0.58 0.44 0.62DJII 0.59 0.59 0.59 0.43 0.61Hang Seng 0.57 0.57 0.57 0.41 0.65Nikkei 0.54 0.54 0.54 0.41 0.62HUF/USD 0.52 0.52 0.52 0.26 0.52EUR/USD 0.42 0.42 0.42 0.42 0.50

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Table 8. Ratios of Jarque–Bera statistics of the alternative models to the respective Jarque–Bera statisticsof the Trend-GARCH model

GARCH(1, 1) AGARCH (1, 1) TGARCH(1, 1) EGARCH (1, 1)

HomeDepot 1700 2000 2000 1800IBM 4300 7400 6700 1300Coca Cola 1900 2100 1600 2000Johnson & Johnson 220 220 280 220Procter & Gamble 86 84 112 70Exxon 1.2 × 106 8.1 × 105 2.0 × 106 8.7 × 105

Dax 1.12 0.98 0.64 0.70DJII 23 20 17 18Hang Seng 160 190 130 130Nikkei 160 220 170 190HUF/USD 1.03 1.26 1.20 0.83EUR/USD 1.24 1.23 1.48 1.20

To provide a second test, which does not rely on a criticizable estimate of the volatility, we alsotest which of the models implies residuals most close to normal. Table 8 displays the ratios of theJarque–Bera statistics for tests of normality of the alternative models to the respective Jarque–Berastatistics of the Trend-GARCH model. Values less than 1 indicate that the alternative model yieldsresiduals more close to normal than the Trend-GARCH model. The simple Trend-GARCH modelas parametrized in Equation (9) has the same number of parameters as the EGARCH, AGARCH,TGARCH models.

From values larger than 1, we see that the residuals of the Trend-GARCH model are far closerto normal than the residuals of the other models for all individual stocks and most stock indices,with exception of the estimates on the DAX time series. For exchange rates, there still is a smalladvantage in favour of the Trend-GARCH model, albeit not as significant.

5. Long Memory Property of Volatility

Finally, we want to address the aspect of persistence of shocks to the volatility process. Forall covariance stationary variants of the GARCH model the impact of a shock to the volatilityprocess decays exponentially. This exponential decay rate, however, is too fast compared toestimates from financial data (see the introduction of Bollerslev et al., 1996 for an extensiveargumentation and literature survey). For this reason, GARCH models tend to produce integratedlike estimates. Engle and Bollerslev (1986) formulated the IGARCH model to account for theseobservations. However, the infinite persistence of shocks and the extreme degree of dependenceon the initial conditions as implied by IGARCH models also contradicts empirical observationsas volatility tends to cluster, i.e. to stick to a high level for a certain period only and then to returnto a lower level. Bollerslev et al. (1996) introduce the class of fractionally integrated GARCHmodels (FIGARCH) to dodge this ‘knife-edge’distinction between exponential decay and infinitepersistence. FIGARCH models show a hyperbolic decay rate, i.e. long but not infinite memory.17

In the case of GARCH(1,1) models as formulated in Equation (2), there is a simple measure forthe persistence p = α + β. For p < 1 the process is covariance stationary. An estimate of p = 1indicates integration. Table 9 gives the estimates of p for the GARCH(1, 1), the TGARCH(1, 1),the Trend-GARCH(3, 1, 1), and the FIGARCH(1, d, 1) model.18

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Table 9. Estimates of p = α + β

GARCH(1, 1) TGARCH(1, 1) Trend-GARCH(3, 1, 1) FIGARCH(1, d, 1)

HomeDepot 0.72 0.65 0.73 −0.33IBM 1.01 1.04 0.58 0.99Coca Cola −0.86 −0.02 0.82 −0.09Johnson & Johnson 0.83 0.66 0.86 0.63Procter & Gamble 0.90 0.94 0.94 0.44Exxon 1.28 0.87 0.66 1.00Dax 0.99 0.96 0.99 0.73DJII 0.99 0.94 0.84 0.89Hang Seng 0.98 0.94 0.53 0.79Nikkei 0.89 0.83 0.85 0.75HUF/USD 1.00 0.96 0.99 0.73EUR/USD 0.99 0.99 0.97 0.86

As already noted above, the estimation of GARCH(1,1) models on financial data sets typicallyyields values of p close to 1 (see Carnero et al., 2004 or Bollerslev et al., 1996). We also seethat the results from the estimates on individual stocks are miscellaneous, while the estimates onstock indices and exchange rates more closely represent the stylized facts of p close to but smallerthan unity. The estimates of the TGARCH(1, 1) and the FIGARCH(1, d, 1) models are lower forthe stock indices and exchange rates and mixed for the individual stocks data sets. For the stockindices and exchange rates, the FIGARCH(1, d, 1) estimates of p are significantly below unity andthe estimated coefficient d of fractional integration lies well between 0 and 1 (see also Table 10).Only the Trend-GARCH model produces stable estimates for each of the series. The estimates ofp are significantly below unity for all individual stocks and indices with exception of the DAX.

The estimates of p = α + β must not be interpreted as appropriate persistence measures forany of the GARCH extensions. The estimates for the Trend-GARCH model below unity do notnecessarily imply covariance stationarity, as part of the shock to the volatility may be transferredto steeper trends and thus in turn increase subsequent volatility. The trend component involvesseveral lags and thus the inference with lagged volatilities and innovations is complex. The deter-mination of an appropriate persistence measure is like the theoretic study of the properties ofthe Trend-GARCH model subject of future research. The sticking point is that the volatilityEquation (9) cannot (at least not without transformations) be viewed as an ARMA process insquared innovations as a linear function of the unsquared innovations enters via the trend.

Thus the interpretation of Table 9 has to be very careful. Yet, two conclusions can be drawnfrom the results. First, GARCH models and extensions like the TGARCH model only insuffi-ciently catch the persistence of empirical volatility processes and Trend-GARCH models mightbe able to account for a part of this phenomenon. Secondly, for individual stocks all traditionalvariants of the GARCH family cannot represent the type of persistence and intertemporal depen-dence of the empirical time series, since the estimated level of integration tends to be 1, althoughthe implications of such estimates also contradict some features of the empirical data sets. TheTrend-GARCH model seems to be able to cope with this problem.

Finally, we compare the estimates from FIGARCH and Trend-FIGARCH models on the sampletime series. The dependence of the future conditional volatility on the current observed trend seemsto catch a part of the long memory property of the volatility process in the analysed time series,which is represented in the simple FIGARCH models. For each of the Trend-FIGARCH models

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Table 10. Estimates of the parameter of fractional integration in FIGARCH and Trend-FIGARCH models and estimate of the squared trend parameterin the variance equation

d2: estimate Estimated1: estimate of of d in of γ ind in FIGARCH Std. error Trend-FIGARCH Std. error p-value Trend-FIGARCH Std. error

Time series model of d1 model of d2 d1 − d2 of d1 − d2 model of d2

HomeDepot 1.00 0.02 0.04 0.01 0.96 2.8 × 10−4 2.28 0.13IBM 0.88 0.03 0.04 0.01 0.84 5.4 × 10−4 2.50 0.14Coca Cola 1.00 0.03 0.67 0.03 0.33 2.2 × 10−4 1.83 0.11Johnson & Johnson 1.00 0.06 0.39 0.03 0.61 1.5 × 10−3 1.28 0.06Procter & Gamble 1.00 0.02 0.01 0.003 0.99 8.0 × 10−6 2.29 0.13Exxon 1.00 0.02 0.02 0.004 0.98 1.4 × 10−5 1.92 0.14Dax 0.46 0.03 0.16 0.01 0.29 1.2 × 10−4 1.65 0.14DJII 0.43 0.05 0.14 0.01 0.30 2.0 × 10−4 1.82 0.14Hang Seng 0.33 0.03 0.13 0.01 0.20 3.3 × 10−5 1.58 0.11Nikkei 0.83 0.04 0.67 0.04 0.16 9.1 × 10−5 0.53 0.02HUF/USD 0.83 0.04 0.15 0.02 0.69 1.1 × 10−3 1.46 0.08EUR/USD 0.30 0.07 0.11 0.03 0.18 2.0 × 10−4 1.37 0.24

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the partial integration parameter is estimated significantly lower than in the standard FIGARCHsetup. The coefficient of fractional integration remains significant. The estimate for the squaredtrend parameter was positive and significant in each time series.19 Table 10 gives the results of theestimations of the parameters of fractional integration in FIGARCH and Trend-FIGARCH modelswith their standard errors, the difference of these estimates and the p-value for the difference.Also,estimates of the squared trend parameters in the variance equation of the Trend-FIGARCH modelswith their standard errors are presented.

The interpretation of these results might be either that a part of the persistence of asset pricevolatilities is carried forward by trend effects and is thus measured correctly by the trend models,or that the estimation procedure of the trend model simply catches up some aspect of the longmemory property of the volatility time series.

We have also run Monte Carlo simulations to estimate a Trend-GARCH(3,1,1) model fromsimulated FIGARCH(1,d,1) time series and vice versa. The model parameters for the simulationsare the estimates from the DJII time series. In both ways, the estimates of the model specificcoefficients are significant, i.e. the FIGARCH estimation finds a significant level of integrationin the simulated Trend-GARCH data (for over 90% of the estimates the coefficient of fractionalintegration does not significantly differ from unity), and the Trend-GARCH estimation findssignificant trend effects.

We summarize our results in the following conclusion. Traditional GARCH models cannotreplicate the persistence of shocks to volatility found in financial data sets. Neither the exponentialrate of decay of simple GARCH models nor the infinite persistence of IGARCH models correctlycatches the long memory property of the volatility time series. The hyperbolic decay rate ofFIGARCH models better suits the data, but still leaves room for improvement. The Trend-GARCHmodel might be useful in complementing this approach, as it seems to be able to replicate someof the still open properties of the volatility process.

6. Summary

The paper empirically tests the impact of short-run price trending on the conditional volatility.We present the Trend-GARCH model, which amends the conditional variance equation with acomponent based on the current trend of the price series. The model is defined at the outsetand compared to alternative GARCH extensions, such as EGARCH, AGARCH, TGARCH, andGARCH-M with respect to their variance equations and the news impact curves. The trend com-ponent of the Trend-GARCH model accounts for volatility effects of trends in the time seriesitself, while the ARCH part of GARCH models is able to account only for trends in the volatilityprocess. Trend-GARCH models connect the past behaviour of time series, the subsequent reactionof market individuals, and thereon changes in the future characteristics of the time series in a waywhich is often suggested by modern microeconomic theory.

The main part of the paper is devoted to an empirical comparison of the alternative GARCHextensions and the Trend-GARCH model based on a sample of 12 daily asset prices series (sixmajor US companies, four stock indices, and two US Dollar exchange rates). We first show, thatTrend-GARCH models may explain the empirical dependence between current trend and futurevariance in these economic time series and that the trend component is empirically relevant. Wethen check for the abilities of the different models to fit the volatility estimates from high frequencydata and to account for the leverage effect and the persistence in the conditional variance.

The Trend-GARCH model better fits the volatility estimates derived from high frequency datathan the EGARCH, AGARCH and TGARCH models. It also proves superior to these models

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in replicating the leverage effect in the conditional variance and in fitting the news impact curve.The asymmetric effects of news to volatility in the Trend-GARCH model is asymmetric since itdepends on the current trend. The effect of news is symmetric given its relation to the trend. Ifthe trend is positive then slightly negative news, which slows the trend down, calms the market.If the trend is negative then slightly positive news, which also slows the trend down, generatesa relatively tranquil market. The asymmetric Trend-GARCH model additionally differentiatesbetween bullish and bearish markets.

The persistence of shocks to volatility found empirically is neither replicated by traditionalGARCH models with an exponential decay rate nor by the infinite persistence of IGARCHmodels. The hyperbolic decay rate of FIGARCH models better suits the data. The Trend-GARCHmodel seems to be useful in complementing this approach. In contrast to the alternative GARCHextensions, the Trend-GARCH model does not only use information from trends in the volatilityseries (ARCH part) but also from trends in the price series itself, which in turn might influencevolatility.

Acknowledgements

We thank two anonymous referees for extensive and very helpful comments. We further thankseminar participants at the Applied Econometrics Association (AEA) Conference in Luxemburg.The author gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft(German Science Foundation).

Notes1 A direct derivation of the Trend-GARCH model from a market microstructure model is left to future research. We

also do not test other direct implications of such dependency, like the relation of volume and trend.2 This aspect is not realized in GARCH-in-Mean models since they only model the impact of the conditional volatility

on the location parameter of the innovation.3 Autoregressive stochastic volatility models are an interesting alternative to GARCH models in replicating several

stylized facts of financial time series (Lehar et al., 2002; Carnero et al., 2004; Sadorsky, 2004).4 Some types of technical analysis, such as candle sticks, are exceptions to that rule.5 Only few approaches to technical trading, such as adverse selection or constant portfolio weights, do not induce trend

following.6 Using absolute trends instead of squared trends doesn’t alter the results qualitatively.7 When looking at the special formulation of the trend component, one might be tempted to generalize this model class

using a conditional volatility process like

ht = ω + β(L)ht + α(L)ε2t + (γ (L)εt )

2 (18)

where γ is a polynomial of degree s. However, the nonlinear structure of (γ (L)εt )2 and the potential identification

problems with the coefficients of α indicate severe difficulties for the applicability of such a model and we thus willnot use this formulation.

8 For exchange rate time series the leverage effect should depend on the inhomogenity of the market microstructure.The exchange rate between two similar countries will be effected symmetrically by news as good news for one halfof the traders is bad news for the other half and vice versa.

9 Different values for the past conditional volatilities and the past squared returns in larger GARCH( p,q) models mayonly shift the NIC on the vertical axis.

10 The choice of the six companies from the DJII sample was random.11 The Hungarian Forint series starts at 17/6/1993 and the Euro series starts at 01/01/1999.12 For this descriptive section, the length of the trend estimate s = 5 is chosen to represent a one week trend (5 working

days).13 The graphs of kernel regressions on simulated data are available from the author upon request.14 Alternativly the daily volatility could be estimated from intraday data. Due to data limitations this approach is

implemented only for the USD-DM exchange rate series in subsection 3.4.

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15 All of the models have the sameARCH part which account for trends in the volatility process. Only the Trend-GARCHmodel includes volatility effects of trends in the time series itself (see discussion in Section 2.1).

16 The complete list of all estimated parameters in all models and all time series is available upon request from theauthor.

17 Carnero et al. (2004) analyzes autoregressive stochastic volatility models as an interesting alternative to some GARCHmodels as these models seems to suit the persistence properties in financial time series well.

18 Recall that p = α + β is an appropriate measure of persistence only for the simple GARCH(1,1) model and not forits extensions.

19 Bauer and Herz (2005) show similar results for all 6 series in their sample of USD exchange rates of industrializedcountries.

Appendix: Model Selection

The data cover daily returns derived from opening stock prices of six major US compa-nies in the DJII (HomeDepot, IBM, Coca Cola, Johnson & Johnson, Procter & Gamble, andExxon), four stock indices (Dax, DJII, Hang Seng, and Nikkei), and two US Dollar exchangerates (to the Hungarian Forint and the Euro). The choice of the six companies from theDJII sample was random. The time series of stocks and indices run from 01/01/1990 to15/07/2004. The Hungarian Forint series starts at 06/17/1993 and the Euro series starts at01/01/1999.

Length s of the Estimated Trend Component

Table A1. Basic model: Trend-GARCH(s, 1, 1) with random walk mean equation, BIC statistics

Lengthof trendestimate s 3 4 5 7 10 15 20 30 50

HomeDepot −17807 −17449 −17442 −17488 −17460 −17228 −17063 −16852 −16352IBM −18593 −18461 −18499 −18403 −18444 −18252 −18167 −17770 −17405Coca Cola −19349 −19729 −20034 −19829 −18903 −19288 −19085 −18690 −18797Johnson &

Johnson−20037 −19983 −19019 −19859 −19298 −19219 −18571 −18770 −18260

Procter &Gamble

−19626 −19963 −19303 −18887 −18459 −19455 −18941 −19309 −18724

Exxon −20897 −21302 −21166 −21001 −20770 −20832 −20475 −20408 −20058Dax −20495 −20486 −20480 −20480 −20518 −20491 −20402 −20344 −20245DJII −24303 −24283 −24288 −24136 −24002 −23965 −23944 −23872 −23735Hang Seng −21394 −20775 −20724 −20357 −20318 −20211 −20176 −20155 −20042Nikkei −20952 −20513 −20233 −20253 −20012 −19967 −19962 −19972 −19626HUF/USD −20089 −20012 −19880 −19881 −19859 −19840 −19811 −19755 −19651EUR/USD −10003 −10024 −9998 −9973 −9965 −9916 −9873 −9792 −9642

Table A2. Basic model: Trend-GARCH(s, 1, 1) with random walk mean equation, average rank of the BICstatistic for the respective model

Length of trend estimate s 3 4 5 7 10 15 20 30 50

Average rank 1.75 2.33 3.50 3.83 4.92 5.33 7.00 7.50 8.33

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86 C. Bauer

Order of the GARCH Models: the Volatility Equation

Table A3. Basic models: GARCH( p, q), TGARCH( p, q), Trend-GARCH(3,p, q) with random walk mean equation, average rank of the BIC statistic for the

respective model

( p, q) GARCH( p, q) TGARCH( p, q) Trend-GARCH(3, p, q)

0,1 11.92 11.08 8.920,2 13.50 12.75 11.000,3 13.67 13.08 11.081,0 10.67 11.83 10.671,1 3.00 3.33 5.251,2 4.00 4.00 4.251,3 5.25 5.42 4.832,0 10.00 11.25 11.502,1 3.42 4.83 5.582,2 7.92 6.50 7.082,3 5.92 5.25 7.333,0 9.58 11.08 10.923,1 4.17 3.92 5.923,2 8.42 7.58 6.083,3 8.58 8.08 9.58

Order of the GARCH Models: the Mean Equation

Table A4. Basic models: GARCH(1, 1), TGARCH(1, 1),Trend-GARCH(3, 1, 1) with ARMA(p,q) mean equation, average

rank of the BIC statistic for the respective model

( p, q) GARCH TGARCH Trend-GARCH

0,0 3.17 3.67 4.080.1 3.83 4.67 4.670.2 5.42 5.67 4.170.3 8.33 8.00 6.081.0 3.75 5.00 5.081.1 6.67 4.75 7.671.2 9.58 6.67 7.421.3 11.00 10.58 10.52.0 5.67 6.00 5.002.1 6.92 7.67 7.672.2 10.67 10.58 11.252.3 12.17 12.25 12.083.0 7.42 7.92 6.173.1 7.58 8.75 8.923.2 11.58 12.00 12.173.3 10.92 10.50 11.75

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