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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
Recommended