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26

chance that multivariate GARCH models can be used to estimate directly the very large covariance matrices that are required to net all the risks in a large trading book. However, it is possible to generate GARCH covariance matrices using only univariate GARCH volatilities, and the chapter ends with an overview of these methods.

The financial community has benefited enormously from the work of many academics. One professor whose pioneering work on GARCH stands out above the rest is Rob Engle (http: weber.ucsd.edu/ ~ mbacci/engle/ index.html). Much of the important development and validation work on different GARCH models is attributable to Rob Engle and his co-authors and numerous research students, particularly Torben Andersen, Tim Bollerslev, Ken Kroner, Joshua Rosenberg and Victor Ng. Since his first seminal article in 1982, Rob Engle has continued to produce path-breaking work in this area. More recently, a number of intriguing new developments and applications of GARCH models have come from Jin-Chuan Duan. His lecture notes, research papers and computer programs, available on his website (www.rotman. utoronto.ca/ ~jcduan), are well worth a visit!

4.1 Introduction to Generalized Autoregressive Conditional Heteroscedasticity

In a generalized autoregressive conditional heteroscedasticity (GARCH) model, returns are assumed to be generated by a stochastic process with time-varying volatility. Instead of modelling the data after they have been collapsed into a single unconditional distribution, a GARCH model introduces more detailed assumptions about the conditional distributions of returns. These conditional distributions change over time in an autocorrelated way - in fact the conditional variance is an autoregressive process (§11.2.1).

4.1.1 Volatility Clustering

Many financial time series display volatility clustering, that is, autoregressive conditional heteroscedasticity. Equity, commodity and foreign exchange markets often exhibit volatility clustering at the daily, even the weekly, frequency, and volatility clustering becomes very pronounced in intra-day data. A typical example of a conditionally heteroscedastic return series is shown in Figure 4.1. Note that two types of news events are apparent. The second volatility cluster shows an anticipated announcement, which turned out to be good news: the market was increasingly turbulent before the announcement, but the large positive return at that time shows that punters were pleased, and the volatility soon decreased. The first cluster of volatility indicates that there is turbulence in the market following an unanticipated piece of bad news.

Volatility clustering implies a strong autocorrelation in squared returns, so a simple method for detecting volatility clustering is to calculate the first-order autocorrelation coefficient in squared returns:

Volatility clustering implies a strong autocorrelation in squared returns, so a simple method for detecting volatility clustering is to calculate the first-order autocorrelation coefficient in squared returns



-0.15

Figure 4.1 Volatility clustering.

1=2

A basic test for the significance of autocorrelation is the Box-Pierce LM test (§11.3.2). To test for conditional heteroscedasticity in the data, this test may be applied to the squared return. However, simply calculating the Box-Pierce statistic is not enough. In fact one might perform a Box-Pierce test that shows insignificant autocorrelation in squared returns even though volatility clustering is present in the data.

To illustrate why this happens, daily data on 20 US stocks from 2 January 1996 to 2 October 2000 has been analysed in Table 4.1. The table gives a number of summary statistics for the daily returns, including the first-order autocorrelation coefficient of squared returns (column 6) and the Box-Pierce statistic for first-order correlation (column 7). Bold type is used to pick out the statistics that indicate a potential problem.

Looking first at columns 6 and 7, there are seven stocks that have low autocorrelation in squared returns (Ford, Hewlett Packard, Microsoft, BankOne, Procter and Gamble, Rockwell and AT&T). But does this really mean that there is no volatility clustering? Actually not. Even though the data are adjusted for dividends and splits, these stocks have at least one very extreme negative return during the data period. How do we know this (other than of course by looking at the data)? The first four sample moments, up to skewness and excess kurtosis. are given in columns 2-5. From the negative

The Box-Pierce test for first-order autocorrelation is asymptotically a chi-squared variable with 1 degree of freedom, so the 1% critical \alue for this test is 6.635. This test is simple but not very robust. Robust but complex test procedures for GARCH models are given in Wooldridge (1991).



Stock

Mean

s.d.

Skewness

Excess

GARCH

GARCH

A-GARCH

A-GARCH

kurtosis*

Auto-

Auto-

correlation

correlation

America Intl Group

0.0012

0.0196

0.2813

1.8360

0.1466

25.7847

-0.0037

0.0162

America Online

0.0026

0.0411

0.1906

1.5125

0.1136

15.4735

-0.0693

5.7641

American Express

0.0013

0.0231

0.0679

1.9543

0.2342

65.8048

-0.0696

5.8184

AT&T

-0.0004

0.0246

-1.4018

20.4217

0.0343

1.4109

0.0270

0.8720

Bank of America

0.0005

0.0228

-0.0441

1.6671

0.1748

36.6869

0.0014

0.0022

BankOne Corp.

0.0002

0.0237

-1.0432

13.8524

0.0278

0.9269

-0.0424

2.1577

Boeing Corp.

0.0004

0.0223

-0.3948

7.3615

0.1459

25.5560

-0.0180

0.3901

Cisco Systems

0.0021

0.0294

-0.1270

1.6547

0.1778

37.9210

-0.1253

18.8389

Citigroup

0.0014

0.0250

0.2046

2.8857

0.1365

22.3706

0.0184

0.4068

Coca Cola Co.

0.0004

0.0197

0.0158

2.5512

0.1833

40.3056

-0.0452

2.4496

Walt Disney

0.0006

0.0213

0.2910

4.0542

0.1594

30.4738

-0.0493

2.9179

Exxon Mobil

0.0007

0.0166

0.1900

1.2463

0.1276

19.5358

-0.0846

8.5854

Ford Motor Co.

0.0003

0.0236

-3.5109

60.3539

0.0159

0.3018

-0.0072

0.0621

General Electric Co.

0.0013

0.0178

0.0352

1.1232

0.1145

15.7423

-0.1093

14.3312

Hewlett Packard

0.0007

0.0287

-0.2401

3.2223

0.0664

5.2907

0.0374

1.6769

Merrill Lynch

0.0014

0.0286

0.1233

1.8895

0.2108

53.3221

-0.0396

1.8812

Microsoft Co.

0.0014

0.0238

-0.4704

4.1963

0.0345

1.4311

0.0100

0.1190

Procter and Gamble

0.0004

0.0221

-3.7453

60.0698

0.0263

0.8322

-0.0382

1.7465

Rockwell Int.

-0.0001

0.0219

-0.9535

11.4014

0.0627

4.7140

-0.0365

1.6010

Unicom Corp.

0.0006

0.0158

0.0429

3.4449

0.1499

26.9722

0.0158

0.2985

*Excess kurtosis is obtained by subtracting 3 from the value of kurtosis, so that the normal distribution has excess kurtosis equal to zero.

skewness and extreme excess kurtosis on these particular stocks it is clear that the apparently low autocorrelation in squared returns is due to one, or even a few, extreme negative returns.

If these outliers were removed from the data, then volatility clustering would be evident from the first-order autocorrelation statistics. For example, the price of Ford stock fell 30% on 8 April 1998, and the effect of removing that one single return on that day is to change the excess kurtosis from 60.35 to 1.79 and the GARCH Lagrange multiplier (LM) statistic from 0.3 to 17.33. Obviously one should either remove or dummy out that return from the historic data set used to estimate the GARCH model parameters. The message to take home for GARCH modelling is: always examine your data carefully before you begin the analysis. The potential saving of time and trouble with non-convergent GARCH models can be considerable.

Excess kurtosis is discussed in detail in §10.1. It is not always due to single outliers, and it may not affect our tests for volatility clustering. For example, Boeing, Disney and Unicom all show signs of significant excess kurtosis, but the GARCH autocorrelation diagnostics are still highly significant.

Always examine your data carefully before you begin the analysis. The potential saving of time and trouble with non-convergent GARCH models can be considerable

Table 4.1: Statistics for 20 US stocks (daily data from 2 January 1996 to 2 October 2000)



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