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4.1.2 The Leverage Effect

If volatility is higher following a negative return than it is following a positive return then the autocorrelation between yesterdays return and todays squared return will be large and negative

In equity markets it is commonly observed that volatility is higher in a falling market than it is in a rising market. The volatility response to a large negative return is often far greater than it is to a large positive return of the same magnitude. The reason for this may be that when the equity price falls the debt remains constant in the short term, so the debt/equity ratio increases. The firm becomes more highly leveraged and so the future of the firm becomes more uncertain. The equity price therefore becomes more volatile. This leverage effect has already been mentioned in §2.2.1 as a secondary cause of the implied volatility skew in equity markets. It also implies an asymmetry in volatility clustering in equity markets: if volatility is higher following a negative return than it is following a positive return then the autocorrelation between yesterdays return and todays squared return will be large and negative.

The asymmetric GARCH tests in the last two columns of Table 4.1 investigate the leverage effect in 20 US stocks. A very simple test of this effect is to compute the first-order autocorrelation coefficient between lagged returns and current squared returns:

1=2 1=2

If this is negative and the corresponding Box-Pierce test (the LM test in the last column) is significantly different from zero, then there is an asymmetry in volatility clustering which will not be captured by a symmetric GARCH model. Instead one of the asymmetric GARCH models should be employed (§4.2.4).

Where implied volatility smiles have noticeable skew effects, these may or may not be indicative of a leverage effect

It is interesting that the asymmetric GARCH tests in Table 4.1 fail to reject the null hypothesis of no asymmetry in many cases. Only three out of the 20 stocks in Table 4.1 show significant leverage effects in the data period: Cisco Systems, General Electric and Exxon Mobil. Using robust test statistics that are much more sophisticated than the simple LM procedure used for Table 4.1, Hagerud (1997) has also found that relatively few Nordic stocks show signs of asymmetric volatility clustering. Only 12 out of his sample of 45 stocks exhibited a noticeable leverage effect. The volatility skew may still be very pronounced in these stocks, so where implied volatility smiles have noticeable skew effects, these may or may not be indicative of a leverage effect.

Table 4.2 examines the results of applying the simple Box-Pierce tests for symmetric and asymmetric GARCH to 12 international equity indices. While all except Taiwan have very significant GARCH effects, only seven out of the 12 indices have asymmetric GARCH effects that are significant at the 1% level.



Table 4.2 LM tests for symmetric and asymmetric GARCH effects in equity indices (daily data from 2 January 1996 to 6 October 2000)

Index

GARCH

A-GARCH

52.9

24.1

AORD

205.4

52.2

44.5

27.7

19.6

FTSE 100

Hang Sang

176.2

43.6

Ibovespa

46.8

30.8

31.8

0.23

Nikkei 225

164.7

77.2

S&P 500

51.8

24.7

Straights Times

67.9

Taiwan

Neither the UK, France, South Africa, Singapore nor Taiwan have had pronounced leverage effects in their equity indices since 1996. However, index options in all these markets have pronounced skews in their volatility smiles.

Literally dozens of different variants of asymmetric GARCH models have been proposed and tested in a vast research literature. Some of these models are outlined in §4.2.4. However, we have just seen that asymmetric GARCH models have a fairly limited practical use. It is a good thing to be able to include the possibility of asymmetry in the GARCH model so that any leverage effect will be captured, but one should do so with caution because the estimation of asymmetric GARCH can be much more difficult than the estimation of symmetric GARCH.

Literally dozens of different variants of asymmetric GARCH models have been proposed and tested in a vast research literature

4.1.3 The Conditional Mean and Conditional Variance Equations

A simple linear regression can provide a model for the conditional mean of a return process. For example, in a factor model regression the expected value of a stock return will change over time, as specified by its relationship with the market return and any other explanatory variables. This expectation is the conditional mean. The classical linear regression model assumes that the unexpected return e„ that is, the error process in the model, is homoscedastic. In other words, the error process has a constant variance V(e,) = a2 whatever the value of the dependent variable (§A.1.1). The fundamental idea in GARCH is to add a second equation to the standard regression model: the conditional variance equation. This equation will describe the evolution of the conditional variance of the unexpected return process, V,(z,) = <j2.

The fundamental idea in GARCH is to add a second equation to the standard regression model: the conditional variance equation

The dependent variable, the input to the GARCH volatility model, is always a return series. Then a GARCH model consists of two equations. The first is the conditional mean equation. This can be anything, but since the focus of



GARCH is on the conditional variance equation it is usual to have a very simple conditional mean equation. Many of the GARCH models used in practice take the simplest possible conditional mean equation r, - + In this case the unexpected return s, is just the mean deviation return, because the constant will be the average of returns over the data period. In some circumstances it is better to use a time-varying conditional mean (recall Figure 1.5), but the modeller must be very careful not to use many parameters in the conditional mean equation otherwise convergence problems are likely (§4.3.3). If there is a significant autocorrelation in returns, you should use an autoregressive conditional mean, and in almost all cases an AR(1) model will suffice (§11.2.1). If there is a structural break, where the mean return jumps to a new level although the volatility characteristics2 of the market are unchanged, then a dummy variable can be included in the conditional mean (§A.4.4).

The second equation in a GARCH model is the conditional variance equation. Different GARCH models arise because the conditional variance equations are specified in different forms; some of the more common GARCH models are overviewed in the next section. There is a fundamental distinction between the symmetric GARCH models that are used to model ordinary volatility clustering and the asymmetric GARCH models that are required to capture leverage effects. In symmetric GARCH the conditional mean and conditional variance equations can be estimated separately, as described in §4.3.2. However, this is not possible for asymmetric GARCH models. Their estimation is much more complex, and for more details the reader is referred to the specific papers cited in §4.2.4.

A GARCH model focuses on the time-varying variance of the conditional distributions of returns. Underlying every GARCH model there is also an unconditional returns distribution. The unconditional distribution of a GARCH process will be stationary under certain conditions on the GARCH parameters, as shown in the next section, and if necessary these conditions can be imposed on the estimation. However, if these constraints are indeed binding, one should really ask whether the GARCH model is appropriate to the data.3

4.2 A Survey of Univariate GARCH Models

Very many different types of GARCH models have been proposed in the academic literature, but not all have found good practical applications. This section gives an overview of some of the better-known GARCH models that are currently used by academics and financial practitioners. The first autoregressive conditional heteroscedasticity (ARCH) model, introduced by

2E.g., the degree of persistence in volatility, and the extent of reaction in volatility to market news (§4.2.2).

3I prefer to estimate GARCH without imposing constraints on the parameters, and if the freely estimated parameters do not satisfy these constraints I would use a different data period or a different GARCH model.



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