Eviews Help

jhh19970724

Sample_Empirical.pdf

Home >Business & Finance homework help >Economics homework help >Eviews Help

Abstract

Last year’s disproportional increase in financial markets’ volatility has emphasised once again

the significance of risk management and asset allocation. Thus, modelling and detecting

short-term volatility patterns have become indispensable for informed decision-making. The

aim of this research is to obtain an adequate return series model with superior data fit and

modelling capabilities to forecast short-term conditional volatility in stock markets. After

identifying stationarity and ARCH effects in all three return series, further examinations have

been conducted, assessing ARCH(1), GARCH(1,1), GARCH-M, and PARCH models with three

criteria. Although the PARCH model demonstrated the highest data fit ability for all indices,

with no contradictions of the underlying assessment criteria, it represented the weakest

performance for heteroscedasticity removal. In opposition, the GARCH(1,1) model removed

heteroscedasticity most effectively in two out of three indices. The most accurate one-month

forecasting results have been obtained for the PARCH model with an exception for the Swiss

Market Index. Further, all indices exhibited similarities in their volatility patterns, indicating

volatility clustering and leverage effects through the attainment of ARCH, GARCH and

leverage parameters, in line with economic developments.

Table of Contents

1 Introduction ................................................................................................................... 1

2 Literature review ............................................................................................................ 3

3 Data ............................................................................................................................... 7

3.1 Data description ............................................................................................................. 7 3.2 Characteristics of the return series ................................................................................ 8

3.2.1 Detecting stationarity ............................................................................................. 8 3.2.2 Investigating structural breaks using Bai-Perron .................................................. 10 3.2.3 Examining ARCH effects ........................................................................................ 12

4 Methodology ................................................................................................................ 14

4.1 GARCH model introduction .......................................................................................... 14 4.2 The GARCH (p,q) and GARCH(1,1) model ..................................................................... 15 4.3 The GARCH-M model ................................................................................................... 16 4.4 The PARCH model ........................................................................................................ 17 4.5 Model evaluation criteria ............................................................................................. 18

5 Empirical results ........................................................................................................... 20

5.1 Data fit comparison ...................................................................................................... 20 5.2 ARCH effect removal .................................................................................................... 27 5.3 Forecasting ................................................................................................................... 28

6 Conclusion .................................................................................................................... 30

7 References ................................................................................................................... 31

8 Appendix ...................................................................................................................... 34

III

List of Tables

Table 1: Stationarity detection with unit root tests ............................................................... 10

Table 2: Bai-Perron multiple breakpoint test applying the global l method .......................... 12

Table 3: ARCH effects with the Breusch-Pagan ARCH test ..................................................... 13

Table 4: Data fit comparison for the SMI ............................................................................... 21

Table 5: Data fit comparison for the SP100 ........................................................................... 23

Table 6: Data fit comparison for the IBOV ............................................................................. 25

Table 7: ARCH-LM test for heteroscedasticity removal ......................................................... 27

Table 8: One-month forecasting error comparison ............................................................... 28

List of Figures

Figure 1: Daily returns of the three indices from 2006 to the end of 2018 ............................. 8

Figure 2: Comparison of SMI conditional variance graphs from 2006 to the end of 2018 .... 22

Figure 3: Comparison of SP100 conditional variance graphs from 2006 to the end of 2018 . 24

Figure 4: Comparison of IBOV conditional variance graphs from 2006 to the end of 2018 .. 26

Figure 5: Volatility forecasts for the three indices for December 2018 ................................. 29

1 Introduction

The ability to model and predict volatility in financial markets has become an indispensable

condition for practitioners, researchers and policymakers, since volatility implies both,

opportunities and risks. Over the last year, excessively high volatility among all asset classes

has been observed within financial markets. This occurrence can mainly be linked to political

and economic incidents such as the long-lasting tariff negotiations between the US and China,

Brexit and the forthcoming recession. The subsequent uncertainty among investors has

triggered heavy fluctuations not only in equity markets but also in real estate, commodities

and fixed income.

Thus, especially under these market conditions, asset allocation, derivative pricing and the

ability to model and forecast volatility have become crucial tasks in daily risk management.

Since superior forecasting results into more adequate pricings of financial asset, it can

significantly improve financial risk management. Consequently, it is important to not only

understand the underlying investment risk but also produce accurate forecasts of the asset

returns variance.

Since short-run volatility can wipe out a substantial amount of an investor’s profits, it plays

a crucial function for not only investment and risk managers but also other market

participants, such as private individuals and policymakers. Hence, the anticipation of market

fluctuations is necessary to determine effective investment strategies and therefore limit the

potential risk exposure. This aim can mainly be achieved by accurate predictions of short-term

volatility patterns, allowing investors to adjust their portfolios and shift their investments

towards more defensive asset classes.

Subsequently, a vast amount of literature from the last decades has described the ability

to model and forecast conditional variances for numerous financial applications. The

underlying development has paved the way for various modifications of well-established time

series models, leading to consistent improvements in the models’ data fit and forecasting

accuracy. Nonetheless, there is still no current consensus among researchers on the most

appropriate model best suited to forecast volatility in stock markets.

This paper contributes to the already existing literature in the field of time-series models

by investigating three stock market indices differing in terms of geographic exposure. Since

the application of autoregressive conditional heteroscedasticity (ARCH) models usually

requires higher-order lags to estimate conditional heteroscedasticity in time series more

accurately, this research paper mainly focusses on the application of generalised ARCH

(GARCH) family models in assessing data fit and forecasting accuracy. As the models consider

two important specifications in stock markets – namely, volatility clustering and excess

kurtosis – they are particularly advantageous for this research.

The remainder of this paper is structured as follows. Section 2 offers a comprehensive

review of the literature and its development. Subsequently, in Section 3, the underlying data

set is analysed and investigated according to stationarity, structural breakpoints and ARCH

effects. Section 4 provides a holistic picture of the methodology applied throughout this paper

and outlines the assessment criteria used for model evaluation. Section 5 discusses the

empirical results for the underlying models, and a conclusion is drawn in the last section of

this research paper.

2 Literature review

This section aims to aims to outline the most relevant past-to-present literature and its

contribution to this paper. During the last decades, a vast amount of literature has explored

different modelling approaches for financial return series.

Mandelbrot (1963) introduced revolutionary statistical procedures to understand stock

market behaviour, such as volatility clustering. Mandelbrot stressed the significance of

including random variables to create a comprehensive picture of volatility patterns. Another

economic phenomenon called the ‘leverage effect’ was discovered by Black (1976). It

describes the non-symmetric reaction of uncertainty to both positive and negative disruptions

in financial markets. These two key findings about the characteristics of return series have

created a basis for future statistical models, allowing for the estimation and prediction of

volatility.

The above stated detection of volatility clustering has been incorporated by Engle (1982)

for the first time. Engle suggested modelling conditional variance fluctuations over certain

time periods and, to that end, introduced the ARCH process. The ARCH model permits the

variance to rely on lagged periods of squared disturbance terms. Engle’s seminal paper has

represented a cornerstone for countless further research papers and model alterations.

Although the ARCH model is advantageous if volatility in stock return series varies less fast,

it experiences difficulties in modelling higher-order ARCH. Bollerslev (1986) successfully

solved these issues by introducing the GARCH model. It includes extra autoregressive terms

and therefore allows for a more flexible lag structure. Numerous research papers have applied

Bollerslev’s GARCH model and its different model specifications for further studies.

For instance, Akgiray (1989) was a pioneer in applying GARCH models to compare

forecasting accuracy and data fit ability among different models. That study analysed daily US

stock market returns. Akgiray concluded that the most precise forecasting results and data fit

were achieved by the GARCH(1,1) model, which delivered slightly better results compared to

more simplistic volatility forecasting models.

Further studies to examine the forecasting ability of random walk and GARCH models in

stock indices have been conducted by McMillan et al. (2000), examining different volatility

frequencies for the FTSE100. The results indicate a slight but consistently superior

performance by GARCH models. Additional evidence for the superior performance of

GARCH(1,1) models has been provided by Brooks and Burke (1998) and Hansen and Lunde

(2005), who predicted volatility in daily exchange rate returns for different currency pairs.

Further research examining the estimation results of GARCH models during stock market

crises has been conducted by Lim and Sek (2013). Their paper profoundly analyses volatility

patterns in the Malaysian stock market during a 20-year period. To assess the accuracy of the

forecasts three different error measurements have been applied, for instance the mean

absolute error. Lim and Sek (2013) clearly stated that conventional GARCH and threshold

GARCH (TGARCH) capture volatility best during pre-crises and GARCH works particularly well

during the crises. A more recent investigation of different volatility patterns in stock markets

has been conducted by Kambouroudis et al. (2016). In the contrary to Lim and Sek (2013),

Kambouroudis et al. stated that it is not sufficient to solely use a simplistic GARCH model to

obtain accurate estimation outputs. The study, including US and European indices over a

twelve-year period, provides evidence that more precise forecasting results are obtained if an

asymmetric GARCH model is applied, using both realized and implied volatility.

As can be seen, there is no current consensus among researchers with respect to the

consistent superior performance of GARCH models. For instance, Harry and Heynen (1994)

conducted an in-depth analysis to predict volatility in financial markets, analysing returns of

seven stocks and five currency pairs over a twelve-year period. Their paper states that superior

model performance is strongly linked to the examined asset class, concluding that the

GARCH(1,1) model only outperformed the other models for currencies. Thus, the authors

confirmed the results of Hansen and Lunde (2005). A more recent study conducted by

McMillan and Speight (2012) provides additional evidence for the more accurate forecasting

capability of GARCH(1,1) models by analysing intra-daily EUR-USD, EUR-GDP and EUR-JPY

exchange rates over a three-year period.

However, GARCH models are applicable to measuring a model’s data fit and forecasting

ability for not only stock markets and foreign exchange but also options and commodities. For

instance, Kanniainen et al. (2014) analysed options on the S&P 500 Index to enhance the

empirical performance of GARCH models. The study managed to reduce the error of the

estimation results significantly and reconfirmed earlier results from Engle and Mustafa (1992).

Another model specification sharing almost the same properties as the GARCH(1,1) is the

GARCH-in-mean (GARCH-M) model introduced by Engle et al. (1987). The augmented model

incorporates an advantage in that the risk premium is no longer assumed to be constant, as

in conventional GARCH models, but is directly connected to the risk in question. Chen et al.

(2001) conducted a profound analysis of returns for four stock exchanges in China to

investigate the predictability of volatility and the relationship between expected returns and

risk. While the researchers stated a high predictability for uncertainty using GARCH models,

they provided no evidence for a connection between expected returns and risk when applying

the GARCH-M. These findings have been strongly supported by Lundblad (2007), who analysed

over 100 years of US equity market data for relationship detection. In opposition to the

previous two papers, the GARCH-M application of Abdalla and Winker (2012) provided

evidence of a positive risk premium for the investigated stock markets in Egypt and Sudan,

indicating a positive relationship between volatility and expected returns.

Nevertheless, it is worth mentioning that the application of GARCH family models is not

always the most appropriate approach for modelling high-frequency data. Pagan and Schwert

(1990) applied alternative econometric models to the conventional GARCH family since they

are not able to measure nonlinear returns in financial markets. To tackle this issue, Nelson

(1991) extended the GARCH model to an exponential GARCH (EGARCH) model, which allows

one to quantify the variance in asymmetric returns. The model has proven its enhanced data

fit and forecasting capabilities according to numerous researchers. For instance, Alberg et al.

(2008) investigated the model’s estimation results in the Tel Aviv Stock Exchange and provided

evidence for the superior performance of EGARCH models as compared to GARCH models.

Similar results comprising stock indices, have been found by the more recent researches of

Dritsaki (2017) and Lin (2018), confirming the achievement of more accurate results for

EGARCH(1,1) compared to GARCH(1,1) models.

In fact, GARCH models are able to take volatility clustering and leptokurtosis into account

but do not have the capability to capture leverage effects. To solve this issue, nonlinear

amplifications of GARCH models have not only been introduced by the priory mentioned

Nelson (1991) but also by Ding et al. (1993), who introduced the asymmetric Power ARCH

(PARCH). A comprehensive study measuring the data fit and forecasting capabilities of PARCH

models was conducted by Brooks (2007), who analysed a vast amount of different stock

markets around the globe over a 10-year period. Brooks’ results provide evidence that PARCH

models generally perform well in emerging markets by examining asymmetric effects. An

additional study by Degiannakis (2004) assessed the estimation results of PARCH models,

among other models, by comparing the daily returns of three European indices. The analysis

of Degiannakis identified that extended PARCH models provide the most accurate one-day-

ahead results and therefore supported the findings of Giot and Laurent (2004), who applied

the PARCH model for value-at-risk estimations of two US stock indices and two currency pairs.

A more recent review of the literature on this topic was conducted by Thorlie et al. (2015),

who investigated daily stock index returns of the S&P 500 from 2002 to 2012. The study

provides additional evidence for the superior data fit and forecasting capabilities of PARCH

models in comparison to the GARCH model and its different model specifications. Moreover,

Thorlie et al. (2015) concluded that the most accurate results are obtained when a skewed

student’s t-distribution is applied.

This paper aims to contribute to the broad literature in the field of modelling and

forecasting volatility in stock markets by analysing the volatility patterns of three stock indices

with different model specifications of the ARCH and GARCH family.

3 Data

This section describes the data set used in this research paper to analyse and forecast

volatility. Moreover, the main characteristics of the return series are investigated to build a

solid basis for the later model selections.

3.1 Data description

To increase the scope and accuracy of this study, three stock market indices from different

regions around the globe are analysed, namely, the Swiss Market Index (SMI), the Standard &

Poor 100 (SP100) and the Ibovspa Brasil Sao Paolo Stock Exchange Index (IBOV). The sample

size for each index is based on daily closing prices between 1 January 2006 and 31 December

2018 and have been gathered from Bloomberg.

During the above time period, 3,263 observations for the SMI; 3,271 observations for the

SP100; and 3,211 observations for the IBOV were collected. As the majority of the test

procedures are conducted using log returns rather than prices, the data set is transformed as

follows:

!" = log((") − log ((",-) (3.1)

where (" represents the closing price of an index at time t and !" signifies the return of an

index. The return series of the three indices are depicted in Figure 1 (below).

Switzerland’s blue-chip companies are represented by the SMI, the most important index

in the country, which includes the 20 largest and most liquid stocks. Since the SMI represents

a price index, it is not corrected for dividends. To capture the performance of large-cap

companies in the US, the SP100 is applied. The SP100 is a subset of the well-known S&P 500

index, consisting of 100 large-cap companies from a broad range of industries, representing a

capitalisation-weighted index. Approximately 70% of the activity in Brazilian stock markets is

captured by the major IBOV index, incorporating the 50 most liquid stocks from a wide range

of industries. Moreover, it represents a weighted total return index and therefore dividends

are treated as reinvestments.

3.2 Characteristics of the return series

This section investigates the return series characteristics of the three sample indices to create

a framework for the development of volatility forecasting models.

3.2.1 Detecting stationarity

First, different unit root tests are conducted to check the series for stationarity. If a series is

stationary in the long run, a mean-reverting process with constant variance can be observed,

as shown in Figure 1. Moreover, stationarity in a return series reflects shocks inevitably

Figure 1: Daily returns of the three indices from 2006 to the end of 2018

The Swiss Market Index

impermanent, due to the mean-reverting process in the long term. Additional evidence for

stationarity can be provided by analysing a correlogram; the autocorrelation decreases to zero

as the length of the lags is augmented (Asteriou and Hall, 2016). Nonetheless, the application

of this technique can be misleading since an almost-stationary process can present nearly the

same patterns (Asteriou and Hall, 2016).

Subsequently, further unit root tests are required to provide definite evidence for the order

of integration. The latter refers to the number of stages a nonstationary series requires to be

differentiated into a stationary series. Thus, stationarity is present if the number of stages is

zero. To enhance the test procedure, this paper uses the augmented Dickey-Fuller (ADF) test

to remove autocorrelation between the error terms (Floros, 2004). The ADF test demands

homoscedasticity and independent disturbance terms (Asteriou and Hall, 2016).

Thus, a Phillips-Perron (PP) test is conducted to enhance the power of the results by using

the Akaike information criterion (AIC) and correcting for t-statistics. Unlike the ADF test, the

PP permits marginal changes in variance. The prior results can be approved by conducting a

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test of the reverse hypothesis (Asteriou and Hall,

2016). Typically, conventional models suggest that time series are stationary in the long run.

However, in the short run, heteroscedasticity and, therefore, non-stationarity are expected.

Simple statistical least-squared models demand the disturbance term to be constant over

time. The latter, also known as homoscedasticity, is a condition of the classic linear regression

model (CLRM), making standard financial models inappropriate for modelling volatility (Engle,

2001).

Therefore, models which are able to deal with changing volatility, such as ARCH family

models, are required. As depicted in Figure 1, financial markets are often disrupted by periods

of high uncertainty, and therefore their time series depict remarkably high volatility followed

by calmer periods. Not only does the magnitude of volatility change but also intervals of

excessive volatility are more likely to be concentrated and followed by intervals of higher

volatility. This occurrence is called ‘volatility clustering’. The unconditional variance of the

return series is used to forecast volatility in long term, whereas conditional variance facilitates

the forecasting of the riskiness of an index (Gokbulut and Pekkaya, 2014, Asteriou and Hall,

2016).

Unit root test results

As mentioned above, different unit root tests are conducted to investigate the series for

stationarity. All correlograms for the different return series show decreasing autocorrelation,

indicating stationarity. Nevertheless, to provide definite evidence for the order of integration,

additional tests are conducted. While the null hypothesis of the ADF and PP tests states a

present unit root and therefore indicate non-stationarity, the reverse hypothesis of the KPSS

test states no presence of a unit root and should support the prior findings.

As depicted in Table 1 (below), it can clearly be stated that stationarity in the three series

is detected. Since the t-statistic for the ADF and PP tests (without trend) is absolutely higher

than the critical value of the 5% significance level, both tests reject the null hypothesis.

Additionally, the KPSS test fails to reject that the series is stationary, due to a lower t-statistic

in comparison to the corresponding critical values.

As all conducted tests support the evidence for stationarity the data series shows a mean-

reverting process with constant variance in the long run, implying that financial disruptions do

not have any effect on the long run returns. These results support a further investigation for

multiple breakpoint test, since the Bai-Perron test performs extraordinary well, with

stationary time series with no trend.

3.2.2 Investigating structural breaks using Bai-Perron

To develop a more accurate data fit and to avoid modelling inaccuracies, Bai and Perron (1998)

introduced a new method for the detection of multiple structural breaks. The method

indicates unexpected fluctuations in the determined regression model parameter over the

Table 1: Stationarity detection with unit root tests

entire time series. These structural breaks can be triggered by sudden changes in technology,

politics or unexpected macroeconomic disruptions such as the financial crisis (Pesaran et al.,

2006, Karanasos et al., 2018).

The Bai-Perron process is mainly applied to linear-regression models, which are estimated

by least squares and include no significant trend in their stationary time-series. (Papell et al.,

2000) Moreover, the consistent model does not only capture pure structural changes but also

investigates partial breaks, where a subset of parameters does not alter. Thus, increasing the

degree of freedom (Bai and Perron, 1998).

While, tests deliver a consistent estimate of the latest structural changes the model is

particularly advantageous, since it permits for heteroscedasticity and autocorrelation in

disturbance terms (Antoshin et al., 2008, Pesaran and Timmermann, 2002).

Moreover, Bai-Perron conducted a sup-Wald test to investigate structural changes by

restricting the distributions of it, and by computing the rate of convergence. While the testing

procedure can be hinged on sequential testing or information criteria, this paper will focus on

sequential testing. The process, examines the significance of the null hypothesis with . breaks

(e. g. zero breaks), against the alternative hypothesis of .+1 breaks (for instance one break)

(Bai and Perron, 2003). This paper relies on the strong Global L breaks method with sequential

estimation to detect further breakpoints (Pesaran and Timmermann, 2002).

Hence, if significant structural breakpoints are detected, the time series should be split

according to the explicit stated date to prevent forecasting errors and increase the accuracy

of the testing procedures.

Testing for structural breakpoints

The Bai-Perron test using the global l method is conducted to investigate the return series of

the different indices for structural breaks. As Table 2 (below) illustrates, the f-statistics are not

statistically significant at the 5% level, as the corresponding critical values are lower for all

tested breaks. Thus, failing to reject the null hypothesis implying that no breaks are

significantly detected in the return series.

Based on these results, the return series of the indices do not need to be split at a particular

date. Hence, further statistical tests can be conducted without splitting the return series. For

completeness, the closest potential breakpoints with the highest f-statistics are included in

the above table.

Together, the previous findings indicate a stationary time series which can be consistently

modelled without splitting the data. Due to stationarity and the presence of constant variance

in the long run, together with volatility clustering (discussed in the stationarity section), short-

run conditional variance with heteroscedasticity is indicated.

3.2.3 Examining ARCH effects

Before ARCH models can be implemented, one must test the series for conditional

heteroscedasticity by either investigating the scatterplot or, more precisely, conducting a

Breusch-Pagan Lagrange multiplier (LM) test. To provide definite evidence for the presence of

ARCH effects, this paper uses the ARCH-LM test (Asteriou and Hall, 2016). To detect changes

in variance, an auxiliary regression (with a constant) on the residuals of the ordinary least-

squared (OLS) mean equation can be run.

To increase the robustness of the ARCH-LM test, a White test must be conducted. If

autoregressive heteroscedasticity is detected, normal CLRMs, such as OLS, should not be

employed. This is mainly due to the biased results of the coefficient matrix, resulting in invalid

t-statistics. To solve this issue, one can apply ARCH models (Asteriou and Hall, 2016).

Table 2: Bai-Perron multiple breakpoint test applying the global l method

ARCH effects results

Since previous findings strongly indicate the presence of ARCH effects, a Breusch-Pagan ARCH

test is employed, using different lag lengths from 15 to 1. For simplicity reasons, not every lag

between 15 and 1 is illustrated in Table 3. The results for the three indices are all statistically

significant at the 1% level. Thus, the null hypothesis of no ARCH effects is rejected, as the f-

statistics are significantly higher than the corresponding critical values.

Notably, the SMI represents a continuous increase in its f-statistics as the lag length

decreases, indicating a strong dependency between the error terms. On the contrary, the

Brazilian IBOV shows an increase in the f-statistics as the lags decrease before significantly

dropping at one lag, representing the weakest ARCH effect. A similar pattern can be observed

for the SP100, with the difference of a slight decrease from four lags to one lag.

Hence, ARCH effects are present in every return series, although the degree differs

between the lags. The latter is important since the detection of ARCH effects requires ARCH

family models to model return series, rather than CLRMs, such as OLS.

Table 3: ARCH effects with the Breusch-Pagan ARCH test

4 Methodology

The underlying section aims to describe appropriate statistical models with adequate

forecasting abilities and a high data fit, based on the return-series characteristics detected in

the previous section.

4.1 GARCH model introduction

One of the first developed models allowing for the presence of ARCH effects is Engle’s ARCH

model. The model introduced in 1982, builds the basis for further GARCH model developments

described throughout this paper. It proposes that the present variances at time t are affected

by historical squared disturbance terms. Additionally, Engle recommended to compute the

mean and variance of a series synchronously rather than separately, if there is uncertainty

about the conditional variance being non-constant (Asteriou and Hall, 2016). A simple ARCH(1)

model can be constructed by permitting the variance to rely on a single lag period of squared

disturbance terms, and can be constructed as follows:

/" 0 = 12 + 1-4",-

0 (4.1)

An ARCH(1) model states that if there was a large disruption in period (t-1), there is a great

possibility for another disruption in time t, leading to a high disturbance term 4" (since 4" is

squared, it needs to be a positive absolute number). Thus, implying that if 4",- 0 is high/low the

following variance of 4" will also be high/low. Therefore, a positive variance is necessarily

connected with positive coefficient for 1’s (Asteriou and Hall, 2016).

However, the model does not necessarily need to rely only on a single lagged parameter

and can therefore simply be augmented to an ARCH(q) model. This model is particularly

advantageous if fluctuations in the series are expected to vary less fast in comparison to an

ARCH(1) model (Asteriou and Hall, 2016). To retain the suitable model, ARCH models with

different orders of integrations must be conducted and compared. This comparison can be

reflected by simultaneously plotting the divergent conditional variance grids (Asteriou and

Hall, 2016).

Despite the advantage of the ARCH(q) model stated above, complications often occur when

it is modelled. This is mainly due to the regular attainment of non-positive estimates for 15

(4.3)

coefficients when a higher order of integration is applied. Furthermore, a major weakness of

the ARCH process regarding Engle was that the pattern is more similar to a moving average

than to an autoregression (Asteriou and Hall, 2016).

4.2 The GARCH (p,q) and GARCH(1,1) model

In 1986, Bollerslev addressed the above stated issues by introducing GARCH model. The

augmented ARCH model permits especially a higher flexibility for the lag construction without

the attainment of non-positive estimates for 15s. In addition to the more dynamic lag

structure, Bollerslev integrated to the ARCH(q) process higher order conditional variance

terms as autoregressive terms. This is particularly advantageous, allowing for a kind of

adaptive learning mechanism (Asteriou and Hall, 2016, Bollerslev, 1986). The elemental

GARCH((,6) model can be represented as follows:

7" = 8 + 9′;" + 4" (4.2)

4"|Ω" ~ ??@ A(0,/" 0)

/" 0 = 12 + DEF/",-

FH-

+ D1I4",I 0

IH-

where 7" represents the mean and /" 0 the variance equation. The variance scaling parameter

(/" 0) does not only rely on the historical value of disruptions (as in ARCH), but also on the

previous values of itself. The latter is expressed by lagged terms (/",- 0 ) with the GARCH

parameter EF, whereas the historical value of the disruptions is represented by the lagged

squared residuals (4",I 0 ), with ARCH the parameter 1I. Hence, a GARCH model can

uncomplicatedly be abbreviated to a classic ARCH(q) model if ( is set equal zero. By

additionally setting 6 to ( equal zero, white noise of the disturbance term is achieved

(Asteriou and Hall, 2016, Bollerslev, 1986).

The most uncomplicated form of GARCH((, 6) is the GARCH(1,1) model with (=1 and 6=1.

The model’s variance equation has the following form:

/" 0 = 12 + E-/",-

0 + 1-4",- 0 (4.4)

this particular form of GARCH(1,1) performs exceptionally well and is additionally

uncomplicated to estimate, since there is only a limited number of unspecified parameters,

(4.5)

namely 12, 1-,E-. Therefore, the loss of degree of freedom can be limited (Asteriou and Hall,

2016).

To provide evidence that the GARCH(1,1) model is an equivalent to the infinite ARCH(6)

process, with geometrically diminishing coefficients the equation (4.4) is modified using a

uncomplicated substitution procedure, stated as follows:

/" 0 = 12 + E-/",-

0 + 1-4",- 0

= 12 + E(12 + E/",0 0 + 1-4",0

0 ) + 1-4",- 0

…

= 12

1 − E + 1- DE

I,-4",I 0

NH-

therefore, solving the issue of higher order ARCH(q) by preventing the estimated 15

coefficients of becoming negative (Asteriou and Hall, 2016). Thus, enhancing the performance

of the model. It might be true that according to literature the GARCH(1,1) is the best

performing model, however, to obtain robust results it is still strongly recommended

conducting comparisons between different order GARCH((, 6) models and the ARCH model

(Asteriou and Hall, 2016).

4.3 The GARCH-M model

Another alteration to the basic GARCH((, 6) model discussed in this paper is the GARCH-M.

This model is taken into consideration because of two main benefits in comparison to the

classic GARCH family. Firstly, traditional GARCH models require a stable average premium for

the risk taken into account, whereas the modified GARCH-M model loosens this restriction.

Moreover, the GARCH-M is a generalisation of the conventional models and therefore permits

to test their validity rather than assuming their correctness (Elyasiani and Mansur, 1998).

The adjusted model incorporates the advantage that it permits the conditional mean to

rely on its individual conditional variance. Therefore, the risk premium is not assumed

constant anymore, as in the conventional GARCH models, but is directly connected to the risk

in question. The augmented model measures uncertainty within financial markets with

(4.7)

volatility (Asteriou and Hall, 2016). Thus, it can be described in the conditional mean function,

as follows:

7" = 8 + 9 O;" + θ/"

0 + 4" (4.6)

4"|Ω" ~ ??@ A(0,/" 0)

/" 0 = 12 + DEF/",F

FH-

+ D1I4",I 0

IH-

The GARCH-M model allows different specifications. In the following equation risk is

captured by standard deviation instead of variance.1 Since both equations share the same

properties only the mean equation is expressed below:

7" = 8 + 9 O;" + θQ/"

0 + 4" (4.8)

The persistence in disruptions to risk and therefore to volatility is a highly important

determinant for the connection between returns and risk. Just long-lasting shocks have an

influence on the returns. Thus, in comparison to the classis models, the GARCH-M model is

the most appropriate model encompassing the relationship between risk and return. Hence,

it is typically used for asset-pricing models like the Capital Asset Pricing Models (CAPM)

(Asteriou and Hall, 2016, Elyasiani and Mansur, 1998).

4.4 The PARCH model

A new model has been introduced in 1993 by Ding. at all. to resolve two insufficient

assumption made by conventional models discussed earlier in this paper. The Monte Carlo

study has provided evidence that ARCH family is able to encompass specific moments of

autocorrelation in financial market returns. The latter property is also captured by the GARCH

model invented by Bollerslev and its specification introduced by Taylor and Schwert (Ding et

al., 1993).

The GARCH model by Bollerslev relies on the hypothesis that there is a direct linear

connection between conditional variance and higher order squared returns, whereas the

model by Taylor and Schwert addresses an equivalent assumption but with the difference of

1 For completeness, both variants of the GARCH-M model are discussed, however testes are conducted using the standard

deviation approach.

(4.9)

absolute lagged returns instead of squared ones. Ding et al. (1993) scrutinized these two

assumptions and therefore introduced the PARCH model to loosen the mentioned

restrictions.

However, the model does not only resolve these issues. While conventional models assume

normally distribution of the data set – which can be examined studying the Histogram with

the Jarque-Berra probability – the PARCH model does not require a normal distributed data

series to efficiently model and forecast returns. For instance, high frequency data is often non-

normally distributed (Brooks et al., 2000). Thus, the PARCH model tackles this issue by using a

broader variety of power transformations, expressed with R in the equation below:

4" = /"S", TℎS!S S" ~ A(0,1) ,

/" V = 12 + D1I(W4",IW − XI4",I)

V + DEF/",F V

FH-

IH-

12 > 0,R ≥ 0,1I ≥ 0,[ = 1,…,(,

−1 < XI < 1,[ = 1,…,(,

EF ≥ 0,? = 1,…,6.

where 1I represents the ARCH and EF the GARCH term. The leverage effect is expressed by the

variable XI while PARCH’s power term is indicated by R (Brooks et al., 2000).

Literature strongly underlines the advantages of using PARCH in specific situations where

the presence of a leverage effects in return series is discovered (Brooks et al., 2000). The

leverage effect describes the non-symmetric reaction of uncertainty to both, positive as well

as negative disruptions in financial markets. Thus, implying a negative correlation between

fluctuations in volatility and stock market returns. In fact, augmented the PARCH models are

the most precise models to forecast volatility one-day ahead (Degiannakis, 2004, Ding et al.,

1993).

4.5 Model evaluation criteria

To assess the data fit ability for the ARCH, GARCH and PARCH models, this paper applies the

AIC and the logarithm-likelihood (log-likelihood) criterion. For the AIC criterion, an absolute

higher value indicates a more accurate data fit, while the log-likelihood criterion is applied to

(4.10)

compare coefficients’ fit among different models. Further, the underlying models are assessed

according to their ARCH effect removal abilities, with an absolute smaller f-statistic indicating

increased ability to remove heteroscedasticity.

An additional evaluation method used in this paper is measuring the preciseness of a

model’s forecast. To assess a model’s forecasting ability, the Mean-Absolute Error (MAE) is

applied. The underlying criterion simply measures the difference between the actual observed

and the forecasted variance in absolute terms. For simplicity’s sake, the conditional variance

is replicated using squared returns as a proxy (Yu, 2002). The underlying formula is expressed

as follows:

^ D[|/̀"

0 − !" 0|]

"H-

where /̀" 0 represents the forecasted variance and !"

0 the squared returns. Applying the MAE

criterion provides an unambiguous picture and does not give relatively high emphasis to large

errors such as the root-mean-squared error (RMSE). Since errors are not squared, outliers

have less of an effect on the estimation outputs (Willmott and Matsuura, 2005). Moreover, to

the MAE criterion a visual comparison is conducted, by evaluating the different volatility

patterns.

5 Empirical results

This section endeavours to identify an adequate return series model for modelling and

forecasting conditional volatility in stock markets. Furthermore, the three models fitting the

data best are further investigated according to their forecasting ability and ARCH effect

removal potential.

5.1 Data fit comparison

A model comparison is conducted using data fit criteria such as the AIC and log-likelihood

criterion. The highest absolute value of the AIC indicates the strongest data fit. To provide

additional evidence, the log-likelihood criterion is applied to compare the coefficients’ fit for

the different models. The criterion should not be employed in isolation.

The following three tables (4, 5 and 6) compare four models applied to an index. The tables

used for the comparison include not only data fit criteria but also the models’ coefficients and

the corresponding z-statistics.

The Swiss Market Index

First, the SMI is analysed. It is important to state that all coefficients in Table 4 are statistically

significant at the 1% level. The table strongly indicates that the PARCH model fits the data

best, since the AIC represents the absolute highest value. This finding is supported by the

greatest log-likelihood criterion. According to these two criteria the best performing model is

followed by the GARCH-M model, which is just slightly better fitting the data as the

GARCH(1,1) model.

The ARCH model clearly showed the weakest data fit ability with respect to the AIC and log-

likelihood criterion compared to the others. This outcome aligns with the literature since ARCH

models often experience difficulties in delivering negative coefficients for 1s as the order of

integration increases. In contrast, GARCH models do not suffer from this issue. This has priory

been demonstrated in Section 3, using the ARCH effect test. Thus, the three most

advantageous models for the SMI are the PARCH, GARCH-M and GARCH(1,1) models.

(4.9)

For clarification, the depicted coefficients for the best-fitting PARCH model are stated in

the formula below, as previously discussed in the methodology:

/" V = 12 + D1I(W4",IW − XI4",I)

V + DEF/",F V

FH-

IH-

where 1I represents the ARCH and EF the GARCH term. The leverage effect is expressed by the

variable XI, while the PARCH’s power term is indicated by R. Focusing on the coefficients, the

GARCH(1,1) and GARCH-M models not only perform similarly in terms of the AIC and log-

likelihood criterion but also feature highly similar coefficients. This occurrence is linked to the

almost identical estimation process applied by these two models.

Taking the PARCH model into consideration, it becomes evident that the number of

coefficients exceeds that of the other models, as the PARCH model also considers leverage

effects and creates an additional ‘power-term’ coefficient.

Table 4: Data fit comparison for the SMI

Thus, the presence of leverage effects might be a reason for the PARCH model’s superior

data fit ability compared to other models. The literature strongly underlines the advantages

of using the PARCH model in specific situations in which leverage effects in return series are

discovered and taken into account.

These findings are captured in the conditional variance graphs depicted in Figure 2 (below).

Since the GARCH(1,1) and GARCH(M) models display almost identical patterns, the model

fitting the data best (PARCH) is described. After a short period of fluctuations, extensive

volatility is detected from mid-2008 to early 2009 and can be directly linked to the financial

crisis. Thereafter, the sovereign debt crisis caused uncertainty in financial markets and

triggered extensive volatility during 2011, leading to the second-highest peak in volatility. The

SMI experienced a shock at the beginning of 2015, mainly due to regulatory actions of the

Swiss National Bank, which discontinued the policy of pegging the Swiss franc to the euro.

Figure 2: Comparison of SMI conditional variance graphs from 2006 to the end of 2018

Further fluctuations occurred around 2016 and can mainly be linked to the Brexit vote and

the unexpected election of President Trump. Moreover, the PARCH model clearly indicates

less volatility than the GARCH(1,1) and GARCH-M models.

The American SP100

Considering the American SP100 index in Table 5, a similar pattern can be observed. As before,

the PARCH model fits the data series best, since the AIC and log-likelihood criteria state the

absolute highest values. The GARCH-M model’s data fit capability is slightly higher again than

that of the GARCH(1,1), according to the two criteria.

A possible reason for this consistent result might be that the model permits the conditional

mean to rely on its individual conditional variance. Therefore, the risk premium is not assumed

to be constant anymore, as in the conventional GARCH models, but is directly connected to

the risk in question. Additionally, the literature suggests that it is the most appropriate model

encompassing the relationship between risk and return. Moreover, positive and significate

ARCH and GARCH parameters can be observed at 1% level, indicating the presence of volatility

Table 5: Data fit comparison for the SP100

clustering while the significant leverage term strongly points towards prevailing leverage

effects.

Figure 3: Comparison of SP100 conditional variance graphs from 2006 to the end of 2018

The SP100 also indicates a certain similarity to the SMI in terms of the constant

underperformance of the ARCH(1) model according to the AIC and log-likelihood criterion.

This outcome is in line with the literature previously discussed in the methodology.

The same findings are represented in the conditional variance graphs in Figure 3. As

compared to the SMI, the SP100 experienced visibly higher volatility during the financial crisis.

A similar pattern for the sovereign debt crises can be observed; these events caused almost

the same level of uncertainty seen in the SMI. The unexpected election of President Trump

triggered slightly higher volatility than that seen in the SMI due to the different geographical

exposure. The slightly lower volatility spikes in 2018 can mainly be explained by tariff

negotiations between the US and China.

The Brazilian IBOV

Continuing the analysis with the IBOV index (Table 6) reveals that the PARCH clearly shows

superior data fit abilities compared to the other models yet again. However, the PARCH

model’s 12 coefficient is not statistically significant for the first time. Nevertheless, a familiar

pattern can be observed in the marginally superior performance of the GARCH-M model

relative to the GARCH(1,1), as both the log-likelihood criterion and AIC are higher in absolute

terms and do not contradict each other.

Parallel to the results for the other two indices, the AIC suggests that the ARCH(1) model is

the weakest in terms of data fit ability. These results are additionally supported by the fact

that the log-likelihood criterion values are the lowest for this model.

Hence, the main findings elaborated in this section strongly align with the literature. The

PARCH model consistently outperforms the other models for all indices. A potential reason for

this outcome is the presence of leverage effects since that model takes these into

consideration. As the evaluation criteria do not contradict each other at any point, the GARCH-

Table 6: Data fit comparison for the IBOV

M model fits the data slightly better than the GARCH(1,1) model, although the margins

between the applied criteria are extremely narrow. As suggested by the literature, the ARCH

models consistently underperform. This outcome can be directly linked to the lower

performance of higher-order ARCH moments.

Figure 4: Comparison of IBOV conditional variance graphs from 2006 to the end of 2018

The previous findings are supported by the conditional variance graphs in Figure 4. In

comparison to the priory described indices, the Brazilian IBOV indicates higher level of

uncertainty for the financial crises. Additionally, the Brazilian index is the only index that

suffered excess volatility during 2017. This strong disruption mainly derived from corruption

accusations against the Brazilian President Temer, which led to great uncertainty among

investors. A volatility pattern similar to that of the other indices can be observed for the

sovereign debt crisis in 2011.

5.2 ARCH effect removal

After identifying the three models best fitting the data, an ARCH LM test is conducted. This

test is especially important since these models, unlike CLRMs, allow heteroscedasticity. The

following test procedure provides an additional criterion for identifying a favourable model

for forecasting.

Table 7 indicates that the f-statistics are all not statistically significant since the

corresponding p-values are higher 5%. Therefore, the null hypothesis stating that no ARCH

effects are present in the indices’ return series is not rejected. Hence, prevailing ARCH effects

are successfully removed by all models.

At first glance, one notes that in contrast to the previous results, the PARCH model has the

highest f-statistics for all indices and is therefore less effective in removing the ARCH effects

than the GARCH(1,1) and GARCH-M models. The latter model most successfully removes

heteroscedasticity for the SP100, while the GARCH(1,1) model performs extraordinarily well

for the SMI and the IBOV indices, delivering the lowest f-statistics. Hence, the GARCH(1,1)

model removes heteroscedasticity most efficiently and represents the best model, in

opposition to the previous findings.

Table 7: ARCH-LM test for heteroscedasticity removal

Table 8: One-month forecasting error comparisonTable 9: ARCH-LM test for heteroscedasticity removal

5.3 Forecasting

After identifying the models best fitting the data and removing heteroscedasticity most

efficiently, the paper now investigates the models’ forecasting performance. To assess the

results obtained from the underlying testing procedure, the paper uses the MAE criterion.

Since the criterion implies an error, the lowest output is preferred. For the variance

forecasting procedure, the volatility of the last month in 2018 is estimated with the underlying

models and compared to the real variance outputs.

Analysing Table 8 illustrates that the PARCH model delivers the most accurate forecasting

results for the SP100 and the Brazilian IBOV, as it yields the lowest MAE. This finding is

completely in line with the results obtained from the data fit analysis, although the model was

the least efficient in the ARCH removal comparison.

However, the GARCH (1,1) consistently underperformed the other two models in the data

fit comparison but delivers more precise forecasting outputs for the SMI. The SMI is the only

index for which the GARCH(1,1) model’s forecasting abilities exceed those of the GARCH-M

model. Moreover, the GARCH(1,1) demonstrates a strong ability to remove heteroscedasticity

for not only the SMI but also the IBOV. Overall, all three models produce respectable

forecasting results, as a relatively small MAE is obtained. Additionally, the best performing

model for each index is visualised in Figure 5.

Table 8: One-month forecasting error comparison

Table 10: One-month forecasting error comparison

Figure 5: Volatility forecasts for the three indices for December 2018

0.0000

0.0004

0.0008

0.0012

03.12 05.12 07.12 09.12 11.12 13.12 15.12 17.12 19.12 21.12 23.12 25.12 27.12

C o

n d

it io

n al

V ar

ia n

ce GARCH(1,1) forecast Swiss Market Index

Realised Volatility Forecasted Volatility Linear (Realised Volatility)

0.0000

0.0003

0.0006

0.0009

03.12 05.12 07.12 09.12 11.12 13.12 15.12 17.12 19.12 21.12 23.12 25.12 27.12

C o

n d

it io

n al

V ar

ia n

PARCH forecast Brazilian IBOV

Realised Volatility Forecasted Volatility Linear (Realised Volatility)

0.0000

0.0008

0.0016

0.0024

03.12 05.12 07.12 09.12 11.12 13.12 15.12 17.12 19.12 21.12 23.12 25.12 27.12 29.12 31.12

C o

n d

it io

n al

V ar

ia n

PARCH forecast American SP100

Realised Volatility Forecasted Volatility Linear (Realised Volatility)

6 Conclusion

This paper has sought to identify an adequate return series model for modelling and

forecasting conditional volatility in stock markets. Since stationarity and ARCH effects have

been detected in the three stock return series, conventional CLRMs cannot deliver accurate

results. Thus, this research focussed on the more appropriate ARCH and GARCH family models,

namely, the ARCH(1), GARCH(1,1), GARCH-M and PARCH models.

Consequently, these models were closely examined according to their data fit ability, with

the best three models further investigated in terms of their heteroscedasticity removal and

one-month forecasting capabilities. For all indices, the PARCH model fit the data best

according to the AIC and log-likelihood criterion. Notably, these two criteria did not once

contradict each other, therefore increasing the collective robustness of the findings. As

suggested by the literature, the ARCH(1) model clearly underperformed all other models in its

data fit. Further ARCH-LM tests demonstrated the successful removal of heteroscedasticity in

the return series for all models. However, in opposition to the previous results, the PARCH

model clearly produced the weakest results, while the GARCH(1,1) model removed

heteroscedasticity most effectively in two of the three indices. Considering these results, the

evaluation of the models’ forecasting ability yielded findings similar to those of the data fit

analysis, as illustrated by the PARCH model’s superior forecasting results for all indices, except

for the SMI; for that index, the GARCH(1,1) model delivered more accurate results. Moreover,

significant leverage parameters implied prevailing leverage effects in all indices, while the

presence of volatility clustering was indicated through ARCH and GARCH parameters. Based

on the modelled volatility pattern, the SMI exhibited the smallest fluctuations, while the

Brazilian IBOV displayed the greatest magnitude, in line with major economic developments.

This research contributes to the literature on time series modelling and forecasting.

Although a potential pattern has been detected, the robustness of the results should be

further enhanced by applying the underlying test procedures to a broader variety of indices.

7 References

Abdalla, S. Z. S. & Winker, P. 2012. Modelling stock market volatility using univariate GARCH models: Evidence from Sudan and Egypt. International Journal of Economics and Finance, 4.

Akgiray, V. 1989. Conditional heteroscedasticity in time series of stock returns: Evidence and forecasts. The Journal of Business, 62, 55-80.

Alberg, D., Shalit, H. & Yosef, R. 2008. Estimating stock market volatility using asymmetric GARCH models. Applied Financial Economics, 18, 1201-1208.

Antoshin, S., Berg, A. & Souto, M. 2008. Testing for structural breaks in small samples. Working Paper, International Monetary Fund, 75.

Asteriou, D. & Hall, S. G. 2016. Applied Econometrics, Palgrave.

Bai, J. & Perron, P. 1998. Estimating and testing linear models with multiple structural changes. Econometrica, 66, 47-78.

Bai, J. & Perron, P. 2003. Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18, 1-22.

Black, F. 1976. Studies of stock market volatility changes. Proceedings of the American Statistical Association, Business and Economics Statistics Section, 177-181.

Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307-327.

Brooks, C. & Burke, S. P. 1998. Forecasting exchange rate volatility using conditional variance models selected by information criteria. Economics Letters, 61, 273-278.

Brooks, R. 2007. Power arch modelling of the volatility of emerging equity markets. Emerging Markets Review, 8, 124-133.

Brooks, R. D., Faff, R. W., Mckenzie, M. D. & Heather, M. 2000. A multi-country study of power ARCH models and national stock market returns. Journal of International Money and Finance, 19, 377–397.

Chen, G.-M., Lee, C. F. & Rui, O. M. 2001. Stock returns and volatility on China’s stock market. The Joumal of Financial Research, 24, 523-543.

Degiannakis, S. 2004. Volatility forecasting: Evidence from a fractional integrated asymmetric power ARCH skewed-t model. Applied Financial Economics, 14, 1333-1342.

Ding, Z., Granger, C. W. J. & Engle, R. F. 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, 83-106.

Dritsaki, C. 2017. An empirical evaluation in GARCH volatility modeling: Evidence from the Stockholm Stock Exchange. Journal of Mathematical Finance, 07, 366-390.

Elyasiani, E. & Mansur, I. 1998. Sensitivity of the bank stock returns distribution to changes in the level and volatility of interest rate: A GARCH-M model. Journal of Banking & Finance, 22, 535-563.

Engle, R. F. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987-1007.

Engle, R. F. 2001. GARCH 101: The use of ARCH/GARCH models in applied econometrics. Journal of Economic Perspectives, 15, 157-168.

Engle, R. F., Lilien, D. M. & Robins, R. P. 1987. Estimating time varying risk premia in the term structure: The Arch-M model. Econometrica, 55, pp. 391-407.

Engle, R. F. & Mustafa, C. 1992. Implied ARCH models from options prices. Journal of Econometrics 52, 289-311.

Floros, C. 2004. Stock returns and inflation in Greece. Euro-American Association of Economic Development, 4.

Giot, P. & Laurent, S. 2004. Modelling daily Value-at-Risk using realized volatility and ARCH type models. Journal of Empirical Finance, 11, 379-398.

Gokbulut, R. I. & Pekkaya, M. 2014. Estimating and forecasting volatility of financial markets using asymmetric GARCH models: An application on Turkish financial markets. International Journal of Economics and Finance, 6.

Hansen, P. R. & Lunde, A. 2005. A forecast comparison of volatility models: Does anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20, 873-889.

Harry, K. M. & Heynen, R. C. 1994. Volatility prediction: A comparison of the stochastic volatility, GARCH(1,1) and Egarch(1,1) models. Journal of Derivatives, 2.

Kambouroudis, D. S., Mcmillan, D. G. & Tsakou, K. 2016. Forecasting stock return volatility: A comparison of GARCH, implied volatility, and realized volatility models. Journal of Futures Markets, 36, 1127-1163.

Kanniainen, J., Lin, B. & Yang, H. 2014. Estimating and using GARCH models with VIX data for option valuation. Journal of Banking & Finance, 43, 200-211.

Karanasos, M., Menla Ali, F., Margaronis, Z. & Nath, R. 2018. Modelling time varying volatility spillovers and conditional correlations across commodity metal futures. International Review of Financial Analysis, 57, 246-256.

Lim, C. M. & Sek, S. K. 2013. Comparing the performances of GARCH-type models in capturing the stock market volatility in Malaysia. Procedia Economics and Finance, 5, 478-487.

Lin, Z. 2018. Modelling and forecasting the stock market volatility of SSE Composite Index using GARCH models. Future Generation Computer Systems, 79, 960-972.

Lundblad, C. 2007. The risk return tradeoff in the long-run: 1836-2003. Journal of Financial Economics, 85, 123-150.

Mandelbrot, B. 1963. New methods in statistical economics. Journal of Political Economy, 71, 421-440.

Mcmillan, D., Speight, A. & Apgwilym, O. 2000. Forecasting UK stock market volatility. Applied Financial Economics, 10, 435-448.

Mcmillan, D. G. & Speight, A. E. H. 2012. Daily FX volatility forecasts: Can the GARCH(1,1) model be beaten using high-frequency data? Journal of Forecasting, 31, 330-343.

Nelson, B. D. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347-370.

Pagan, A. R. & Schwert, G. W. 1990. Alternative models for conditional stock volatility. Journal of Econometrics, 45, 267-290.

Papell, D. H., Murray, C. J. & Ghiblawi, H. 2000. The structure of unemplyment. The Review of Economics and Statistics, 82, 309–315.

Pesaran, H. M., Pettenuzzo, D. & Timmermann, A. 2006. Forecasting time series subject to multiple structural breaks. Review of Economic Studies, 73, 1057-1084.

Pesaran, M. H. & Timmermann, A. 2002. Market timing and return prediction under model instability. Journal of Empirical Finance, 9, 495–510.

Thorlie, M. A., Song, L., Amin, M. & Wang, X. 2015. Modeling and forecasting of stock index volatility with APARCH models under ordered restriction. Statistica Neerlandica, 69, 329-356.

Willmott, C. J. & Matsuura, K. 2005. Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30, 79-82.

Yu, J. 2002. Forecasting volatility in the New Zealand stock market. Applied Financial Economics, 12, 193-202.

8 Appendix

Histogram SMI

Histogram SP100

Histogram IBOV

100

150

200

250

300

350

400

4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

Series: PRICE Sample 1/02/2006 12/28/2018 Observations 3263

Mean 7666.802 Median 7978.960 Maximum 9611.610 Minimum 4307.670 Std. Dev. 1241.540 Skewness -0.454689 Kurtosis 1.990193

Jarque-Bera 251.0713 Probability 0.000000

100

200

300

400

500

300 400 500 600 700 800 900 1000 1100 1200 1300

Series: PRICE Sample 1/03/2006 12/31/2018 Observations 3271

Mean 749.0350 Median 679.6800 Maximum 1302.800 Minimum 322.1300 Std. Dev. 227.2748 Skewness 0.582715 Kurtosis 2.416348

Jarque-Bera 231.5430 Probability 0.000000

100

150

200

250

300

350

30000 40000 50000 60000 70000 80000 90000

Series: PRICE Sample 1/02/2006 12/28/2018 Observations 3211

Mean 57399.29 Median 56852.84 Maximum 89820.09 Minimum 29435.11 Std. Dev. 11866.02 Skewness 0.270145 Kurtosis 2.891720

Jarque-Bera 40.62412 Probability 0.000000

Bai-Perron IBOV – Global l Method

Bai-Perron SMI – Global l Method

Bai-Perron SP100 – Global l Method

GARCH(1,1) SMI

GARCH(1,1) SP100

GARCH(1,1) IBOV

GARCH-M SMI

GARCH-M SP100

GARCH-M IBOV

PARCH SMI

PARCH SP100

PARCH IBOV