Volatility of stock returns_Yuxiang Ou

Volatility of stock returns: a comparison between the United

States and BRICS

Yuxiang Ou

Abstract

In the paper, we use different types of GARCH models to investigate the volatility of

stock returns for the United States and BRICS countries. The statistical fit suggests that

volatility of stock returns, which varies among the six countries, is mostly asymmetric. There

is not enough evidence that the volatility leads to a higher stock return.

Keywords: stock returns; GARCH; T-GARCH; E-GARCH; GARCH-in-Mean

1 Introduction

The global growth of stock markets, especially for the emerging ones, has attracted

increasing attentions of researchers. Many developing countries, including India and China,

are reforming regulations and laws to stimulate stock market development. As the overseas

investment becomes more and more popular, the stock market performances exhibit

dependence or correlation across different countries. But due to some political or historical

factors, there still exist many distinctions. This paper aims to examine the differences among

the developing countries and compares them to one typical developed country – the United

States. Regarding the developing world, we focus on five representative countries – Brazil,

Russia, India, China, and South Africa. Actually, they form an association called BRICS and

agree to meet annually at formal summits to promote economic cooperation and

development.

To account for stock market performance, volatility, the variance of stock returns, should

be emphasized. In reality, it is more often to witness a specific period of commonly high

volatility or an ordinarily less volatile period. This clustering phenomenon conveys that the

volatility is not normally distributed. We then need to allow for this heteroskedasticity when

performing model estimations. Given that, GARCH-type models appear to be the appropriate

model selections.

GARCH model is expanded by Bollerslev (1986) from ARCH model. As is known to us

all, volatility is likely to rise during periods of downtrend and likely to fall during uptrend.

Since neither GARCH and ARCH model can capture this asymmetry, we need extensive

models like T-GARCH model (Zakoian (1994) and Glosen et al. (1994)) and E-GARCH

model (Nelson (1991)). Moreover, we might have to consider a GARCH-in-Mean model

(Engle et al. (1987)) because in the financial markets risk and the expected return are

correlated, and as a result of that, in the mean equation of the GARCH model there should be

a reference to variance or standard deviation. This portion of return is usually named as “risk

premium”.

To deliver a complete discussion of the topic, the remainder of the paper is organized as

follows: section 2 describes the data selection and visualizes the trend in stock prices and

returns; section 3 presents the techniques of our empirical research; section 4 shows the

results and corresponding analysis and eventually section 5 gives the conclusions.

2 Data

The study uses daily stock market indices at closing times as collected from

`http://finance.yahoo.com/stock-center/. S&P 100 Index, BOVESPA Index, RTS Index, BSE

SENSITIVE Index, Shanghai Composite Index, and iShares MSCI South Africa are the

measures of US, Brazilian, Russian, Indian, Chinese, and South African stock market

performances, respectively. Considering that South Africa officially became a member nation

on December 24th, 2010, we collect data right after the entry of South Africa, from December

27th, 2010 to December 3rd, 2015. Since data can be log-transformed to approach normal

distribution and improve interpretability, we will use log value of the stock price quotes other

than the original data. Then the stock returns are derived from first differences in logarithmic

stock prices.

Before further investigation, it would be helpful to have a glance at the data and get a first

impression of the trend.

Figure 1: Trend of stock prices and returns

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

I II III IV I II III IV I II III IV I II III IV I II III IV

2011 2012 2013 2014 2015

logP_US

-8

-6

-4

-2

0

2

4

6


2011 2012 2013 2014 2015

r_US

10.6

10.7

10.8

10.9

11.0

11.1

11.2


2011 2012 2013 2014 2015

logP_Brazil

-10

-8

-6

-4

-2

0

2

4

6


2011 2012 2013 2014 2015

r_Brazil

6.4

6.6

6.8

7.0

7.2

7.4

7.6

7.8


2011 2012 2013 2014 2015

logP_Russia

-15

-10

-5

0

5

10

15


2011 2012 2013 2014 2015

r_Russia

9.6

9.7

9.8

9.9

10.0

10.1

10.2

10.3

10.4


2011 2012 2013 2014 2015

logP_India

-8

-6

-4

-2

0

2

4


2011 2012 2013 2014 2015

r_India

7.4

7.6

7.8

8.0

8.2

8.4

8.6


2011 2012 2013 2014 2015

logP_China

-12

-8

-4

0

4

8


2011 2012 2013 2014 2015

r_China

3.90

3.95

4.00

4.05

4.10

4.15

4.20

4.25

4.30


2011 2012 2013 2014 2015

logP_South Africa

-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0


2011 2012 2013 2014 2015

r_South Africa

Figure 1 shows differing trends of stock prices and returns among the six countries. Stock

market of the United States grows increasingly after a drop in late 2011, while stock markets

of the other five developing countries witness a stagnation or even a downtrend during the

same period. Comparing the vertical coordinates in the stock return diagrams, we find that

the stock returns in Russia and South Africa have the largest range (i.e. most variability). To

confirm this, we collect statistical characteristics of stock returns of all the markets in table 1:

Table 1: Summary statistics on stock returns

US Brazil Russia India China South

Africa

Observation 1244 1240 1222 1220 1196 1245

Mean (%) 0.040294 -0.032416 -0.063536 0.020238 0.01981 -0.020148

Maximum

(%) 4.322944 4.976031 13.24619 3.703462 7.412341 8.552488

Minimum

(%) -6.443033 -8.430746 -13.25455 -6.119711 -8.872906 -9.270363

Std. Dev.

(%) 0.949147 1.444521 1.968825 1.05312 1.50812 1.75995

Skewness -0.458072 -0.058483 -0.202607 -0.161823 -0.761266 -0.208609

Kurtosis 7.727651 4.410845 9.505865 4.591115 8.955003 4.836085

Jarque-

Bera 1202.015 103.5485 2163.474 134.0167 1882.712 183.9114

Probability 0.000 0.000 0.000 0.000 0.000 0.000

Comparison in the standard deviations tells that stock markets of Russia and South Africa

are the most volatile while, on the opposite side, the United States stock market embraces

relative stability. What’s more, the average stock return is the highest in the United States and

the lowest in Russia, exactly as what we have seen in the trend diagrams.

3 Empirical techniques

3.1 Definition of stock returns

To examine the volatility of stock return, we need to define it first. Normally, a return 𝑟" at

time t is defined as:

𝑟" =$%&'($%

$% (1)

where 𝑝" is the price at time t.

However, for financial data (e.g. stock price) analysis, it is more common to define it in an

alternative way:

𝑟" = 𝑙𝑜𝑔𝑝"-. − 𝑙𝑜𝑔𝑝" = 𝑙𝑜𝑔 $%&'$%

(2)

There exist plenty of theories supporting this definition. For example, approximate raw-log

equality claims that when returns are very small (common for trades with short holding

durations), the following approximation satisfies:

𝑙𝑜𝑔 1 + 𝑟 ≈ 𝑟, 𝑟 ≪ 1 (3)

that is,

𝑙𝑜𝑔𝑝"-. − 𝑙𝑜𝑔𝑝" = 𝑙𝑜𝑔 $%&'$%

= 𝑙𝑜𝑔 1 + $%&'($%$%

≈ $%&'($%$%

= 𝑟" (4)

as long as $%&'($%$%

≪ 1. From the data listed before, we know that the stock return is always

less than 1 (100%). Note that the maximum stock return is merely 13.25% in Russia. The

approximation condition satisfies all the time.

3.2 Unit root test

In statistics, a unit root test tests whether a time series variable is non-stationary or not

using an autoregressive model. Two well-known and widely-used tests are Augmented

Dickey-Fuller test (ADF) and Phillips-Perron test (PP). Both tests use the existence of a unit

root as the null hypothesis.

ADF test is applied to the model

Δ𝑦" = 𝛼 + 𝛽𝑡 + 𝛾𝑦"(. + 𝛿.∆𝑦"(. + ⋯+ 𝛿$(.Δ𝑦"($-. + 𝜀" (5)

The unit root test is carried out under the following hypothesis statements:

𝐻@: 𝛾 = 0𝐻.: 𝛾 < 0 (6)

If the test statistic is less than the critical value, then the null hypothesis is rejected and no

unit root is present. In that case, we are able to maintain that the time series variable is

stationary. Otherwise, if we fail to reject the null hypothesis, the variable is non-stationary and

we would not be able to make valid estimating predictions.

The PP test also builds on the Dickey-Fuller test of the null hypothesis 𝜌 = 0 in Δ𝑦" =

𝜌𝑦"(. + 𝑢", but considers a higher order of autocorrelation. Despite that the PP test makes a

non-parametric correction to the t-test statistic, the process of hypothesis testing is quite similar

to the ADF test.

3.3 GARCH models

In econometrics, autoregressive conditional heteroskedasticity (ARCH) models are used to

characterize and model time series when the error terms have a characteristic size or variance.

If an autoregressive moving average model (ARMA model) is assumed for the error variance,

the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.

GARCH models are helpful in modeling financial time series that exhibit time-varying

volatility clustering.

Generally, the GARCH (p, q) model (where p is the order of the ARCH terms 𝜀F and q is

the order of the GARCH terms 𝜎F) is given by

𝑦" = 𝑥"I𝑏 + 𝜀" (7)

𝜀"~𝒩 0, 𝜎"F (8)

M%NOP-Q'R%S'

N -⋯-QTR%STN -U'M%S'

N -⋯-UVM%SVN

OP- QWR%SWNT

WX' - UWM%SWNV

WX' (9)

where (7) is called mean equation and (9) is named as variance equation.

To construct a complete GARCH model, we need to estimate the best fitting model for both

mean equation and variance equation.

For mean equation, the primary problem is the determination of the lag length. In order to

find out the best length, we resort to Akaike information criterion (AIC) and Bayesian

information criterion (BIC) and choose the model with the lowest AIC and BIC values.

Note that (7) assumes a constant conditional mean of the time series variable, which may

not be true in practice, especially for stock prices. As we all can see, returns of financial assets

are closely correlated with investment risks (i.e. volatility of assets’ prices). Thus, conditional

mean might be based on volatility. To deal with this problem, the GARCH-in-Mean (GARCH-

M) model adds a heteroskedasticity term into the mean equation, so that mean equation

becomes

𝑦" = 𝑥"I𝑏 + 𝜆𝜎" + 𝜀" (10)

where 𝜎" can be replaced by other variance terms like 𝜎"F or 𝑙𝑜𝑔(𝜎"F). When 𝜆 is statistically

significant and larger than 0, we can argue that a higher risk will contribute to a correspondingly

higher return of financial assets.

For variance equation, we need to take different effects - symmetric or asymmetric - of

variance into consideration. Note that the normal variance equation given by (9) shows a

symmetric effect. In reality, however, it is often observed that a downtrend in the stock markets

is more likely to be followed by higher volatility than an uptrend, with all other things being

equal. This is the asymmetric case we need to account for. To address the asymmetry, there are

two methods we can employ, that is, T-GARCH model and E-GARCH model. Both of them

make correction on variance equation by adding some terms to include the impact of the sign

of the past residuals. Specifically, for a T-GARCH (1, 1) model, variance equation becomes

𝜎"F = 𝜔 + 𝛼𝜀"(.F + 𝛾𝐷"(.𝜀"(.F + 𝛽σ"(.F (11)

where 𝐷"(. = 1 for 𝜀"(. < 0 and 𝐷"(. = 0 otherwise. When 𝛾 is statistically significant and

larger than 0, we can argue that a negative shock (i.e.𝜀"(. < 0) has a greater (𝛼 + 𝛾 verses 𝛼)

impact on volatility.

And for a E-GARCH (1, 1) model, the variance equation is specified as

𝑙𝑜𝑔(𝜎"F) = 𝜔 + 𝛿.R%S'

M%S'N+ 𝛿F

R%S'

M%S'N+ 𝛽𝑙𝑜𝑔(𝜎"(.F ) (12)

which will show asymmetry if 𝛿F ≠ 0. When 𝛿F is statistically significant and less than 0, we

can argue that a negative shock (i.e.𝜀"(. < 0) has a greater ( aN

M%S'N𝜀"(. > 0 when 𝜀"(. < 0)

impact on volatility.

Besides, all the GARCH model estimations in the paper employ maximum likelihood

procedure and are conducted using EViews 9 Student Version.

4 Empirical results

4.1 Stationarity and unit root tests

4.1.1 Test in levels

It is necessary to test for the stationarity of data first, which is the fundamental issue that

underlies our estimation model and analysis. Non-stationarity tends to have a strong impact

on a time series variable’s behavior and properties, making it hard to model by econometric

theories. And if two non-stationary variables are trending over time, a regression of one of

the other could have a high R-squared even if the two are totally unrelated. Stationarity is

then the key to prevent this spurious regression and its unreliable results. More importantly,

the standard assumptions for asymptotic analysis will not be valid in the case that the

variables in the regression model are not stationary. In other words, the usual t-ratios will not

follow a t-distribution, so we cannot validly perform hypothesis tests about the regression

parameters.

We now apply two commonly used unit root tests - augmented Dickey-Fuller test (ADF

test) and Phillips and Perron test (PP test) – to check the stationarity of stock price. The stock

price quotes are transformed in the logarithm patterns. In order to show the distinctions, we

test the stock price in levels and then in first differences to make a comparison.

Table 2 covers the statistics of ADF test and PP test of stock price data in levels.

Table 2: Unit Root Statistics: Levels

Countries

(logP)

Augmented Dickey-Fuller test Phillips and Perron test

Without

Intercept

and

Trend

With

Intercept

With

Intercept

and

Trend

Without

Intercept

and

Trend

With

Intercept

With

Intercept

and

Trend

US 1.408674 -0.982054 -3.400660 1.785238 -0.820824 -3.073463

Brazil -0.770953 -2.504864 -3.541492* -0.827843 -2.437436 -3.534302*

Russia -1.135369 -0.985595 -3.022216 -1.100477 -1.050914 -3.134209

India 0.684869 -0.776499 -3.047777 0.651552 -0.729360 -2.980023

China 0.476748 -0.705285 -1.291054 0.452012 -0.744191 -1.344911

South

Africa -0.487342 -3.678861** -3.693273* -0.536814 -3.738057** -3.775949*

Note:

i. * implies that the null hypothesis of the existence of a unit root is rejected at a 5%

significance level.

ii. ** implies that the null hypothesis of the existence of a unit root is rejected at a 1%

significance level.

From the chart, we hardly observe significance in the unit root test statistics among the six

countries’ stock price indices in levels. Unit root does exist in stock market performance of

the United States, Russia, India and China, no matter the auxiliary regression is run with or

without intercept and trend. This means that the stock price is non-stationary for these

countries, thus we cannot perform valid estimation models based on the price data in levels.

For the other two countries, i.e. Brazil and South Africa, we find statistically significance

only in some specific occasions. For instance, ADF test statistic for South Africa’s stock

price is significant at 1% level when the auxiliary regression model contains a constant

intercept.

4.1.2 Test in first differences

To continue on our research, we need to consider the stock price data in first differences

instead of that in levels. The results are collected as table 3.

Table 3: Unit Root Statistics: First differences

Countries

(logP)

Augmented Dickey-Fuller test Phillips and Perron test

Without

Intercept

and Trend

With

Intercept

With

Intercept

and Trend

Without

Intercept

and Trend

With

Intercept

With

Intercept

and Trend

US -36.73693*** -36.78577*** -36.77106*** -37.37899*** -37.72022*** -37.70187***

Brazil -35.42399*** -35.42609*** -35.41192*** -35.49195*** -35.51751*** -35.50268***

Russia -32.55357*** -32.57079*** -32.56199*** -32.53130*** -32.57368*** -32.56490***

India -32.12671*** -32.12532*** -32.12651*** -32.12902*** -32.11398*** -32.11474***

China -25.17208*** -25.16898*** -25.22167*** -31.59249*** -31.58491*** -31.63648***

South

Africa -38.76171*** -38.75127*** -38.74059*** -39.77776*** -39.77663*** -39.77171***

Note:

i. * implies that the null hypothesis of the existence of a unit root is rejected at a 5% significance

level.

ii. ** implies that the null hypothesis of the existence of a unit root is rejected at a 1% significance

level.

iii. *** implies that the null hypothesis of the existence of a unit root is rejected at a 1‰ significance

level.

Table 2 demonstrates a common statistically significance at 1‰ level in the unit root test

statistics for all countries and auxiliary regression models. The results lay the foundation for

the following analysis based on the stock price data in first differences. Note that first

differences in the stock prices are defined as the stock returns (see “Data” section), therefore,

the stock returns data is stationary. As a result, we can estimate the stock returns with

empirical models, test for coefficients’ significance and make justifiable statements.

4.2 GARCH models

4.2.1 Lag length selection

To undertake a GARCH estimation, we need to create a best fitting model for the mean

equation at the very first stage, i.e. to select a proper lag length for the stock returns. As is

mentioned before, we conform to Akaike information criterion (AIC) rule and Bayesian

information criterion (BIC) rule. Using US stock returns data for an example, we run

GARCH (1, 1) models with lag lengths from 1 to 10 and obtain AIC and BIC values as

below:

Table 4: Lag length selection for US stock returns GARCH(1,1) estimation

Lag length Akaike Information criterion (AIC) Bayesian information criterion (BIC)

1 2.460919*** 2.481536***

2 2.464273 2.489029

3 2.467130 2.496030

4 2.469149 2.502200

5 2.464367 2.501573

6 2.467886 2.509253

7 2.470330 2.515864

8 2.473486 2.523191

9 2.474429 2.528312

10 2.477127 2.535192

Note: *** denotes the lowest value in all lengths.

Table 4 shows that the model with lag 1 provides the best fitting performance. In that case,

we should choose AR (1) to construct mean equation of the GARCH model for US data.

Similarly, we can prove that lag 1 is preferred for the rest of the countries’ stock returns data.

4.2.2 GARCH

In the previous part, we assume that the GARCH (1, 1) is the best setting for the GARCH

model. Now there is a need to check whether the order of ARCH/GARCH terms is best to be

set both at 1.

The original GARCH (1, 1) estimation results in figure 2:

Figure 2: US stock returns GARCH (1, 1) estimation results

By comparison, results of GARCH (1, 2) and GARCH (2, 1) are listed as follows:



From figure 3 we notice that coefficient of GARCH (-2) term is not statistically significant

at 5% level. While figure 4 shows that the coefficients of all variance equation terms are

statistically significant, the Schwarz criterion value is higher than that in GARCH (1, 1)

model nonetheless. To sum up, we abandon these two model structures (i.e. leave alone

higher orders of ARCH/GARCH terms in variance equation) and focus on GARCH (1, 1)

instead. The process applies for the other five countries’ cases as well.

4.2.3 T-GARCH and E-GARCH

To demonstrate the asymmetric effect of past variance, we should use T-GARCH model

and E-GARCH model in the estimation. Still using US stock returns for an example, we

derive the following T-GARCH (1, 1) results and E-GARCH (1, 1) results:

Figure 5: US stock returns T-GARCH (1, 1) estimation results

Figure 6: US stock returns E-GARCH (1, 1) estimation results

Both figure 5 and figure 6 show statistic significance of all the variables in variance

equation. Note that the coefficient of “RESID(-1)^2*(RESID(-1)<0)” (in figure 5) is positive

(0.348143) and the coefficient “C(5)” (in figure 6) is negative, we can claim that the previous

negative shock (𝜀"(. < 0) will have a greater impact on the volatility of stock returns,

making the investment in stock market more risky. This result conforms to the empirical

theory as well as the realistic market performances.

Following the same procedure, we construct T-GARCH and E-GARCH models for the

other five countries and obtain results respectively. The result is listed in the next section

after we discuss about GARCH-in-Mean model.

4.2.4 GARCH-in-Mean

Finally, the study considers a modification in the mean equation of a standard GARCH

model – to include a standard deviation or variance term. This modification is important in

the analysis of stock returns as the stock prices are commonly expected to contain a risk

premium. A higher risk usually leads to a higher price for a financial asset (e.g. stock equity).

This information should also be included in our estimation model. In that case, a GARCH-in-

Mean is required to test the stock returns data.

Figure 7: US stock returns GARCH-in-Mean (1, 1) estimation results

As figure 7 shows, the standard deviation term (“@SQRT(GARCH)”) is significant at 5%

level. The positive coefficient tells that a higher risk tends to increase the stock returns, which

matches our expectation about the model and also fits reality.

Applying the method to the other countries and then GARCH models gives the following

results:

Table 5: Summary of statistic significance of different GARCH models

Countries

GARCH T-GARCH E-GARCH

Best fitting

model GARCH GARCH-

in-Mean

T-

GARCH

T-

GARCH-

in-Mean

E-

GARCH

E-

GARCH-

in-Mean

US *** * *** - *** - E-GARCH

Brazil *** - - - *** - E-GARCH

Russia *** - - - *** - E-GARCH

India *** - - - *** - E-GARCH

China *** - - - - - GARCH

South

Africa *** * - ** *** **

E-GARCH-

in-Mean

Note:

i. * implies that the coefficients in variance equation/coefficient of GARCH-in-Mean term are

statistically significant at 5% level.

ii. ** implies that the coefficients in variance equation/coefficient of GARCH-in-Mean term are

statistically significant at 1% level.

iii. *** implies that the coefficients in variance equation/coefficient of GARCH-in-Mean term

are statistically significant at 1‰ level.

iv. - implies that the coefficients in variance equation/coefficient of GARCH-in-Mean term are

not statistically significant at 5% level or lower.

v. All the GARCH models listed here consider 1 order for both ARCH and GARCH terms.

That is, they are either GARCH(1,1), T-GARCH(1,1), E-GARCH(1,1), or GARCH-in-

Mean(1,1), T-GARCH-in-Mean(1,1), E-GARCH-in-Mean(1,1).

From table 5, we know that almost all GARCH and E-GARCH models are valid to

estimate the stock returns data while T-GARCH and GARCH-in-Mean models only applies

to a few countries. Among the statistically significant model options for each country, we

intend to find out the best model using the AIC and BIC rules again. It turns out that GARCH

(1, 1) is the best fit for China and E-GARCH-in-Mean (1, 1) for South Africa. For the other

four countries, E-GARCH (1, 1) is the best fitting model. Therefore, we observe a common

asymmetric effect of previous shock on the volatility.

5 Concluding remarks

This paper explores the volatility of stock markets for six countries, US and BRICS. It

attempts to determine whether the volatility is symmetric or asymmetric, and whether the

volatility should be included to estimate the stock returns. Intuitively, from the trend figures

of stock prices and returns for the six countries, we may put that stock markets of BRICS are

typically more volatile than US, with Russian and South African markets being the most

unstable. It is easy to observe that the average stock return rate during the recent five years is

highest in the US as well. According to unit root test (ADF test and PP test), we find that the

logarithmic stock prices in levels are nearly non-stationary while the data in first differences

are stationary. We, therefore, can use the first differences of logarithmic stock prices to

calculate and estimate stock returns. The AIC and BIC rules support that AR (1) model

should be employed in building the mean equation of GARCH for all countries’ stock return

estimations. To represent the impact of volatility on the average stock returns, we add

standard deviation (GARCH-in-Mean model) to the mean equation; To explain the impact of

previous shock on the volatility, we add one more variable incorporating the effect of the sign

of the shock to the variance equation (T-GARCH and E-GARCH models). To sum up, the

volatility of stock returns for most of the countries is asymmetric but can only be described

by E-GARCH (1, 1) model. As GARCH-in-Mean model is valid only for US and South

African markets, there is not enough evidence to support that the volatility should be included

when determining the stock price.

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Chittedi, Krishna Reddy. "Global stock markets development and integration: With special

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Panait, Iulian, and Ecaterina Oana Slăvescu. "Using GARCH-in-mean model to investigate

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2012." Theoretical and Applied Economics 5.5 (2012): 55.

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Volatility of stock returns_Yuxiang Ou