In this chapter, there is description of informations used in this survey, which are The Hang Seng Index ( HSI ) , The Hongkong and Shanghai Banking Corporation Holding plc ( HSBC Hldgs ) and Oriental Press Group Limited. After that, methodological analysiss of anticipations based on past volatilities and GARCH category conditional volatility theoretical accounts are presented. Finally, I will speak about how to measure the prediction public presentations in harmonizing to choose the best theoretical accounts. Now, allow me present the informations analyzed in my survey.

## 3.1 Data description

## 3.1.1 The Hang Seng Index ( HSI )

Global investors treat the Hang Seng Index ( HSI ) as an index of the public presentation of the stock market in Hong Kong. The HSI was publically launched on 24 November 1969 and is one of the earliest stock market indexes in Hong Kong. It is now maintained by Hang Seng Indexes Company Limited, which is a subordinate of Hang Seng Bank, one of the component companies. The HSI is a freefloat-adjusted market capitalisation weighted stock market index.

: The current monetary value at Day t

: The shutting monetary value at Day t-1

IS: The figure of issued portions

FAF: Freefloat-adjusted Factor, which is between 0 and 1

Californium: Cap factor, which is between 0 and 1

It consists of 43 component companies which represent around 60 % of the entire market capitalisation of the Hong Kong Stock Exchange ( HKEX ) . In order to demo the monetary value motions of the major sectors of the market clearer, the HSI component stocks are grouped into four sub-indexes: Properties, Utilities, Commerce and Industry, and Finance.

Normally, qualified possible components are companies with a primary listing on the Main Board of the Hong Kong Stock Exchange ( HKEX ) . In recent old ages, more and more mainland China companies listed on the Hong Kong Stock Exchange ( HKEX ) , they can besides be one of the qualified possible components if they meet several conditions. The company ‘s entire ordinary portion capital must be in the signifier of H portions, the portions of the company incorporated in mainland China that are traded on the Hong Kong Stock Exchange. And besides, the company needs to finish the procedure of Share Reform foremost. As a consequence, the company has no unlisted portion capital. As the market capitalisation, the turnover ranking and the fiscal public presentations of the companies may alter in different periods, the list of constitutional stocks is reviewed quarterly.

In this paper, the day-to-day shutting Hang Seng Index and the hebdomadal Hang Seng Index are used. The information set runing from 1 July 1997 to 30 June 2008 is retrieved from Datastream, a U.K. incorporated informations service company. The information set is partitioned into the in-sample appraisal periods 1997 – 2007, the out-of-sample prognosis periods 2007 – 2008. This separation provides 2467 and 522 in-sample observations for the day-to-day series and hebdomadal series severally. For out-of-sample observations, there are 246 for day-to-day series and 52 for hebdomadal series. In order to obtain more accurate consequences, public vacations and particular incidents taking the public proclamation of non-trading, such as the black signal of the Rainstorm Warning System and no. 8 storm force wind signal of Hong Kong ‘s Tropical Cyclone Warning Signals.

During this period, there were rather a batch of economic events impacting the stock market in Hong Kong. Hong Kong is one of the victims of 1997 Asiatic Financial Crisis. After the HSI peaked at 16820, it was attacked by international speculators, taking to a 50 % dip. In 1998, the Hong Kong authorities intercession on stock market to buy component portions supported the market and the HSI rebounded to 18000. The rise in both the involvement rate and the rough oil monetary value burst the dot com bubble in 2000. The HSI dropped to 14000 and so fluctuated between 14000 and 16000. 911 Incident occurred in 2001. The HSI kept falling and the lower limit was 8894. The stock market started to retrieve after Mainland and Hong Kong Closer Economic Partnership Arrangement ( CEPA ) was signed on 29 June 2003. In August 2007, mainland China declared the program to let portion of citizens to put on Hong Kong market straight. In October, stimulated by the intelligence of AH stock fudging plan, the HSI interruption through 30000. Unfortunately, the HSI was affected by the fallen concatenation of major stock markets in the universe and dropped 2061.23 on 22 January 2008.

During this period, the mean point of the HSI is 14313.39 and the base divergence is 4563.68. The lowest point in these 11 old ages was 6660.42 which occurred on 13 August 1998. 31638.22 was the highest point and occurred on 30 October 2007.

## 3.1.2 HSBC hldgs

Besides the HSI, one of the component stocks is indiscriminately selected. At the beginning of the choice procedure, each component stock is assigned a figure, 1 – 43. Then, a figure, between 1 – 43, is indiscriminately generated by Excel. If the assigned figure of the company matches the generated figure, that company will be selected and studied in this paper. The Hongkong and Shanghai Banking Corporation Holding plc ( HSBC Hldgs ) is the selected component company and belongs to the Finance Sub-indexes.

HSBC hldgs is one of the most favorite stocks in Hong Kong. In 1865, it was founded in Hong Kong in March and in Shanghai one month subsequently. The bank ‘s headquarter was in Hong Kong until 1992. Then it moved to London as a status of the acquisition of Midland Bank in UK, and besides due to the handover of Hong Kong ‘s sovereignty. Presently, HSBC hldgs is both the universe ‘s largest banking and fiscal services group and the 5th largest stock in the Hong Kong Stock Exchange ( HKEX ) by market capitalisation. HSBC hldgs is besides listed on the Bermuda, New York, London and Paris Stock Exchanges. It is a component of the FTSE 100 Index and the largest company listed on the FTSE. In 2010, CEO Michael Geoghegan moved to Hong Kong since HSBC hldgs focuses on Asia more now.

Similarly, both day-to-day and hebdomadal shutting monetary values of HSBC hldgs runing from 1 July 1997 to 30 June 2008 are retrieved from Datastream. The information set is besides divided into the in-sample appraisal periods 1997 – 2007 and the out-of-sample prognosis periods 2007 – 2008. The figure of in-sample observations and out-of-sample observations for both day-to-day and hebdomadal series are besides the same as the HSI series and the non-trading yearss are excluded. For in-sample observations, there are 2467 for day-to-day series and 522 for hebdomadal series. For out-of-sample observations, there are 246 for day-to-day series and 52 for hebdomadal series.

During this period, the mean shutting monetary value of HSBC hldgs is 96.92 and the base divergence is 23.26. The lowest shutting monetary value in these 11 old ages was 40.48 which occurred on 21 September 1998. 140.586 was the highest shutting monetary value and occurred on 15 October 2007.

## 3.1.3 Oriental Press Group Limited

In this paper, another stock which is non the component stock is besides studied. The stock is selected indiscriminately like the chosen component stock. Oriental Press Group Limited is selected and is incorporated in Hong Kong.

The group takes portion in the publication of day-to-day newspapers, Oriental Daily News and The Sun, and The Sun Racing Journal. Oriental Daily News was foremost published in 1969. Today, the paper is at the top in the day-to-day circulation and has a record readership of around 3,100,000. Unlike Oriental Daily News, The Sun marks at a younger and fresher reader groups. The Sun was published since 1999, and can besides be found outside Hong Kong. By sing a batch of Hong Kong people populating in North America and in order to develop the market of The Sun, there is a North America version in New York and Toronto. The Sun Racing Journal was established in 1991. It is one of the major Equus caballus rushing magazines in Hong Kong.

Like the HSI and HSBC Hldgs, both day-to-day and hebdomadal shutting monetary values of Oriental Press Group Limited runing from 1 July 1997 to 30 June 2008 are retrieved from Datastream. The information set is besides divided into the in-sample and the out-of-sample with the same period as the above two series. The figure of in-sample observations and out-of-sample observations for both day-to-day and hebdomadal series are besides equal to as that of the HSI series and HSBC hldgs series.

During this period, the mean shutting monetary value of Oriental Press Group Limited is 1.499 and the base divergence is 0.657. The lowest shutting monetary value in these 11 old ages was 0.533 which occurred on 23 June 1998. 3.175 was the highest shutting monetary value and occurred on 14 January 2004.

## 3.1.4 Transformation of Datas

As this paper is analyzing the volatility, I focus on the returns of the HSI and the stocks instead than the index and shutting monetary values. Therefore, a transmutation of informations is needed before patterning. Datas are transformed into day-to-day returns by taking the first difference of the natural logarithm of the day-to-day index and shutting monetary values.

: The day-to-day return

: The current monetary value at Day t

: The shutting monetary value at Day t-1

Average peers to 0.00138 and stand divergence is 0.0176.

Average peers to 0.00163 and stand divergence is 0.0169.

Average peers to -0.00032 and stand divergence is 0.0315.

## 3.2 Methodology

## Stationarity

If a clip series can be estimated utilizing clip series theoretical accounts, it must be a stationary procedure, as non-stationary procedure can non be studied straight. If the joint chance distribution of clip series informations supports changeless when clip is traveling, this information set can be said to be stationary and the mean and discrepancy do non alter over clip. On the contrary, a non-stationary procedure whose joint chance maintain changing in the period. The mean and discrepancy vary at different clip points. However, normally after a non-stationary series has been transformed, such as differencing or log-difference, the series become stationary.

Consequently, it is necessary to prove whether the information set is under a stationary procedure before any farther analyzing. The Augmented Dickey and Fuller ( ADF ) trial is employed in this paper. The ADF trial is an improved version of the Dickey and Fuller ( DF ) trial. Both trials ‘ nothing hypotheses are the series with a unit root, which means the series is non-stationary. The DF trial is merely based on a simple arrested development.

where the remainders are followed the Dickey-Fuller Distribution.

The hypothesis testing is

The trial statistic is

After that, comparing the trial statistic with the relevant critical value for the Dickey-Fuller trial, if the value is larger than the critical value, the void hypothesis can be rejected, and the theoretical account is stationary ; frailty poetry.

The ADF trial is based on a more complicated more.

where is a changeless, is the coefficient of a deterministic tendency and is the summing up of the lag orders of the autoregressive procedure.

## Historical mean

Probably, the historical mean provides the easiest manner to calculate the volatility. All the in-sample observations are used and every bit weighted to calculate.

where is the figure of out-of-sample observations

## Traveling norm

Under traveling mean method, the prognosis of the volatility is given by an unweighted norm of the in-sample observations. Unlike the historical mean method, non all the in-sample observations are used. The norm is based on the in-sample observations over a peculiar clip interval of fixed length.

where T is called the moving mean period or ‘rolling window ‘ . Three different lengths are considered for each frequence. For day-to-day informations, three months, six months and one twelvemonth are chosen. For monthly informations, six months, one twelvemonth and two old ages are chosen.

## Random walk

Both historical mean and traveling mean methods assume that the volatility will be in stable or alteration decelerating with a tendency. But, if the volatility fluctuates erratically, the best prognosis of the volatility of following period is the current existent volatility.

## Exponential smoothing

Under exponential smoothing method, the prognosis of the volatility is a leaden norm of the old existent observation and the old prognosis.

where is the smoothing factor, and must be between 0 and 1. If is equal to zero, the prognosis precisely equals the anterior existent observation and the exponential smoothing method becomes random walk. When approaches one, the predating prognosis is the bulk. The value of smoothing factor is determined by minimising the in-sample amount of squared mistakes. The estimated smoothing factors for day-to-day informations are between 0.64 and 0.95. For hebdomadal informations, the values lie in the scope 0.88 to 0.92.

## Exponentially leaden traveling norm ( EWMA )

The exponential weighted moving mean ( EWMA ) method is similar to the exponential smoothing method. The lone difference is that the anterior existent volatility is replaced by a moving mean method.

where is a smoothing factor lying between 0 and 1. Like the exponential smoothing method, the EWMA method becomes random walk if the smoothing factor is equal to nothing. When the factor is acquiring closer and closer one, the traveling norm will be weighted heavier and heavier. For day-to-day informations, the estimated values of factors are in the scope 0.001 to 0.53. The smoothing factors lie between 0.001 and 0.4 for hebdomadal informations.

## Generalized autoregressive conditional heteroscedasticity ( GARCH )

If the discrepancy of the information is changeless, the procedure is called homoscedastic. However, a batch of clip series informations in the existent universe have time-varying discrepancy. In 1982, Engle proposed the autoregressive conditional heteroscedasiticity ( ARCH ) theoretical account to cover with clip series informations. Four old ages subsequently, Bollerslev suggested the GARCH theoretical account. A GARCH ( P, Q ) theoretical account is

where, , and. There are joint appraisals of both the conditional mean and the conditional discrepancy procedure in the GARCH theoretical account. The error term, , follows a normal distribution with nothing mean and the discrepancy which varies with clip. indicates the continuity of dazes to volatility. If the amount is equal to one, the GARCH theoretical account becomes the integrated-GARCH ( IGARCH ) theoretical account.

## Threshold-GARCH ( TGARCH )

In the GARCH theoretical account, it assumes that dazes with equal magnitude have the same impacts on the volatility no affair which mark dazes are, positive or negative. However, the different mark of the daze may hold a different impact on the volatility. In order to parameterize this thought, the TGARCH theoretical account is one of a figure of ways. The TGARCH theoretical account was proposed by Glosten, Jagannathan and Runkle in 1993. The theoretical account is express as

where if, and if. Consequently, a positive daze has an impact of on the volatility, and a negative daze has an impact of on the volatility. If is greater than zero, the negative daze has an larger impact on the volatility and frailty versa. quantifies the continuity of dazes.

## Asymmetric-GARCH ( AGARCH )

The AGARCH theoretical account is another manner to parameterize the thought that different mark of daze with equal magnitude has a different impact on the volatility. The AGARCH theoretical account looks rather similar to the TGARCH theoretical account and is expressed as

If is a positive figure, the positive daze has a smaller impact on the volatility and frailty versa.

## Exponential-GARCH ( EGARCH )

The EGARCH theoretical account ( Nelson, 1991 ) does non hold the non-negativity restraint on the parametric quantity, and, in the above three GARCH theoretical accounts. And besides, the EGARCH theoretical account is the 3rd asymmetric sort of GARCH theoretical accounts in this paper. The theoretical account is expressed as

where

The positive value of indicates that the positive daze has an larger impact on the volatility and frailty versa.

## 3.3 Forecast rating

After the volatility has been forecasted, the predictors need to be evaluated to see how little the prognosis mistakes are. There are several loss maps for the prognosis mistake. The pick of loss maps wholly depends on investors ‘ penchants.

## Symmetrical loss map

The average mistake ( ME ) , average absolute mistake ( MAE ) and root mean squared mistake ( RMSE ) are the easiest and commonest ways to mensurate the prediction public presentation.

The consequence of the ME can be an index demoing the way of overprediction or underprediction on norm. As the mistakes by overprediction and underprediction offset each other, it is non surprising that the ME statistic normally is the lowest 1 among the others. Unlike the ME, the MAE does non hold the job of the countervailing consequence. The RMSE provides a measuring if the investor prefers to enforce a more to a great extent punishment upon the larger forecast mistakes.

## Asymmetrical loss map

As mentioned before, all investors have their single penchant for the loss maps. It is likely that investors have asymmetric loss maps instead than symmetric loss maps. Mentioning the past research ( Pagan and Schwert, 1990 ; Brailsford and Faff, 1996 ) , the mean assorted mistake is employed in this paper. First, the mean assorted mistake with a more to a great extent punishment on the prognosis mistakes by underpredicting is

Second, the mean assorted mistake punishing the overpredicted prognosis mistakes move to a great extent is

where O is the figure of overpredictions and U is the figure of underpredication in the out-of-sample prognosiss.