A Better Asymmetric Model of Changing Volatility in Stock and Exchange Rate Returns: Trend-GARCH

Abstract

The impact of short run price trending on the conditional volatility is tested empirically. A new family of conditionally heteroscedastic models with a trend-dependent conditional variance equation: The Trend-GARCH model is described. Modern microeconomic theory often suggests the connection between the past behaviour of time series, the subsequent reaction of market individuals, and thereon changes in the future characteristics of the time series. Results reveal important properties of these models, which are consistent 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-GARCH model 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 and in fitting the volatility estimates from high frequency data. In addition, we show that the leverage effect is dependent on the current trend, i.e. it differentiates between bullish and bearish markets. Furthermore, trend effects can account for a significant part of the long memory property of asset price volatilities.

Keywords:

• GARCH
• trend
• volatility
• news impact curve
• leverage effect
• persistence

Acknowledgements

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

Notes

¹A 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.

²This 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.

³Autoregressive 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).

⁴Some types of technical analysis, such as candle sticks, are exceptions to that rule.

⁵Only few approaches to technical trading, such as adverse selection or constant portfolio weights, do not induce trend following.

⁶Using absolute trends instead of squared trends doesn’t alter the results qualitatively.

⁷When looking at the special formulation of the trend component, one might be tempted to generalize this model class using a conditional volatility process like

• where is a polynomial of degree s. However, the nonlinear structure of ( and the potential identification problems with the coefficients of α indicate severe difficulties for the applicability of such a model and we thus will not use this formulation.

⁸For 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 half of the traders is bad news for the other half and vice versa.

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

¹⁰The choice of the six companies from the DJII sample was random.

¹¹The Hungarian Forint series starts at 17/6/1993 and the Euro series starts at 01/01/1999.

¹²For this descriptive section, the length of the trend estimate s=5 is chosen to represent a one week trend (5 working days).

¹³The graphs of kernel regressions on simulated data are available from the author upon request.

¹⁴Alternativly 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.

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

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

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

¹⁸Recall that is an appropriate measure of persistence only for the simple GARCH(1,1) model and not for its extensions.

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