TESTING CAPITAL ASSET PRICING MODEL AND VOLATILITY

ABSTRACT

The tests on CAPM have been conducted to test beta, intercept, linearity and the

residual variance. The beta estimates are obtained by taking into account

volatility as usually financial time series data go thorough some phases of

volatility followed by periods of tranquillity. As a result the test for volatility has

also been conducted. Two different data sets were used according to the beta

estimates obtained from EGARCH and GARCH - M. However CAMP does not

hold in either of data sets, as the residual variance is found to affect returns. The

result on beta is inconsistent as a determinant of returns, as one data set

(EGARCH) found no systematic effects, whereas the other data set (GARCH –

M) found to affect returns.

1. INTRODUCTION

Financial researchers have paid considerable attention during the last few years

to the new equity markets that have emerged around the world. The new interest

has been spurred by the large and often extraordinary returns offered by these

markets. Investors all over the world use a plethora of models in their portfolio

selection process and in their attempt to asses the risk exposure to different

assets. One of the most important developments in the modern capital theory is

the Capital Asset Pricing Model (CAPM). CAPM is a financial theory that

describes the relationship between risk and return and serves as model for the

pricing of risky securities. CAPM suggests that high expected returns are

associated with high levels of risk. CAPM postulates that the expected return on

an asset above the risk free rate is linearly related to the non – diversifiable risk

as measured by the assets beta.

The purpose of the paper is to examine weather the CAPM hold for the monthly

data of 12 securities. The following section is the background to the theory of the

CAPM which is followed literature survey on some of the past studies on CAPM.

The following section presents the methodology, results and then concludes.

2. BACKGROUND TO THE THEORY

A fundamental idea of modern finance is that an investor needs a financial

incentive to take a risk. CAPM describes the relationship between risk and

expected return, and serves as a model for the pricing of the risky securities

(Galagedera et al) The CAPM asserts that the only risk that is priced by the

rational investors is systematic risk because the risk cannot be eliminated by

diversification. In its simplest form, the theory predicts that that the expected

return on an asset above the risk free rate is proportional to the non diversifiable,

an asset’s systematic risk referred as beta (β) (Bollerslev, Engle and Wooldridge

1988). According to the CAPM, a stock with a β<1 has less risk than the market

portfolio and therefore has lower expected excess return than the market

portfolio. In contrast a stock with a β>1 is riskier than the portfolio and thus

commands a higher expected excess return (Stock and Watson 2007).

Moreover, in the words of Verbeek (2000), CAPM is an equilibrium model which

assumes that all investors compose their asset portfolios on the basis of trade off

between the expected returns and the variance of the return, given by the beta,

which represents the single risk factor (Eun 1994), on their portfolio, a portfolio

that gives maximum expected return for a given level of risk.

Since its introduction in the early 1960’s, CAPM has been one of the most

challenging topics in financial economics and has provided a simple and

compelling theory of asset market pricing for more than 20 years. Almost any

manager who wants to undertake a project must justify his decision partly based

on CAPM. The model provides for a firm to calculate the return that its investors

demand. The model attempts to show how to assess the risk of the cash flows of

a potential investment project, to estimate projects cost of capital and the

expected rate of return that investors will demand if they are to invest in a project.

3. PREVIOUS WORK

There are numerous research work done by authors to test the beta and return

relationship.

Gürsoy and Rejapova (2007) tested the CAPM in the case Turkish equity

markets by regressing weekly risk premiums on 20 beta portfolios from the

period 1995 to 2004. Research findings by using the Fama and Macbeth

methodology framework found no meaningful relationship between systematic

risk beta and average weekly premiums in order to conclude the validity of the

CAPM.

Tsopoglou, Papanastasiou and Mariola (2006) examined the CAPM for the

Greek securities market using data for 100 stocks listed in the Athens stock

exchange from the period January 1998 to December 2002. The characteristics

line for each stock estimated with EGARCH in order to comfront with

misspecification. In order to improve the precision of the beta estimates the

authors have used portfolio returns and betas. The article found no evidence of

CAPM for the time period examined; the beta for each portfolio wasn’t

significantly different from zero. The model was linear and the residual variance

of each portfolio did not offer explanatory power.

Hin (2002) tested the Sharpe – Litner – Mossin CAPM on Japanese equity

markets using monthly data from the Japanese stock exchange from the period

1952 to 1986. At 5 percent level of significance both the alpha and beta portfolios

were different zero. The author mentioned that the lack of diversification as the

main reason for the empirical invalidity of the CAPM in the Japanese stock

market.

Using the data from Caracas stock exchange, Gonzalez (2001) found evidence

that CAPM should not be used in order to predict the stock returns. To his

findings, he found that beta and returns relationship to be linear and found that

factors other than betas provide explanatory power to predict returns.

Empirical investigation carried out by Sauer and Murphy (1992) found evidence

beta and return relation in German stock markets and found the unconditional

CAPM provide better explanatory power than the conditional CAPM.

Andor, Ormos and Szabó (1999) using a monthly data of 17 securities find that

CAPM acceptably describes the Hungarian Capital market.

The above literature provides a mixed support for CAPM. There have been

numerous reasons that have been levelled at CAPM to indicate its validity. For

instance, one of the arguments put against CAPM is that beta, which represents

the volatility coefficient on stock does not only explain expected returns; there are

firm specific characteristics like size, equity value, leverage ratio etc. Secondly,

one assumption of the CAPM is that the betas of the each individual stock are

time invariant. Empirical evidence on stock returns is based on the argument that

the volatility of the stock returns is constantly changing, hence one must refer to

time varying to conditional mean, variance and covariance that change

depending on the current information. The most widely used methods to estimate

the conditional variance of the stocks is called GARCH (General Autoregressive

Conditional Hetroscedasticity).

The lack of empirical support for CAPM has lead researchers to find and test

alternative theories to examine the beta and return relationship.

However, there have been classical supports for the theories after its

introduction in the early 1960’s. In 1972, Black, Jensen and Scholes1 found

supportive evidence of CAPM using monthly observations. Another classical

empirical study found support of the CAPM was by Fama and Macbeth2.

Moreover, they used the squared beta to test for linearity and also investigated

weather the volatility can explain any cross sectional variations not captured by

the beta alone. (Tsopoglou, Papanastasiou and Mariola 2006).

1 Black, F., Jensen, M.C. and Scholes, M, 1972, The Capital asset pricing model: Some empirical tests, Studies in the theory of Capital Markets, pp. 79 – 21, New York: Praeger. 2 Fama, E.F. and Macbeth, J. 1973, Risk, return and equilibrium: Empirical tests, Journal of Political Economy, 81, pp. 607 – 636.

4. HYPOTHESES AND DATA

HYPOTHESES

The early evidence of the model was largely supportive of CAPM, but recent

findings by researchers have doubted its validity. The following hypotheses have

been formulated in order to test the CAPM for the given data set of 12 stocks:

THE NULL HYPOTHESES

1. Ho: The intercept (gamma [ ]) in the CAPM, ex post security market

line, is not significantly different from zero

2. Ho: There is no significant positive relationship between betas and risk

premiums (excess returns on securities)

3. Ho: There are no non linearity ( ) in the security market line or in the

CAPM equation.

4. Ho: The residual variance is not significant ( ).

DATA

The data for purpose of testing the CAPM consist of a monthly data of individual

12 securities, 90 – day Treasury bills interest rate, and a portfolio or a market

index, from January 2000 to 1st February 2007.

5. METHODOLOGY

Testing the CAPM empirically consist of two stages. In the time series

regression, from the characteristics line estimation, the betas for each individual

security are obtained. The second stage consist the cross sectional regression

which is also called the security market line. In this regression the betas obtained

from the first stage time series regression or from the characteristics line is used

as the independent variable.

5.1 TIME SERIES REGRESSION

Time series are typically studied in the context of homescedastic processes. In

analysis of financial time series data the disturbance variances are less stable

than usually assumed; many financial time series go through occasional periods

of high volatility associated with, say, financial crises (shocks), interspersed with

extended periods of comparative stability. In analysing models of finance, large

and small forecast errors or ‘shocks’ appear to occur in clusters and the

histogram of shocks has fatter tails than would be expected, suggesting a form of

hetroscedasticity in which the variance of the forecast error depends on the size

of the of the preceding disturbance. In other words the variance today is

conditional on the variance in recent periods. Engle has suggested the

Autoregressive Conditional Hetroscedasticity, or ARCH model as an alternative

to the usual time series process (Greene 2000; Stewart 2005) to model to

estimate the conditional variance of stocks and stock index returns.

ARCH accounts for three stylised facts associated with time series of asset

prices and associated returns:

Conditional variances change over time, sometimes quite substantially.

There is volatility clustering – large (small) changes in unpredictable

returns tend to be followed by large (small) changes of either sign.

The unconditional distribution of returns has ‘fat’ tails giving a relatively

large probability of ‘outliers’ relative to the normal distribution.

(Patterson 2000)

There are several ARCH type family models. These ARCH type family models

have found useful in capturing the certain non linear features of financial time

series. In particular they are capable of producing heavy tailed distributions and

clusters of outliers (Cao and Tsay 1992). Hence in order to correct for non linear

ties and obtain accurate estimation of the betas these ARCH type family model is

used whenever there is ARCH effect or presence of volatility clustering.

The most widely used ARCH type family model is GARCH (General

Autoregressive Conditional Hetroscedasticity). Analogous to ARCH, GARCH

avoids the problem setting long lags of the squared error terms in the modelling

of conditional variance (q in ARCH[q] determines the number of lags). GARCH in

the literature is sometimes denoted as GARCH (q, p) process, where p stands for

number of autoregressive terms, and q, the number of moving average or error

term in the model.

5.1.1 GARCH ( q , p ) 1

In their most general form, the univariate GARCH models make the conditional

variance at time t a function of exogenous and lagged endogenous variables,

past residuals and conditional variances, time, parameters. Formally, let ( ) be a

sequence of prediction errors, a vector of parameters, a vector of

exogenous and lagged endogenous variables and , the variance of , given

information at time t,

=

( ) i.i.d with E ( ) = 0, var ( ) = 1

= h ( , , …, , ,…, xt, t,)

1 For more detailed explanation see Enders (2004) or Patterson (2000)

The most widely used GARCH models make h, a linear function of lagged

conditional variances and squared past residuals by defining:

= + + …+ + + …+

From a theoretical point of view, these models present a crucial property:

linearity. This is because they imply an ARMA equation for the squared

innovation process , which allows for a complete study of the distributional

properties of ( ). In additional to an adequate model of dependence volatility,

GARCH models also take into account of the fact that stock returns are fat tailed

(Enders 2004).

One important drawback of the GARCH procedure is that the choice of

quadratic form for the conditional variance has got important consequences as

far as the time paths of the solution processes are concerned. The time paths are

characterized by periods of high volatility (corresponding to high past values of

the error, of any sign) and other periods when it is low. The impact of the past

values on the innovation on the current volatility is only a function of their

magnitude. However this is not true in the financial context. Typically, volatility

tends to be higher after decrease than after an equal increase. These

‘asymmetry’ is another feature of the financial time series. The choice of a

symmetric (quadratic) form for the conditional variance prevents the modelling of

such phenomenon (Rabemananjara and Zakoian 1993). An additional drawback

of the GARCH process is in its incapability to take into account cyclical or any

non – linear behaviour in the volatility (Rabemananjara and Zakoian 1993).

5.1.2 EGARCH The ARCH and the GARCH models cannot capture some of the important

features of the data. The most interesting feature not addressed by these models

is the leverage or asymmetric effects. Statistically, this effect occurs when an

unexpected drop in share price (bad news) increases predictable volatility more

than an unexpected increase in price (good news) of similar magnitude. This

effect suggests that a symmetry constraint on the conditional variance function in

past ’s (shocks) is appropriate. One method proposed to capture such

asymmetric effects is the exponential EGARCH model developed by Nelson in

1991 (Engle and Ng 1993).

The EGARCH (p, q) model is:

ln ( ) = + +

where N (0, 1)

Specifying the function as the logarithm of ensures positivity (so even if the

product on the right hand side is negative, the antilog must be positive). Dividing

the innovations by the conditional standard deviation results in standardised

shocks thus, the effect of these terms depends upon their relative size (Chen and

Kuan 2002). The EGARCH model is asymmetric because the level of is

included with a coefficient . Since this coefficient is typically negative, positive

return shocks generate less volatility then negative return shocks, all else being

equal (Engle and Ng 1993).

5.1.3 GARCH in mean

One extension of the ARCH model is the ARCH – M or ARCH in mean model,

which not only models the hetroscedasticity process, but also includes the

resulting measure of volatility in the regression. At its simplest, the square root of

the conditional variance that is the conditional standard deviation is included in

the regression function (Najand 2002) i.e.

1

An extension to ARCH in Mean is to specify the hetroscedasticity as the GARCH

in mean, and then add the conditional variance or some function of it to the

specification of the mean function. The resulting model is known as GARCH – M.

The above GARCH, EGARCH and GARCH – M2 is used in order to obtain

accurate estimation of the betas of each security and to confront with

misspecification.

The beta was estimated by regressing each stock’s monthly returns against the

market index according to the following equation which is also known as the

characteristics line:

Where,

is the return on stock i (I = 1, 2 …12)is the rate of return on market free interest rate.is the rate of return on market index.

is the beta of stock i is the corresponding random disturbance term in the equation.

is the excess return on each stock, i, which could also be expressed as .

5.2 CROSS SECTION REGRESSION

To test CAPM, empirically, it consists of two stages. First is the time series

regression where the betas are estimated. The estimated betas are used to test 1 In some applications the log of t has also been used. 2 ARCH, GARCH, EGARCH and GARCH – M are estimated using the Maximum likelihood method rather than OLS. For technical details, refer to Greene (2000)

the CAPM equation, which is called the Security Market Line (SML) that plots

the relationship between average returns of all the securities against the

estimated beta. The slope of the line is given by the average market premium

i.e. market returns less the risk free rate of returns. The CAPM equation is a

cross sectional regression of average returns on the estimated beta coefficients.

The relation and thus the CAPM equation to be estimated is as follows:

Where,

is the zero beta rate, the expected return on an asset which has a beta of

zero

is the market price of the risk excess market returns, the risk premium for

bearing one unit of beta risk.

A similar methodology to test CAPM was used by Manjunatha, Mallikarjunappa

and Begum (2007) for their study of CAPM in the Indian securities market.

In order to test for non linearity, the squared beta termed ( = ) is added to

regression.

Finally, in order to examine weather residual variance of each security affects

excess returns, an additional term was included, which represents the non

systematic risk. ( )

Market index as well all the stocks in the data are expressed in logarithmic forms.

The risk free rate, i.e. 90 day TB rate was adjusted to express it in monthly rates.

All the variables are expressed in returns.

6. EMPIRICAL RESULTS

The first part of the methodology required for the estimation of the betas by

regressing the excess individual returns on excess market returns. To obtain

betas estimates, ARCH/GARCH effects will be tested and used in order to

correct for non linearity and obtain accurate estimation of the betas.

6.1 TESTING FOR ARCH/GARCH EFFECTS USING LM TEST

The mechanism to test for ARCH/GARCH effects is to first save the residuals

from an OLS regression. An auxiliary regression is then run by regressing the

squared residuals on lagged variables of squared residuals. A joint significance

test is used using the Lagrange Multiplier (LM) method. If the p value from the chi

square values exceeds the level of significance then the null hypothesis of no

joint significance is rejected (Johnston and DiNardo 1997).

Appendix A6.1 produces the result from the LM test results. The results

indicate that excess stock returns of six stocks, namely, r13 – r16, r19 and r24

does not reject the null at less than 10 percent level of significance. There is

presence of ARCH/GARCH effects in stocks, r17, r18 and r20 – r23, albeit r23

indicates ARCH/GARCH effect at lag 3 at 10 percent level of significance.

For the purpose of informal testing, the volatility of excess monthly returns

(volatility of the series) has been depicted for the stocks r17, r18 and r20 – r23.

The figures show clusters of large positive squared residuals hence showing the

evidence of ARCH effect.

r17

r18

0

.02

.04

.06

0 20 40 60 80Months (2000 - 2007)

Volatility of monthly excess returns0

.1.2

.3.4

0 20 40 60 80Months (2000 - 2007)

Volatility of monthly excess returns

r20

r21

0

.01

.02

.03

.04

.05

0 20 40 60 80Months (2000 - 2007)

Volatility monthly excess returns0

.02

.04

.06

.08

0 20 40 60 80

Months (2000 - 2007)

Volatility of monthly excess returns

r22

r23

0

.00

5.0

1.0

15

.02

0 20 40 60 80Months (2000 - 2007)

Volatility of monthly excess returns0

.00

5.0

1.0

15

.02

.02

5

0 20 40 60 80Months (2000 - 2007)

Volatility of monthly excess returns

6.2 ESTIMATES OF BETA1

EGARCH and GARCH in Mean were used whenever necessary in order to

correct for non linearity and obtain accurate estimates of the betas.

Results from r17 i.e. stock 17 in excess returns were estimated using EGARCH

(1, 1). Results are given in Appendix A6.2. The standard errors reported are semi

robust standard errors, using the log pseudo log likelihood2. However, since the

LM test found evidence of ARCH effects in r17, the asymmetric does not appear

significant by not taking into account the non robust standard errors. The

regression is presented in appendix A6.2.

The beta estimates from stock 18, in excess returns indicated the presence of

EGARCH (2, 2) and GARCH – M (2, 2) effects. The beta and regression results

are produced in the appendix A6.2.

Stock 20, showed evidence of either EGARCH or GARCH – M effects but there

is a presence of GARCH (1, 1) and GARCH (2, 2) effects where the latter is

estimated with semi robust standard errors. The results are reported in A6.2. By

taking a likelihood ratio test it is possible to determine the correct model. By

taking GARCH (2, 2) as the unrestricted model, the likelihood ratio from the

restricted and the unrestricted models are obtained which are 103.6405 and

103.6792 respectively. The numbers of degrees of freedom are 2. Therefore by

using the following test statistic

2 (LRUR - LRR)

and the setting the null hypothesis as the restricted model as the true model, the

value of the test statistic is 0.0774. The critical value at 5% is 5.991 thus not

rejecting the null hypothesis. So, GARCH (1, 1) is chosen.

1 Results of GARCH, EGARCH or GARCH – M which were found to be either insignificant or when failed to converge was not put in this essay for the purpose of space and convenience. The results that were found to be significant using appropriate commands are only reported and mentioned. 2 For details on log pseudo likelihood refer to Greene (2000)

EGARCH (2, 1) and EGARCH (2, 2) estimates for r21 are shown in A6.2. By

using the same procedure of the LR test, the test statistic is 2.3, which at 1 d.f

and 5 % level cannot reject the null. There was no GARCH – M effect on r21,

either the sigma term was not significant and the maximum likelihood failed to

converge.

EGARCH (1, 2) estimates for r22 beta show that the asymmetry term is just

significant at 10 percent. There is however no GARCH – M effects for r22 (A6.2).

EGARCH (2, 2) and GARCH – M (1, 2) effects are evident in stock r23. Using

semi – robust standard errors, at the asymmetric term was significant at 10

percent, indicating presence of asymmetric shocks. The sigma term in the

GARCH – M is not significant at 10 percent, but it is at 15 percent level of

significance. Therefore it has 85 percent probability of committing the type 1

error.

Results for r24 indicate EGARCH (2, 1) and GARCH – M (2, 1) effect, where

the GARCH – M effect is estimated with semi robust standard errors. The results

are printed in A6.2. The sigma term in the GARCH – M is not significant at 5 or at

10 percent but it is significant at 15 percent level.

6.3 CROSS SECTIONAL REGRESSION (SML ESTIMATION)1

From the above estimates two data sets can be obtained for the purpose of

testing CAPM. One data set refers to beta estimates obtained from EGARCH (p,

q). Stocks, r17, r18, r21, r22, r23 and r24 all indicated the presence of

asymmetric effects. Whereas another data set refers to GARCH – M betas which

were found in stocks r18, r23 and r24.

Table 1

1 The results produce t test statistics as the data is small enough to use z or normal distribution.

EGARCH BetasStock Betas er rvr13 1.078633 0.0251805 0.0050329r14 2.153493 0.006106765 0.012697r15 0.465501 0.017616486 0.0029812r16 0.604111 0.018057879 0.0029938r17 0.8869735 0.000480388 0.0051066r18 2.009042 -0.0178997 0.019153r19 0.369359 0.003245031 0.0023071r20 1.004637 0.002575246 0.0056798r21 1.004174 -0.006224486 0.0083853r22 0.1234361 0.007939993 0.0019744r23 0.1951622 0.01544458 0.0030372r24 0.1845129 -0.001812384 0.0019968

Where rv is the residual variance which represents the non systematic risk and er is the average excess returns on each stock or security.

Table 2

GARCH - M BetasStock Betas er rvr13 1.078633 0.0251805 0.0050329r14 2.153493 0.006106765 0.012697r15 0.465501 0.017616486 0.0029812r16 0.604111 0.018057879 0.0029938r17 0.8869735 0.000480388 0.0051066r18 2.811368 -0.0178997 0.0167935r19 0.369359 0.003245031 0.0023071r20 1.004637 0.002575246 0.0056798r21 1.004174 -0.006224486 0.0083853r22 0.1234361 0.007939993 0.0019744r23 0.346241 0.01544458 0.0031282r24 0.2054688 -0.001812384 0.0018572

The CAPM cross sectional regression from the values and betas from table 1 are

produced in Appendix A6.3. In the first cross sectional regression, a regression is

run by regressing the average excess returns of each security on betas and

square of the betas. The result is produced in A6.3.1. The non linearity

assumption of the CAMP is rejected as the square of the beta; bsq is significantly

not different from zero thus not rejecting the null hypothesis i.e. = 0, at 5, 10

and 15 percent level of significance. However, estimated betas, which

represents the assets non diversifiable is not significantly different from zero,

implying that beta does not explain the average excess returns and thus

accepting the null hypothesis that . The result of the F test indicates that

overall the regression is of not a good fit to the data.

In order test weather residual risk affects the average excess returns; an

additional independent variable was added. If CAMP is valid, then the residual

risk or residual variance for each security should not be different from zero and

thus accepting the null that . The regression result is produced in A6.3.2.

The beta square, betas, and intercept remain insignificant at 5 percent, but the

intercept which should not be different from zero cannot reject the null at 10

percent. The residual variance is significant at 5 percent indicating that the non

market, diversifiable risk or idiosyncratic risk strongly influences the expected

return. Based on the diagnostics test, there was no indication of any

misspecification of the model and hetroscedasticity.

In order to test the CAPM using the GARCH – M beta estimates, a cross

sectional regression is run using the square of the beta, bsq, beta and the

intercept. The result is produced in A6.3.3. The model is linear, but both beta and

the intercept value cannot reject the null hypothesis that they are different from

zero. By using the residual variance as the additional explanatory variable

(A6.3.4), the resultant F test statistic becomes significant at 10 percent and both

the beta and the residual variance of each security are significantly different from

zero and thus reject their corresponding null hypothesis i.e. given in section 5.2

at 10 percent level of significance, indicating that there is a presence of both

systematic as well as non systematic risk. Based on the diagnostic tests there

was no evidence of misspecification or the presence of hetroscedasticity as in

both of the cases the null hypothesis of no misspecification and hetroscedasticity

can be rejected.

7. CONCLUSION

The paper examined the validity of the CAPM using data on 12 securities and

tested for their volatility using a family of ARCH type models which found the

presence of volatility effects with asymmetric and in mean effects

The result of this paper indicates that the result on beta which represents the

volatility coefficient in the market for is mixed. Using beta estimates from two

different cross sectional regression; one cross sectional regression from EGACH

estimates of the beta and another from GARCH – M betas, indicates that, beta in

the former regression does not explain expected returns in the market, whereas

the beta in the latter regression provides a better explanatory power to the

investors in making portfolio decisions. The linearity assumption of the CAMP is

not rejected in either of the regressions. However, in both of the regressions,

CAMP is invalid since the residual term which represents the idiosyncratic risk in

the market is found to influence returns in the market. The intercept term which

should not be different from zero is significant in the cross sectional regression

with EGARCH betas, albeit at 10 percent.

The results not in favour of CAPM can arise in possible source. First, forming

portfolio excess returns and measuring portfolio betas can help to diversify the

firm specific part of returns and hence improving the precision of the betas.

Forming portfolios though requires a larger data set.

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APPENDIX

A6.1 TEST FOR ARCH/GARCH EFFECTS

APPENDIX A6.2 BETA ESTIMATES

r17 from EGARCH (1, 1) using robust standard errors

r17, EGARCH (1, 1) estimates using non robust standard errors

r18, EGARCH (2, 2)

r18, GARCH – M (2, 2)

r20, GARCH (1, 1)

r20, GARCH (2, 2) using robust

r21, EGARCH (2, 1) and EGARCH (2, 2)

r22, EGARCH (1, 2)

r23, EGARCH (2, 2) robust

r23, GARCH – M (1, 2)

r24, EGARCH (2, 1)

r24, GARCH – M (2, 1)

APPENDIX A6.3

A.6.3.1

A.6.3.2

A6.3.3

A6.3.4