Stock monetary value index is one of the most interesting subject of world since, its volatility significantly affects fiscal society and a whole economic system, particularly in an emerging market like Vietnam. Thus, calculating the stock monetary value index is necessary for investors and market directions to hold its general hereafter form. In this survey, the day-to-day Vietnamese stock monetary value index ( Vnindex ) is obtained from the site of Hochiminh stock exchange from 28th July 2000 to 29th July 2012 utilizing Box-Jenkins methodological analysis and Generalized Autoregressive Conditional Heteroscedasticity ( GARCH ) attack. ARIMA ( 1,1,1 ) and GARCH ( 1,1 ) are appropriate theoretical accounts to calculating Vnindex. Based on the survey, although the information besides proves that the GARCH ( 1,1 ) theoretical account is better tantrum than ARIMA ( 1,1,1 ) theoretical account, the truth step of ARIMA ( 1,1,1 ) is smaller than those 1s of GARCH ( 1,1 ) . Therefore, it is required to utilize both ARIMA ( 1,1,1 ) GARCH ( 1,1 ) in calculating Vnindex.

In history of world, one of the most hard and ambitious procedure is anticipation the hereafter. It can be considered as an art of making statements about events whose existent 1s have non yet been happened. Forecasting hereafter is really critical for many stakeholders in different industries. For illustration, husbandmans would wish to cognize the hereafter rainfall form in order to decently seed their seeds while, fiscal investors expect to cognize the future public presentation of assorted market stocks to maximize their net income. Fiscal analysts use prognosiss to do fiscal programs and gaining outlook. Investors invest their in stocks with the outlook that they will acquire a positive final payment. Hence holding a good cognition about portion monetary value motion in the hereafter serves the involvement of fiscal professionals and investors. This cognition about the hereafter boosts their assurance by manner of confer withing and puting.

There are assorted of import variables in the economic system and societal of a state that can be forecasted such as gross domestic merchandise ( GDP ) , rising prices, unemployment rate, birth rate, mortality rate and import/export amongst others. Forecasting and analysing utilizing clip series method has been one of the most considerable field in recent old ages. This method has been used in agribusiness for rice outputs ( Shabri et al. , 2009 ) and the monetary value of chocolate cean ( Assis et al. , 2010 ) . In concern, it was besides employed in foretelling exchange rate ( Zhang, 2001 ) , ( Fahimifard et al. , 2009 ) and rough oil monetary values ( Kumar, 1992 ) .

During the last two decennaries, the surveies of clip series foretelling in portion monetary values, bond output and exchange rate motions were classified into two classs viz. : cardinal analysis ( Edward 1998, Kenneth 1994 ) utilizing macro economic variables and proficient analysis utilizing historical informations and graphs ( Harvey 1990, Diebold 1989 ) . Stock monetary values are excessively high volatility esteeming to the alterations in basicss ( Elena and Storis, 2009 ) . In the proficient analysis attack, merely historical clip series informations like the ex-date stock monetary value is used to happen the tendency and volatility of a procedure. If the investors can make their owned anticipations on the motion of the stock monetary values, they will do a determination faithfully to maximise their net income, to protect themselves from the loss or cut down it every bit much as possible.

This survey focal point on prediction clip series of stock index, Vietnamese stock index ( Vnindex ) , by using Autoregressive Integrated Moving Average ( ARIMA ) theoretical account and Generalized Autoregressive Conditional Heteroscedastic ( GARCH ) theoretical account. Although, Box-Jenkins methodological analysis method is powerful and flexible, it is non able to manage the volatility that is present in the information series. The volatility of Vnindex can be handled by utilizing GARCH theoretical account. The prognosiss obtained from the GARCH will be evaluated with the benchmark is ARIMA prognosis consequences.

Backgroud survey

This survey concentrates on the usage of 2 standard theoretical accounts in calculating future motion of Vnindex. While clip series prognosis utilizing ARIMA theoretical accounts was combined and shown by Box-Jenkins in 1970, the construct of conditional heteroscedasticity was foremost introduced by the Nobel Prize victor, Robert Engle in 1982. Both of this attack is really popular in calculating clip series informations. Bollerslev ( 1986 ) had modified Engle ‘s ARCH theoretical account into a more generalised theoretical account called GARCH theoretical account with is a simplified theoretical account to ARCH theoretical account but more powerful. The simple GARCH theoretical account is able to observe the fiscal volatility in a clip tendency.

Statement of the job

The monetary value of stock is extremely volatile throughout clip particularly, in developing markets like Vietnam. Since stock monetary value variableness does critical consequence on the whole economic system, the anticipation of future stock monetary value index becomes important.

This survey will research the undermentioned inquiry: which method between Box-Jenkins and GARCH performs better in calculating VNindex? ”

Aim of the survey

The aims of this survey follows as:

+ Estimating suited Box-Jenkins and GARCH theoretical accounts for calculating Vietnamese stock index.

+ Measuring the public presentation of both theoretical accounts in foretelling Vietnamese stock index.

+ Forecasting utilizing EViews package.

Scope of the survey

This survey focuses on the Box-Jenkins and GARCH theoretical accounts to calculate stock monetary value index in Vietnam. Since the stock index volatility is the chief concern, the survey uses merely day-to-day informations. The information is obtained from the site of Hochiminh stock exchange from 28th July 2000 to 29th June 2012. and analyzed by utilizing the EVIEWS.

Significance of the survey

Due to the high volatility of Vnindex, the appraisal of the clip series theoretical account must be able to observe its volatility. In add-on, the exactly Box-Jenkins and GARCH theoretical accounts will be determined when calculating the volatility of Vnindex. The procedure will be done with the assistance of Eview package. As a consequence of this survey, a theoretical account and package that can be used to calculate a volatile clip series can be proposed.

Outline of the survey

This survey looks at two standard methods of calculating Vnindex. It is organized into five chapters as follows:

+ Chapter 1: Introduction – this subdivision brings a background that constitutes the suitableness for the survey within the context of antecedently published survey.

+ Chapter 2: Literature reappraisal – in this subdivision utilizing the related literature in the field, a comprehensive theoretical model is developed. First, the development and function of Vietnamese stock exchange will be reviewed. Then, the volatility in stock market and clip series theoretical accounts are besides generated. Finally, relevant researches about the volatility of stock markets utilizing Box-Jenkins method and GARCH theoretical accounts are mentioned

+ Chapter 3: Methodology. In this portion, the methodological analysis applied for prediction of the same set of informations utilizing ARIMA and GARCH theoretical accounts are good explained.

+ Chapter 4: Analysis. This chapter presents the analysis of the same information sets utilizing the ARIMA and GARCH theoretical accounts. A comparing between the ARIMA and GARCH theoretical accounts are besides made.

+ Chapter 5: Decisions and suggestions – the whole survey are summarized and concluded with some suggestions for future survey.

Chapter 2

LITERATURE REVIEW

The literature reappraisal provides an insight position on the clip series anticipation. The layout of this chapter is as follows: foremost, the birth and development of the Vietnamese stock market is summarized. Second, the function and volatility of the stock index in the emerging market like Vietnam will be examined. Hence, in this subdivision, some of import causes impacting the significance alterations of the stock index will be discussed. Third, a treatment on two theoretical accounts ARIMA and GARCH that are used in this survey will besides be presented. While, clip series prognosis utilizing ARIMA theoretical accounts was combined by Box-Jenkins ( 1970 ) , the construct of conditional heteroscedasticity was foremost introduced by Engle ( 1982 ) . Those two indispensable clip series theoretical accounts are widely applied in assorted Fieldss. Finally, based on those surveies from other research workers ‘ plants, stock index monetary values forecast utilizing Box-Jenkins methodological analysis and GARCH attack is highlighted since these are the concentration of this survey.

2.1 Vietnamese stock exchange overview

The Vietnamese Stock Exchange, officially known as the Securities Trading Centre ( STC ) , located in Ho Chi Minh City, was launched on July 28th 2000. There were merely 22 listed companies with a market capitalisation of $ 144 million. Over five old ages of operation, the figure of listed companies have rised to 32 with a entire market capitalization of USD 398.96 million ( State securities commissionA of vietnam web site ) . In 2006, the index climbed 145 % and rose additionaly 60 % in the first three months of 2007. Although the market has significantly grown over the period, its graduated table is still really little. The universe fiscal crisis in 2008 lead the capitalisation down to 18 % from 50 % of Vietnamese GDP. Fortunately, the capitalisation rose to about 40 % GDP when the universe economic system started recover in 2009. The period from 2008 to 2009 witnessed a about dual addition in investing history from 500,000 to about 900,000. This is singular comparing with about 400 active histories in the beginning of the market.

Basically, Vietnamese stock market is supervised and managed by the State Securities Commission that is a portion of the Ministry of Finance. A jurisprudence on securities was adopted in June of 2006 to ease the development of the securities market quickly and sustainably. In 2010, Vietnamese State Securities Commission independently separated from the Ministry of Finance. Hence, Vietnamese stock markets operates more freely and efficaciously.

Although, Vietnamese stock market developed in quality and measure, investors are still limited in their ability to merchandise securities. In 2010, merely approximately 60 % of the market capitalisation of Vietnam ‘s stock market which valued at $ 33.2 billion, are freely traded.A This is because the authorities still holds commanding portions in big companies such as Vietcombank, Vinamilk, and Baoviet Holdings, the biggest insurance and fiscal services house in Vietnam.

2.2 Role of the emerging stock market

The function of fiscal development to the economic growing of developing states like Vietnam is one of the popular subjects of economic experts. By and large, well-developed stock markets are expected to speed up economic growing, by increasing domestic nest eggs and the quality of investings ( Singh, 1997 ) . They provide an extra fiscal instrument that may raise the nest eggs rate for an person ( Levine and Zervos, 1996 ) . Furthermore, in states that have developed stock markets, since companies are less dependent on bank funding, the hazard of recognition crunch is decreased ( Capasso, 2003 ) . Hence, stock markets tend to positively act upon economic growing through persons ‘ nest eggs, and better funding for houses.

Although, empirical grounds shows the being of a strong positive correlativity between stock market and economic growing, there are some statements such as Bencivenga and Smith ( 1991 ) , Adjasi and Biekpe ( 2006 ) , Naceur and Ghazouani ( 2007 ) A who looked at developing states. However, recent surveies concentrate on stock market as an engine of economic growing in emerging states like Vietnam. Levine and Zervos ( 1998 ) , for case, found a positive and important relationship between stock market development and long tally growing. Mohtadi and Agarwal ( 2004 ) use a dynamic panel method to analyze the relationship between economic growing and stock market development in 21 emerging markets over 21years. Their consequences besides suggest a positive correlativity between several indexs of the stock market public presentation and economic growing both straight and indirectly by hiking investing behaviour. Hence, in developing states, stock markets play an of import function in economic growing and development by helping nest eggs and altering hard currency flow from rescuers to investors.

2.3 Volatility of stock monetary value in the emerging market

In developing states such as Vietnam, although the high growing of fiscal markets really attractive, the volatility of return can be a major stumbling block for investors. Several documents analyzing the behaviour of liberalized stock exchanges ( Borenzstein and Gelos 2000 ; Froot, et al 1999 ) have found strong grounds of crowding, tendency chasing and impulse trading, all of which can take to increase the volatility of portion monetary values. It besides affects the economic system due to its consequence on consumer disbursement ( Starr-McCluer,1998, and Poterba, 2000 ) . A lessening in the stock market will weaken client ‘s assurance and drive down consumer ‘s disbursement and frailty versa. The volatility of stock market may besides straight influences concern investing ( Zuliu, 1995 ) and economic growing ( Levine and Zervos, 1996 ) . An addition in volatility can take to raise hazard of buying equity and to alter in financess to cut down hazardous assets. Thus the support cost of houses will be raised and investors will alter to purchase well-known houses ‘ stocks alternatively of new houses.

Theoretically, stock monetary values can alter due to universe macroeconomic variables and universe events. Fama ( 1981 ) investigates the strong relationship between stock monetary values and existent activity, rising prices and pecuniary policies. Hamao ( 1988 ) and Lee ( 1992 ) show that rising prices dramatically affected stock return. Beside, a negative relationship between both long term and short term involvement rates and stock returns is found by Gallic et Al in 1987. The exchange rate affects stock monetary values in a similar manner to the rising prices factor. A positive correlativity between stock monetary values andrevaluation of the US dollar is found by Aggarwal ( 1981 ) . Mukherjee and Naka ( 1995 ) besides find that in Japan and Indonesia exchange rate positively affects stock monetary values. Bilson et Al ( 1999 ) suggests that the exchange rate is one of the most influential factor among the other variables such as money supply, existent activity and goods monetary values play a small function.

By and large, universe events such as war, natural catastrophes or terrorist act significantly influence stock monetary values. They frequently occur in concatenation reactions and in both direct and indirect manner. After the terrorist onslaughts on September 11th 2001, the universe witnessed that many investors, particularly in the United States, halt or merchandise less and concentrate on less hazard stock and bond. Wars can be a clear illustration of an indirect influence on stock markets. The occurrence of a war lead to raise the monetary value of military equipment and arms makers. This addition in bend accretes the stock ‘s value of companies providing military equipments and engineering. This is similar to raise the demand for natural resources that would raise the monetary value of stocks of mining companies every bit good as natural resource processing 1s.

Another indispensable factor impacting the stock monetary value is dividend policy. The correlativity between dividend policy and volatility of the stock monetary value is enquired by many different research workers ( Baskin, 1989 ; Allen and Rachim, 1996 ) . This impact can be summarized that if there is an addition in dividend paid among the stockholders, the monetary values of the portion will travel up due to leveling demand. In contract, the monetary values of the portions will fall, if managers decide to administer less dividend among the stockholders.

2.4 Time Series Forescasting Models

The prognosis of stock monetary value is indispensable non merely for investors but besides for economic contrivers because it plays an of import function in the economic system of states. There are assorted types of theoretical account used to calculate clip series informations. However, in this survey, the two most popular 1s, ARIMA and GARCH, are applied.

In 19th century, Yule ( 1927 ) foremost conducted the impression of randomness by suggesting that every clip procedure can be considered as the realisation of a stochastic procedure. It can be seen as the birth of clip series prognosis. Since so, the construct of autoregressive ( AR ) and traveling norm ( MA ) theoretical accounts was determined continuing from that footing thought. Box and Jenkins ( 1970 ) combined the current cognition to organize a standard attack for clip series prognosis in their celebrated book Time series analysis: prediction and control ” . This book important impacts modern clip series analysis and prognosis in the both theory and pattern. The success of Box-Jenkins method is recorded on fact that the many theoretical accounts can make so adequately without necessitating assorted estimated parametric quantities in the concluding pick of the theoretical account. However, in the beginning of calculating utilizing ARIMA theoretical account, research workers faced a important job for in selecting of a theoretical account that there was no algorithm to find an alone theoretical account. Since so, numberous techniques and methods have been developed and proposed to add mathematical truth to the gauging procedure of the ARIMA theoretical account. The two necessity of them is Akaike ‘s information standard ( AIC ) and Bayesian information standard ( BIC ) or Schwarz standard ( SIC ) .

The autoregressive incorporate traveling norm ( ARIMA ) theoretical accounts provide methodological analysis and attack for clip series parametric quantity analysing and foretelling of average returns ( Box Jenkins, 1976 ) . This attack is one of a big household of quantitative prediction methods developed in the Fieldss of operations research, direction scientific disciplines and statistics. It is particularly appropriate for short-run anticipation because of its focal point on the recent yesteryear instead than the distant yesteryear ( Pankratz, 1983 ) . The last few decennaries have witnessed a well addition in suggesting ARIMA theoretical accounts to calculate univariate clip series informations. It was confirmed that ARIMA theoretical accounts are suited for agricultural predicting ( Fatimah and Roslan, 1986 ) . They besides were employed to calculate rice outputs ( Shabri et al. , 2009 ) , Cocoa Bean Price ( Assis et al. , 2010 ) . Box-Jenkins transportation map theoretical accounts are used by Liu ( 1991 ) to analyze the dynamic relationships between US gasolene monetary values, rough oil monetary values and the stock of gasolene. Kumar ( 1992 ) compares the prognosis truth of rough oil monetary values obtained utilizing clip series theoretical accounts with the truth of future monetary value prognosiss. Chinn et Al ( 2005 ) see the correlativity between topographic point and hereafters monetary values of energy trade goods. One of them was rough oil monetary values that was expected by ARIMA ( 1,1,1 ) . Furthermore, ARIMA theoretical accounts are a low research cost method compared with econometric theoretical accounts ( Shamsudin et al. , 1992 ) . Although ARIMA theoretical accounts are widely employed in many practical applications, it can non capture nonlinear forms of complex clip series when nonlinearity exists.

A chief characteristic of fiscal clip series is that a big return is likely to be followed by another big return which means there are high volatility show periods. This phenomenon is defined as volatility constellating in econometrics and finance. Autoregressive Conditional Heteroskedasticity ( ARCH ) theoretical accounts points that a clip series informations relates its ain lagged informations. The attack was foremost introduced by Engle ( 1982 ) to foretell UK rising prices. In this category of theoretical accounts, dynamic alterations in conditional discrepancy can be described as a deterministic map of past returns.

ARCH theoretical accounts were extended by Bollerslev ( 1986 ) into Generalized Autoregressive Conditional Heteroskedasticity ( GARCH ) theoretical accounts that portion many belongingss and supply better consequence. Sabbatini and Linton ( 1998 ) study that the simple GARCH ( 1,1 ) theoretical account determine good parametric quantities for the day-to-day returns of the Swiss market index. Fahimifard et Al ( 2009 ) use R2, MAD And RMSE standards to compare the prognosis truth of exchange rate determined by ARIMA and GARCH. Their survey shows that GARCH theoretical accounts outperforms AIRMA theoretical accounts. These theoretical accounts have been loosely employed to calculate several clip series informations, including stock monetary values ( Schwert 1989, Hamilton and Susmel, 1994, Cho and Engle 1999 ) , exchange rates ( West and Cho 1994, Campa and Chang 1997 ) and involvement rates ( Edwards 1998, Boscher et al 2000 ) . It is reported that there is a negative correlativity between returns and conditional discrepancy of the following period ‘s returns. A negative or positive returns are comparatively connected to upward or downward alterations of the conditional volatility. Engle and Ng ( 1993 ) refer this phenomenon to asymmetric volatility. The consequence of dissymmetries on the out-of-sample prognosis ability of assorted GARCH theoretical accounts is found by Awartani and Corradi ( 2005 ) . In 2006, Zhou et al employed ARIMA and GARCH to construct a web traffic calculating theoretical account. They besides found that in footings of calculating truth, ARIMA/GARCH is better than the Fractional Autogressive Intergrated Moving Average ( FARIMA ) . The assorted ARIMA/GARCH theoretical account besides were confirmed that it outperformed ARIMA and GARCH for foretelling Tawau Cocoa bean monetary values ( Assis, 2010 ) .

2.5 Relevant research in stock monetary value.

Since Box-Jenkins method is widely used, there are many surveies using them for calculating stock monetary value volatility. The volatility of S & A ; P500 index is foremost examined by Poterba and Summers ( 1986 ) utilizing AR ( 1 ) procedure. It is besides described by Gallic et Al ( 1987 ) who applied ARIMA ( 0,1,3 ) theoretical account. Schwert ( 1990 ) estimates a additive AR ( 12 ) theoretical account for stock returns utilizing day-to-day informations of S & A ; P composite portfolio. Recently, Al-Shiab ( 2006 ) employs ARIMA ( 4,1,5 ) to foretell the Amman stock exchange daily general index. Although the usage of such additive theoretical accounts rises, they are non strong to explicate chief characteristics of volatility series. Many empirical groundss show that, for illustration, the returns have a inclination to exhibit outliers. This means a big discrepancy is likely to be followed by another big discrepancy. Those drawbacks of additive theoretical account motivate research workers to utilize nonlinear theoretical accounts such as ARCH household theoretical accounts in clip series prediction.

The ARCH category theoretical accounts including the ARCH theoretical accounts of Engle ( 1982 ) , the generalized ARCH ( GARCH ) theoretical accounts of Bollerslev ( 1986 ) and exponential GARCH ( EGARCH ) theoretical accounts of Nelson ( 1991 ) are the most common nonlinear theoretical accounts used in the finance survey. They are appropriate to use to fiscal clip series informations that have bunchs of outliers and heavy tailed distributions. Akgiray ( 1989 ) , in his survey using GARCH ( 1,1 ) theoretical account to day-to-day stock returns, suggests that GARCH is the best theoretical account depicting and calculating stock market volatility. The prognostic ability of additive and non-linear theoretical accounts in forecast day-to-day S & A ; P 500 hereafters index volatility is observed by Najand ( 2002 ) . This survey shows that nonlinear GARCH theoretical accounts dominate additive theoretical accounts by utilizing the RMSE and the MAPE truth steps.

McMillan et Al ( 2000 ) surveies United Kingdom ( UK ) stock market volatility utilizing monthly, hebdomadal and day-to-day informations. Their consequence suggests that GARCH and traveling mean theoretical accounts bring marginally superior day-to-day volatility prognosiss. The truth of day-to-day stock monetary values volatility forecast in New Zealand market of nine different theoretical accounts is evaluated by Yu ( 2002 ) . This survey confirms that the stochastic volatility theoretical accounts provide the best consequence among the other theoretical accounts. It besides shows that the public presentation of ARCH category theoretical accounts strongly depend on the signifier chosen.

In general, the volatility of stock returns in well-developed stock markets are examined many times. The major stock markets in the United State are consider by Poterba and Summer ( 1986 ) , Akgiray ( 1989 ) and Najand ( 2002 ) . Similarly, the 1s in UK are observed by Dimson and Marsh ( 1990 ) , McMillan, Speight and Gwilym, ( 2000 ) . Asia Pacific stock markets, for illustration Japan ( Tse, 1991 ) , Singapore ( Tse and Tung, 1992 ) , Australia ( Brailsford and Faff, 1996 ) , New Zealand ( Yu, 2002 ) in bend are besides inspected. However, there are few studis about on immature stock markets like the 1s in Amman ( Al-shiab, 2006 ) and Pakistan ( Rashid and Ahmad, 2008 ) stock exchange.

In this paper, the ARIMA theoretical accounts and GARCH ( 1,1 ) are applied to calculate the day-to-day stock monetary value index in the comparatively developing stock exchange market, viz. Vietnamese Stock Exchange.

Chapter 3

Methodology

This chapter describes the clip series attacks to gauge and calculate stock monetary values including types of prediction, patterning techniques and brief description of foretelling theoretical accounts used for optimisation of mistake prosodies. From the literature, the stock monetary values can be estimated and expected by assorted statistical attacks. However, in this survey, ARIMA and GARCH theoretical accounts are applied utilizing the historical information to calculate for the hereafter Vnindex.

Autoregressive integrated traveling norm ( ARIMA ) theoretical accounts can calculate accurately based on of historical stationary clip series forms. Since stock monetary values in the emerging market like Viet Nam fluctuate over the clip, a heteroscedasticity attack shall be tested for the full information series. Hence, GARCH theoretical account is besides used to capture volatility constellating in stock monetary values clip series. Then both forecast consequences of two theoretical accounts are compared utilizing mistake statistics.

3.1 Box-Jenkins attack

In this portion, the prognostic theoretical account used to foretell stock returns will be outlined. Over the last decennaries, faculty members have used a huge graduated table off different theoretical account specifications to prove for stock return anticipation. Some use non-parametric methods or let prognostic variables to exhibit non-linear forms. However, specification becomes a critical issue when sing these types of theoretical accounts. Goyal and Welch ( 2008 ) besides indicate out that some of these theoretical accounts are bound to work both in in-sample and out-of-sample. Box and Jenkins ( 1976 ) proposed the Autoregressive Integrated Moving Average ( ARIMA ) theoretical accounts which have been used for a broad assortment of clip series calculating application. They are employed in instances where information is a stationary procedure or its non-stationary characteristic can be removed. By and large, Box-Jenkins methodological analysis consists 4 stairss affecting theoretical account designation, appraisal, theoretical account checking and prediction.

3.1.1 Fiscal clip series and stationary procedure

Fiscal clip series yt can be describe as:

Yt= { Yt-1, Yt-2, .. , Y0 }

A procedure is stationary when its mean, discrepancy changeless and covariance between two 2 times merely depend on their distance and the hold and independ on the existent clip which covariance is calculated. The stationarity of the clip series is tested by utilizing correlograms, Q-statistics or Augmented Dickey Fuller ( ADF ) unit root trial.

The correlograms involve autocorrelation maps ( ACF ) and partial autocorrelation maps ( PACF ) . Figure 3.1 shows an illustration of correlograms and Q-statistics.

Figure 3.1 An illustration of correlogram and Q-statistics from Eviews

In extra, Q-statistic or ADF trial can be used to look into the stationary of a procedure.

3.1.1.1 Q-statistics

The Q-statistics at slowdown K is a trial standard for the void hypothesis that there is no autocorrelation up to order k. This statistic can be estimated as:

( 3.1 )

Where is the j-th autocorrelation

N is the figure of these observations

Under the void hypothesis, Q-statistic is asymptotically distributed as a I‡2 with grades of freedom equal to the figure of autocorrelations. If the chance of Q-statistic at any slowdown is less than 0.05 so the void hypothesis is rejected for all slowdowns. ( Mishra et al, 2010 ) . Therefore, the procedure is stationary.

3.1.1.2 Augmented Dickey-Fuller Test

The stationary of a information set can be tested by the augmented Dickey-Fuller ( ADF ) trial. This trial is the improved version of Dickey-Fuller ( 1979 ) trial of the model that provides a parametric rectification for higher-order correlativity by presuming that the procedure follows an AR ( P ) procedure. This trial besides addes Ps lagged difference footings of the dependant variable Y to the right-hand side of the arrested development as follow:

( 3.2 )

The void hypothesis of this trial is H0: = 0 informations is non-stationary which mens that differencing must be done to do it stationary. The alternate hypothesis H1: & lt ; 0 which means that the information is likely to stationary and can be analyzed without differencing the information. The trial statistic is computed as

( 3.3 )

Where is the estimation of coefficient

Se ( ) is the coefficient criterion mistake

This trial statistic is compared with the relevant critical value from Dickey-Fuller trial. The void hypothesis will be rejected if the alsolute value of trial statistic less than the critical value.

3.1.2 Autoregressive-moving-average theoretical account ( ARMA )

3.1.2.1 Autoregressive theoretical account ( AR )

An autoregressive procedure is a difference equation which shows the current value of a series as map of the old values. This means the value of discrepancy at clip T yt depends linearly on its past values. The order term P determines how many past values are to be included in the difference equation to gauge the current value. A difference equation relates a variable Yt at clip T with its old values ( Horvath, et Al, 2006 ) . The AR ( P ) theoretical account with pth order AR of a clip series is written:

Yt= c + I†1yt-1 + I†2yt-2 + aˆ¦+ I†t-pyt-p+ Iµt ( 3.4 )

=

The coefficients I†i can be estimated by ordinary least squares regression while degree Celsius is a changeless and Iµt is white noise. The theoretical account remainsA stationary when the value of parametric quantity.

The least squares method is applied to gauge the value of I† . It minimizes the amount of square of mistakes for the ascertained values with regard to I† .

( 3.5 )

Solving equation ( 3.5 ) to acquire the expected value of I† :

From the estimated value of, distribution of mistake footings can be represented

Replacing the estimated value of and distribution of mistake informations to the equation ( 3.5 ) , the theoretical account can be fitted.

3.1.2.2 Moving-Average theoretical account ( MA )

A clip series procedure is affected by random dazes in the noisy environment. Therefore, the random dazes of old values act upon the current value of series. In traveling mean theoretical accounts, the current value of discrepancy at clip T yt depends linearly on the lagged value of the residuary in the same period and the old. The MA ( Q ) theoretical account is written

( 3.6 )

The parametric quantities can be estimated by a dedicated ARMA modeling plan, is the expected valu of Yt which is frequently assumed to be 0, and are white noise mistake term.

3.1.2.3 Autoregressive Moving-Average theoretical account ( ARMA )

The autoregressive Moving Avergae ( ARMA ) theoretical account contains 2 theoretical accounts AR ( P ) and MA ( Q ) . The autoregressive theoretical account and moving mean theoretical account can be used to come close any stationary procedure. Uniting two equations ( 3.4 ) and ( 3.6 ) , ARMA theoretical account of order P and Q is created:

( 3.7 )

3.1.3 Autoregressive Intergrated Moving Average Process – ARIMA ( P, vitamin D, Q )

The ARMA theoretical account assumes that the clip series informations is stationary. However, the existent procedures are non stationary in nature. A differencing procedure is applied to tranforming a stationary clip series to non-stationary. The ARMA theoretical account whose clip series is made stationary by differencing, is called as Autoregressive Intergrated Moving Average ( ARIMA ) theoretical account.

Deducing from the ARMA theoretical account, the ARIMA theoretical account besides consists p order of autoregressive theoretical account and q order of traveling mean theoretical account. However, it besides involves an extra parametric quantity viz. , 500 order times of differencing. This theoretical account takes and applies historical informations to an autoregressive ( AR ) procedure that contains memory of old events. An Intergrated ( I ) procedure makes informations go stationary for easy anticipation and a Moving Average ( MA ) procedure of prognosis mistakes. Box-Jenkins has specified four stairss for constructing an ARIMA theoretical account:

Model designation: finding values of P, vitamin D, Q

Model appraisal: gauging parametric quantities of the theoretical account

Model checking: sing whether theoretical account tantrums informations or non ; if non see another 1

Forecasting utilizing the best selected theoretical account

3.1.3.1 Model designation

Box-Jenkins method can non be applied to a non stationary clip series procedure. Therefore, sing whether the series is stationary or non, is one of of import undertakings. This undertaking can be done by utilizing the graph of ACF and PACF or cheking through Augmented Dickey-Fuller or Phillips-Perron ( PP ) Unit Root Test.

The autocorrelation at slowdown K is the correlativity coefficient for values of the series K periods apart. While, the partial autocorrelation at slowdown K is the correlativity of Y values that are thousand periods apart after taking the correlativity from the intervening slowdown. In EView, the autocorrelation and partial autocorrelation at lag K of a procedure is computed by

( 3.8 )

( 3.9 )

Where is mean of series

is estimated autocorrelation at slowdown K

A stationary procedure has a graph of ACF which cuts off rapidly or dies down rapidly. In the antonym, if the graph of ACF dies down really easy, the clip series values will be considered non-stationary. The values of ACF and PACF are compared with to see the significance of them. The behaviour of the ACF and PACF of theoretical accounts can be generalized as table 3.1.

Model

ACF

PACF

AR ( P )

Dies down

Cut of after slowdown P

MA ( Q )

Cut of after slowdown Q

Dies down

ARMA ( P, Q )

Dies down

Dies down

Table 3.1: The behaviour of ACF and PACF for each of the general theoretical accounts

In this attack, differencing can be done non-stationary clip series until it becomes stationary. The figure of differences required to accomplish stationary is denoted by term vitamin D of ARIMA ( P, vitamin D, Q ) .

Harmonizing to the rule of parsimoniousness, simple theoretical accounts are preferred to complex theoretical accounts when all things being equal ( Hanke et al. , 2001 ) . It is comparatively easy to happen a theoretical account with big figure of parametric quantities that fits the limited information well. However, calculating from this theoretical account is likely to be hapless because there are excessively much fluctuation in the informations due to random mistake is being modelled. The accomplishment is to develop the simplest theoretical account that bringe an equal description of the major characteristics of the informations.

3.1.3.2 Model appraisal

Since a possible theoretical account has been chosen, the parametric quantities for that theoretical account must be determined. They can be estimated by utilizing the least square arrested development method. This method minimizes the amount of squares of the fitting map. Once the least squares estimations and their standard mistakes are found, values of parametric quantities can be constructed and interpreted in the usual manner. Parameters that are judged dramatically different from nothing are retained in the fitted theoretical account ; parametric quantities that are non important are dropped from the theoretical account.

3.1.3.3 Model checking

In this phase, the estimated theoretical account must be checked for adequateness. Basically, a theoretical account is adequacy when the remainders has the normal distribution and should be random. The Ljung-box Q statistic based on the the size of the residuary autocorrelation secret plan is one common trial for entropy. The Ljung-Box trial can be written:

H0: The procedure is independently distributed

H1: The procedure is non independently distributed.

The trial statistic Q is:

( 3.10 )

Where:

rk ( vitamin E ) = the residuary autocorrelation at slowdown K

n = figure of remainders

m = figure of clip slowdowns includes in the trial

If the p-value of Q-statistic is little ( less than important value ) , the theoretical account is inadequacy. Then, a new or modified theoretical account must be considered and checked until the satisfied 1s has been found. Diagnostic checking plays an of import function when two simple viing theoretical accounts may adequately depict the informations and a pick may be made on the footing of the nature of the prognosiss.

3.1.3.4 Forecasting

Once a best theoretical account has been estimated, it will be used for to anticipate the future values of a series utilizing the old values. Although the fittest ARIMA theoretical accounts frequently involve differences, the anticipation for the original series can be computed straight from it. Forecast can be in-sample or out-sample. In this survey I am aware of the in-sample prognosis. This measure besides of import since the determinations made today will depend on the consequences of prognosis.

3.2 Standard additive arrested development

The standard additive arrested development theoretical account may be written as:

Lolo = I?0 + xiI?i + Iµi ; i= 1,2, aˆ¦ , N

Lolo: additive combination of parametric quantity

elevens: independent variable

I?i: coefficients

Iµi: mistake footings

3.2.1 Arrested development Consequences

The illustration of coefficient consequences that are conducted in EView will be illustrated in figure 3.2

Figure 3.2 An illustration of equation end product from EViews

3.2.1.1 Arrested development Coefficients

The column labelled Coefficient ” in Figure 3.2 represents the estimated coefficients. The least squares coefficients I? are estimated by the standard ordinary least square expression:

I’= ( X’X ) -1X’y

In such sort of theoretical account, the coefficient measures the fringy part between the independent and dependent variables. Generaly, in the arrested development consequences in EView, the coefficient of the variable degree Celsius is a changeless the arrested development. It is the base-level of the prognosis because all of the other independent variables can be zero. The other coefficients are described as the incline of the correlativity between the corresponding independent and dependent variables, presuming all other variables do non alter.

3.2.1.2 Standard Mistakes

The Std. Mistake ” column in Figure 3.2 shows the determined standard mistakes of the estimated coefficients. They measure the statistical dependability of the coefficients. Hence, the larger the standard mistakes, the more statistical noise is present in the estimations. They are defined as the square roots of the diagonal facets of the coefficient covariance matrix.

3.2.1.3 t-statistics and chance

The t-statistic is an of import step used to prove the hypothesis that a coefficient value is zero. It is defined as the ratio of an estimated coefficient to its standard mistake. This term determines the chance of the value coefficient is equal nothing. If normalcy can merely keep asymptotically, so z-statistic will be used alternatively of t-statistic.

The chance in the last column is the p-value or the fringy significance degree. The p-value is a standard step used to reject or accept the hypothesis that the true coefficient is zero. The p-value are computed from a t-distribution with n-k grades of freedom.

3.2.2 Arrested development statistics

All arrested development statistics is shown in the estimated equation below arrested development consequence as the illustration in Figure 3.2. These are R-squared, adjusted R-squared mistake of the arrested development, sum-of-squared remainders, log likeliness, Durbin-Watson statistic, Akaike Information Criterion, Schwarz Information Criterion and F-statistic.

3.2.2.1 R-squared

The R-squared ( R2 ) statistic determines the success of the arrested development in the anticipation of dependant variable value within the sample. A R2 is the proportion of variableness in a procedure that is computed for by the statistical theoretical account. R2 statistic will be one and zero if it fits no better than the simple mean of the dependant variable. This means consequences of the arrested development is absolutely suited for the information set. R2 is estimated in EViews as:

( 3.11 )

Where is the mean of the dependant variable.

3.2.2.2 Adjusted R-squared

R2 has a job is that it ne’er falls even more regressors have being added. Adjusted R2 ( is created to work out this job. If regressors are added to the theoretical account, its will decreased. value can be negative for ill fitting theoretical accounts. Its absolute value is normally less than or equal to that ofA R2. The adjustedA R2A is defined as:

( 3.12 )

3.2.2.3 Standard Error of the Arrested development

The standard mistake of the arrested development can be determined by utilizing estimated discrepancy of the remainders. It is computed as:

( 3.13 )

3.2.2.4 Sum-of-squared Remainders

The sum-of-squared remainders is a drumhead measuring that can be employed in assorted of statistical computations.

( 3.14 )

3.2.2.5 Durbin-Watson Statistic

The Durbin-Watson ( DW ) statistic is a trial statistic applied to observe the consecutive correlativity in the remainders. This statistic is computed as

( 3.15 )

By and large, the value of DW statistic ever lies between 0 and 4. If the DW statistic equal nothing, there will be no consecutive correlativity. The consecutive correlativity is positive when the statistic less than 2. In the antonym, if DW value greater than 2, the consecutive correlativity is negative.

3.2.2.6 Mean and Standard Deviation

The mean and standard divergence of the dependant variable Y can be estimated by utilizing the standard formulars:

( 3.16 )

( 3.17 )

3.2.2.7 Log likeliness

A likeliness ratio testA is aA statistic trial applied to compare the tantrum of two theoretical accounts, the nothing and the alternate theoretical account. When the logarithm of the likeliness ratio is used, this statistic becomes a log likeliness ratio statistic. It is know as the difference between the log likeliness values of the restricted and unrestricted equation. The log likeliness is evaluated in Eview as:

( 3.18 )

3.2.2.8 Akaike Information Criterion

The Akaike information standard is a step for the fittest of statistical theoretical account. Akaike ( 1973 ) suggests this standard in chosing theoretical account procedure. The thought is minimising the negative likeliness penalised by the figure of parametric quantities. The smaller the values of the AIC is, the better theoretical account is. The AIC is computed as:

( 3.19 )

Where cubic decimeter is the log likehood

3.2.2.9 Schwarz Information Criterion

The Scharz Information Criterion ( SIC ) is a standard for theoretical account choice among a specific set of theoretical accounts. It is an alternate critera to AIC that enforces a larger punishment for extra coefficient.

( 3.20 )

3.2.2.10 F-Statistic

The F-statistic in figure 3.2 is is a standard statistic used to prove of the hypothesis that all the incline coefficients in a arrested development is zero. For OLS theoretical accounts, F-statistic is computed as:

( 3.21 )

The p-value below the F-statistic is the fringy significance degree of the F -test. If the P -value is less than the significance degree we are proving, say 0.05, we reject the void hypothesis that all incline coefficients are zero.

3.3 Generalized Autoregressive Conditional Heteroskedasticity ( GARCH )

The white noise mistake term in the ARMA theoretical account is defined as ARCH theoretical account by Engle ( 1982 ) in his celebrated paper Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation ” . This term can be shown as

( 3.22 )

Where is clip depent standard divergence and can be modeled by ARCH ( Q ) procedure

( 3.23 )

Bollerslev ( 1986 ) generalized ARCH theoretical accounts by adding the lagged conditional discrepancy. GARCH theoretical account allow for autoregressive and moving mean elements in the heteroscedastic discrepancy. The mistake processes in GARCH can be written as

( 3.24 )

( 3.25 )

Where are invariables

Since is a white noise procedure, the conditional and unconditioned agencies of are equal zero and the conditional discrepancy of is

( 3.26 )

The GARCH ( P, Q ) allows both AR and MA constituents in the heteroscedastic discrepancy. If p = 0 and q = 1, so GARCH ( P, Q ) will be GARCH ( 0,1 ) or ARCH ( 1 ) . Therefore, if all values of equal nothing, the GARCH ( P, Q ) is tantamount to the ARCH ( Q ) theoretical account. The advantage of the GARCH theoretical account is that it is much easier to place and gauge a higher order ARCH theoretical account with more penurious GARCH representation. It is perfectly true because all coefficients ( I± , I? ) when gauging an ARCH or GARCH theoretical account must be positive. Furthermore, all characteristic roots in the discrepancy equation must lie inside the unit circle to guarantee that the discrepancy is finite. Therefore, More penurious theoretical accounts will intend few restricted coefficient.

An of import characteristic of GARCH is that the conditional discrepancy of the perturbations of the yt series is an ARIMA procedure. Therefore, the remainders expected from a fitted ARIMA theoretical account should demo their characteristic forms. For illustration, if a fitted ARIMA theoretical account of the series is appropriate, the correlogream of ACF and PACF will demo the remainders should be a white-noise procedure. However, if a GARCH theoretical account is more desirable, the ACF of the squared remainders should assist place the procedure.

3.3.1 Garch ( 1,1 )

GARCH ( 1,1 ) is the most popular GARCH theoretical accounts since it is easy to use to many informations series. This theoretical account ‘s specification is:

yt = Aµ + Iµt

Iµt = I?tI?t

I?t ~ iid N ( 0,1 )

I?t2=I±0 + I±1 Iµ2t-1 + I?1I?2t-1

I±0 & gt ; 0 ; I±1, I?1 a‰? 0

Where yt is the dependent variable over period T

Aµ is a changeless mean

I?t2 conditional discrepancy

The conditional discrepancy contains a changeless term ( I±0 ) , volatility from the old period, measured as the slowdown of the squared remainder from the average equation ( Iµ2t-1 ) ; last period ‘s prognosis discrepancy ( I?2t-1 ) . The last status ensures non-negative conditional discrepancies, while the status 0 & lt ; I±1 + I?1 & lt ; 1 ensures that the discrepancy of the unconditioned return is stationary and finite. Since the intercept I±0 is ever positive, it allows analyst to pattern the volatility as mean-reverting. If the volatility is high, it will be given to fall over clip, and if the volatility is low, it will be given to lift over clip. ( Dowd, 2002 ) . This specification frequently appears in a fiscal context, where bargainers and agents predict this period ‘s discrepancy by organizing a leaden norm of a long term norm ( I±0 ) , information about volatility witnessed in the old period ( Iµ2t-1 ) and the forecasted discrepancy from last period ( I?2t-1 ) . If the plus return is all of a sudden big in either the upward or the downward way this period, the estimation of the discrepancy will be increased in the following period.

In fiscal Fieldss, it is common to happen GARCH slowdown coefficient ( I?1 ) over 0.7 and GARCH return coefficients ( I±1 ) ever less than 0.25. The size of these parametric quantities represent the form of the ensuing volatility clip series. A high value of I?1 agencies that volatility is relentless and it takes a long clip to alter, while a high value of I±1 show that volatility is peaky and sensitive with market alterations ( Alexander, 1998 ) .

3.3.2 Parameter Appraisal

The OLS method understating the residuary amount of squares, which depend merely on the conditional average equation and do non related to conditional covariance, can non be employed to gauge GARCH theoretical accounts. Another technique known as maximal likeliness is applied in order to gauge GARCH theoretical accounts.

By and large, this attack focal point on happening the most likely values of the parametric quantities given the existent information. In peculiarly, a log likeliness map is formed and the values of the parametric quantities that maximise it are fought. It includes three stairss: foremost, finding the appropriate equations for the mean and the discrepancy, so, specifing the log-likelihood map to maximize under normalcy premise for the perturbations. Finally, maximizing the log likeliness map and bring forthing parametric quantity values that building their standard mistakes ( Brooks, 2008 ) .

The log likeliness map in this method is estimated as:

( 3.27 )

Harmonizing to the equation ( 3.27 ) , log likelihood map can be maximised by understating, since there is a negative mark in the map and nlog ( 2Iˆ ) is a changeless. Understating these footings is non a simple undertaking for time-varying discrepancies. Fortunately, EViews are able to choose the parametric quantity values that maximize this log likeliness map.

3.3.3 Diagnostic Checking

In general, a fitted GARCH theoretical account gaining control all dynamic elements of the theoretical account of the mean and the theoretical account of the discrepancy. Its estimated remainders should besides be serially uncorrelated and should non expose any staying conditional volatility. This means each residuary utilizing its conditional criterion divergence, ; and the resulting series should hold a mean and a discrepancy of integrity ( Enders, 2004 ) .

The Ljung-Box Q- statistics of the serie and the squared standardised remainders are formed to prove the theoretical account of the mean and the staying GARCH effects severally. If there are any consecutive correlativities in the, the theoretical account of the mean will non be decently specified. Therefore, the void hypothesis that Q-statistic equal nothing will non be rejected. If there is no staying GARCH effects, the void hypothesis that the sample values of the Q-statistics are equal to zero, should non be rejected. Hence, the belongingss of the serie should mime those of a white noise procedure.

3.3.4 Forecasting

The one-step-ahead prognosis of the conditional discrepancy is simple to obtain. The following period discrepancy will be represented as:

( 3.28 )

Taking the conditional expection will give:

( 3.29 )

Since the expected value of equal the expected value of E ( ) = E ( ( Brook,2008 ) so equation ( 3.29 ) equivalent

( 3.30 )

Day t+2 and t+3 will be:

On twenty-four hours t+n in the hereafter, we have:

Replacing with the value of so the equation above will be:

The forecasted discrepancy for the whole hebdomad would merely be the amount of the i¬?ve day-to-day discrepancy prognosiss. If the standard divergence is the needed volatility estimation instead than the discrepancy, merely take the square root of the discrepancy prognosiss. Note besides, nevertheless, that standard divergences are non linear. Hence, if day-to-day criterion divergences are the needed volatility step, they must be squared to turn them to discrepancies. Then the discrepancies would be added and the square root taken to obtain a hebdomadal criterion divergence.

3.4 Accurate prediction trials

There are assortment ways to measure the truth of prediction theoretical accounts. In this survey, intend absolute mistake, root mean squared mistake, and intend absolute per centum mistake will be used as rating standards. These error statistics are used to compare the truth of ARIMA and GARCH theoretical accounts in this survey.

3.4.1 Mean Absolute Error

The average absolute mistake ( MAE ) which represents the typical mistake is a truth step used to find how close prognosiss or anticipations are to the existent results. MAE is estimated from:

( 3.31 )

Where is forecasted value of existent value

3.4.2 Root Mean Squared Error

The root mean squared mistake is one of the most popular dimensioned statistics to analyze the public presentations of calculating theoretical account. RMSE is calculated as:

( 3.32 )

Similarly to MAE, RMSE depends on the graduated table of the dependant variable. However, with the fixed Numberss of mistake ( n ) , RMSE is ever larger than MAE because of the square root of the figure of mistakes. Althought, MAE and RMSE can be used together to analyze the mistakes in a set of prognosiss. MAE is extremely recommended to utilize because it is natural step of mean mistake every bit good as unambiguous ( Willmott, C.J, Matssura, K 2005 ) .

3.4.3 Mean Absolute Percentage Mistake

Mean absolute Percentage Error is the norm of absolute per centum sum by which prognosiss differ from results. It can be estimated form the undermentioned equation:

( 3.33 )

The MAPE is about the same as MAE except that it expresses truth as per centum. This statistic is appreciated when sample size is little, forecast mistakes for each period can be presented in the per centum signifier. It will be helpful in doing comparing among prognosiss from different units of step.

Chapter 4

Analysis AND RESULTS

4.1 Introduction

Forecasting stock monetary value volatility is really of import to fiscal market. In this chapter the consequences of the public presentation refering both calculating methods and mistake prosodies are analysed. First, using 4 stairss Box-Jenkin attack to find, estimation and prognosis by ARIMA theoretical account, followed by sing Garch theoretical account. Both theoretical accounts are analysed by utilizing EView package. Finally, public presentations of these two theoretical accounts for calculating day-to-day stock monetary value informations are besides compared.

4.2 Data Management

In this survey, Vietnamese stock monetary value index ( Vnindex ) is obtained from the site of Hochiminh stock exchange from 28th July 2000 to 30th July 2012. The informations are divided into two parts. The first portion that is from 28th July 2000 to 29th June 2012, is used to place and gauge ARIMA and GARCH theoretical accounts. The 2nd one varies from 30th June 2023 to 30th July 2012. This information set is a bendmark for the out-of-sample prognosis consequence anticipating from appropriate ARIMA and GARCH theoretical accounts.

4.3 Vnindex clip series

The Vnindex will be determined the tendency of the seires of being changeless, additive or non-linear. The information from 28th July 2000 to 29th June 2012 will be illustrated in Figure 4.1.

Figure 4.1 The clip series for Vnindex

By and large, Vnindex have fluctuated in the scope from 100 to 1200. It can be considered that this procedure ‘s features is non-linear ( Fan et al. , 2008 ) . At the beginning at 100 on from 28th July 2000, Vnindex increased to about 600 before it declined to about the starting point at the terminal of 2003. The 2004-2007 period witnesses a important up tendency of Vnindex which was at extremum on March 2007. However, it started to dramatically diminish from the terminal of the same twelvemonth because of the universe fiscal crisis. This index has fluctuated around 400 from 2010 until now.

Figure 4.2 Histogram and normality trial on Vnindex

From the histogram, it can be seen that a great figure of observations are located around 390. As summarized in Figure 4.2, the mean and standard divergence of vnindex are 409.82 and 235.47 severally. The value for lopsidedness is 1.3351 and kurtosis is 4.4612 which imply that the graph is asymmetric and leptokurtosis. Jarque-Bera trial indicates that we do non reject void hypothesis of being normal distribution at 5 % significance degree.

4.4 Stationary procedure

Box-Jenkins attack is merely applied for stationary clip series. The unit roots trial such as ADF and PP trials can be used to find the stationary of a series. Initially, the stationary of the original Vnindex must be checked foremost.

Null Hypothesis: VNINDEX has a unit root

Exogenous: Changeless

Lag Length: 5 ( Automatic based on SIC, MAXLAG=27 )

t-Statistic

A A Prob. *

Augmented Dickey-Fuller trial statistic

-1.743874

A 0.4090

Test critical values:

1 % degree

-3.432494

5 % degree

-2.862373

10 % degree

-2.567258

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.1 ADF trial for Vnindex

Accordin to postpone 4.1, the ADF trial statistic is a?’1.7439 which is greater than test critical values of a?’3.4333, a?’2.8624 and a?’2.5673 at 1 % , 5 % and 10 % significance degrees. The p-value of 0.4090 strongly disagrees that the series is stationary. Therefore, the void hypothesis that the information set is a non-stationary procedure, is accepted.

Null Hypothesis: VNINDEX has a unit root

Exogenous: Changeless

Bandwidth: 23 ( Newey-West utilizing Bartlett meat )

Adj. t-Stat

A A Prob. *

Phillips-Perron trial statistic

-1.807177

A 0.3774

Test critical values:

1 % degree

-3.432490

5 % degree

-2.862371

10 % degree

-2.567257

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.2 PP trial for Vnindex

Vnindex besides was tested with PP trial and the consequence are shown in Table 4.2. A similar illation is obtained where the PP trial statistic of a?’1.1807 is greater than its trial critical values at 1 % , 5 % and 10 % significance degrees. Since the nonreversible p-value is 0.3774, the void hypothesis of being non-stationary is accepted. Therefore, this clip series need to be differenced to obtain a stationary series.

Because the original Vnindex is a non-stationary clip series, the stationarity of the first order difference of Vnindex in bend will be checked. However before taking difference, natural logarithms are applied on Vnindex. Harmonizing to ( Lutkepohl, H.A andA Xu, F.,2012 ) , forecasts yt which can be described as yt = logxt, so change overing the consequence to x1 can be more precise than direct foretelling on crosstalk if yt has a more stable discrepancy than crosstalk. Hence, taking first order difference of natural logarithms of a procedure to make the return of Vnindex.

Ln ( yt ) – Ln ( yt-1 ) = ln ( yt/yt-1 ) = per centum return in twenty-four hours T ( 4.1 )

Null Hypothesis: Roentgen has a unit root

Exogenous: Changeless

Lag Length: 4 ( Automatic based on SIC, MAXLAG=27 )

t-Statistic

A A Prob. *

Augmented Dickey-Fuller trial statistic

-18.81547

A 0.0000

Test critical values:

1 % degree

-3.432494

5 % degree

-2.862373

10 % degree

-2.567258

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.3 ADF for Vnindex return

The ADF trial for Vnindex ‘s return is shown in Table 3.3. From Table 3.3, the t-statistic for the first lagged difference series is a?’18.81547 which is much smaller than the 1 % significance degree of trial critical value. Hence, the void hypothesis that states that the first order difference for natural logarithms of Vnindex series is non-stationary, is rejected.

Null Hypothesis: Roentgen has a unit root

Exogenous: Changeless

Bandwidth: 21 ( Newey-West utilizing Bartlett meat )

Adj. t-Stat

A A Prob. *

Phillips-Perron trial statistic

-40.56328

A 0.0000

Test critical values:

1 % degree

-3.432491

5 % degree

-2.862371

10 % degree

-2.567257

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.4 PP trial for the return of Vnindex

The PP trial on the return of Vnindex monetary values is shown in Table 3.4. It gives a similar consequence to ADF trial above. The adjusted t-statistic for the first order lagged difference of natural logarithms series is a?’40.5632, which is really little compared to the 1 % degree of trial critical value of a?’3.4324. The p-value equal nothing indicates that the PP trial is important. Therefore, we make the same illation as in the ADF trial where the return of Vnindex is stationary.

Figure 4.3 The return of Vnindex

The return of Vnindex which is the first lagged difference from the natural logarithms of informations clip series, is shown in Figure 4.3. It can be found that the return of Vnindex is stationary since about of the return values are allocated around nothing. However, some spikes which appear in the figure, represents a high volatility period.

Figure 4.4 Histogram and normality trial of Vnindex ‘s return

Figure 4.4 illustrates the histogram and normalcy distribution trial statistics, including mean, average, maximal and minimal values, standard divergence, lopsidedness, kurtosis, and Jarque-Bera trial of first Vnindex ‘s return. The histogram is centred and peaked at nothing. The average is equal to 0 because most of the values after first lagged difference autumn within the intervals of nothing. The values of lopsidedness and kurtosis are severally a?’0.1707 and 5.1735 which show that the distribution is somewhat asymmetric and extremely leptokurtosis.

Harmonizing to the information obtained above, It can

In history of world, one of the most hard and ambitious procedure is anticipation the hereafter. It can be considered as an art of making statements about events whose existent 1s have non yet been happened. Forecasting hereafter is really critical for many stakeholders in different industries. For illustration, husbandmans would wish to cognize the hereafter rainfall form in order to decently seed their seeds while, fiscal investors expect to cognize the future public presentation of assorted market stocks to maximize their net income. Fiscal analysts use prognosiss to do fiscal programs and gaining outlook. Investors invest their in stocks with the outlook that they will acquire a positive final payment. Hence holding a good cognition about portion monetary value motion in the hereafter serves the involvement of fiscal professionals and investors. This cognition about the hereafter boosts their assurance by manner of confer withing and puting.

There are assorted of import variables in the economic system and societal of a state that can be forecasted such as gross domestic merchandise ( GDP ) , rising prices, unemployment rate, birth rate, mortality rate and import/export amongst others. Forecasting and analysing utilizing clip series method has been one of the most considerable field in recent old ages. This method has been used in agribusiness for rice outputs ( Shabri et al. , 2009 ) and the monetary value of chocolate cean ( Assis et al. , 2010 ) . In concern, it was besides employed in foretelling exchange rate ( Zhang, 2001 ) , ( Fahimifard et al. , 2009 ) and rough oil monetary values ( Kumar, 1992 ) .

During the last two decennaries, the surveies of clip series foretelling in portion monetary values, bond output and exchange rate motions were classified into two classs viz. : cardinal analysis ( Edward 1998, Kenneth 1994 ) utilizing macro economic variables and proficient analysis utilizing historical informations and graphs ( Harvey 1990, Diebold 1989 ) . Stock monetary values are excessively high volatility esteeming to the alterations in basicss ( Elena and Storis, 2009 ) . In the proficient analysis attack, merely historical clip series informations like the ex-date stock monetary value is used to happen the tendency and volatility of a procedure. If the investors can make their owned anticipations on the motion of the stock monetary values, they will do a determination faithfully to maximise their net income, to protect themselves from the loss or cut down it every bit much as possible.

This survey focal point on prediction clip series of stock index, Vietnamese stock index ( Vnindex ) , by using Autoregressive Integrated Moving Average ( ARIMA ) theoretical account and Generalized Autoregressive Conditional Heteroscedastic ( GARCH ) theoretical account. Although, Box-Jenkins methodological analysis method is powerful and flexible, it is non able to manage the volatility that is present in the information series. The volatility of Vnindex can be handled by utilizing GARCH theoretical account. The prognosiss obtained from the GARCH will be evaluated with the benchmark is ARIMA prognosis consequences.

Backgroud survey

This survey concentrates on the usage of 2 standard theoretical accounts in calculating future motion of Vnindex. While clip series prognosis utilizing ARIMA theoretical accounts was combined and shown by Box-Jenkins in 1970, the construct of conditional heteroscedasticity was foremost introduced by the Nobel Prize victor, Robert Engle in 1982. Both of this attack is really popular in calculating clip series informations. Bollerslev ( 1986 ) had modified Engle ‘s ARCH theoretical account into a more generalised theoretical account called GARCH theoretical account with is a simplified theoretical account to ARCH theoretical account but more powerful. The simple GARCH theoretical account is able to observe the fiscal volatility in a clip tendency.

Statement of the job

The monetary value of stock is extremely volatile throughout clip particularly, in developing markets like Vietnam. Since stock monetary value variableness does critical consequence on the whole economic system, the anticipation of future stock monetary value index becomes important.

This survey will research the undermentioned inquiry: which method between Box-Jenkins and GARCH performs better in calculating VNindex? ”

Aim of the survey

The aims of this survey follows as:

+ Estimating suited Box-Jenkins and GARCH theoretical accounts for calculating Vietnamese stock index.

+ Measuring the public presentation of both theoretical accounts in foretelling Vietnamese stock index.

+ Forecasting utilizing EViews package.

Scope of the survey

This survey focuses on the Box-Jenkins and GARCH theoretical accounts to calculate stock monetary value index in Vietnam. Since the stock index volatility is the chief concern, the survey uses merely day-to-day informations. The information is obtained from the site of Hochiminh stock exchange from 28th July 2000 to 29th June 2012. and analyzed by utilizing the EVIEWS.

Significance of the survey

Due to the high volatility of Vnindex, the appraisal of the clip series theoretical account must be able to observe its volatility. In add-on, the exactly Box-Jenkins and GARCH theoretical accounts will be determined when calculating the volatility of Vnindex. The procedure will be done with the assistance of Eview package. As a consequence of this survey, a theoretical account and package that can be used to calculate a volatile clip series can be proposed.

Outline of the survey

This survey looks at two standard methods of calculating Vnindex. It is organized into five chapters as follows:

+ Chapter 1: Introduction – this subdivision brings a background that constitutes the suitableness for the survey within the context of antecedently published survey.

+ Chapter 2: Literature reappraisal – in this subdivision utilizing the related literature in the field, a comprehensive theoretical model is developed. First, the development and function of Vietnamese stock exchange will be reviewed. Then, the volatility in stock market and clip series theoretical accounts are besides generated. Finally, relevant researches about the volatility of stock markets utilizing Box-Jenkins method and GARCH theoretical accounts are mentioned

+ Chapter 3: Methodology. In this portion, the methodological analysis applied for prediction of the same set of informations utilizing ARIMA and GARCH theoretical accounts are good explained.

+ Chapter 4: Analysis. This chapter presents the analysis of the same information sets utilizing the ARIMA and GARCH theoretical accounts. A comparing between the ARIMA and GARCH theoretical accounts are besides made.

+ Chapter 5: Decisions and suggestions – the whole survey are summarized and concluded with some suggestions for future survey.

Chapter 2

LITERATURE REVIEW

The literature reappraisal provides an insight position on the clip series anticipation. The layout of this chapter is as follows: foremost, the birth and development of the Vietnamese stock market is summarized. Second, the function and volatility of the stock index in the emerging market like Vietnam will be examined. Hence, in this subdivision, some of import causes impacting the significance alterations of the stock index will be discussed. Third, a treatment on two theoretical accounts ARIMA and GARCH that are used in this survey will besides be presented. While, clip series prognosis utilizing ARIMA theoretical accounts was combined by Box-Jenkins ( 1970 ) , the construct of conditional heteroscedasticity was foremost introduced by Engle ( 1982 ) . Those two indispensable clip series theoretical accounts are widely applied in assorted Fieldss. Finally, based on those surveies from other research workers ‘ plants, stock index monetary values forecast utilizing Box-Jenkins methodological analysis and GARCH attack is highlighted since these are the concentration of this survey.

2.1 Vietnamese stock exchange overview

The Vietnamese Stock Exchange, officially known as the Securities Trading Centre ( STC ) , located in Ho Chi Minh City, was launched on July 28th 2000. There were merely 22 listed companies with a market capitalisation of $ 144 million. Over five old ages of operation, the figure of listed companies have rised to 32 with a entire market capitalization of USD 398.96 million ( State securities commissionA of vietnam web site ) . In 2006, the index climbed 145 % and rose additionaly 60 % in the first three months of 2007. Although the market has significantly grown over the period, its graduated table is still really little. The universe fiscal crisis in 2008 lead the capitalisation down to 18 % from 50 % of Vietnamese GDP. Fortunately, the capitalisation rose to about 40 % GDP when the universe economic system started recover in 2009. The period from 2008 to 2009 witnessed a about dual addition in investing history from 500,000 to about 900,000. This is singular comparing with about 400 active histories in the beginning of the market.

Basically, Vietnamese stock market is supervised and managed by the State Securities Commission that is a portion of the Ministry of Finance. A jurisprudence on securities was adopted in June of 2006 to ease the development of the securities market quickly and sustainably. In 2010, Vietnamese State Securities Commission independently separated from the Ministry of Finance. Hence, Vietnamese stock markets operates more freely and efficaciously.

Although, Vietnamese stock market developed in quality and measure, investors are still limited in their ability to merchandise securities. In 2010, merely approximately 60 % of the market capitalisation of Vietnam ‘s stock market which valued at $ 33.2 billion, are freely traded.A This is because the authorities still holds commanding portions in big companies such as Vietcombank, Vinamilk, and Baoviet Holdings, the biggest insurance and fiscal services house in Vietnam.

2.2 Role of the emerging stock market

The function of fiscal development to the economic growing of developing states like Vietnam is one of the popular subjects of economic experts. By and large, well-developed stock markets are expected to speed up economic growing, by increasing domestic nest eggs and the quality of investings ( Singh, 1997 ) . They provide an extra fiscal instrument that may raise the nest eggs rate for an person ( Levine and Zervos, 1996 ) . Furthermore, in states that have developed stock markets, since companies are less dependent on bank funding, the hazard of recognition crunch is decreased ( Capasso, 2003 ) . Hence, stock markets tend to positively act upon economic growing through persons ‘ nest eggs, and better funding for houses.

Although, empirical grounds shows the being of a strong positive correlativity between stock market and economic growing, there are some statements such as Bencivenga and Smith ( 1991 ) , Adjasi and Biekpe ( 2006 ) , Naceur and Ghazouani ( 2007 ) A who looked at developing states. However, recent surveies concentrate on stock market as an engine of economic growing in emerging states like Vietnam. Levine and Zervos ( 1998 ) , for case, found a positive and important relationship between stock market development and long tally growing. Mohtadi and Agarwal ( 2004 ) use a dynamic panel method to analyze the relationship between economic growing and stock market development in 21 emerging markets over 21years. Their consequences besides suggest a positive correlativity between several indexs of the stock market public presentation and economic growing both straight and indirectly by hiking investing behaviour. Hence, in developing states, stock markets play an of import function in economic growing and development by helping nest eggs and altering hard currency flow from rescuers to investors.

2.3 Volatility of stock monetary value in the emerging market

In developing states such as Vietnam, although the high growing of fiscal markets really attractive, the volatility of return can be a major stumbling block for investors. Several documents analyzing the behaviour of liberalized stock exchanges ( Borenzstein and Gelos 2000 ; Froot, et al 1999 ) have found strong grounds of crowding, tendency chasing and impulse trading, all of which can take to increase the volatility of portion monetary values. It besides affects the economic system due to its consequence on consumer disbursement ( Starr-McCluer,1998, and Poterba, 2000 ) . A lessening in the stock market will weaken client ‘s assurance and drive down consumer ‘s disbursement and frailty versa. The volatility of stock market may besides straight influences concern investing ( Zuliu, 1995 ) and economic growing ( Levine and Zervos, 1996 ) . An addition in volatility can take to raise hazard of buying equity and to alter in financess to cut down hazardous assets. Thus the support cost of houses will be raised and investors will alter to purchase well-known houses ‘ stocks alternatively of new houses.

Theoretically, stock monetary values can alter due to universe macroeconomic variables and universe events. Fama ( 1981 ) investigates the strong relationship between stock monetary values and existent activity, rising prices and pecuniary policies. Hamao ( 1988 ) and Lee ( 1992 ) show that rising prices dramatically affected stock return. Beside, a negative relationship between both long term and short term involvement rates and stock returns is found by Gallic et Al in 1987. The exchange rate affects stock monetary values in a similar manner to the rising prices factor. A positive correlativity between stock monetary values andrevaluation of the US dollar is found by Aggarwal ( 1981 ) . Mukherjee and Naka ( 1995 ) besides find that in Japan and Indonesia exchange rate positively affects stock monetary values. Bilson et Al ( 1999 ) suggests that the exchange rate is one of the most influential factor among the other variables such as money supply, existent activity and goods monetary values play a small function.

By and large, universe events such as war, natural catastrophes or terrorist act significantly influence stock monetary values. They frequently occur in concatenation reactions and in both direct and indirect manner. After the terrorist onslaughts on September 11th 2001, the universe witnessed that many investors, particularly in the United States, halt or merchandise less and concentrate on less hazard stock and bond. Wars can be a clear illustration of an indirect influence on stock markets. The occurrence of a war lead to raise the monetary value of military equipment and arms makers. This addition in bend accretes the stock ‘s value of companies providing military equipments and engineering. This is similar to raise the demand for natural resources that would raise the monetary value of stocks of mining companies every bit good as natural resource processing 1s.

Another indispensable factor impacting the stock monetary value is dividend policy. The correlativity between dividend policy and volatility of the stock monetary value is enquired by many different research workers ( Baskin, 1989 ; Allen and Rachim, 1996 ) . This impact can be summarized that if there is an addition in dividend paid among the stockholders, the monetary values of the portion will travel up due to leveling demand. In contract, the monetary values of the portions will fall, if managers decide to administer less dividend among the stockholders.

2.4 Time Series Forescasting Models

The prognosis of stock monetary value is indispensable non merely for investors but besides for economic contrivers because it plays an of import function in the economic system of states. There are assorted types of theoretical account used to calculate clip series informations. However, in this survey, the two most popular 1s, ARIMA and GARCH, are applied.

In 19th century, Yule ( 1927 ) foremost conducted the impression of randomness by suggesting that every clip procedure can be considered as the realisation of a stochastic procedure. It can be seen as the birth of clip series prognosis. Since so, the construct of autoregressive ( AR ) and traveling norm ( MA ) theoretical accounts was determined continuing from that footing thought. Box and Jenkins ( 1970 ) combined the current cognition to organize a standard attack for clip series prognosis in their celebrated book Time series analysis: prediction and control ” . This book important impacts modern clip series analysis and prognosis in the both theory and pattern. The success of Box-Jenkins method is recorded on fact that the many theoretical accounts can make so adequately without necessitating assorted estimated parametric quantities in the concluding pick of the theoretical account. However, in the beginning of calculating utilizing ARIMA theoretical account, research workers faced a important job for in selecting of a theoretical account that there was no algorithm to find an alone theoretical account. Since so, numberous techniques and methods have been developed and proposed to add mathematical truth to the gauging procedure of the ARIMA theoretical account. The two necessity of them is Akaike ‘s information standard ( AIC ) and Bayesian information standard ( BIC ) or Schwarz standard ( SIC ) .

The autoregressive incorporate traveling norm ( ARIMA ) theoretical accounts provide methodological analysis and attack for clip series parametric quantity analysing and foretelling of average returns ( Box Jenkins, 1976 ) . This attack is one of a big household of quantitative prediction methods developed in the Fieldss of operations research, direction scientific disciplines and statistics. It is particularly appropriate for short-run anticipation because of its focal point on the recent yesteryear instead than the distant yesteryear ( Pankratz, 1983 ) . The last few decennaries have witnessed a well addition in suggesting ARIMA theoretical accounts to calculate univariate clip series informations. It was confirmed that ARIMA theoretical accounts are suited for agricultural predicting ( Fatimah and Roslan, 1986 ) . They besides were employed to calculate rice outputs ( Shabri et al. , 2009 ) , Cocoa Bean Price ( Assis et al. , 2010 ) . Box-Jenkins transportation map theoretical accounts are used by Liu ( 1991 ) to analyze the dynamic relationships between US gasolene monetary values, rough oil monetary values and the stock of gasolene. Kumar ( 1992 ) compares the prognosis truth of rough oil monetary values obtained utilizing clip series theoretical accounts with the truth of future monetary value prognosiss. Chinn et Al ( 2005 ) see the correlativity between topographic point and hereafters monetary values of energy trade goods. One of them was rough oil monetary values that was expected by ARIMA ( 1,1,1 ) . Furthermore, ARIMA theoretical accounts are a low research cost method compared with econometric theoretical accounts ( Shamsudin et al. , 1992 ) . Although ARIMA theoretical accounts are widely employed in many practical applications, it can non capture nonlinear forms of complex clip series when nonlinearity exists.

A chief characteristic of fiscal clip series is that a big return is likely to be followed by another big return which means there are high volatility show periods. This phenomenon is defined as volatility constellating in econometrics and finance. Autoregressive Conditional Heteroskedasticity ( ARCH ) theoretical accounts points that a clip series informations relates its ain lagged informations. The attack was foremost introduced by Engle ( 1982 ) to foretell UK rising prices. In this category of theoretical accounts, dynamic alterations in conditional discrepancy can be described as a deterministic map of past returns.

ARCH theoretical accounts were extended by Bollerslev ( 1986 ) into Generalized Autoregressive Conditional Heteroskedasticity ( GARCH ) theoretical accounts that portion many belongingss and supply better consequence. Sabbatini and Linton ( 1998 ) study that the simple GARCH ( 1,1 ) theoretical account determine good parametric quantities for the day-to-day returns of the Swiss market index. Fahimifard et Al ( 2009 ) use R2, MAD And RMSE standards to compare the prognosis truth of exchange rate determined by ARIMA and GARCH. Their survey shows that GARCH theoretical accounts outperforms AIRMA theoretical accounts. These theoretical accounts have been loosely employed to calculate several clip series informations, including stock monetary values ( Schwert 1989, Hamilton and Susmel, 1994, Cho and Engle 1999 ) , exchange rates ( West and Cho 1994, Campa and Chang 1997 ) and involvement rates ( Edwards 1998, Boscher et al 2000 ) . It is reported that there is a negative correlativity between returns and conditional discrepancy of the following period ‘s returns. A negative or positive returns are comparatively connected to upward or downward alterations of the conditional volatility. Engle and Ng ( 1993 ) refer this phenomenon to asymmetric volatility. The consequence of dissymmetries on the out-of-sample prognosis ability of assorted GARCH theoretical accounts is found by Awartani and Corradi ( 2005 ) . In 2006, Zhou et al employed ARIMA and GARCH to construct a web traffic calculating theoretical account. They besides found that in footings of calculating truth, ARIMA/GARCH is better than the Fractional Autogressive Intergrated Moving Average ( FARIMA ) . The assorted ARIMA/GARCH theoretical account besides were confirmed that it outperformed ARIMA and GARCH for foretelling Tawau Cocoa bean monetary values ( Assis, 2010 ) .

2.5 Relevant research in stock monetary value.

Since Box-Jenkins method is widely used, there are many surveies using them for calculating stock monetary value volatility. The volatility of S & A ; P500 index is foremost examined by Poterba and Summers ( 1986 ) utilizing AR ( 1 ) procedure. It is besides described by Gallic et Al ( 1987 ) who applied ARIMA ( 0,1,3 ) theoretical account. Schwert ( 1990 ) estimates a additive AR ( 12 ) theoretical account for stock returns utilizing day-to-day informations of S & A ; P composite portfolio. Recently, Al-Shiab ( 2006 ) employs ARIMA ( 4,1,5 ) to foretell the Amman stock exchange daily general index. Although the usage of such additive theoretical accounts rises, they are non strong to explicate chief characteristics of volatility series. Many empirical groundss show that, for illustration, the returns have a inclination to exhibit outliers. This means a big discrepancy is likely to be followed by another big discrepancy. Those drawbacks of additive theoretical account motivate research workers to utilize nonlinear theoretical accounts such as ARCH household theoretical accounts in clip series prediction.

The ARCH category theoretical accounts including the ARCH theoretical accounts of Engle ( 1982 ) , the generalized ARCH ( GARCH ) theoretical accounts of Bollerslev ( 1986 ) and exponential GARCH ( EGARCH ) theoretical accounts of Nelson ( 1991 ) are the most common nonlinear theoretical accounts used in the finance survey. They are appropriate to use to fiscal clip series informations that have bunchs of outliers and heavy tailed distributions. Akgiray ( 1989 ) , in his survey using GARCH ( 1,1 ) theoretical account to day-to-day stock returns, suggests that GARCH is the best theoretical account depicting and calculating stock market volatility. The prognostic ability of additive and non-linear theoretical accounts in forecast day-to-day S & A ; P 500 hereafters index volatility is observed by Najand ( 2002 ) . This survey shows that nonlinear GARCH theoretical accounts dominate additive theoretical accounts by utilizing the RMSE and the MAPE truth steps.

McMillan et Al ( 2000 ) surveies United Kingdom ( UK ) stock market volatility utilizing monthly, hebdomadal and day-to-day informations. Their consequence suggests that GARCH and traveling mean theoretical accounts bring marginally superior day-to-day volatility prognosiss. The truth of day-to-day stock monetary values volatility forecast in New Zealand market of nine different theoretical accounts is evaluated by Yu ( 2002 ) . This survey confirms that the stochastic volatility theoretical accounts provide the best consequence among the other theoretical accounts. It besides shows that the public presentation of ARCH category theoretical accounts strongly depend on the signifier chosen.

In general, the volatility of stock returns in well-developed stock markets are examined many times. The major stock markets in the United State are consider by Poterba and Summer ( 1986 ) , Akgiray ( 1989 ) and Najand ( 2002 ) . Similarly, the 1s in UK are observed by Dimson and Marsh ( 1990 ) , McMillan, Speight and Gwilym, ( 2000 ) . Asia Pacific stock markets, for illustration Japan ( Tse, 1991 ) , Singapore ( Tse and Tung, 1992 ) , Australia ( Brailsford and Faff, 1996 ) , New Zealand ( Yu, 2002 ) in bend are besides inspected. However, there are few studis about on immature stock markets like the 1s in Amman ( Al-shiab, 2006 ) and Pakistan ( Rashid and Ahmad, 2008 ) stock exchange.

In this paper, the ARIMA theoretical accounts and GARCH ( 1,1 ) are applied to calculate the day-to-day stock monetary value index in the comparatively developing stock exchange market, viz. Vietnamese Stock Exchange.

Chapter 3

Methodology

This chapter describes the clip series attacks to gauge and calculate stock monetary values including types of prediction, patterning techniques and brief description of foretelling theoretical accounts used for optimisation of mistake prosodies. From the literature, the stock monetary values can be estimated and expected by assorted statistical attacks. However, in this survey, ARIMA and GARCH theoretical accounts are applied utilizing the historical information to calculate for the hereafter Vnindex.

Autoregressive integrated traveling norm ( ARIMA ) theoretical accounts can calculate accurately based on of historical stationary clip series forms. Since stock monetary values in the emerging market like Viet Nam fluctuate over the clip, a heteroscedasticity attack shall be tested for the full information series. Hence, GARCH theoretical account is besides used to capture volatility constellating in stock monetary values clip series. Then both forecast consequences of two theoretical accounts are compared utilizing mistake statistics.

3.1 Box-Jenkins attack

In this portion, the prognostic theoretical account used to foretell stock returns will be outlined. Over the last decennaries, faculty members have used a huge graduated table off different theoretical account specifications to prove for stock return anticipation. Some use non-parametric methods or let prognostic variables to exhibit non-linear forms. However, specification becomes a critical issue when sing these types of theoretical accounts. Goyal and Welch ( 2008 ) besides indicate out that some of these theoretical accounts are bound to work both in in-sample and out-of-sample. Box and Jenkins ( 1976 ) proposed the Autoregressive Integrated Moving Average ( ARIMA ) theoretical accounts which have been used for a broad assortment of clip series calculating application. They are employed in instances where information is a stationary procedure or its non-stationary characteristic can be removed. By and large, Box-Jenkins methodological analysis consists 4 stairss affecting theoretical account designation, appraisal, theoretical account checking and prediction.

3.1.1 Fiscal clip series and stationary procedure

Fiscal clip series yt can be describe as:

Yt= { Yt-1, Yt-2, .. , Y0 }

A procedure is stationary when its mean, discrepancy changeless and covariance between two 2 times merely depend on their distance and the hold and independ on the existent clip which covariance is calculated. The stationarity of the clip series is tested by utilizing correlograms, Q-statistics or Augmented Dickey Fuller ( ADF ) unit root trial.

The correlograms involve autocorrelation maps ( ACF ) and partial autocorrelation maps ( PACF ) . Figure 3.1 shows an illustration of correlograms and Q-statistics.

Figure 3.1 An illustration of correlogram and Q-statistics from Eviews

In extra, Q-statistic or ADF trial can be used to look into the stationary of a procedure.

3.1.1.1 Q-statistics

The Q-statistics at slowdown K is a trial standard for the void hypothesis that there is no autocorrelation up to order k. This statistic can be estimated as:

( 3.1 )

Where is the j-th autocorrelation

N is the figure of these observations

Under the void hypothesis, Q-statistic is asymptotically distributed as a I‡2 with grades of freedom equal to the figure of autocorrelations. If the chance of Q-statistic at any slowdown is less than 0.05 so the void hypothesis is rejected for all slowdowns. ( Mishra et al, 2010 ) . Therefore, the procedure is stationary.

3.1.1.2 Augmented Dickey-Fuller Test

The stationary of a information set can be tested by the augmented Dickey-Fuller ( ADF ) trial. This trial is the improved version of Dickey-Fuller ( 1979 ) trial of the model that provides a parametric rectification for higher-order correlativity by presuming that the procedure follows an AR ( P ) procedure. This trial besides addes Ps lagged difference footings of the dependant variable Y to the right-hand side of the arrested development as follow:

( 3.2 )

The void hypothesis of this trial is H0: = 0 informations is non-stationary which mens that differencing must be done to do it stationary. The alternate hypothesis H1: & lt ; 0 which means that the information is likely to stationary and can be analyzed without differencing the information. The trial statistic is computed as

( 3.3 )

Where is the estimation of coefficient

Se ( ) is the coefficient criterion mistake

This trial statistic is compared with the relevant critical value from Dickey-Fuller trial. The void hypothesis will be rejected if the alsolute value of trial statistic less than the critical value.

3.1.2 Autoregressive-moving-average theoretical account ( ARMA )

3.1.2.1 Autoregressive theoretical account ( AR )

An autoregressive procedure is a difference equation which shows the current value of a series as map of the old values. This means the value of discrepancy at clip T yt depends linearly on its past values. The order term P determines how many past values are to be included in the difference equation to gauge the current value. A difference equation relates a variable Yt at clip T with its old values ( Horvath, et Al, 2006 ) . The AR ( P ) theoretical account with pth order AR of a clip series is written:

Yt= c + I†1yt-1 + I†2yt-2 + aˆ¦+ I†t-pyt-p+ Iµt ( 3.4 )

=

The coefficients I†i can be estimated by ordinary least squares regression while degree Celsius is a changeless and Iµt is white noise. The theoretical account remainsA stationary when the value of parametric quantity.

The least squares method is applied to gauge the value of I† . It minimizes the amount of square of mistakes for the ascertained values with regard to I† .

( 3.5 )

Solving equation ( 3.5 ) to acquire the expected value of I† :

From the estimated value of, distribution of mistake footings can be represented

Replacing the estimated value of and distribution of mistake informations to the equation ( 3.5 ) , the theoretical account can be fitted.

3.1.2.2 Moving-Average theoretical account ( MA )

A clip series procedure is affected by random dazes in the noisy environment. Therefore, the random dazes of old values act upon the current value of series. In traveling mean theoretical accounts, the current value of discrepancy at clip T yt depends linearly on the lagged value of the residuary in the same period and the old. The MA ( Q ) theoretical account is written

( 3.6 )

The parametric quantities can be estimated by a dedicated ARMA modeling plan, is the expected valu of Yt which is frequently assumed to be 0, and are white noise mistake term.

3.1.2.3 Autoregressive Moving-Average theoretical account ( ARMA )

The autoregressive Moving Avergae ( ARMA ) theoretical account contains 2 theoretical accounts AR ( P ) and MA ( Q ) . The autoregressive theoretical account and moving mean theoretical account can be used to come close any stationary procedure. Uniting two equations ( 3.4 ) and ( 3.6 ) , ARMA theoretical account of order P and Q is created:

( 3.7 )

3.1.3 Autoregressive Intergrated Moving Average Process – ARIMA ( P, vitamin D, Q )

The ARMA theoretical account assumes that the clip series informations is stationary. However, the existent procedures are non stationary in nature. A differencing procedure is applied to tranforming a stationary clip series to non-stationary. The ARMA theoretical account whose clip series is made stationary by differencing, is called as Autoregressive Intergrated Moving Average ( ARIMA ) theoretical account.

Deducing from the ARMA theoretical account, the ARIMA theoretical account besides consists p order of autoregressive theoretical account and q order of traveling mean theoretical account. However, it besides involves an extra parametric quantity viz. , 500 order times of differencing. This theoretical account takes and applies historical informations to an autoregressive ( AR ) procedure that contains memory of old events. An Intergrated ( I ) procedure makes informations go stationary for easy anticipation and a Moving Average ( MA ) procedure of prognosis mistakes. Box-Jenkins has specified four stairss for constructing an ARIMA theoretical account:

Model designation: finding values of P, vitamin D, Q

Model appraisal: gauging parametric quantities of the theoretical account

Model checking: sing whether theoretical account tantrums informations or non ; if non see another 1

Forecasting utilizing the best selected theoretical account

3.1.3.1 Model designation

Box-Jenkins method can non be applied to a non stationary clip series procedure. Therefore, sing whether the series is stationary or non, is one of of import undertakings. This undertaking can be done by utilizing the graph of ACF and PACF or cheking through Augmented Dickey-Fuller or Phillips-Perron ( PP ) Unit Root Test.

The autocorrelation at slowdown K is the correlativity coefficient for values of the series K periods apart. While, the partial autocorrelation at slowdown K is the correlativity of Y values that are thousand periods apart after taking the correlativity from the intervening slowdown. In EView, the autocorrelation and partial autocorrelation at lag K of a procedure is computed by

( 3.8 )

( 3.9 )

Where is mean of series

is estimated autocorrelation at slowdown K

A stationary procedure has a graph of ACF which cuts off rapidly or dies down rapidly. In the antonym, if the graph of ACF dies down really easy, the clip series values will be considered non-stationary. The values of ACF and PACF are compared with to see the significance of them. The behaviour of the ACF and PACF of theoretical accounts can be generalized as table 3.1.

Model

ACF

PACF

AR ( P )

Dies down

Cut of after slowdown P

MA ( Q )

Cut of after slowdown Q

Dies down

ARMA ( P, Q )

Dies down

Dies down

Table 3.1: The behaviour of ACF and PACF for each of the general theoretical accounts

In this attack, differencing can be done non-stationary clip series until it becomes stationary. The figure of differences required to accomplish stationary is denoted by term vitamin D of ARIMA ( P, vitamin D, Q ) .

Harmonizing to the rule of parsimoniousness, simple theoretical accounts are preferred to complex theoretical accounts when all things being equal ( Hanke et al. , 2001 ) . It is comparatively easy to happen a theoretical account with big figure of parametric quantities that fits the limited information well. However, calculating from this theoretical account is likely to be hapless because there are excessively much fluctuation in the informations due to random mistake is being modelled. The accomplishment is to develop the simplest theoretical account that bringe an equal description of the major characteristics of the informations.

3.1.3.2 Model appraisal

Since a possible theoretical account has been chosen, the parametric quantities for that theoretical account must be determined. They can be estimated by utilizing the least square arrested development method. This method minimizes the amount of squares of the fitting map. Once the least squares estimations and their standard mistakes are found, values of parametric quantities can be constructed and interpreted in the usual manner. Parameters that are judged dramatically different from nothing are retained in the fitted theoretical account ; parametric quantities that are non important are dropped from the theoretical account.

3.1.3.3 Model checking

In this phase, the estimated theoretical account must be checked for adequateness. Basically, a theoretical account is adequacy when the remainders has the normal distribution and should be random. The Ljung-box Q statistic based on the the size of the residuary autocorrelation secret plan is one common trial for entropy. The Ljung-Box trial can be written:

H0: The procedure is independently distributed

H1: The procedure is non independently distributed.

The trial statistic Q is:

( 3.10 )

Where:

rk ( vitamin E ) = the residuary autocorrelation at slowdown K

n = figure of remainders

m = figure of clip slowdowns includes in the trial

If the p-value of Q-statistic is little ( less than important value ) , the theoretical account is inadequacy. Then, a new or modified theoretical account must be considered and checked until the satisfied 1s has been found. Diagnostic checking plays an of import function when two simple viing theoretical accounts may adequately depict the informations and a pick may be made on the footing of the nature of the prognosiss.

3.1.3.4 Forecasting

Once a best theoretical account has been estimated, it will be used for to anticipate the future values of a series utilizing the old values. Although the fittest ARIMA theoretical accounts frequently involve differences, the anticipation for the original series can be computed straight from it. Forecast can be in-sample or out-sample. In this survey I am aware of the in-sample prognosis. This measure besides of import since the determinations made today will depend on the consequences of prognosis.

3.2 Standard additive arrested development

The standard additive arrested development theoretical account may be written as:

Lolo = I?0 + xiI?i + Iµi ; i= 1,2, aˆ¦ , N

Lolo: additive combination of parametric quantity

elevens: independent variable

I?i: coefficients

Iµi: mistake footings

3.2.1 Arrested development Consequences

The illustration of coefficient consequences that are conducted in EView will be illustrated in figure 3.2

Figure 3.2 An illustration of equation end product from EViews

3.2.1.1 Arrested development Coefficients

The column labelled Coefficient ” in Figure 3.2 represents the estimated coefficients. The least squares coefficients I? are estimated by the standard ordinary least square expression:

I’= ( X’X ) -1X’y

In such sort of theoretical account, the coefficient measures the fringy part between the independent and dependent variables. Generaly, in the arrested development consequences in EView, the coefficient of the variable degree Celsius is a changeless the arrested development. It is the base-level of the prognosis because all of the other independent variables can be zero. The other coefficients are described as the incline of the correlativity between the corresponding independent and dependent variables, presuming all other variables do non alter.

3.2.1.2 Standard Mistakes

The Std. Mistake ” column in Figure 3.2 shows the determined standard mistakes of the estimated coefficients. They measure the statistical dependability of the coefficients. Hence, the larger the standard mistakes, the more statistical noise is present in the estimations. They are defined as the square roots of the diagonal facets of the coefficient covariance matrix.

3.2.1.3 t-statistics and chance

The t-statistic is an of import step used to prove the hypothesis that a coefficient value is zero. It is defined as the ratio of an estimated coefficient to its standard mistake. This term determines the chance of the value coefficient is equal nothing. If normalcy can merely keep asymptotically, so z-statistic will be used alternatively of t-statistic.

The chance in the last column is the p-value or the fringy significance degree. The p-value is a standard step used to reject or accept the hypothesis that the true coefficient is zero. The p-value are computed from a t-distribution with n-k grades of freedom.

3.2.2 Arrested development statistics

All arrested development statistics is shown in the estimated equation below arrested development consequence as the illustration in Figure 3.2. These are R-squared, adjusted R-squared mistake of the arrested development, sum-of-squared remainders, log likeliness, Durbin-Watson statistic, Akaike Information Criterion, Schwarz Information Criterion and F-statistic.

3.2.2.1 R-squared

The R-squared ( R2 ) statistic determines the success of the arrested development in the anticipation of dependant variable value within the sample. A R2 is the proportion of variableness in a procedure that is computed for by the statistical theoretical account. R2 statistic will be one and zero if it fits no better than the simple mean of the dependant variable. This means consequences of the arrested development is absolutely suited for the information set. R2 is estimated in EViews as:

( 3.11 )

Where is the mean of the dependant variable.

3.2.2.2 Adjusted R-squared

R2 has a job is that it ne’er falls even more regressors have being added. Adjusted R2 ( is created to work out this job. If regressors are added to the theoretical account, its will decreased. value can be negative for ill fitting theoretical accounts. Its absolute value is normally less than or equal to that ofA R2. The adjustedA R2A is defined as:

( 3.12 )

3.2.2.3 Standard Error of the Arrested development

The standard mistake of the arrested development can be determined by utilizing estimated discrepancy of the remainders. It is computed as:

( 3.13 )

3.2.2.4 Sum-of-squared Remainders

The sum-of-squared remainders is a drumhead measuring that can be employed in assorted of statistical computations.

( 3.14 )

3.2.2.5 Durbin-Watson Statistic

The Durbin-Watson ( DW ) statistic is a trial statistic applied to observe the consecutive correlativity in the remainders. This statistic is computed as

( 3.15 )

By and large, the value of DW statistic ever lies between 0 and 4. If the DW statistic equal nothing, there will be no consecutive correlativity. The consecutive correlativity is positive when the statistic less than 2. In the antonym, if DW value greater than 2, the consecutive correlativity is negative.

3.2.2.6 Mean and Standard Deviation

The mean and standard divergence of the dependant variable Y can be estimated by utilizing the standard formulars:

( 3.16 )

( 3.17 )

3.2.2.7 Log likeliness

A likeliness ratio testA is aA statistic trial applied to compare the tantrum of two theoretical accounts, the nothing and the alternate theoretical account. When the logarithm of the likeliness ratio is used, this statistic becomes a log likeliness ratio statistic. It is know as the difference between the log likeliness values of the restricted and unrestricted equation. The log likeliness is evaluated in Eview as:

( 3.18 )

3.2.2.8 Akaike Information Criterion

The Akaike information standard is a step for the fittest of statistical theoretical account. Akaike ( 1973 ) suggests this standard in chosing theoretical account procedure. The thought is minimising the negative likeliness penalised by the figure of parametric quantities. The smaller the values of the AIC is, the better theoretical account is. The AIC is computed as:

( 3.19 )

Where cubic decimeter is the log likehood

3.2.2.9 Schwarz Information Criterion

The Scharz Information Criterion ( SIC ) is a standard for theoretical account choice among a specific set of theoretical accounts. It is an alternate critera to AIC that enforces a larger punishment for extra coefficient.

( 3.20 )

3.2.2.10 F-Statistic

The F-statistic in figure 3.2 is is a standard statistic used to prove of the hypothesis that all the incline coefficients in a arrested development is zero. For OLS theoretical accounts, F-statistic is computed as:

( 3.21 )

The p-value below the F-statistic is the fringy significance degree of the F -test. If the P -value is less than the significance degree we are proving, say 0.05, we reject the void hypothesis that all incline coefficients are zero.

3.3 Generalized Autoregressive Conditional Heteroskedasticity ( GARCH )

The white noise mistake term in the ARMA theoretical account is defined as ARCH theoretical account by Engle ( 1982 ) in his celebrated paper Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation ” . This term can be shown as

( 3.22 )

Where is clip depent standard divergence and can be modeled by ARCH ( Q ) procedure

( 3.23 )

Bollerslev ( 1986 ) generalized ARCH theoretical accounts by adding the lagged conditional discrepancy. GARCH theoretical account allow for autoregressive and moving mean elements in the heteroscedastic discrepancy. The mistake processes in GARCH can be written as

( 3.24 )

( 3.25 )

Where are invariables

Since is a white noise procedure, the conditional and unconditioned agencies of are equal zero and the conditional discrepancy of is

( 3.26 )

The GARCH ( P, Q ) allows both AR and MA constituents in the heteroscedastic discrepancy. If p = 0 and q = 1, so GARCH ( P, Q ) will be GARCH ( 0,1 ) or ARCH ( 1 ) . Therefore, if all values of equal nothing, the GARCH ( P, Q ) is tantamount to the ARCH ( Q ) theoretical account. The advantage of the GARCH theoretical account is that it is much easier to place and gauge a higher order ARCH theoretical account with more penurious GARCH representation. It is perfectly true because all coefficients ( I± , I? ) when gauging an ARCH or GARCH theoretical account must be positive. Furthermore, all characteristic roots in the discrepancy equation must lie inside the unit circle to guarantee that the discrepancy is finite. Therefore, More penurious theoretical accounts will intend few restricted coefficient.

An of import characteristic of GARCH is that the conditional discrepancy of the perturbations of the yt series is an ARIMA procedure. Therefore, the remainders expected from a fitted ARIMA theoretical account should demo their characteristic forms. For illustration, if a fitted ARIMA theoretical account of the series is appropriate, the correlogream of ACF and PACF will demo the remainders should be a white-noise procedure. However, if a GARCH theoretical account is more desirable, the ACF of the squared remainders should assist place the procedure.

3.3.1 Garch ( 1,1 )

GARCH ( 1,1 ) is the most popular GARCH theoretical accounts since it is easy to use to many informations series. This theoretical account ‘s specification is:

yt = Aµ + Iµt

Iµt = I?tI?t

I?t ~ iid N ( 0,1 )

I?t2=I±0 + I±1 Iµ2t-1 + I?1I?2t-1

I±0 & gt ; 0 ; I±1, I?1 a‰? 0

Where yt is the dependent variable over period T

Aµ is a changeless mean

I?t2 conditional discrepancy

The conditional discrepancy contains a changeless term ( I±0 ) , volatility from the old period, measured as the slowdown of the squared remainder from the average equation ( Iµ2t-1 ) ; last period ‘s prognosis discrepancy ( I?2t-1 ) . The last status ensures non-negative conditional discrepancies, while the status 0 & lt ; I±1 + I?1 & lt ; 1 ensures that the discrepancy of the unconditioned return is stationary and finite. Since the intercept I±0 is ever positive, it allows analyst to pattern the volatility as mean-reverting. If the volatility is high, it will be given to fall over clip, and if the volatility is low, it will be given to lift over clip. ( Dowd, 2002 ) . This specification frequently appears in a fiscal context, where bargainers and agents predict this period ‘s discrepancy by organizing a leaden norm of a long term norm ( I±0 ) , information about volatility witnessed in the old period ( Iµ2t-1 ) and the forecasted discrepancy from last period ( I?2t-1 ) . If the plus return is all of a sudden big in either the upward or the downward way this period, the estimation of the discrepancy will be increased in the following period.

In fiscal Fieldss, it is common to happen GARCH slowdown coefficient ( I?1 ) over 0.7 and GARCH return coefficients ( I±1 ) ever less than 0.25. The size of these parametric quantities represent the form of the ensuing volatility clip series. A high value of I?1 agencies that volatility is relentless and it takes a long clip to alter, while a high value of I±1 show that volatility is peaky and sensitive with market alterations ( Alexander, 1998 ) .

3.3.2 Parameter Appraisal

The OLS method understating the residuary amount of squares, which depend merely on the conditional average equation and do non related to conditional covariance, can non be employed to gauge GARCH theoretical accounts. Another technique known as maximal likeliness is applied in order to gauge GARCH theoretical accounts.

By and large, this attack focal point on happening the most likely values of the parametric quantities given the existent information. In peculiarly, a log likeliness map is formed and the values of the parametric quantities that maximise it are fought. It includes three stairss: foremost, finding the appropriate equations for the mean and the discrepancy, so, specifing the log-likelihood map to maximize under normalcy premise for the perturbations. Finally, maximizing the log likeliness map and bring forthing parametric quantity values that building their standard mistakes ( Brooks, 2008 ) .

The log likeliness map in this method is estimated as:

( 3.27 )

Harmonizing to the equation ( 3.27 ) , log likelihood map can be maximised by understating, since there is a negative mark in the map and nlog ( 2Iˆ ) is a changeless. Understating these footings is non a simple undertaking for time-varying discrepancies. Fortunately, EViews are able to choose the parametric quantity values that maximize this log likeliness map.

3.3.3 Diagnostic Checking

In general, a fitted GARCH theoretical account gaining control all dynamic elements of the theoretical account of the mean and the theoretical account of the discrepancy. Its estimated remainders should besides be serially uncorrelated and should non expose any staying conditional volatility. This means each residuary utilizing its conditional criterion divergence, ; and the resulting series should hold a mean and a discrepancy of integrity ( Enders, 2004 ) .

The Ljung-Box Q- statistics of the serie and the squared standardised remainders are formed to prove the theoretical account of the mean and the staying GARCH effects severally. If there are any consecutive correlativities in the, the theoretical account of the mean will non be decently specified. Therefore, the void hypothesis that Q-statistic equal nothing will non be rejected. If there is no staying GARCH effects, the void hypothesis that the sample values of the Q-statistics are equal to zero, should non be rejected. Hence, the belongingss of the serie should mime those of a white noise procedure.

3.3.4 Forecasting

The one-step-ahead prognosis of the conditional discrepancy is simple to obtain. The following period discrepancy will be represented as:

( 3.28 )

Taking the conditional expection will give:

( 3.29 )

Since the expected value of equal the expected value of E ( ) = E ( ( Brook,2008 ) so equation ( 3.29 ) equivalent

( 3.30 )

Day t+2 and t+3 will be:

On twenty-four hours t+n in the hereafter, we have:

Replacing with the value of so the equation above will be:

The forecasted discrepancy for the whole hebdomad would merely be the amount of the i¬?ve day-to-day discrepancy prognosiss. If the standard divergence is the needed volatility estimation instead than the discrepancy, merely take the square root of the discrepancy prognosiss. Note besides, nevertheless, that standard divergences are non linear. Hence, if day-to-day criterion divergences are the needed volatility step, they must be squared to turn them to discrepancies. Then the discrepancies would be added and the square root taken to obtain a hebdomadal criterion divergence.

3.4 Accurate prediction trials

There are assortment ways to measure the truth of prediction theoretical accounts. In this survey, intend absolute mistake, root mean squared mistake, and intend absolute per centum mistake will be used as rating standards. These error statistics are used to compare the truth of ARIMA and GARCH theoretical accounts in this survey.

3.4.1 Mean Absolute Error

The average absolute mistake ( MAE ) which represents the typical mistake is a truth step used to find how close prognosiss or anticipations are to the existent results. MAE is estimated from:

( 3.31 )

Where is forecasted value of existent value

3.4.2 Root Mean Squared Error

The root mean squared mistake is one of the most popular dimensioned statistics to analyze the public presentations of calculating theoretical account. RMSE is calculated as:

( 3.32 )

Similarly to MAE, RMSE depends on the graduated table of the dependant variable. However, with the fixed Numberss of mistake ( n ) , RMSE is ever larger than MAE because of the square root of the figure of mistakes. Althought, MAE and RMSE can be used together to analyze the mistakes in a set of prognosiss. MAE is extremely recommended to utilize because it is natural step of mean mistake every bit good as unambiguous ( Willmott, C.J, Matssura, K 2005 ) .

3.4.3 Mean Absolute Percentage Mistake

Mean absolute Percentage Error is the norm of absolute per centum sum by which prognosiss differ from results. It can be estimated form the undermentioned equation:

( 3.33 )

The MAPE is about the same as MAE except that it expresses truth as per centum. This statistic is appreciated when sample size is little, forecast mistakes for each period can be presented in the per centum signifier. It will be helpful in doing comparing among prognosiss from different units of step.

Chapter 4

Analysis AND RESULTS

4.1 Introduction

Forecasting stock monetary value volatility is really of import to fiscal market. In this chapter the consequences of the public presentation refering both calculating methods and mistake prosodies are analysed. First, using 4 stairss Box-Jenkin attack to find, estimation and prognosis by ARIMA theoretical account, followed by sing Garch theoretical account. Both theoretical accounts are analysed by utilizing EView package. Finally, public presentations of these two theoretical accounts for calculating day-to-day stock monetary value informations are besides compared.

4.2 Data Management

In this survey, Vietnamese stock monetary value index ( Vnindex ) is obtained from the site of Hochiminh stock exchange from 28th July 2000 to 30th July 2012. The informations are divided into two parts. The first portion that is from 28th July 2000 to 29th June 2012, is used to place and gauge ARIMA and GARCH theoretical accounts. The 2nd one varies from 30th June 2023 to 30th July 2012. This information set is a bendmark for the out-of-sample prognosis consequence anticipating from appropriate ARIMA and GARCH theoretical accounts.

4.3 Vnindex clip series

The Vnindex will be determined the tendency of the seires of being changeless, additive or non-linear. The information from 28th July 2000 to 29th June 2012 will be illustrated in Figure 4.1.

Figure 4.1 The clip series for Vnindex

By and large, Vnindex have fluctuated in the scope from 100 to 1200. It can be considered that this procedure ‘s features is non-linear ( Fan et al. , 2008 ) . At the beginning at 100 on from 28th July 2000, Vnindex increased to about 600 before it declined to about the starting point at the terminal of 2003. The 2004-2007 period witnesses a important up tendency of Vnindex which was at extremum on March 2007. However, it started to dramatically diminish from the terminal of the same twelvemonth because of the universe fiscal crisis. This index has fluctuated around 400 from 2010 until now.

Figure 4.2 Histogram and normality trial on Vnindex

From the histogram, it can be seen that a great figure of observations are located around 390. As summarized in Figure 4.2, the mean and standard divergence of vnindex are 409.82 and 235.47 severally. The value for lopsidedness is 1.3351 and kurtosis is 4.4612 which imply that the graph is asymmetric and leptokurtosis. Jarque-Bera trial indicates that we do non reject void hypothesis of being normal distribution at 5 % significance degree.

4.4 Stationary procedure

Box-Jenkins attack is merely applied for stationary clip series. The unit roots trial such as ADF and PP trials can be used to find the stationary of a series. Initially, the stationary of the original Vnindex must be checked foremost.

Null Hypothesis: VNINDEX has a unit root

Exogenous: Changeless

Lag Length: 5 ( Automatic based on SIC, MAXLAG=27 )

t-Statistic

A A Prob. *

Augmented Dickey-Fuller trial statistic

-1.743874

A 0.4090

Test critical values:

1 % degree

-3.432494

5 % degree

-2.862373

10 % degree

-2.567258

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.1 ADF trial for Vnindex

Accordin to postpone 4.1, the ADF trial statistic is a?’1.7439 which is greater than test critical values of a?’3.4333, a?’2.8624 and a?’2.5673 at 1 % , 5 % and 10 % significance degrees. The p-value of 0.4090 strongly disagrees that the series is stationary. Therefore, the void hypothesis that the information set is a non-stationary procedure, is accepted.

Null Hypothesis: VNINDEX has a unit root

Exogenous: Changeless

Bandwidth: 23 ( Newey-West utilizing Bartlett meat )

Adj. t-Stat

A A Prob. *

Phillips-Perron trial statistic

-1.807177

A 0.3774

Test critical values:

1 % degree

-3.432490

5 % degree

-2.862371

10 % degree

-2.567257

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.2 PP trial for Vnindex

Vnindex besides was tested with PP trial and the consequence are shown in Table 4.2. A similar illation is obtained where the PP trial statistic of a?’1.1807 is greater than its trial critical values at 1 % , 5 % and 10 % significance degrees. Since the nonreversible p-value is 0.3774, the void hypothesis of being non-stationary is accepted. Therefore, this clip series need to be differenced to obtain a stationary series.

Because the original Vnindex is a non-stationary clip series, the stationarity of the first order difference of Vnindex in bend will be checked. However before taking difference, natural logarithms are applied on Vnindex. Harmonizing to ( Lutkepohl, H.A andA Xu, F.,2012 ) , forecasts yt which can be described as yt = logxt, so change overing the consequence to x1 can be more precise than direct foretelling on crosstalk if yt has a more stable discrepancy than crosstalk. Hence, taking first order difference of natural logarithms of a procedure to make the return of Vnindex.

Ln ( yt ) – Ln ( yt-1 ) = ln ( yt/yt-1 ) = per centum return in twenty-four hours T ( 4.1 )

Null Hypothesis: Roentgen has a unit root

Exogenous: Changeless

Lag Length: 4 ( Automatic based on SIC, MAXLAG=27 )

t-Statistic

A A Prob. *

Augmented Dickey-Fuller trial statistic

-18.81547

A 0.0000

Test critical values:

1 % degree

-3.432494

5 % degree

-2.862373

10 % degree

-2.567258

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.3 ADF for Vnindex return

The ADF trial for Vnindex ‘s return is shown in Table 3.3. From Table 3.3, the t-statistic for the first lagged difference series is a?’18.81547 which is much smaller than the 1 % significance degree of trial critical value. Hence, the void hypothesis that states that the first order difference for natural logarithms of Vnindex series is non-stationary, is rejected.

Null Hypothesis: Roentgen has a unit root

Exogenous: Changeless

Bandwidth: 21 ( Newey-West utilizing Bartlett meat )

Adj. t-Stat

A A Prob. *

Phillips-Perron trial statistic

-40.56328

A 0.0000

Test critical values:

1 % degree

-3.432491

5 % degree

-2.862371

10 % degree

-2.567257

*MacKinnon ( 1996 ) nonreversible p-values.

Table 4.4 PP trial for the return of Vnindex

The PP trial on the return of Vnindex monetary values is shown in Table 3.4. It gives a similar consequence to ADF trial above. The adjusted t-statistic for the first order lagged difference of natural logarithms series is a?’40.5632, which is really little compared to the 1 % degree of trial critical value of a?’3.4324. The p-value equal nothing indicates that the PP trial is important. Therefore, we make the same illation as in the ADF trial where the return of Vnindex is stationary.

Figure 4.3 The return of Vnindex

The return of Vnindex which is the first lagged difference from the natural logarithms of informations clip series, is shown in Figure 4.3. It can be found that the return of Vnindex is stationary since about of the return values are allocated around nothing. However, some spikes which appear in the figure, represents a high volatility period.

Figure 4.4 Histogram and normality trial of Vnindex ‘s return

Figure 4.4 illustrates the histogram and normalcy distribution trial statistics, including mean, average, maximal and minimal values, standard divergence, lopsidedness, kurtosis, and Jarque-Bera trial of first Vnindex ‘s return. The histogram is centred and peaked at nothing. The average is equal to 0 because most of the values after first lagged difference autumn within the intervals of nothing. The values of lopsidedness and kurtosis are severally a?’0.1707 and 5.1735 which show that the distribution is somewhat asymmetric and extremely leptokurtosis.

Harmonizing to the information obtained above, It can