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School of Economics and Finance
Queen Mary University of London
Modell ing and Forecasting Volatil ity of the BRICS Stock
Markets: Evidence Using GARCH Models.
By
ANDRIA PERATITI
090448124
20/08/2013
Supervisors:
Dr. Leone Leonida
Dr. Dario Maimone Ansaldo Patti
Word Count: 5983
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Abstract
A number of previous studies have used ARCH and GARCH models in order to conclude on
the most appropriate model for estimating and forecasting volatility. In attempt to contribute to
literature, this study is devoted to find the most accurate heteroskedastic model for estimating
and forecasting volatility in the five main emerging economies: Brazil, Russia, India, China and
South Africa (BRICS) using daily equity indices. The empirical investigation is conducted
using various models from the GARCH-family. The models employed in this dissertation are
the GARCH (1,1), TGARCH (1,1,1), EGARCH (1,1) and the GARCH-M (1,1) with both
standard deviation and conditional variance in the mean equation.
The findings reveal that the TGARCH (1,1,1) model followed by EGARCH (1,1) model is
best suited for estimating and forecasting volatility. The results also suggest the presence of
leverage effects in the data. GARCH-M (1,1) is inappropriate for both modelling and
forecasting volatility in the emerging economies, as the risk-reward relationship is not detected.
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Table of contents:
ABSTRACT………………………………......……………...….…………….……2
TABLE OF CONTENTS………………………………..………..…….………..…3
ACKNOWLEDGEMENT………………………………..………………….…......5
SECTION I: INTRODUCTION………………………….………....…...…………6
SEDTION 2: LITERATURE REVIEW……………………..……...……...……....7
SECTION 3: DATA………………………...……..……………......…………...….9
SECTION 4: METHODOLOGY…………….…………………...………………..9
4.1: DATA BEHAVIOUR…………………..………………………………9
4.2: ESTIMATING VOLATILITY MODELS…………………..………...10
4.3: FORECASTING VOLATILITY………………..……………….…….12
SECTION 5: ESTIMATION RESULTS…………………..………………….......15
5.1: PRELIMINARY ANALYSIS………………………..….………….....15
5.2: MODEL ESTIMATION………………..………………………..….....19
5.3: FORECASTING EVALUATION……………..…………….………...24
SECTION 6: CONCLUSIONS………..………….……………………………….28
REFERENCE………..………………………………………….…………………29
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List of tables:
TABLE 1: Descriptive statistics for daily return series.............................................................. 15
TABLE 2: Autocorrelation test……………...……………………………………………….…19
TABLE 3.Test for “ARCH-effects” for daily returns……………………….……...……….….19
TABLE 4A: Coefficient estimates of GARCH (1,1)………………………...…….…...………20
TABLE 4B: Coefficient estimates of GARCH-M(1,1) with the conditional standard deviation
term in the mean……………………………………...…………………….……...….………...21
TABLE 4C: Coefficient estimates of GARCH-M (1,1) with the conditional variance term in the
mean………………………………..…………………………….……...……………………….21
TABLE 4D: Coefficient estimates of TGARCH(1,1,1)………………...……………………...22
TABLE 4E: Coefficient estimates of EGARCH(1,1)……………...…………………………..22
TABLE 5: Akaike Information Criterion………………………………………………..……..23
TABLE 6: Diagnostics in the standardized residuals………………………………………..…23
TABLE 7: RMSE measures from forecasting daily equity return volatility…………………...25
TABLE 8: MAPE measures from forecasting daily equity return volatility………………..….25
TABLE 9: MAE measures from forecasting daily equity return volatility.……………………25
List of figures:
Figures 1a-1e: Histograms of daily returns…………….……………...….……………….........16 Figures 2a-2e: Performance of daily returns………………..……...…..…………………...17-18
Figures 3a-3e: Plots of proxy against forecasted volatility…………..……..………………26-27
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Acknowledgement
The author wishes to express appreciation to her supervisor Dr Leone Leonida and her
Teaching Assistant Mr Davide Cafaro for their support during the process of this
dissertation.
Special thanks should go to Dr Dario Maimone Ansaldo Patti for his valuable guidance
and help.
The author also wants to thank five people important to her, who know who they are, for
their encouragement and love throughout this year.
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1: Introduction
Many academics and researchers have extensively discussed the importance of volatility
forecasting over the years. Akgiray (1989), states that volatility forecast is significant as it
provides evidence for the usefulness of the GARCH-type models as evaluation instruments of
the stock market. Volatility is an input in the Black-Scholes-Merton pricing formula for
determining the price of call and put options trading on an exchange. Hence, an accurate
forecast would be useful for pricing financial securities correctly. Tsay (2005), in his book,
explains more reasons for the importance of modelling and forecasting volatility. As he states,
another financial function of volatility is that it offers a simple method for calculating the Value
at Risk (VaR) in risk management. In addition it has a leading role in asset allocation under the
mean-variance relationship for investment decisions. Finally, the Volatility Index of market has
recently become a financial instrument (VIX). Therefore, forecasting the volatility of time
series makes parameter estimation more efficient as well as the interval forecast more accurate.
In finance, volatility is measured by standard deviation σ or variance σ2 from a size sample n
as! !! = !
!!! !! − !)!!!!! (1)
where µ is the mean return.
The purpose of this dissertation is to examine some of the linear, non-linear and asymmetric
econometric models for modelling and forecasting volatility of equity returns in BRICS and
conclude upon whether the predictive ability of a model outperforms the forecasting power of
the other models. Furthermore this study is interested on whether there has been a
chronological improvement in the literature’s models’ ability to forecast volatility. BRICS is
the abbreviation given to the five main emerging countries in the world: Brazil, Russia, India,
China and South Africa since 2010. Although the literature that focuses on the forecasting
ability of volatility models in developed economies is immense, little has been found for the
case of developing economies and even less for these five major emerging economies of the
world. The models tested in this paper are the Generalised ARCH (GARCH), Threshold
GARCH (TGARCH), Exponential GARCH (EGARCH) and the GARCH-in-mean (GARCH-
M) with both standard deviation and conditional variance in the mean equation.
This study is structured as follows: Section 1 is the introduction, which presents the aims
and motivation of this dissertation. It also mentions some important background information
about the topic. Section 2 discusses and analyses the main literature available for this topic. A
clarification of how this dissertation fits within the literature is presented. Section 3 describes
the data and Section 4 the methodology engaged in this dissertation. Section 5 demonstrates
and analyses the empirical findings. Section 6 is the conclusion part, which reviews the results
for the superiority of a certain model over the others.
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2.#Literature#Review#There is extensive literature that focuses on the evaluation of different models for modelling
and forecasting volatility in developed and emerging economies. This dissertation only focuses
on the GARCH-family models and more specifically on the performance of the linear, non-
linear and asymmetric GARCH models for the emerging economies of Brazil, Russia, India,
China and South Africa.#Song et al (1998) examined the relationship between returns and volatility for the emerging
Chinese stock markets. The main conclusion is that GARCH-M (1,1) specification explains
return series in the best way. The existence of time-varying risk premium in the emerging
markets of Latin America has also been observed by De Santis and Imrohoroglu (1997).
In the context of modelling volatility, Haroutounian and Price (2001) and Siourounis (2002)
concluded that the linear GARCH model estimates volatility in the most suitable way.
Alagidede and Panagiotidis (2009) studied the stock returns for 7 markets in Africa, including
South Africa. They concluded that GARCH, GARCH-M and EGARCH-M models estimate the
conditional variance properly.#Several researches questioned the superiority of the linear GARCH model in terms of its
forecasting power. Akgiray (1989) concluded that the GARCH (1,1) model is superior to the
other models for forecasting monthly US stock indices. West and Cho (1995) agree with the
choice of the linear model when forecasting volatility using dollar exchange rates. On the other
hand, Tse (1991) and Tse and Tung (1992), studying the Japanese and Singaporean stock
market respectively, concluded that Exponentially Weighted Moving Average (EWMA) model
has more forecasting ability than the ARCH/GARCH models. #The forecasting power of ARCH-type models was examined by Hansen and Lunde (2005).
They compared 330 ARCH-type models using DM–$ exchange rate data and IBM return data.
They found evidence of superiority of GARCH (1,1) model to more sophisticated models when
the evaluation is based on exchange rates, and inferiority of GARCH (1,1) modek when the
analysis is based on IBM returns. Finally, the study suggests that a model is superior to other
models for out-of-sample evaluation when it accommodates a leverage effect.
Although Dimson and Marsh (1990) did not examine the volatility forecast ability of the
(G)ARCH- family models, their conclusion gives an important implication for the forecasting
power of complex GARCH models. They proposed that more complex non-linear and non-
parametric models are more probable to underperform than parsimonious linear models. They
recommended the Exponential Smoothing model and the Regression model for forecasting
quarterly volatility.
Huang (2011) presents an extensive assessment of volatility models by investigating 31
markets. The findings of this assessment showed that for both the developed and developing
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economies, the GARCH models provide weak forecasting ability. In addition, the Stochastic
Volatility model gives the most accurate forecasts.#Some studies have shown that asymmetric models produce weak forecasts. Examples
include Brooks (1998) in the case of New York Stock Exchange and Franses and Van Dijk
(1996) in the case of five developed countries. Similarly, Xu (1999) who focused his research
on the Chinese market concluded that the symmetric model outperforms the EGARCH and
GJR-GARCH models. He observed that generally it is hard to use any GARCH-type models
for forecasting purposes as volatility in Shanghai is mostly influenced by governmental policy
on stock markets.#Gokcan (2000) extended Franses and Van Dijk (1996) study by investigating seven
emerging countries including Brazil. He concluded that GARCH (1,1) model has stronger
forecasting power than the EGARCH model, even though the return series are asymmetric.
McMilan et al (2000) using UK daily, weekly and monthly indices and Day and Lewis (1992)
using S&P 100 index have shown the superiority of the GARCH model when compared to the
EGARCH model.
On the other hand, there is vast literature suggesting that the role of asymmetry in volatility
forecasting is very important. Poon and Granger (2003) and Liu and Huang (2010) concluded
that asymmetric GARCH models perform better than the linear GARCH model. Similarly,
Awartani and Corradi (2005) employed daily S&P-500 Composite Price Index and suggested
that for one-step-ahead and longer time horizons, asymmetric GARCH models outperform the
GARCH (1,1) model. Furthermore, Carvalhal and Mendes (2008) tested the forecasting
performance of seven econometric models for the emerging markets in Latin America and Asia
including Brazil and India. They found that for in-sample estimation TGARCH and EGARCH
outperform the other models whilst the ARMA model has the best forecasting ability for out-
of-sample estimations.
Moreover, many authors showed that EGARCH model gives more accurate results than the
predictions given by the other asymmetric GARCH models. (Pagan and Schwert, 1990; Alberg
et al, 2008; Miron and Tudor, 2010; Shamiri and Isa, 2009; Chong et al, 1999)
Along the same lines, Brailsford and Faff (1993,1996) and Forte and Manera (2002) found
that for the Australian returns and for the data of ten European Stock Markets respectively,
GJR-GARCH model provides smaller forecasting errors. Finally, Engle and Ng (1993) when
examined the Japanese Stock Market concluded that GJR-GARCH and EGARCH models give
the most accurate forecasts.!
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3. Data
The data extracted for the purpose of this dissertation comprise 2610 daily equity
observations for each of the following countries: Brazil, Russia, India, and South Africa and
1891 daily equity observations for China. The data available for Brazil, Russia, India and South
Africa cover a period of 9 years and start from 1/4/2002 to 30/3/2012, while for China the
available data start from 31/12/2004 to 30/3/2012 and cover a period of 7 years and 3 months.
The sample period is divided into two sub-samples. The in-sample includes all the equity index
returns for the period of 1/4/2002 to 31/3/2011 for Brazil, Russia, India and South Africa and
for the period of 31/12/2004 to 31/3/2011 for China. These data are used for estimating the
models. The out-of-sample includes daily returns for the period of 1/4/2011 to 30/3/2012 for all
the countries, and is used to investigate the volatility forecasting power of the models.
The following daily equity indices of the BRICS Stock Exchange markets have been used:
Brazil’s BM&FBOVESPA, Russia’s Moscow Interbank Currency Exchange (MICEX), India’s
Bombay Stock Exchange (BSE), China Security Index 300 (CSI 300) and South Africa’s
FTSE/JSE. The data has been extracted from the Macrobond data set of the Excel add-in
function.
The high-frequency data turns out to be highly predictable and more accurate for ex-post
interdaily volatility evaluation (Andersen and Bollerslev, 1998). The use of daily observations
for examining the volatility forecasting ability of different models in the emerging economies is
compatible with literature (Song et al, 1998; Xu, 1999; Siourounis, 2002; Su and Knowes,
2006; Huang, 2011).
4. Methodology
4.1. Data Behaviour
Before conducting the main analysis and in order to make the time series stationary, the
daily returns Rt are defined as the first log-difference of each index’s value for two successive
days times 100. Hence, the continuously compounded daily returns (Rt) at time t are calculated
as:
!! = 100 !"#(!!)− !"#!(!!!!) = 100 !"# !!!!!!
(2)
for t = 1,…,2609 for all the countries except China and for t=1,…1890 for the latter. Pt is the
equity index at time t, for t =0,…,2610 for all the countries except China and for t =0,…,1891
for China.
Mandelbrot (1963) stated, “Large changes tend to be followed by large changes - of either
sign- and small changes tend to be followed by small changes ”. This is referred by academics
as volatility clustering and is one of the apparent patterns of financial time series, the so-called
stylized facts about volatility. Cont (2001) suggests that the other main features that
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characterize return series are: skewness, leptokurtosis, and volatility persistence and volatility
clustering. It is necessary that a preliminary analysis be conducted in order to decide if the
GARCH-family models can be employed since theory suggests that these models capture the
features stated above. Preliminary analysis is conducted by inspecting the descriptive statistics,
the absolute return series, and the variance of the daily returns and the autocorrelation of the
squared returns.#
4.2. Estimating Volatility Models Engle (2001) stated in his paper: “ARCH and GARCH models treat heteroskedasticity as a
variance to be modeled”. Therefore, it is important to compute the Engle (1982) test to
determine the presence of “ARCH effects” in the error terms. This can also be thought as
testing for the presence of conditional heteroskedasticity in the error terms. The first step is to
run the linear ARMA (1,1) model and obtain the residuals !!. Then to test for ARCH of order
five the residuals are squared and regressed on five lags as shown in the following equation:
!!! = !! + !!!!!!!! + !!!!!!!! +⋯+ !!!!!!! + !! (3)
where !! is an error term. Both the LM-test and the F-test are used to check for
heteroskedasticity. If heteroskedasticity is detected then the ARCH estimation method should
be used instead of the OLS method.
In the presence of heteroskedasticity, the five models used to estimate the conditional
variance are GARCH (1,1), TGARCH (1,1,1), EGARCH (1,1) and GARCH-M (1,1) with both
standard deviation and conditional variance in the mean equation. These models have been
extensively used in the literature due to their uncomplicatedness and verified ability to forecast
volatility. The statistical software used for estimating and forecasting the models is Eviews 7. When explaining the ARCH model, Engle (1982) stated that “for real processes one might
expect better forecast intervals if additional information from the past were allowed to affect
the forecast variance; a more general class of models is desirable” This has inspired Bollerslev
(1986) and Taylor (1986) to extend the ARCH model to the Generalised ARCH (GARCH)
model by allowing the conditional variance process simulate the ARMA process. One of the
reasons that the linear GARCH model is so popular is that it can effectively capture both the
volatility-clustering effect and the excess kurtosis in return series. The model is more
parsimonious in the estimation of the parameters and so less likely to violate the non-negativity
constraint. GARCH (1,1) is the most common model used for equity return data and has
conditional variance equation:
ℎ! = !! + !!!!!!! + !ℎ!!! (4)
where !! is the ARCH coefficient, ! is the GARCH coefficient and !! is the residual at time t.
The advantage of using it comes from the fact that it does not consider only the information
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about volatility from the previous period (!!!!!!! ) but also information coming from the fitted
values of the last period’s variance (!ℎ!!!). Chou (1988) proposed that when the sum of the
ARCH and GARCH parameters !! + ! is unitary then the shocks to volatility persist
infinitely and the model does not determine the unconditional variance. The fact that the linear GARCH model does not allow for any relationship between the
variance and the mean of returns, non-linear GARCH models have been suggested as a
consequence of the perceived problems. The GARCH-in-mean (GARCH-M) model suggested
by Engle, Lilien and Robins (1987) includes the conditional variance as another regressor to the
mean equation and thus allows ℎ! to have mean effects. The GARCH-M model is given by the
specification:
!! = ! + ! ℎ!!! + !! (5.1)
and
ℎ! = !! + !!!!!!! + !ℎ!!! (5.2)
where, !! ∼! 0, ℎ! , !! is the equity index return and µ is the mean. The parameter δ is
called the time varying risk premium. If δ is positive and statistically significant then, the
model is able to explain the volatility in the returns and it should be employed for forecasting
purposes.
In equity markets, stock prices are negatively correlated with volatility. This tendency of
volatility to increase more following a negative shock than following a positive one of the same
magnitude is named leverage effect (Black, 1976). To account for this phenomenon two
asymmetric models that capture this tendency of volatility are also estimated: TGARCH and
EGARCH. By estimating and testing the significance of the asymmetric terms one can
conclude on whether TGARCH and EGARCH models fit the data appropriately. The Threshold GARCH (TGARCH) of Glosten, Jagannathan and Runkle (1993) and
Zakonian (1994) accounts for possible asymmetries with the addition of a dummy variable in
the simple GARCH model. The conditional variance equation of TGARCH (1,1,1) is given by:
ℎ! = !! + !!!!!!! + !ℎ!!! + !!!!!! !!!! (6)
where !!!! is a binary dummy variable taking the value of 1 if !!!! < 0 and zero otherwise. As
Enders (2004), states in his book !!!! = 0 is a threshold and any innovations below or above
this have different effects on volatility. When the market moves upwards (!! > 0), conditional
variance is influenced by !!, while when the market moves downwards (!! < 0), volatility is
changed by !! + !. This leads to the implication that in the presence of leverage effects, ! is
strictly greater than zero and so higher volatility is observed when negative shocks happen.
When !!takes any other value than zero, the changes in the market are asymmetric. The Exponential GARCH (EGARCH) proposed by Nelson (1991) estimates the natural
logarithm of conditional variance. Therefore, even if the parameters are negative, ℎ! will
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always be positive. Thus, there is no need to artificially impose non-negativity constraints on
the model’s parameters, except ! < 1. Finally, the logarithm of the conditional variance
implies that, the leverage effect is exponential, in comparison with the TGARCH that is
quadratic. There are various ways to express the conditional variance equation of EGARCH
(1,1). One possible specification is:
!" ℎ! = !! + !"# ℎ!!! + ! !!!!!!!!
+ !! !!!!!!!!
− !! (7)
The parameter ! captures the asymmetry of the returns. Since asymmetries are allowed
under the EGARCH formulation, and the relationship between volatility and returns is
negative, in the presence of leverage effect ! is strictly smaller than zero. In order to decide on the appropriateness of the GARCH models the Akaike’s Information
Criterion (AIC) is used. For each model’s estimation table produced by Eviews software, the
corresponding AIC value is given. The most appropriate model minimises the value of the
Information Criterion. In order to estimate the Information Criterion value, Eviews software
uses the formula:
!"# = −2! ! + !!! (8)
where l is the log likelihood, T is the sample size and k is the number of the parameters.
To distinguish the best fit data model out of the appropriate GARCH models, diagnosis tests
are conducted. This study follows the diagnosis tests employed by Song et al (1998). The
standardised residual series of a suitable fit GARCH model should follow a normal distribution
and hence to have skewness and kurtosis coefficients close to zero and three respectively.
Therefore, the p-value of the Jarque-Bera statistic should be larger than 0.05 in order to not
reject, at 5% level, the null hypothesis that the errors are normally distributed. The Ljung-Box
cumulative statistic up to twelve lags is also employed to test the presence of autocorrelation in
the standardised residuals. The best fit model should fail to reject the null hypothesis of no
autocorrelation in the standardised residuals up to twelve lags.
4.3. Forecasting Volatility
Once the models with significant parameters have been estimated, they can be used to
derive one-year-ahead volatility forecasts. The idea behind forecasting volatility using the
GARCH-family models is that by theory, the conditional variance of the value of the return
series at each time period, given its prior values, is identical to the conditional variance of the
disturbance term for the same period, given its prior values (Brooks, 2008). This fact along
with the ability of the GARCH models to explain movements in the conditional variance of the
disturbance term infers that the forecasts of the conditional variance of each model are the
variance forecasts of the return series (Brook, 2008). Forecasts are produced for the whole of
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the out-of-sample period from 1/4/2011 to 30/3/2012 for all the five countries. The method
used for the forecasts is dynamic. In other words, multi-step-ahead procedure is carried out
starting on the first period of the out-of-sample and generating forecasts for one-, two-, and up
to s-steps-ahead, so that the forecasting interval is for the next s periods. It is important to note
that if the residual terms are not available for the forecast procedure of this dissertation, the
terms are replaced by their actual values. For the case of GARCH (1,1) model, with conditional variance equation:
ℎ!= !!+!!!!!!! + !ℎ!!! (9.1)
where all the coefficient estimates are known and the data are available up to and including
time T, the one-step-ahead forecast is calculated as:
ℎ!!!= !!+!!!!! + !ℎ! (9.2)
Given ℎ!!!, the two-step-ahead forecast is defined by the equation:
!ℎ!!! = !! + !!ℎ!!! + !ℎ!!!= !! + (!! + !)!ℎ!!! (9.3)
Finally, the s-step-ahead forecast is produced by:
!ℎ!!! = !! (!! + !)!!!!!!!!! + (!! + !)!!!ℎ!!! (9.4)
where s ≥ 2. The other models produce forecasts similarly. In order to determine the accuracy of a forecast, evaluation tools must be employed.
According to Bollerslev et al (1994), the most important forecast evaluation measure is the
economic loss function. However, it is usually unavailable so statistical loss function is used
instead. Following Chu and Freund (1996) and Brailsford and Faff (1996), the one-year-ahead
forecasts are assessed by the Root Mean Squared Error (RMSE), the Mean Absolute Error
(MAE) and the Mean Absolute Percentage Error (MAPE) defined as:
!"#! = !!!(!!!!)
(!!!! − ℎ!!!)!!!!!! (10)
!"# = ! !!!(!!!!)
!!!! − ℎ!!!!!!!! (11)
!"#$ = ! !""!!(!!!!)
!!!!!!!!!!!!!
!!!!! (12)
where ℎ!!!!is the s-step-ahead volatility forecasts of the equity index at time t, !! is the actual
volatility at time t, T is the total sample size and T1 is the first out-of sample forecast
observation. RMSE is an orthodox error statistic that gives a harsh penalty on large forecast
errors. MAE is a conventional forecast criterion that gives the same weight to all forecast
errors, while MAPE is interpreted as a percentage error and its value cannot get negative.
Makridakis (1993) claims that MAPE is the most accurate statistical loss function. In terms of
analysis, the lower the value of RMSE and MAE is, the more accurate the model is in terms of
forecasting, while the closer to 100% the value of MAPE is, the better the model predicts
volatility. Finally, in order to see how well each model tracks price changes, the forecasted volatility is
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compared to the actual volatility. Due to the fact that actual volatility is unobservable, squared
daily returns are used as a proxy of actual volatility. Squared returns are an unbiased but very
noisy estimator of ex post volatility (Lopez, 2001). Since the squared daily returns are only
used for comparison reason, their imprecise characteristic does not imply that the inferences
made are incorrect because they are unbiased. The choice of squared daily returns as a proxy of
actual volatility is consistent with literature. Brailsford and Faff (1996) and Awartani and
Corradi (2005) have also used this evaluation method.
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5. Estimation Results
5.1Preliminary Analysis
By inspecting the summary statistics presented in Table 1 it can be seen that the mean and
the median of all the five countries is positive and not significantly different from zero,
implying that equity returns increase slightly as time passes. Furthermore, the large difference
between the minimum and the maximum values of the returns indicate that large changes in
volatility often occur in the BRICS countries. De Santis and Imrohoroglu (1997) when
examining the volatility of the emerging economies including Brazil, India and China found
similar results. High volatility, measured by standard deviation, is accompanied by high
average daily returns. This is a common observation in the emerging economies and is
consistent with Shamiri and Isa (2009). In fact, South Africa has the lowest mean and standard
deviation value while Russia is the most volatile country with the highest average returns.
The Gaussian distribution of the return series is not a phenomenon in the emerging markets’
data (Choudry, 1996; Gokcan, 2000; Kurma, 2006). This study is consistent with literature
observing leptokurtic and skewed residuals. In a Gaussian distribution the coefficients of
skewness and kurtosis should be zero and three correspondingly. The coefficients of kurtosis,
which are much larger than three, indicate the leptokurtic distribution of the residuals, while the
negative, but very close to zero, coefficients of skewness imply that market falls more frequent
than it rises. So, the equity returns follow a non-normal distribution mostly because of the
excess kurtosis presented in the residuals rather than skewness. Finally, the null hypothesis of
the Jarque- Bera statistic that the residuals follow a normal distribution is strongly rejected at
1% significance level as the p-value in all cases is zero. This means that the implications made
for coefficient estimates could be incorrect. The fact that the sample is large makes this concern
less severe.
Non-normality is also supported by visually inspecting the histograms (see Figures 1a-1e).
As it can be seen none of them has a bell-shape, instead they are all asymmetric with a longer
tail on the left. Bolleslev et al (1994) stated that the non-constant variance of the time-
dependent series produces leptokurtosis, which links volatility clustering with non-normal
distribution.
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Figure 1a. Histogram of daily returns for Figure 1b. Histogram of daily returns for
Brazil (1/4/2002-31/3/2011) Russia (1/4/2002-31/3/2011)
Figure 1c. Histogram of daily returns for Figure 1d. Histogram of daily returns for
India (1/4/2002-31/3/2011) China (31/12/2004-31/3/2011)
Figure 1e. Histogram of daily returns for
South Africa (1/4/2002-31/3/2011)
Another important feature of financial returns is the volatility clustering. As it can be seen
from Figures 2a-2e volatility in BRICS occurs in bursts. The daily return series of South Africa
appears to be the most volatile in this period of time. In general slight tranquility is recorded for
all the countries over the time-span. Inevitably due to the U.S housing boom far more volatility
has been recorded in mid-2008 till 2009 for all the countries with many large positive and
negative returns in such a small time interval. Finally, the non-constant variance suggests the
consideration of heteroskedasticity when estimating the models.
0
100
200
300
400
500
600
700
800
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
Series: DAILYRETURNSSample 4/01/2002 3/31/2011Observations 2349
Mean 0.030391Median 0.025384Maximum 5.940376Minimum -5.253256Std. Dev. 0.818362Skewness -0.094643Kurtosis 8.005594
Jarque-Bera 2455.860Probability 0.000000
0
100
200
300
400
500
600
-3 -2 -1 0 1 2 3
Series: DAILYRETURNSSample 4/01/2002 3/31/2011Observations 2349
Mean 0.019835Median 0.009648Maximum 2.967956Minimum -3.292249Std. Dev. 0.572863Skewness -0.138375Kurtosis 6.372683
Jarque-Bera 1120.824Probability 0.000000
0
50
100
150
200
250
300
350
400
-4 -3 -2 -1 0 1 2 3 4
Series: DAILYRETURNSSample 12/31/2004 3/31/2011Observations 1629
Mean 0.031203Median 0.037013Maximum 3.878630Minimum -4.210558Std. Dev. 0.859568Skewness -0.427795Kurtosis 5.687656
Jarque-Bera 539.9815Probability 0.000000
0
100
200
300
400
500
600
700
800
-8 -6 -4 -2 0 2 4 6 8 10
Series: DAILYRETURNSSample 4/01/2002 3/31/2011Observations 2349
Mean 0.033084Median 0.031808Maximum 10.95557Minimum -8.971256Std. Dev. 1.025332Skewness -0.178730Kurtosis 18.79540
Jarque-Bera 24431.78Probability 0.000000
0
200
400
600
800
1,000
-4 -2 0 2 4 6
Series: DAILYRETURNSSample 4/01/2002 3/31/2011Observations 2349
Mean 0.031867Median 0.036548Maximum 6.944362Minimum -5.128660Std. Dev. 0.712594Skewness -0.089064Kurtosis 11.27464
Jarque-Bera 6704.572Probability 0.000000
! 17!
Figure 2a. Daily returns of Brazil (1/4/2002-31/3/2011)
Figure 2b. Daily returns of Russia (1/4/2002-31/3/2011)
Figure 2c. Daily returns of India (1/4/2002-31/3/2011)
! 18!
Figure 2d. Daily returns of China (31/12/2004-31/3/2011)
Figure 2e. Daily returns of South Africa (1/4/2002-31/3/2011)
Finally, Table 2 reports the Ljung-Box Q-statistic with the relevant probabilities for the first,
sixth and twelfth lags under the joint null hypothesis of no serial correlation in the squared
returns up to each lag. The table also presents the autocorrelation coefficients at each lag. It is
obvious that small positive first-order autocorrelations are present in all the countries. These
results are consistent with the results found by Gokcan (2000) for the case of seven emerging
markets and validate the employment of GARCH-type specifications for the estimation of
volatility models. The actual correlograms can be found in Appendix 1.
! 19!
5.2 Model Estimation
After the ARMA(1,1) model has been estimated, the presence of conditional
heteroskedasticity in the residuals is examined in order to decide whether the ARCH estimation
method is more suitability than the OLS method for modeling the data. The values of the F-test
and the LM-test are synopsized in Table 3. Actual tables can be found in Appendix 2.
Both the LM-test and the F-statistic, in all the emerging markets have a probability value of
0. Therefore, the null hypothesis of homoscedasticity of errors, denoted as
H0:γ1=γ2=γ3=γ4=γ5=0, is rejected at 1% significance level, suggesting the presence of ARCH
effects in the residuals and so the presence of volatility clustering in the returns. Hence
GARCH specifications should be employed for modeling the volatility of the BRICS Exchange
market. Ortij (2001) and Chong et al (1999) have also found evidence of ARCH effects in other
emerging economies.
Tables 4A-4E synopsize the coefficient estimates along with their probabilities and standard
errors for each model for all the five countries together. These findings are used in the
! 20!
empirical analysis in order to establish the superiority of a model for estimating volatility.
Actual Tables can be found in Appendix 3.
For GARCH (1,1) model, the constant (!!), the lagged squared residual (!!) and the lagged
conditional variance (β) in the conditional variance equation are positive and highly significant
for all the countries. The mean coefficients (µ) are also statistically significant at 1% level.
Since all the parameters are positive, the non-negativity condition of the GARCH models is
satisfied. The fact that the value of the ARCH parameter is much smaller than the value of the
GARCH parameter signifies high persistence with little expected deviations. The high
persistence of shocks to conditional variance can be seen by the fact that the sum of the
persistence parameters (α1 + β) is close to one. De Santis and Imrohoroglu (1997) and Song et
al (1998) found similar results regarding the high persistence of shocks to volatility in the case
of other emerging economies. The above observations are presented in the Table 4A:
Tables 4B and 4C display the findings for the GARCH-M (1,1) model with conditional
standard deviation and conditional variance term in the mean respectively. Generally, the
coefficient estimates in the volatility equation of the GARCH-M (1,1) model in both cases are
highly statistically significant. On the other hand, the standard deviation and the conditional
variance in the mean equation for both GARCH-M models correspondingly are not statistically
significant. Although δ is in most cases positive, it is highly insignificant in all the five
countries for both models and so no sign of time varying risk premium exists. Choundry (1996)
also failed to indicate the presence of risk premium when examining six emerging economies
including India. Concluding, GARCH-M model is not significant in explaining the volatility in
the BRICS returns.
! 21!
!
As it has been observed in the preliminary analysis, the error terms of daily returns follow a
non-linear distribution. The presence of skewness implies the asymmetry in the data. TGARCH
(1,1,1) and EGARCH (1,1) are the asymmetric models employed to capture this phenomenon.
Tables 4D and 4E present the coefficient estimates of both asymmetric models.
The asymmetry term (γ) is positive in the TGARCH (1,1,1) model for all the countries.
Hence, leverage effect is present implying that negative shocks will lead to higher conditional
variance than positive shocks of the same sign. The non-negativity constraints are completely
satisfied in the TGARCH model as !! >0, !! >0, ! > 0 and !! + γ ≥ 0 and so the model is
acceptable. For the EGARCH (1,1), γ is negative in all five countries indicating the asymmetric
influence of news on volatility. Variance rises more after negative returns than after positive
returns. The persistence parameter (!) in the EGARCH model is too large indicating that
volatility moves slowly as time passes. In addition, the absolute value of the intercept (!!) in
! 22!
the volatility equation of the EGARCH model is much higher than for the other models tested.
This is explained by the fact that the response variable is logarithmic. China is the only
exception having an insignificant asymmetric coefficient. This indicates that the leverage effect
is not present in the Chinese stock returns. Song et al (1998) found similar evidence for the
Chinese market.
From Tables 4A-4E it is concluded that GARCH (1,1), TGARCH (1,1,1) and EGARCH
(1,1) models are suitable for explaining volatility in the returns of the five main emerging
countries. The GARCH-M (1,1) models are inappropriate for modeling the volatility in the
BRICS due to the insignificant δ parameter.
In Table 5, the Akaike’s Information Criterion is presented for all the five models for each
emerging country in order to evaluate the relative quality of the models. Results show that
TGARCH model has the smallest value of information criterion in the case of four out of five
! 23!
countries followed by EGARCH model. GARCH(1,1) model outperforms the other models
only in the case of China followed by GARCH-M with standard deviation in the mean equation
model. This is consistent with the finding of Song et al (1998) for China stating that GARCH-
M is the most suitable model for capturing the volatility of the country.
In order to distinguish the best fit model out of the three appropriate models, diagnosis tests
of the standardised and squared standardised residuals are conducted. A summary of the
diagnosis tests is presented in Table 6. The actual histograms and correlograms can be found in
Appendix 4.
! 24!
By comparing the descriptive statistics of the daily returns presented in Table 1 with the
descriptive statistics derived from the diagnostics of the standardised residuals it is concluded
that although the skewness coefficients are still very close to zero, they have slightly increased
in absolute values, indicating the weakness of the estimating models to capture the asymmetry
of the returns. On the other hand, the coefficient of kurtosis has decreased more than half its
value in almost all cases suggesting that the three GARCH-type models are able to capture
leptokurtosis better than asymmetry in the residual distribution. Still they are not very close to
three. Jarque-Bera’s critical value has also decreased dramatically, although the null hypothesis
that the standardised residuals follow a normal distribution is rejected at 1% significance level.
This implies that the non-normality problem is less severe in the standardized residuals, even
though it still exists. The Ljung-Box cumulative statistic indicates that up to twelve lags the
null hypothesis of no autocorrelation in the squared residuals cannot be rejected at 1% in
almost all the cases. De Santis and Imrohoroglu (1997) and Liu and Hung (2010) also showed
that the best fitted models are sufficient to correct serial correlation of the return series in the
conditional variance equation. The only exception is India when both the TGARCH and
EGARCH models are used. In both cases the null hypothesis is rejected at 1% significance
level.
After close inspection of all the diagnostics it is observed that the TGARCH (1,1,1) model
outperforms the other models, although EGARCH (1,1) fits the data in a sufficient way.
GARCH (1,1) is inferior to the other two models. Carvalhal and Mendes (2008) when
examining the emerging economies of Latin America and Asia, including Brazil and India
found similar results. They proposed TGARCH and EGARCH as the best models for in-sample
estimation.
5.3. Forecasting Evaluation:
After estimating the models, an out-of-sample forecast evaluation is performed. The best
fitted models are then used to make dynamic forecasts from 1/4/2011- 30/3/2012. Tables 7-9
present the evaluation tools used in this dissertation for volatility forecasting. Each table
evaluates the volatility performance in terms of each error statistic. The last row at the bottom
of each table gives each model’s ranking compared to the other modules. The actual graphs for
the dynamic forecasts are available in Appendix 5. RMSE, MAPE and MAE mutually support
that TGARCH (1,1,1) model has the highest volatility forecasting power, followed by
EGARCH (1,1). GARCH (1,1) underperforms the other two models. Poon and Granger (2005)
and Liu and Hung (2010) also found that TGARCH followed by EGARCH give the most
accurate volatility forecasts. In addition, consistent with De Santis and Imrohoroglu (1997) this
study concludes that the best fitted model also gives the most accurate volatility forecasts. This
! 25!
observation contradicts the conclusion of Shamiri and Isa (2009) that the best fitted model
based on AIC does not necessarily give the best volatility forecasts in terms of MSE and MAE.
Finally, in order to assess the performance of the three models, one needs to compare
forecasted values with the squared returns (used as a proxy of actual volatility). The plots of
the proxy against forecasted volatility derived from each model for each country are presented
below (see Figures 3a-3e). The TGARCH model is observed to track variations in the market
volatility more accurately in Brazil, Russia and South Africa. For India, EGARCH model
appears to follow the actual volatility pattern more precisely while for China none of the
models has a better forecasting power than any other model. Therefore, it is confirmed that the
use of squared returns, as a proxy for the actual volatility is consistent with the models’ ranking
given by the statistical loss functions.
! 26!
Figure 3a. Proxy against forecasted volatility in Brazil
Figure 3b. Proxy against forecasted volatility in India
Figure 3c. Proxy against forecasted volatility in Russia
0
2
4
6
8
10
12
14
2011-01-04 3/30/2012
EQUITYRETURNS2 EGARCH GARCH TGARCH
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
2011-01-04 3/30/2012
EQUITYRETURNS2 EGARCHGARCH TGARCH
0
2
4
6
8
10
12
14
2011-01-04 3/30/2012
EQUITYRETURNS2 EGARCHGARCH TGARCH
! 27!
Figure 3d. Proxy against forecasted volatility in South Africa
Figure 3e. Proxy against forecasted volatility in China
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2011-01-04 3/30/2012
EQUITYRETURNS2 EGARCHGARCH TGARCH
0
1
2
3
4
5
2011-01-04 3/30/2012
EQUITYRETURNS2 EGARCHGARCH TGARCH
! 28!
6. Conclusions
This dissertation uses daily equity indices in order to find the best heteroskedastic model for
estimating and forecasting volatility in the five main emerging economies: Brazil, Russia,
India, China and South Africa. The preliminary analysis of the data set finds similar
characteristics to those found in the literature for many emerging economies. Non-normality,
skewness, leptokurtosis, leverage effect and ARCH effects exist in the time series of all the five
stock markets.
GARCH (1,1), GARCH-M (1,1) with both standard deviation and conditional variance in
the mean equation, TGARCH (1,1,1) and EGARCH (1,1) have been employed for modeling
volatility. GARCH-M is insignificant in explaining the volatility in the returns, as no sign of
time varying risk premium exists in the return series of all the five countries. The linear
GARCH (1,1) is sufficient in explaining the conditional variance. TGARCH followed by
EGARCH outperform the other models in estimating volatility. Both models provide strong
evidence of asymmetry in the return series.
Several significant observations have emerged from this study in the context of the one-
year-ahead forecasting performance of the GARCH (1,1), TGARCH (1,1,1) and EGARCH
(1,1) for the equity indices of the BRICS countries. RMSE, MAPE and MAE statistical loss
functions decisively and mutually indicate that the TGARCH followed by EGARCH give the
most accurate forecasts. The simple GARCH model has a weak forecasting ability. These
results demonstrate that in the presence of non-normal distribution and leverage effect, possible
asymmetries of the equity returns have to be taken into account and be modeled in order to
achieve more accurate volatility forecasts.
Relating these finding to the main argument outlined in the introduction it is concluded that
there has been a chronological improvement in the models. TGARCH of Glosten, Jagannathan
and Runkle (1993) and Zakonian (1994) and EGARCH of Nelson (1991) have been proposed
after GARCH and GARCH-M models have been developed in 1986 and 1987 respectively.
Furthermore, this paper contradicts what Shamiri and Isa (2009) stated; that the best fitted
model based on AIC does not necessarily give the best volatility forecasts in terms of MSE and
MAE. Finally, the comparison between the out-of-sample forecasts and the squared daily
returns (used as a proxy for actual volatility) supports that TGARCH gives the most accurate
volatility forecasts for the BRICS countries.
! 29!
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! 36!
Appendix 2 Tests for “ARCH-effects” Brazil
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
India !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Heteroskedasticity!Test:!ARCH! ! !! ! ! ! !! ! ! ! !F"statistic! 117.6681!!!!!Prob.!F(5,2337)! 0.0000!
Obs*R"squared! 471.2211!!!!!Prob.!Chi"Square(5)! 0.0000!! ! ! ! !! ! ! ! !! ! ! ! !
Test!Equation:! ! ! !Dependent!Variable:!RESID^2! ! !Method:!Least!Squares! ! !Date:!07/27/13!!!Time:!02:02! ! !Sample!(adjusted):!4/09/2002!3/31/2011! !Included!observations:!2343!after!adjustments! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! t"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.229888! 0.038290! 6.003888! 0.0000!
RESID^2("1)! 0.028931! 0.020309! 1.424562! 0.1544!RESID^2("2)! 0.283431! 0.020232! 14.00929! 0.0000!RESID^2("3)! 0.064245! 0.021022! 3.056093! 0.0023!RESID^2("4)! 0.090082! 0.020232! 4.452408! 0.0000!RESID^2("5)! 0.190012! 0.020309! 9.356026! 0.0000!
! ! ! ! !! ! ! ! !R"squared! 0.201119!!!!!Mean!dependent!var! 0.670143!
Adjusted!R"squared! 0.199409!!!!!S.D.!dependent!var! 1.774122!S.E.!of!regression! 1.587408!!!!!Akaike!info!criterion! 3.764640!Sum!squared!resid! 5888.924!!!!!Schwarz!criterion! 3.779388!Log!likelihood! "4404.276!!!!!Hannan"Quinn!criter.! 3.770012!F"statistic! 117.6681!!!!!Durbin"Watson!stat! 2.047231!Prob(F"statistic)! 0.000000! ! ! !
! ! ! ! !! ! ! ! !! ! ! ! !! ! ! ! !
Heteroskedasticity!Test:!ARCH! ! !! ! ! ! !! ! ! ! !F"statistic! 39.26449!!!!!Prob.!F(5,2337)! 0.0000!
Obs*R"squared! 181.5732!!!!!Prob.!Chi"Square(5)! 0.0000!! ! ! ! !! ! ! ! !! ! ! ! !
Test!Equation:! ! ! !Dependent!Variable:!RESID^2! ! !Method:!Least!Squares! ! !Date:!07/27/13!!!Time:!02:07! ! !Sample!(adjusted):!4/09/2002!3/31/2011! !Included!observations:!2343!after!adjustments! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! t"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.264250! 0.036552! 7.229465! 0.0000!
RESID^2("1)! 0.135969! 0.020552! 6.615774! 0.0000!RESID^2("2)! 0.080924! 0.020665! 3.915889! 0.0001!RESID^2("3)! 0.061325! 0.020694! 2.963379! 0.0031!RESID^2("4)! 0.086977! 0.020666! 4.208720! 0.0000!RESID^2("5)! 0.113448! 0.020553! 5.519744! 0.0000!
! ! ! ! !! ! ! ! !R"squared! 0.077496!!!!!Mean!dependent!var! 0.506826!
Adjusted!R"squared! 0.075522!!!!!S.D.!dependent!var! 1.612987!S.E.!of!regression! 1.550883!!!!!Akaike!info!criterion! 3.718084!Sum!squared!resid! 5621.042!!!!!Schwarz!criterion! 3.732832!Log!likelihood! "4349.735!!!!!Hannan"Quinn!criter.! 3.723456!F"statistic! 39.26449!!!!!Durbin"Watson!stat! 2.011017!Prob(F"statistic)! 0.000000! ! ! !
! ! ! ! !! ! ! ! !
! 37!
Russia !!
South Africa:
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!
Heteroskedasticity!Test:!ARCH! ! !! ! ! ! !! ! ! ! !F"statistic! 65.23262!!!!!Prob.!F(5,2337)! 0.0000!
Obs*R"squared! 286.9521!!!!!Prob.!Chi"Square(5)! 0.0000!! ! ! ! !! ! ! ! !! ! ! ! !
Test!Equation:! ! ! !Dependent!Variable:!RESID^2! ! !Method:!Least!Squares! ! !Date:!07/27/13!!!Time:!02:09! ! !Sample!(adjusted):!4/09/2002!3/31/2011! !Included!observations:!2343!after!adjustments! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! t"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.483846! 0.092189! 5.248441! 0.0000!
RESID^2("1)! 0.032939! 0.020632! 1.596486! 0.1105!RESID^2("2)! 0.092775! 0.020588! 4.506333! 0.0000!RESID^2("3)! 0.268337! 0.019918! 13.47206! 0.0000!RESID^2("4)! 0.073323! 0.020588! 3.561501! 0.0004!RESID^2("5)! 0.071687! 0.020632! 3.474508! 0.0005!
! ! ! ! !! ! ! ! !R"squared! 0.122472!!!!!Mean!dependent!var! 1.049623!
Adjusted!R"squared! 0.120595!!!!!S.D.!dependent!var! 4.401020!S.E.!of!regression! 4.127128!!!!!Akaike!info!criterion! 5.675598!Sum!squared!resid! 39806.55!!!!!Schwarz!criterion! 5.690346!Log!likelihood! "6642.963!!!!!Hannan"Quinn!criter.! 5.680970!F"statistic! 65.23262!!!!!Durbin"Watson!stat! 2.002379!Prob(F"statistic)! 0.000000! ! ! !
! ! ! ! !! ! ! ! !
Heteroskedasticity!Test:!ARCH! ! !! ! ! ! !! ! ! ! !F"statistic! 106.5867!!!!!Prob.!F(5,2337)! 0.0000!
Obs*R"squared! 435.0844!!!!!Prob.!Chi"Square(5)! 0.0000!! ! ! ! !! ! ! ! !! ! ! ! !
Test!Equation:! ! ! !Dependent!Variable:!RESID^2! ! !Method:!Least!Squares! ! !Date:!07/27/13!!!Time:!02:10! ! !Sample!(adjusted):!4/09/2002!3/31/2011! !Included!observations:!2343!after!adjustments! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! t"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.109831! 0.017247! 6.368069! 0.0000!
RESID^2("1)! 0.043375! 0.020200! 2.147334! 0.0319!RESID^2("2)! 0.169394! 0.020091! 8.431516! 0.0000!RESID^2("3)! 0.126455! 0.020225! 6.252296! 0.0000!RESID^2("4)! 0.110287! 0.020091! 5.489490! 0.0000!RESID^2("5)! 0.215514! 0.020195! 10.67178! 0.0000!
! ! ! ! !! ! ! ! !R"squared! 0.185695!!!!!Mean!dependent!var! 0.327829!
Adjusted!R"squared! 0.183953!!!!!S.D.!dependent!var! 0.760519!S.E.!of!regression! 0.687017!!!!!Akaike!info!criterion! 2.089643!Sum!squared!resid! 1103.047!!!!!Schwarz!criterion! 2.104391!Log!likelihood! "2442.016!!!!!Hannan"Quinn!criter.! 2.095015!F"statistic! 106.5867!!!!!Durbin"Watson!stat! 2.022714!Prob(F"statistic)! 0.000000! ! ! !
! ! ! ! !! ! ! ! !
! 38!
China !! !!
!
!
!
Heteroskedasticity!Test:!ARCH! ! !! ! ! ! !! ! ! ! !F"statistic! 15.02217!!!!!Prob.!F(5,1617)! 0.0000!
Obs*R"squared! 72.04311!!!!!Prob.!Chi"Square(5)! 0.0000!! ! ! ! !! ! ! ! !! ! ! ! !
Test!Equation:! ! ! !Dependent!Variable:!RESID^2! ! !Method:!Least!Squares! ! !Date:!07/27/13!!!Time:!02:12! ! !Sample!(adjusted):!1/11/2005!3/31/2011! !Included!observations:!1623!after!adjustments! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! t"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.463800! 0.051027! 9.089238! 0.0000!
RESID^2("1)! 0.093188! 0.024860! 3.748474! 0.0002!RESID^2("2)! 0.059480! 0.024821! 2.396396! 0.0167!RESID^2("3)! 0.085808! 0.024773! 3.463827! 0.0005!RESID^2("4)! 0.108942! 0.024821! 4.389167! 0.0000!RESID^2("5)! 0.025385! 0.024860! 1.021087! 0.3074!
! ! ! ! !! ! ! ! !R"squared! 0.044389!!!!!Mean!dependent!var! 0.739485!
Adjusted!R"squared! 0.041434!!!!!S.D.!dependent!var! 1.594011!S.E.!of!regression! 1.560638!!!!!Akaike!info!criterion! 3.731757!Sum!squared!resid! 3938.351!!!!!Schwarz!criterion! 3.751690!Log!likelihood! "3022.321!!!!!Hannan"Quinn!criter.! 3.739153!F"statistic! 15.02217!!!!!Durbin"Watson!stat! 2.003634!Prob(F"statistic)! 0.000000! ! ! !
! ! ! ! !! ! ! ! !
! 39!
Appendix 3
Brazil GARCH (1,1)
!
TGARCH (1,1,1)
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:41! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!10!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.050336! 0.014436! 3.486874! 0.0005!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.012145! 0.002477! 4.903506! 0.0000!
RESID("1)^2! 0.066246! 0.007554! 8.769335! 0.0000!GARCH("1)! 0.912919! 0.009687! 94.24617! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000594!!!!!Mean!dependent!var! 0.030391!
Adjusted!R"squared! "0.000594!!!!!S.D.!dependent!var! 0.818362!S.E.!of!regression! 0.818605!!!!!Akaike!info!criterion! 2.202799!Sum!squared!resid! 1573.429!!!!!Schwarz!criterion! 2.212611!Log!likelihood! "2583.188!!!!!Hannan"Quinn!criter.! 2.206373!Durbin"Watson!stat! 2.015135! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:46! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!14!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*RESID("1)^2*(RESID("1)<0)!+!!!!!!!!!C(5)*GARCH("1)! ! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.032628! 0.014231! 2.292715! 0.0219!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.017199! 0.002658! 6.470974! 0.0000!
RESID("1)^2! 0.008557! 0.009054! 0.945141! 0.3446!RESID("1)^2*(RESID("1)<0)! 0.105895! 0.015670! 6.757694! 0.0000!
GARCH("1)! 0.906299! 0.010513! 86.20643! 0.0000!! ! ! ! !! ! ! ! !R"squared! "0.000007!!!!!Mean!dependent!var! 0.030391!
Adjusted!R"squared! "0.000007!!!!!S.D.!dependent!var! 0.818362!S.E.!of!regression! 0.818365!!!!!Akaike!info!criterion! 2.184574!Sum!squared!resid! 1572.506!!!!!Schwarz!criterion! 2.196838!Log!likelihood! "2560.782!!!!!Hannan"Quinn!criter.! 2.189040!Durbin"Watson!stat! 2.016317! ! ! !
! ! ! ! !! ! ! ! !
! 40!
GARCH-M (1,1) with conditional standard deviation term in the mean !
GARCH-M (1,1) with conditional variance term in the mean !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:47! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!12!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !@SQRT(GARCH)! 0.086015! 0.082436! 1.043421! 0.2968!
C! "0.007193! 0.057198! "0.125758! 0.8999!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.012050! 0.002485! 4.848533! 0.0000!
RESID("1)^2! 0.066214! 0.007562! 8.756670! 0.0000!GARCH("1)! 0.913114! 0.009691! 94.22366! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.003135!!!!!Mean!dependent!var! 0.030391!
Adjusted!R"squared! "0.003562!!!!!S.D.!dependent!var! 0.818362!S.E.!of!regression! 0.819819!!!!!Akaike!info!criterion! 2.203265!Sum!squared!resid! 1577.425!!!!!Schwarz!criterion! 2.215529!Log!likelihood! "2582.735!!!!!Hannan"Quinn!criter.! 2.207732!Durbin"Watson!stat! 2.008513! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:48! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!11!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !GARCH! 0.032925! 0.040399! 0.814995! 0.4151!
C! 0.034866! 0.024157! 1.443347! 0.1489!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.012162! 0.002487! 4.891149! 0.0000!
RESID("1)^2! 0.066332! 0.007557! 8.777821! 0.0000!GARCH("1)! 0.912790! 0.009692! 94.18150! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.003091!!!!!Mean!dependent!var! 0.030391!
Adjusted!R"squared! "0.003518!!!!!S.D.!dependent!var! 0.818362!S.E.!of!regression! 0.819800!!!!!Akaike!info!criterion! 2.203448!Sum!squared!resid! 1577.355!!!!!Schwarz!criterion! 2.215712!Log!likelihood! "2582.950!!!!!Hannan"Quinn!criter.! 2.207915!Durbin"Watson!stat! 2.008938! ! ! !
! ! ! ! !! ! ! ! !
! 41!
EGARCH (1,1) !
India
GARCH (1,1) !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:49! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!13!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!LOG(GARCH)!=!C(2)!+!C(3)*ABS(RESID("1)/@SQRT(GARCH("1)))!+!C(4)!!!!!!!!!*RESID("1)/@SQRT(GARCH("1))!+!C(5)*LOG(GARCH("1))!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.034654! 0.013918! 2.489842! 0.0128!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C(2)! "0.102269! 0.010867! "9.410755! 0.0000!
C(3)! 0.112875! 0.012949! 8.717205! 0.0000!C(4)! "0.081947! 0.009954! "8.232149! 0.0000!C(5)! 0.975215! 0.003691! 264.1855! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000027!!!!!Mean!dependent!var! 0.030391!
Adjusted!R"squared! "0.000027!!!!!S.D.!dependent!var! 0.818362!S.E.!of!regression! 0.818373!!!!!Akaike!info!criterion! 2.189856!Sum!squared!resid! 1572.537!!!!!Schwarz!criterion! 2.202120!Log!likelihood! "2566.986!!!!!Hannan"Quinn!criter.! 2.194322!Durbin"Watson!stat! 2.016277! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!21:58! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!14!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!!!!
! ! ! !! ! ! ! !C! 0.057705! 0.010531! 5.479346! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.009447! 0.001519! 6.219851! 0.0000!
RESID("1)^2! 0.129080! 0.009769! 13.21303! 0.0000!GARCH("1)! 0.855960! 0.010429! 82.07120! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001315!!!!!Mean!dependent!var! 0.031867!
Adjusted!R"squared! "0.001315!!!!!S.D.!dependent!var! 0.712594!S.E.!of!regression! 0.713063!!!!!Akaike!info!criterion! 1.798566!Sum!squared!resid! 1193.861!!!!!Schwarz!criterion! 1.808378!Log!likelihood! "2108.416!!!!!Hannan"Quinn!criter.! 1.802140!Durbin"Watson!stat! 1.895439! ! ! !
! ! ! ! !! ! ! ! !
! 42!
TGARCH (1,1,1) !
!
GARCH-M (1,1) with conditional standard deviation term in the mean !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:00! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!19!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*RESID("1)^2*(RESID("1)<0)!+!!!!!!!!!C(5)*GARCH("1)! ! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.044933! 0.010639! 4.223515! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.011148! 0.001525! 7.308421! 0.0000!
RESID("1)^2! 0.066579! 0.008844! 7.528052! 0.0000!RESID("1)^2*(RESID("1)<0)! 0.113797! 0.015303! 7.436239! 0.0000!
GARCH("1)! 0.854015! 0.011113! 76.84696! 0.0000!! ! ! ! !! ! ! ! !R"squared! "0.000336!!!!!Mean!dependent!var! 0.031867!
Adjusted!R"squared! "0.000336!!!!!S.D.!dependent!var! 0.712594!S.E.!of!regression! 0.712714!!!!!Akaike!info!criterion! 1.786579!Sum!squared!resid! 1192.693!!!!!Schwarz!criterion! 1.798843!Log!likelihood! "2093.337!!!!!Hannan"Quinn!criter.! 1.791046!Durbin"Watson!stat! 1.897294! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:00! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!14!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !@SQRT(GARCH)! 0.012400! 0.062045! 0.199849! 0.8416!
C! 0.051441! 0.032170! 1.599003! 0.1098!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.009438! 0.001530! 6.169297! 0.0000!
RESID("1)^2! 0.129063! 0.009769! 13.21154! 0.0000!GARCH("1)! 0.856007! 0.010432! 82.05274! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001710!!!!!Mean!dependent!var! 0.031867!
Adjusted!R"squared! "0.002137!!!!!S.D.!dependent!var! 0.712594!S.E.!of!regression! 0.713355!!!!!Akaike!info!criterion! 1.799401!Sum!squared!resid! 1194.331!!!!!Schwarz!criterion! 1.811666!Log!likelihood! "2108.397!!!!!Hannan"Quinn!criter.! 1.803868!Durbin"Watson!stat! 1.894390! ! ! !
! ! ! ! !! ! ! ! !
! 43!
GARCH-M (1,1) with conditional variance term in the mean !
!
EGARCH (1,1) !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:01! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!15!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !GARCH! 0.005809! 0.041729! 0.139215! 0.8893!
C! 0.056054! 0.015309! 3.661414! 0.0003!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.009445! 0.001523! 6.201872! 0.0000!
RESID("1)^2! 0.129090! 0.009782! 13.19714! 0.0000!GARCH("1)! 0.855960! 0.010437! 82.00816! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001599!!!!!Mean!dependent!var! 0.031867!
Adjusted!R"squared! "0.002025!!!!!S.D.!dependent!var! 0.712594!S.E.!of!regression! 0.713316!!!!!Akaike!info!criterion! 1.799408!Sum!squared!resid! 1194.198!!!!!Schwarz!criterion! 1.811672!Log!likelihood! "2108.405!!!!!Hannan"Quinn!criter.! 1.803875!Durbin"Watson!stat! 1.894689! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:02! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!16!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!LOG(GARCH)!=!C(2)!+!C(3)*ABS(RESID("1)/@SQRT(GARCH("1)))!+!C(4)!!!!!!!!!*RESID("1)/@SQRT(GARCH("1))!+!C(5)*LOG(GARCH("1))!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.045210! 0.009926! 4.554535! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C(2)! "0.219798! 0.014892! "14.75900! 0.0000!
C(3)! 0.246481! 0.015356! 16.05139! 0.0000!C(4)! "0.078695! 0.009464! "8.314786! 0.0000!C(5)! 0.967800! 0.004088! 236.7151! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000351!!!!!Mean!dependent!var! 0.031867!
Adjusted!R"squared! "0.000351!!!!!S.D.!dependent!var! 0.712594!S.E.!of!regression! 0.712719!!!!!Akaike!info!criterion! 1.794411!Sum!squared!resid! 1192.711!!!!!Schwarz!criterion! 1.806675!Log!likelihood! "2102.536!!!!!Hannan"Quinn!criter.! 1.798877!Durbin"Watson!stat! 1.897267! ! ! !
! ! ! ! !! ! ! ! !
! 44!
Russia GARCH (1,1)
!
TGARCH (1,1,1) !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:05! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!18!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.073671! 0.014437! 5.102947! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.023585! 0.002573! 9.166555! 0.0000!
RESID("1)^2! 0.112289! 0.008214! 13.67002! 0.0000!GARCH("1)! 0.859839! 0.009823! 87.53445! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001568!!!!!Mean!dependent!var! 0.033084!
Adjusted!R"squared! "0.001568!!!!!S.D.!dependent!var! 1.025332!S.E.!of!regression! 1.026135!!!!!Akaike!info!criterion! 2.423101!Sum!squared!resid! 2472.333!!!!!Schwarz!criterion! 2.432913!Log!likelihood! "2841.932!!!!!Hannan"Quinn!criter.! 2.426674!Durbin"Watson!stat! 1.957857! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:06! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!17!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*RESID("1)^2*(RESID("1)<0)!+!!!!!!!!!C(5)*GARCH("1)! ! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.054339! 0.014991! 3.624845! 0.0003!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.027131! 0.002690! 10.08455! 0.0000!
RESID("1)^2! 0.065396! 0.009427! 6.936912! 0.0000!RESID("1)^2*(RESID("1)<0)! 0.086921! 0.013812! 6.293157! 0.0000!
GARCH("1)! 0.855319! 0.010101! 84.68018! 0.0000!! ! ! ! !! ! ! ! !R"squared! "0.000430!!!!!Mean!dependent!var! 0.033084!
Adjusted!R"squared! "0.000430!!!!!S.D.!dependent!var! 1.025332!S.E.!of!regression! 1.025552!!!!!Akaike!info!criterion! 2.414446!Sum!squared!resid! 2469.525!!!!!Schwarz!criterion! 2.426710!Log!likelihood! "2830.767!!!!!Hannan"Quinn!criter.! 2.418913!Durbin"Watson!stat! 1.960083! ! ! !
! ! ! ! !! ! ! ! !
! 45!
GARCH-M (1,1) with conditional standard deviation term in the mean !
!
GARCH-M (1,1) with conditional variance term in the mean
!
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:07! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!19!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !@SQRT(GARCH)! "0.025478! 0.059549! "0.427846! 0.6688!
C! 0.091550! 0.043138! 2.122275! 0.0338!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.023490! 0.002553! 9.202283! 0.0000!
RESID("1)^2! 0.112224! 0.008227! 13.64067! 0.0000!GARCH("1)! 0.860061! 0.009784! 87.90886! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000373!!!!!Mean!dependent!var! 0.033084!
Adjusted!R"squared! "0.000799!!!!!S.D.!dependent!var! 1.025332!S.E.!of!regression! 1.025741!!!!!Akaike!info!criterion! 2.423896!Sum!squared!resid! 2469.384!!!!!Schwarz!criterion! 2.436160!Log!likelihood! "2841.866!!!!!Hannan"Quinn!criter.! 2.428363!Durbin"Watson!stat! 1.961000! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:07! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!14!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !GARCH! "0.001140! 0.025473! "0.044744! 0.9643!
C! 0.074284! 0.019693! 3.772026! 0.0002!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.023571! 0.002570! 9.170235! 0.0000!
RESID("1)^2! 0.112261! 0.008241! 13.62250! 0.0000!GARCH("1)! 0.859887! 0.009819! 87.57402! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001395!!!!!Mean!dependent!var! 0.033084!
Adjusted!R"squared! "0.001822!!!!!S.D.!dependent!var! 1.025332!S.E.!of!regression! 1.026265!!!!!Akaike!info!criterion! 2.423952!Sum!squared!resid! 2471.907!!!!!Schwarz!criterion! 2.436216!Log!likelihood! "2841.932!!!!!Hannan"Quinn!criter.! 2.428419!Durbin"Watson!stat! 1.958290! ! ! !
! ! ! ! !! ! ! ! !
! 46!
EGARCH (1,1) !
South Africa GARCH (1,1)
!
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:08! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!24!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!LOG(GARCH)!=!C(2)!+!C(3)*ABS(RESID("1)/@SQRT(GARCH("1)))!+!C(4)!!!!!!!!!*RESID("1)/@SQRT(GARCH("1))!+!C(5)*LOG(GARCH("1))!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.079204! 0.013425! 5.899732! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C(2)! "0.165156! 0.008909! "18.53785! 0.0000!
C(3)! 0.203937! 0.011133! 18.31749! 0.0000!C(4)! "0.059014! 0.007668! "7.696206! 0.0000!C(5)! 0.963906! 0.003445! 279.7743! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.002024!!!!!Mean!dependent!var! 0.033084!
Adjusted!R"squared! "0.002024!!!!!S.D.!dependent!var! 1.025332!S.E.!of!regression! 1.026369!!!!!Akaike!info!criterion! 2.447690!Sum!squared!resid! 2473.460!!!!!Schwarz!criterion! 2.459954!Log!likelihood! "2869.812!!!!!Hannan"Quinn!criter.! 2.452157!Durbin"Watson!stat! 1.956964! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:13! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!10!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.038741! 0.009152! 4.232824! 0.0000!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.004169! 0.001206! 3.456168! 0.0005!
RESID("1)^2! 0.089833! 0.010522! 8.537332! 0.0000!GARCH("1)! 0.897655! 0.011691! 76.78300! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001090!!!!!Mean!dependent!var! 0.019835!
Adjusted!R"squared! "0.001090!!!!!S.D.!dependent!var! 0.572863!S.E.!of!regression! 0.573175!!!!!Akaike!info!criterion! 1.435795!Sum!squared!resid! 771.3863!!!!!Schwarz!criterion! 1.445606!Log!likelihood! "1682.341!!!!!Hannan"Quinn!criter.! 1.439368!Durbin"Watson!stat! 1.924627! ! ! !
! ! ! ! !! ! ! ! !
! 47!
TGARCH (1,1,1) !
!
GARCH-M (1,1) with conditional standard deviation term in the mean !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:14! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!10!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*RESID("1)^2*(RESID("1)<0)!+!!!!!!!!!C(5)*GARCH("1)! ! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.021768! 0.009406! 2.314238! 0.0207!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.004624! 0.000963! 4.800061! 0.0000!
RESID("1)^2! 0.003378! 0.008936! 0.378023! 0.7054!RESID("1)^2*(RESID("1)<0)! 0.122208! 0.015336! 7.968468! 0.0000!
GARCH("1)! 0.917689! 0.010236! 89.65531! 0.0000!! ! ! ! !! ! ! ! !R"squared! "0.000011!!!!!Mean!dependent!var! 0.019835!
Adjusted!R"squared! "0.000011!!!!!S.D.!dependent!var! 0.572863!S.E.!of!regression! 0.572866!!!!!Akaike!info!criterion! 1.411744!Sum!squared!resid! 770.5555!!!!!Schwarz!criterion! 1.424008!Log!likelihood! "1653.093!!!!!Hannan"Quinn!criter.! 1.416210!Durbin"Watson!stat! 1.926703! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:15! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!11!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !@SQRT(GARCH)! 0.025990! 0.070027! 0.371151! 0.7105!
C! 0.027478! 0.031743! 0.865644! 0.3867!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.004156! 0.001208! 3.440469! 0.0006!
RESID("1)^2! 0.089493! 0.010499! 8.524200! 0.0000!GARCH("1)! 0.897999! 0.011688! 76.82848! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001554!!!!!Mean!dependent!var! 0.019835!
Adjusted!R"squared! "0.001980!!!!!S.D.!dependent!var! 0.572863!S.E.!of!regression! 0.573429!!!!!Akaike!info!criterion! 1.436586!Sum!squared!resid! 771.7438!!!!!Schwarz!criterion! 1.448850!Log!likelihood! "1682.270!!!!!Hannan"Quinn!criter.! 1.441053!Durbin"Watson!stat! 1.923263! ! ! !
! ! ! ! !! ! ! ! !
! 48!
GARCH-M (1,1) with conditional variance term in the mean !
!
EGARCH (1,1) !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:16! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!11!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !GARCH! 0.027363! 0.056526! 0.484076! 0.6283!
C! 0.033149! 0.014749! 2.247586! 0.0246!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.004157! 0.001208! 3.441754! 0.0006!
RESID("1)^2! 0.089574! 0.010502! 8.529350! 0.0000!GARCH("1)! 0.897921! 0.011690! 76.81001! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.001800!!!!!Mean!dependent!var! 0.019835!
Adjusted!R"squared! "0.002227!!!!!S.D.!dependent!var! 0.572863!S.E.!of!regression! 0.573500!!!!!Akaike!info!criterion! 1.436545!Sum!squared!resid! 771.9338!!!!!Schwarz!criterion! 1.448809!Log!likelihood! "1682.222!!!!!Hannan"Quinn!criter.! 1.441012!Durbin"Watson!stat! 1.922430! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/17/13!!!Time:!22:16! ! !Sample:!4/01/2002!3/31/2011! ! !Included!observations:!2349! ! !Convergence!achieved!after!15!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!LOG(GARCH)!=!C(2)!+!C(3)*ABS(RESID("1)/@SQRT(GARCH("1)))!+!C(4)!!!!!!!!!*RESID("1)/@SQRT(GARCH("1))!+!C(5)*LOG(GARCH("1))!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.023470! 0.009351! 2.509714! 0.0121!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C(2)! "0.112740! 0.013556! "8.316755! 0.0000!
C(3)! 0.112926! 0.014785! 7.637921! 0.0000!C(4)! "0.089265! 0.009485! "9.411674! 0.0000!C(5)! 0.983190! 0.003324! 295.8149! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000040!!!!!Mean!dependent!var! 0.019835!
Adjusted!R"squared! "0.000040!!!!!S.D.!dependent!var! 0.572863!S.E.!of!regression! 0.572874!!!!!Akaike!info!criterion! 1.412913!Sum!squared!resid! 770.5777!!!!!Schwarz!criterion! 1.425177!Log!likelihood! "1654.466!!!!!Hannan"Quinn!criter.! 1.417379!Durbin"Watson!stat! 1.926647! ! ! !
! ! ! ! !! ! ! ! !
! 49!
China GARCH (1,1)
!
TGARCH (1,1,1) !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/18/13!!!Time:!15:01! ! !Sample!(adjusted):!1/03/2005!3/31/2011! !Included!observations:!1629!after!adjustments! !Convergence!achieved!after!13!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.043202! 0.015909! 2.715628! 0.0066!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.005876! 0.001459! 4.027214! 0.0001!
RESID("1)^2! 0.053578! 0.006478! 8.270552! 0.0000!GARCH("1)! 0.939561! 0.006802! 138.1332! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000195!!!!!Mean!dependent!var! 0.031203!
Adjusted!R"squared! "0.000195!!!!!S.D.!dependent!var! 0.859568!S.E.!of!regression! 0.859652!!!!!Akaike!info!criterion! 2.357067!Sum!squared!resid! 1203.095!!!!!Schwarz!criterion! 2.370316!Log!likelihood! "1915.831!!!!!Hannan"Quinn!criter.! 2.361983!Durbin"Watson!stat! 1.976861! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/18/13!!!Time:!15:02! ! !Sample!(adjusted):!1/03/2005!3/31/2011! !Included!observations:!1629!after!adjustments! !Convergence!achieved!after!14!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(2)!+!C(3)*RESID("1)^2!+!C(4)*RESID("1)^2*(RESID("1)<0)!+!!!!!!!!!C(5)*GARCH("1)! ! !
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.041580! 0.017511! 2.374520! 0.0176!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.006318! 0.001525! 4.142036! 0.0000!
RESID("1)^2! 0.050936! 0.008940! 5.697736! 0.0000!RESID("1)^2*(RESID("1)<0)! 0.005861! 0.009360! 0.626217! 0.5312!
GARCH("1)! 0.938327! 0.006912! 135.7476! 0.0000!! ! ! ! !! ! ! ! !R"squared! "0.000146!!!!!Mean!dependent!var! 0.031203!
Adjusted!R"squared! "0.000146!!!!!S.D.!dependent!var! 0.859568!S.E.!of!regression! 0.859631!!!!!Akaike!info!criterion! 2.358155!Sum!squared!resid! 1203.036!!!!!Schwarz!criterion! 2.374716!Log!likelihood! "1915.717!!!!!Hannan"Quinn!criter.! 2.364299!Durbin"Watson!stat! 1.976958! ! ! !
! ! ! ! !! ! ! ! !
! 50!
GARCH-M (1,1) with conditional standard deviation term in the mean !
GARCH-M (1,1) with conditional variance term in the mean !
!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/18/13!!!Time:!15:03! ! !Sample!(adjusted):!1/03/2005!3/31/2011! !Included!observations:!1629!after!adjustments! !Convergence!achieved!after!12!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !@SQRT(GARCH)! "0.039905! 0.078664! "0.507290! 0.6120!
C! 0.070706! 0.056291! 1.256096! 0.2091!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.005794! 0.001447! 4.005065! 0.0001!
RESID("1)^2! 0.053400! 0.006465! 8.259962! 0.0000!GARCH("1)! 0.939856! 0.006766! 138.9062! 0.0000!
! ! ! ! !! ! ! ! !R"squared! 0.000283!!!!!Mean!dependent!var! 0.031203!
Adjusted!R"squared! "0.000331!!!!!S.D.!dependent!var! 0.859568!S.E.!of!regression! 0.859711!!!!!Akaike!info!criterion! 2.358153!Sum!squared!resid! 1202.519!!!!!Schwarz!criterion! 2.374715!Log!likelihood! "1915.716!!!!!Hannan"Quinn!criter.! 2.364298!Durbin"Watson!stat! 1.978927! ! ! !
! ! ! ! !! ! ! ! !
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/18/13!!!Time:!15:04! ! !Sample!(adjusted):!1/03/2005!3/31/2011! !Included!observations:!1629!after!adjustments! !Convergence!achieved!after!13!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!GARCH!=!C(3)!+!C(4)*RESID("1)^2!+!C(5)*GARCH("1)!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !GARCH! "0.017057! 0.047282! "0.360759! 0.7183!
C! 0.052066! 0.029001! 1.795318! 0.0726!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C! 0.005829! 0.001452! 4.014640! 0.0001!
RESID("1)^2! 0.053441! 0.006472! 8.257545! 0.0000!GARCH("1)! 0.939762! 0.006779! 138.6266! 0.0000!
! ! ! ! !! ! ! ! !R"squared! 0.000149!!!!!Mean!dependent!var! 0.031203!
Adjusted!R"squared! "0.000466!!!!!S.D.!dependent!var! 0.859568!S.E.!of!regression! 0.859769!!!!!Akaike!info!criterion! 2.358222!Sum!squared!resid! 1202.681!!!!!Schwarz!criterion! 2.374783!Log!likelihood! "1915.772!!!!!Hannan"Quinn!criter.! 2.364366!Durbin"Watson!stat! 1.978525! ! ! !
! ! ! ! !! ! ! ! !
! 51!
EGARCH (1,1) !
!
!!!!!!!!!!!!!!!!!!!!!!!!
Dependent!Variable:!DAILYRETURNS! !Method:!ML!"!ARCH!(Marquardt)!"!Normal!distribution!Date:!07/18/13!!!Time:!15:04! ! !Sample!(adjusted):!1/03/2005!3/31/2011! !Included!observations:!1629!after!adjustments! !Convergence!achieved!after!22!iterations! !Presample!variance:!backcast!(parameter!=!0.7)!LOG(GARCH)!=!C(2)!+!C(3)*ABS(RESID("1)/@SQRT(GARCH("1)))!+!C(4)!!!!!!!!!*RESID("1)/@SQRT(GARCH("1))!+!C(5)*LOG(GARCH("1))!
! ! ! ! !! ! ! ! !Variable! Coefficient! Std.!Error! z"Statistic! Prob.!!!! ! ! ! !! ! ! ! !C! 0.056466! 0.016346! 3.454546! 0.0006!! ! ! ! !! ! ! ! !! Variance!Equation! ! !! ! ! ! !! ! ! ! !C(2)! "0.092805! 0.009333! "9.944144! 0.0000!
C(3)! 0.119974! 0.012553! 9.557482! 0.0000!C(4)! "0.003857! 0.006311! "0.611251! 0.5410!C(5)! 0.987821! 0.002696! 366.3351! 0.0000!
! ! ! ! !! ! ! ! !R"squared! "0.000864!!!!!Mean!dependent!var! 0.031203!
Adjusted!R"squared! "0.000864!!!!!S.D.!dependent!var! 0.859568!S.E.!of!regression! 0.859940!!!!!Akaike!info!criterion! 2.359964!Sum!squared!resid! 1203.900!!!!!Schwarz!criterion! 2.376526!Log!likelihood! "1917.191!!!!!Hannan"Quinn!criter.! 2.366109!Durbin"Watson!stat! 1.975539! ! ! !
! ! ! ! !! ! ! ! !
! 52!
Appendix 4
Histograms and correlograms of standardized residuals GARCH (1,1) Brazil
India
0
50
100
150
200
250
300
350
400
-5 -4 -3 -2 -1 0 1 2 3 4
Series: RESID01Sample 4/01/2002 3/30/2012Observations 2349
Mean -0.026355Median -0.033941Maximum 3.950321Minimum -5.623176Std. Dev. 0.999928Skewness -0.282704Kurtosis 4.054041
Jarque-Bera 140.0285Probability 0.000000
0
100
200
300
400
500
600
-4 -3 -2 -1 0 1 2 3 4 5 6 7
Series: RESID01Sample 4/01/2002 3/30/2012Observations 2349
Mean -0.039104Median -0.035634Maximum 6.940723Minimum -4.165294Std. Dev. 0.999444Skewness -0.219400Kurtosis 4.986628
Jarque-Bera 405.1278Probability 0.000000
! 53!
Russia
0
100
200
300
400
500
600
700
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.036350Median -0.049991Maximum 4.866119Minimum -6.750265Std. Dev. 0.999725Skewness -0.326369Kurtosis 5.509360
Jarque-Bera 658.0094Probability 0.000000
! 54!
South Africa
China
0
50
100
150
200
250
300
350
-4 -3 -2 -1 0 1 2 3 4
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.037725Median -0.045994Maximum 4.006832Minimum -4.239149Std. Dev. 0.999216Skewness -0.257043Kurtosis 3.539939
Jarque-Bera 54.40075Probability 0.000000
0
40
80
120
160
200
240
280
-4 -3 -2 -1 0 1 2 3 4 5
Ser ies: Standardized ResidualsSample 1/03/2005 3/31/2011Observations 1629
Mean -0.008210Median -0.006701Maximum 5.070198Minimum -4.697691Std. Dev. 1.000961Skewness -0.388135Kurtosis 5.092895
Jarque-Bera 338.2078Probability 0.000000
! 55!
TGARCH (1,1,1) Brazil
0
50
100
150
200
250
300
350
400
-5 -4 -3 -2 -1 0 1 2 3 4
Series: RESID02Sample 4/01/2002 3/30/2012Observations 2349
Mean 0.045246Median 0.036093Maximum 4.171531Minimum -5.519152Std. Dev. 0.999250Skewness -0.261429Kurtosis 3.992470
Jarque-Bera 123.1637Probability 0.000000
! 56!
India
Russia
0
100
200
300
400
500
600
-4 -2 0 2 4 6 8
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.012439Median -0.016272Maximum 8.000987Minimum -4.324972Std. Dev. 1.000110Skewness -0.117059Kurtosis 5.687931
Jarque-Bera 712.5086Probability 0.000000
0
100
200
300
400
500
600
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.010250Median -0.026332Maximum 4.755611Minimum -6.736079Std. Dev. 1.000342Skewness -0.302130Kurtosis 5.390525
Jarque-Bera 595.0546Probability 0.000000
! 57!
South Africa
0
40
80
120
160
200
240
280
320
-4 -3 -2 -1 0 1 2 3
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.004075Median -0.021329Maximum 3.409517Minimum -4.312074Std. Dev. 0.999725Skewness -0.274564Kurtosis 3.430010
Jarque-Bera 47.61137Probability 0.000000
! 58!
China
EGARCH (1,1) Brazil
0
40
80
120
160
200
240
280
-4 -3 -2 -1 0 1 2 3 4 5
Ser ies: Standardized ResidualsSample 1/03/2005 3/31/2011Observations 1629
Mean -0.005753Median -0.005128Maximum 5.024099Minimum -4.742286Std. Dev. 1.000897Skewness -0.395559Kurtosis 5.095053
Jarque-Bera 340.4008Probability 0.000000
0
100
200
300
400
500
600
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Series: RESID03Sample 4/01/2002 3/30/2012Observations 2349
Mean -0.005855Median -0.010844Maximum 4.519906Minimum -5.769279Std. Dev. 1.000007Skewness -0.256675Kurtosis 4.081930
Jarque-Bera 140.3626Probability 0.000000
! 59!
India
0
100
200
300
400
500
600
-4 -2 0 2 4 6 8
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.013341Median -0.016046Maximum 7.511284Minimum -4.599721Std. Dev. 1.000061Skewness -0.166962Kurtosis 5.473087
Jarque-Bera 609.5329Probability 0.000000
! 60!
Russia
South Africa
0
100
200
300
400
500
600
700
-8 -6 -4 -2 0 2 4 6
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.043673Median -0.055094Maximum 5.507766Minimum -7.649230Std. Dev. 0.999360Skewness -0.303153Kurtosis 6.488993
Jarque-Bera 1227.419Probability 0.000000
0
50
100
150
200
250
300
350
-4 -3 -2 -1 0 1 2 3
Ser ies: Standardized ResidualsSample 4/01/2002 3/31/2011Observations 2349
Mean -0.005555Median -0.022774Maximum 3.363872Minimum -4.525791Std. Dev. 0.999423Skewness -0.276483Kurtosis 3.420130
Jarque-Bera 47.20327Probability 0.000000
! 61!
China
0
50
100
150
200
250
300
-5 -4 -3 -2 -1 0 1 2 3 4 5
Ser ies: Standardized ResidualsSample 1/03/2005 3/31/2011Observations 1629
Mean -0.024462Median -0.025215Maximum 4.862561Minimum -4.916523Std. Dev. 1.000744Skewness -0.370151Kurtosis 5.000087
Jarque-Bera 308.7222Probability 0.000000
! 62!
Appendix 5
Dynamic forecasts GARCH (1,1) Brazil India
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.673544Mean Absolute Error 0.486456Mean Abs. Percent Error 110.9524Theil Inequality Coefficient 0.933879 Bias Proportion 0.008076 Variance Proportion NA Covariance Proportion NA
.2
.3
.4
.5
.6
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.548043Mean Absolute Error 0.433957Mean Abs. Percent Error 107.2480Theil Inequality Coefficient 0.912273 Bias Proportion 0.019310 Variance Proportion NA Covariance Proportion NA
.2
.3
.4
.5
.6
.7
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
! 63!
Russia South Africa
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.749315Mean Absolute Error 0.542109Mean Abs. Percent Error 132.1336Theil Inequality Coefficient 0.917809 Bias Proportion 0.019023 Variance Proportion NA Covariance Proportion NA
.3
.4
.5
.6
.7
.8
.9
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.474256Mean Absolute Error 0.350371Mean Abs. Percent Error 232.6630Theil Inequality Coefficient 0.926333 Bias Proportion 0.004526 Variance Proportion NA Covariance Proportion NA
.22
.24
.26
.28
.30
.32
.34
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
! 64!
China TGARCH (1,1,1) Brazil
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.570715Mean Absolute Error 0.415328Mean Abs. Percent Error 133.1922Theil Inequality Coefficient 0.937398 Bias Proportion 0.024054 Variance Proportion NA Covariance Proportion NA
.2
.3
.4
.5
.6
.7
.8
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
-1.6
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.672184Mean Absolute Error 0.484751Mean Abs. Percent Error 103.2577Theil Inequality Coefficient 0.955453 Bias Proportion 0.004058 Variance Proportion NA Covariance Proportion NA
.25
.30
.35
.40
.45
.50
.55
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
! 65!
India Russia
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.546415Mean Absolute Error 0.431281Mean Abs. Percent Error 103.1457Theil Inequality Coefficient 0.929320 Bias Proportion 0.013456 Variance Proportion NA Covariance Proportion NA
.20
.25
.30
.35
.40
.45
.50
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.746894Mean Absolute Error 0.539961Mean Abs. Percent Error 120.7923Theil Inequality Coefficient 0.937031 Bias Proportion 0.012653 Variance Proportion NA Covariance Proportion NA
.3
.4
.5
.6
.7
.8
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
! 66!
South Africa China
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.473417Mean Absolute Error 0.349718Mean Abs. Percent Error 168.7918Theil Inequality Coefficient 0.956400 Bias Proportion 0.000995 Variance Proportion NA Covariance Proportion NA
.21
.22
.23
.24
.25
.26
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.570466Mean Absolute Error 0.414993Mean Abs. Percent Error 131.4307Theil Inequality Coefficient 0.939491 Bias Proportion 0.023201 Variance Proportion NA Covariance Proportion NA
.3
.4
.5
.6
.7
.8
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
! 67!
EGARCH (1,1)
Brazil India
-2
-1
0
1
2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.672316Mean Absolute Error 0.484914Mean Abs. Percent Error 104.0549Theil Inequality Coefficient 0.952896 Bias Proportion 0.004450 Variance Proportion NA Covariance Proportion NA
.2
.3
.4
.5
.6
.7
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
Forecast of Variance
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.546447Mean Absolute Error 0.431337Mean Abs. Percent Error 103.2308Theil Inequality Coefficient 0.928937 Bias Proportion 0.013573 Variance Proportion NA Covariance Proportion NA
.25
.30
.35
.40
.45
.50
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
! 68!
Russia South Africa
-2
-1
0
1
2
3
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.750098Mean Absolute Error 0.542766Mean Abs. Percent Error 135.4392Theil Inequality Coefficient 0.912583 Bias Proportion 0.021070 Variance Proportion NA Covariance Proportion NA
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.473474Mean Absolute Error 0.349764Mean Abs. Percent Error 175.1267Theil Inequality Coefficient 0.953238 Bias Proportion 0.001234 Variance Proportion NA Covariance Proportion NA
.21
.22
.23
.24
.25
.26
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
! 69!
China ! !!
-3
-2
-1
0
1
2
3
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M3
2011 2012
DAILYRETURF – 2 S.E.
Forecast: DAILYRETURFActual: DAILYRETURNSForecast sample: 4/01/2011 3/30/2012Included observations: 261Root Mean Squared Error 0.572922Mean Absolute Error 0.418242Mean Abs. Percent Error 147.9798Theil Inequality Coefficient 0.920958 Bias Proportion 0.031559 Variance Proportion NA Covariance Proportion NA
0.2
0.4
0.6
0.8
1.0
1.2
1.4
M4 M5 M6 M7 M8 M9 M10 M11 M12 M1 M2 M32011 2012
Forecast of Variance
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