0 -I-,-,-,-,-

Jan-96 Jan-97 Jan-98 Jan-99 Jan-00

G) S&P 500

120-,-

100 J

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(k) Straits Times - All Data - - - Post October 97

70" 60-

10-1

oj-,-,-,-,-

Jan-96 Jan-97 Jan-98 Jan-99 Jan-00

(I) Taiwan

Figure 4.7 GARCH( 1,1) volatility: (a) AEX; (b) AORD; (c) CAC; (d) DAX; (e) FTSE 100; (0 Hang Seng; (g) Ibovespa; (h) Nikkei 225; (i) Johannesburg; (j) S&P 500: (k) Straits Times; (1) Taiwan.

and their long-run volatility forecasts are correspondingly a little higher (but more uncertain). Currently the reaction and persistence parameters in the USA (S&P 500) and Japan (Nikkei 225) are very similar, but the higher estimate of omega in Japan yields a long run volatility of 23.7% for Japan, compared with 19.6% for the USA based on the same sample period.

On the basis of these results, and looking at the graphs in Figure 4.7, the indices that are most reactive in volatility seem to be in Australia (AORD), Hong Kong (Hang Seng), Brazil (Ibovespa), South Africa (Johannesberg) and Taiwan. The equity markets in Hong Kong, Singapore and Brazil have been generally rather volatile since 1998. For example, having fallen 17.2% on 10 September 1998, the Brazilian Ibovespa then proceeded to recover, and in the early part of 1999 some quite spectacular gains were made: on 15 January 1999 it rose 28.8%. Singapore has been less fortunate, although a gain of 12.87% was made on a single day on 2 February 1998. It is not surprising, therefore, that the long-run volatility forecasts for 6 October 2000 that are based on these data come in at around 50% for Brazil and Hong Kong.14 The long-run volatility forecast for Taiwan is much lower because, although it often experiences volatilities in the 60%> range, it has the least persistent volatility. This type of over-reactive and low persistent volatility is a well-documented feature of Asian-Pacific equity and currency markets (Alexander and Thillainathan, 1995).

The Australian equity market volatility seems to have more in common with that of the South African equity market than the other Asian-Pacific markets, except that both were badly affected by the Asian crisis in 1997. Australia and South Africa were uncharacteristically volatile during that time: on 28 October 1997 the Australian index fell by 7.5% and the South African index fell by 11.85%. And volatility in both markets, particularly South Africa, has increased since that time.

Now suppose it is 6 October 2000 and you want to make a volatility forecast a year ahead. You consider that it is very unlikely that the spectacular losses associated with the Asian crisis will be repeated during your forecast horizon, and you also believe that the relatively tranquil equity markets experienced in Australia and South Africa before the 1997 mini-crash will not be seen again for at least another year. Then there is a good reason to choose the starting date for the data to be after the mini-crash in equities of the summer of 1997. In fact the South African equity index GARCH model is not well specified when data before the mini-crash are included in the sample: the results for South Africa (and in fact for Singapore) in the upper part of the table correspond to an I-GARCH model, which has no long-run volatility because volatility is a random walk (§4.1.2). However, when the data period starts on 1 November

14Long-run forecasts are not available for Singapore because the estimate of a + f5 is 0.999, close enough to 1 for the GARCH model to be integrated. This indicates that the model is not well specified for the data and should be re-run using a different historic period, as has been done for South Africa and Australia, or a different (asymmetric) GARCH model.

In this case there is a good reason to choose the starting date for the data to be after the mini-crash in equities of the summer of 1997

1997, the GARCH model for South Africa is well specified, and it gives a long-run volatility forecast of about 24% on 6 October 2000. The last two rows of Table 4.7 show that if the Australian and South African GARCH models were estimated on data from 1 November 1997, instead of from 2 January 1996, they would tell quite a different story for 6 October 2000.

The results for Australia and South Africa in Table 4.7 are improved when one only uses about 3 years of data, rather than the original sample of almost 5 years. Thus you may well ask whether it would be even better to use only 2 years of data instead of 3. However, it is important to understand that a certain minimum amount of data will be necessary for the likelihood function to be well defined. Usually a few years of daily data are necessary to ensure proper convergence of the model and, even if there are sufficient data to avoid convergence problems, parameter estimates may lack robustness if too few observations are used. That is, as the data window is rolled forward day by day, the estimates of the GARCH parameters might lack stability. Therefore it is always a good idea to check the robustness of GARCH parameter estimates by doing some rolling GARCH regressions.

A certain minimum amount of data will be necessary for the likelihood function to be well defined

Figure 4.8 shows the GARCH(1, 1) parameter estimates and corresponding long-run GARCH volatility forecasts for five of the US stocks from Table 4.2. Instead of using the entire data period from 1 January 1996 to 2 October 2000 to estimate the GARCH model as in Table 4.3, only 4 years of data were used. The GARCH model was rolled weekly from December 1999 until the end of the data set, each time recording the parameter estimates, and these are shown in Figure 4.8.

From Figure 4.8a it is evident that the GARCH omega constants for Microsoft and America Online declined significantly during the estimations, but for other stocks the GARCH constants were relatively stable. Looking at Figures 4.8b and 4.8c, America Online started off with a higher alpha and lower beta, that is, a high reaction but low persistence, giving a spiky volatility. However, by the end of the period its volatility characteristics had changed to become more similar to those of the other stocks.

The reaction coefficients (alpha) and the persistence coefficients (beta) shown in Figures 4.8b and 4.8c are more variable in all stocks. After the technology bubble burst most American stocks seem to have settled down again sometime during the second quarter of 2000, Microsoft being an exception. At the beginning of the data period the Microsoft GARCH models gave incredibly high reaction and low persistence volatilities, but this may have had something to do with the legal battle between Microsoft and the US Monopolies Commission. Indeed, all the GARCH parameters for Microsoft underwent considerable change leading up to the court ruling to split in April 2000. Around that time the reaction coefficient for Microsoft was extremely variable, and this has fed through to the long-term volatility forecasts shown in Figure 4.8d.