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8/2/2019 2011s4C359 TMA2 Camile a Rowe
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Disclaimer
I hereby declare that the work containedin this paper is entirely my own and that everyattempt has been madeto acknowledge all references and other sources.
8/2/2019 2011s4C359 TMA2 Camile a Rowe
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ASSIGNMENT 2
Full Name: Camile Arlene Rowe
Student Reference Number: 080261603
Module Code: 2011s4C359
Assignment: TMA2
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Part a)
Consider a stochastic process, ty . The process, ty is said to have no serial correlation if
the auto-covariance or equivalently the autocorrelation function is zero for all lags except at a
lag, 0s . To put this mathematically, there is no serial correlation in the process ty if the
following conditions are true:
2 0cov( , )
0 otherwise
ts t t s
sy y
(1.1)
1 0
0 otherwisevar
s
s t
s
y
(1.2)
where s is the autocovariance as a function of the lag s ; s is the autocorrelation function;
cov ,t t sy y is the covariance between ty and t sy and; 2var t ty is the variance of ty .
The Ljung Box Q statistic can be used to test that there is no serial correlation in ty . The
Ljung Box Q statistic tests the joint hypothesis that all m , s coefficients are simultaneously
equal to zero where m is the maximum number of lags being considered, that is, it tests the null
hypothesis, 0H given by:
0 1 2
1
: 0
: at least one of 0, 1, ,
m
s
H
H s m
where 1H is the alternative hypothesis. The test is conducted by forming the Q-statistic which
is given by:
2
1
2m
k
k
Q T TT k
(1.3)
If we assume that ty is white noise, that is, normally distributed with mean 0 and variance2 ,
then the sample autocorrelation function, s , is approximately normally distributed with mean
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zero and variance1
Tand the Qstatistic which is the sum of squares of independent standard
normal variates will follow a 2 distribution with m degrees of freedom. At confidence level,
the null-hypothesis may be rejected if 21Q , that is, if the Q-statistic is greater than the
critical value at the 1 confidence interval.
The Q-statistic will be used here to test whether there is serial correlation in the weekly
returns for the shares of Barclays. The data used is the historical weekly share prices between
January 1, 2003 and March 9, 2011. From this data, the weekly returns are found by using the
formula:
1
1
t tt
t
P Pr
P
(1.4)
We will assume that the return is described by the following:
t tr u (1.5)
where is the mean and tu is a zero mean white noise residual. The plot and distribution of tr
is shown in Figure 1.1 and 1.2 respectively. The fact that a large return is followed by large return
in the following period does indicate that there is indeed serial correlation in the returns. In
Figure 1.2, the large value of the Jarque-Bera test statistic for tr indicates that the assumption of
normality of the returns does not really hold but nonetheless, we will proceed with the Ljung
Box test for serial correlation.
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Figure 1.1: Plot of the Weekly Returns of Barclays shares
Figure 1.2: Distribution of weekly returns for Barclays shares.
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
50 100 150 200 250 300 350 400
BARCLAYS_RETURNS
0
40
80
120
160
200
-0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Series: BARCLAYS_RETURNSSample 1 428Observations 427
Mean 0.003293Median 0.000629Maximum 1.072266Minimum -0.477551Std. Dev. 0.093372Skewness 3.748485Kurtosis 52.14457
Jarque-Bera 43970.20
Probability 0.000000
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The Q-statistic for various lag lengths can be found in Eviews by using the correlogram
function. The autocorrelation and partial autocorrelation functions along with the Qstatistic
and associated p-values for various values of lag (up to 36 lags) are shown in Figure 1.2 below.
For a lag length of k in the table, the Qstatistic follows a 2( )k (a chi-squared distribution with
k degrees of freedom).
As indicated by the p-values for the Qstatistics, for all lag lengths (except 1s and 5s ),
the null hypothesis of no autocorrelation may be rejected at the 1% confidence level. This
indicates that there is serial correlation in the returns for Barclays shares.
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Date: 07/20/11 Time: 12:47
Sample: 1 428
Included observations: 427
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
*|. | *|. | 1 -0.077 -0.077 2.5433 0.111
*|. | *|. | 2 -0.151 -0.158 12.400 0.002
.|. | .|. | 3 0.064 0.040 14.179 0.003
.|. | .|. | 4 0.009 -0.006 14.213 0.007
.|. | .|. | 5 -0.030 -0.015 14.614 0.012
.|* | .|* | 6 0.115 0.113 20.347 0.002
.|. | .|* | 7 0.065 0.080 22.204 0.002
.|* | .|* | 8 0.087 0.143 25.529 0.001
.|. | .|. | 9 -0.060 -0.030 27.133 0.001
*|. | *|. | 10 -0.089 -0.076 30.576 0.001
.|. | .|. | 11 0.048 0.010 31.609 0.001
.|. | *|. | 12 -0.059 -0.098 33.167 0.001
.|. | .|. | 13 0.030 0.020 33.565 0.001
.|. | *|. | 14 -0.024 -0.084 33.825 0.002
.|. | .|. | 15 0.059 0.066 35.395 0.002
.|. | .|. | 16 -0.030 -0.017 35.787 0.003
.|. | .|. | 17 0.024 0.066 36.050 0.005
*|. | *|. | 18 -0.106 -0.085 41.061 0.001
.|. | .|. | 19 0.070 0.064 43.286 0.001
.|. | .|. | 20 0.056 0.059 44.716 0.001
.|. | .|. | 21 -0.022 -0.003 44.941 0.002
.|. | .|* | 22 0.069 0.087 47.084 0.001
*|. | *|. | 23 -0.074 -0.109 49.588 0.001
*|. | *|. | 24 -0.101 -0.082 54.238 0.000
.|. | .|. | 25 0.007 -0.050 54.259 0.001
.|. | *|. | 26 -0.043 -0.100 55.098 0.001
.|. | .|. | 27 0.015 0.008 55.197 0.001
.|. | .|. | 28 0.071 0.004 57.509 0.001*|. | .|. | 29 -0.068 0.010 59.649 0.001
.|. | .|. | 30 0.005 0.026 59.661 0.001
.|. | .|. | 31 -0.032 0.024 60.130 0.001
.|. | .|. | 32 -0.030 0.010 60.548 0.002
.|. | .|. | 33 -0.008 -0.016 60.577 0.002
.|. | .|. | 34 -0.011 -0.030 60.631 0.003
.|. | .|. | 35 0.014 -0.020 60.725 0.004
.|* | .|. | 36 0.093 0.072 64.795 0.002
Figure 1.3: The Correlogram of the Returns for Barclays Shares
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Part b)
Figure 2.2 and 2.1 below shows the plot and Correlogram for the squared returns for
Barclays shares respectively. The Plot of the squared returns does indicate that there is serial
correlation in the returns since a large squared return in the previous periods seem to cause a
large squared return in the subsequent periods.
The Qstatistics also indicates strong evidence of serial correlation since for all lag
lengths up to lag 36, the Qstatistic is much greater than the critical value of the pertinent 2
distribution.
It is hardly surprising that there is strong evidence of serial correlation in squared
returns for Barclays shares since the square returns provides a measure of the volatility of the
stocks returns and there is some empirical evidence that volatility is autocorrelated. In other
words, empirical evidence suggests that the current level of volatility of a stocks returns tends
to be positively correlated with its levels in periods immediately preceding the current period,
that is, volatility occurs in clusters or bursts. Models such as ARCH and GARCH tries to model
this often observed phenomena in stock behaviour.
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Date: 07/20/11 Time: 13:10
Sample: 1 428
Included observations: 427
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|** | .|** | 1 0.222 0.222 21.164 0.000
.|* | .|* | 2 0.150 0.106 30.866 0.000
.|. | .|. | 3 0.047 -0.007 31.801 0.000
.|. | .|. | 4 -0.001 -0.026 31.801 0.000
.|. | .|. | 5 0.065 0.070 33.608 0.000
.|. | .|. | 6 0.028 0.007 33.957 0.000
.|* | .|* | 7 0.144 0.130 42.998 0.000
.|** | .|** | 8 0.350 0.316 96.548 0.000
.|* | .|. | 9 0.124 -0.027 103.26 0.000
.|. | .|. | 10 0.062 -0.050 104.95 0.000
.|. | .|. | 11 0.063 0.059 106.70 0.000
.|. | .|. | 12 0.008 -0.010 106.73 0.000
.|. | .|. | 13 0.027 -0.021 107.05 0.000
.|. | .|. | 14 0.033 0.034 107.54 0.000
.|. | .|. | 15 0.026 -0.056 107.85 0.000
.|* | .|. | 16 0.141 0.019 116.73 0.000
.|. | .|. | 17 0.011 -0.042 116.78 0.000
.|. | .|. | 18 0.002 -0.025 116.79 0.000
.|. | .|. | 19 0.016 -0.008 116.90 0.000
.|. | .|. | 20 0.005 0.017 116.91 0.000
.|. | .|. | 21 0.009 -0.014 116.94 0.000
.|. | .|. | 22 0.010 -0.001 116.99 0.000
.|. | .|. | 23 0.012 0.009 117.05 0.000
.|. | .|. | 24 0.047 -0.003 118.04 0.000
.|. | .|. | 25 -0.006 -0.004 118.06 0.000
.|. | .|. | 26 -0.003 0.017 118.06 0.000
.|. | .|. | 27 0.037 0.037 118.70 0.000
.|. | .|. | 28 0.021 0.010 118.90 0.000
.|. | .|. | 29 -0.003 -0.019 118.91 0.000
.|. | .|. | 30 -0.008 -0.014 118.93 0.000
.|. | .|. | 31 -0.004 0.002 118.94 0.000
.|. | .|. | 32 -0.008 -0.027 118.97 0.000
.|. | .|. | 33 -0.005 0.013 118.98 0.000
.|. | .|. | 34 -0.005 -0.002 118.99 0.000
.|. | .|. | 35 -0.006 -0.042 119.01 0.000
.|. | .|. | 36 0.001 -0.006 119.01 0.000
Figure 2.2: The Correlogram of the Squared Returns for Barclays Shares
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Figure 2.2: The Plot ofSquared Returns for Barclays Shares
0.0
0.2
0.4
0.6
0.8
1.0
1.2
50 100 150 200 250 300 350 400
RETURNS_SQ
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Part c)
The assumption that the variance of the errors (residuals) in a time series model is
constant, that is, that homoscedacity holds, is often not true for financial time-series data and
thus, more realistic models (than the Classical Linear Regression Model), which does not
assume that the variance is constant is often used to model this type of data. One such class of
models is the ARCH models. An ARCH (q ) model may be described by the following:
1 2 2 3 3
2 2 2 20 1 1 2 2
t t t n nt t
t t t q t q
y x x x u
u u u
(1.6)
where tu is the residual which is assumed to be normally distributed with mean zero and have a
conditional variance 2 1 2var | , ,t t t tu u u . In the ARCH( q ) model, the conditional variance
of the residuals at any point in time is assumed to be dependent upon the weighted sum of up to
lag q squared residuals.
Before estimating an ARCH model, one has to first test the data for ARCH effects. To
formally test for ARCH effects in the returns, we first regress the squared residuals, 2tu , from the
mean equation (1.7) on their lagged values:
t tr u (1.7)
In general, we choose q lags and run the following regression:
2 2 2 20 1 1 1 2t t t q t q tu u u u (1.8)
where t is an error term. ARCH effects are present if the null hypothesis 0H given by
0 1 2
1
: 0
: 0 for at least one 1,2, ,
q
k
H
H k q
(1.9)
is rejected. The alternative hypothesis, 1H , is that at least one of the coefficients of the lag
squared residuals is non-zero. The test statistic for ARCH effects is given by 2TR where Tis the
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number of observations and 2R is the coefficient of multiple correlation which is obtained from
the regression results. The test statistic is distributed as a 2 distribution with q degrees of
freedom thus, the null hypothesis may be rejected if the test statistic, 2 21 ( )TR q .
The Eviews results for the regression in (1.8) when 5q is shown in Figure 3.1.
Dependent Variable: U_SQ
Method: Least Squares
Date: 07/20/11 Time: 19:23
Sample (adjusted): 7 428
Included observations: 422 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.005837 0.003055 1.910872 0.0567
U_SQ(-1) 0.203383 0.048900 4.159159 0.0000
U_SQ(-2) 0.112646 0.049865 2.258991 0.0244
U_SQ(-3) -0.012009 0.050167 -0.239386 0.8109
U_SQ(-4) -0.041232 0.049865 -0.826877 0.4088
U_SQ(-5) 0.072366 0.048900 1.479882 0.1397
R-squared 0.067232 Mean dependent var 0.008776
Adjusted R-squared 0.056021 S.D. dependent var 0.062639
S.E. of regression 0.060859 Akaike info criterion -2.746385
Sum squared resid 1.540808 Schwarz criterion -2.688873
Log likelihood 585.4873 Hannan-Quinn criter. -2.723658
F-statistic 5.996891 Durbin-Watson stat 2.000907Prob(F-statistic) 0.000023
Figure 3.1: Regression of squared residuals
From the results, 2 0.0672R and the number of observations included in the regression was
422T , thus, the test statistic, 2 0.0672 422 28.3584TR . At the 1% significance level, the
critical value is 15.1, thus the null hypothesis may be rejected which indicates that there are
ARCH effects in the return series. The Fstatistic for the regression also indicates that the
hypothesis that all the coefficients are jointly equal to zero may be rejected, which further
supports the fact that there are ARCH effects in the data. Using the t statistics, only the first lag
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of the squared residuals is significant in the regression. The test can be automatically carried
out in Eviews as shown by the results in Figure 3.2.
Heteroskedasticity Test: ARCH
F-statistic 5.996891 Prob. F(5,416) 0.0000
Obs*R-squared 28.37193 Prob. Chi-Square(5) 0.0000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 07/20/11 Time: 21:01
Sample (adjusted): 7 428
Included observations: 422 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.005837 0.003055 1.910872 0.0567
RESID^2(-1) 0.203383 0.048900 4.159159 0.0000
RESID^2(-2) 0.112646 0.049865 2.258991 0.0244
RESID^2(-3) -0.012009 0.050167 -0.239386 0.8109
RESID^2(-4) -0.041232 0.049865 -0.826877 0.4088
RESID^2(-5) 0.072366 0.048900 1.479882 0.1397
R-squared 0.067232 Mean dependent var 0.008776
Adjusted R-squared 0.056021 S.D. dependent var 0.062639
S.E. of regression 0.060859 Akaike info criterion -2.746385
Sum squared resid 1.540808 Schwarz criterion -2.688873
Log likelihood 585.4873 Hannan-Quinn criter. -2.723658
F-statistic 5.996891 Durbin-Watson stat 2.000907
Prob(F-statistic) 0.000023
Figure 3.2: Eviews ARCH effects test
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Part d)
A GARCH model is an extension of the ARCH model since the conditional variance is
parameterised to depend not only on lags of the squared error but also on lags of the conditional
variance. In the general case, the conditional variance, t , in a GARCH( ,p q ) model is
dependent upon q lags of the squared error andp lags of the conditional variance as given by:
2 2 2 2 20 1 1 2 2 1 1
2 22 2
t t t q t q t
t p t p
u u u
(1.10)
For a GARCH(1,1), we need to estimate the conditional mean and the condition variance
equations. Since there is some serial correlation in the returns for Barclays shares, we will
assume that the mean equation is an ARMA(2,2) process, that is, we will assume that the mean
equation is given by:
1 1 2 2 1 1 2 2t t t t t tr r r u u u (1.11)
The GARCH(1,1) conditional variance equation is described by:
2 2 20 1 1 1 1t t tu (1.12)
The coefficients of the model may be estimated jointly by using the maximum likelihood
method. Under the normality assumption for the disturbances, the log-likelihood function to
maximise is
2
1 1 2 22
21 1
1 1log 2 log
2 2 2
T Tt t t
tt t t
r r rTL
(1.13)
The model may be estimated in Eviews by choosing the ARCH estimation method. The results
are shown in Figure 4.1. The model derived from this result is given by:
1 2 1 2
2 21 1
0.52268 0.4368 0.3796 0.5750
0.00005 0.17462 0.8261
t t t t t t
t t t
r r r u u u
u
(1.14)
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All the coefficients in the conditional variance equation are statistically significant. The
parameter 1 in (1.12) measures the reaction of the conditional volatility to market shocks, a
large value indicates that the model is sensitive to market shocks. The value obtained for 1 is
0.1746 which indicates that the volatility is fairly sensitive to market shocks. The coefficient 1
in (1.12) indicates the persistence of the volatility, with a high value indicating that the volatility
takes a long while to die out. For the results obtained, 1 0.8261 which indicates that the
volatility is persistent. The sum, 1 1 1 , which indicates that the model is an integrated
GARCH model suggesting that shocks have a permanent impact on volatility.
Dependent Variable: BARCLAYS_RETURNSMethod: ML - ARCH (Marquardt) - Normal distributionDate: 08/02/11 Time: 11:33Sample (adjusted): 4 428Included observations: 425 after adjustmentsConvergence achieved after 13 iterationsMA Backcast: 2 3Presample variance: backcast (parameter = 0.7)GARCH = C(6) + C(7)*RESID(-1)^2 + C(8)*GARCH(-1)
Variable Coefficient Std. Error z-Statistic Prob.
C 0.002050 0.001182 1.733491 0.0830AR(1) -0.522593 0.042835 -12.20000 0.0000
AR(2) 0.436796 0.040640 10.74784 0.0000MA(1) 0.379644 0.005133 73.96126 0.0000MA(2) -0.575017 0.005057 -113.7000 0.0000
Variance Equation
C 5.57E-05 1.86E-05 2.997636 0.0027RESID(-1)^2 0.174637 0.026906 6.490659 0.0000GARCH(-1) 0.826143 0.021025 39.29246 0.0000
R-squared 0.012524 Mean dependent var 0.003530Adjusted R-squared 0.003120 S.D. dependent var 0.093470S.E. of regression 0.093324 Akaike info criterion -3.083305Sum squared resid 3.657906 Schwarz criterion -3.007031Log likelihood 663.2024 Hannan-Quinn criter. -3.053173
F-statistic 0.760976 Durbin-Watson stat 1.889367Prob(F-statistic) 0.620397
Inverted AR Roots .45 -.97Inverted MA Roots .59 -.97
Figure 4.1: Estimation of the GARCH(1,1) model using ML method
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Part e)
The E-GARCH model or the exponential GARCH model may be described by the
following1:
12 2 10 1 1 2 21 1
2ln ln ttt t
t t
uu
(1.15)
In comparison to the GARCH model, the E-GARCH model has some useful features; it ensures
that the conditional variance is always positive without having to impose a restriction on the
coefficients as is required for the GARCH model. In addition, it is useful for detecting
asymmetry in the volatility, that is, it can be used to detect whether negative or positive past
errors have the same impact on volatility.
Estimating the E-GARCH model involves the estimation of the conditional mean
equation for the log return and the conditional variance equation given in(1.15). Before doing the
estimation for the returns on Barclays shares , the Correlogram of the log of the returns for
Barclays shares will be examined. This is shown in Figure 5.1. As indicated by the Qstatistics,
there is evidence of serial correlation in the log returns.
We will use an ARMA(2,2) model for the mean equation:
1 1 2 2 1 1 2 2t t t t t tr r r u u u (1.16)
where the return in this case is the log return. The Eviews test shown in Figure 5.2 for ARCH
effects confirms the presence of ARCH effects in the log returns so the GARCH or one of its
variants is possible a good model to fit the data. The conditional mean equation given in (1.16)
and the conditional variance model in (1.15) will be estimated simultaneously using maximum
likelihood estimation as shown in the results of Figure 5.3.
1There are more than one forms for the E-GARCH
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Date: 07/20/11 Time: 22:49
Sample: 1 428
Included observations: 427
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|. | .|. | 1 -0.058 -0.058 1.4432 0.230
*|. | *|. | 2 -0.155 -0.159 11.790 0.003
.|. | .|. | 3 0.049 0.031 12.839 0.005
.|. | .|. | 4 -0.001 -0.021 12.840 0.012
.|. | .|. | 5 -0.009 0.002 12.875 0.025
.|* | .|* | 6 0.146 0.145 22.141 0.001
.|. | .|. | 7 0.035 0.055 22.672 0.002
.|. | .|. | 8 0.007 0.062 22.696 0.004
*|. | *|. | 9 -0.097 -0.095 26.831 0.001
*|. | *|. | 10 -0.078 -0.090 29.536 0.001.|. | .|. | 11 0.017 -0.032 29.659 0.002
*|. | *|. | 12 -0.081 -0.132 32.538 0.001
.|. | .|. | 13 0.024 0.002 32.787 0.002
.|. | .|. | 14 0.010 -0.028 32.836 0.003
.|* | .|* | 15 0.110 0.169 38.229 0.001
.|. | .|. | 16 -0.026 0.035 38.540 0.001
.|. | .|* | 17 0.006 0.079 38.556 0.002
*|. | *|. | 18 -0.103 -0.093 43.302 0.001
.|. | .|. | 19 0.062 0.038 45.029 0.001
.|. | .|. | 20 0.053 0.008 46.296 0.001
.|. | .|. | 21 0.022 -0.023 46.510 0.001
.|. | .|. | 22 0.061 0.051 48.184 0.001
*|. | *|. | 23 -0.080 -0.101 51.113 0.001
*|. | *|. | 24 -0.120 -0.073 57.708 0.000.|. | .|. | 25 -0.010 -0.050 57.751 0.000
.|. | *|. | 26 -0.045 -0.089 58.671 0.000
.|. | .|. | 27 -0.037 -0.043 59.300 0.000
.|. | .|. | 28 0.050 -0.002 60.430 0.000
*|. | .|. | 29 -0.068 -0.013 62.556 0.000
.|. | .|. | 30 0.004 0.029 62.563 0.000
.|. | .|. | 31 -0.006 0.049 62.580 0.001
.|. | .|. | 32 -0.027 0.005 62.914 0.001
.|. | .|. | 33 -0.007 0.009 62.940 0.001
.|. | .|. | 34 0.011 -0.035 63.001 0.002
.|. | .|. | 35 0.022 -0.018 63.237 0.002
.|* | .|. | 36 0.106 0.057 68.482 0.001
Figure 5.1: The Correlogram of the Log Returns for Barclays Shares
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Heteroskedasticity Test: ARCH
F-statistic 31.62209 Prob. F(5,416) 0.0000
Obs*R-squared 116.2191 Prob. Chi-Square(5) 0.0000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 08/01/11 Time: 23:00
Sample (adjusted): 7 428
Included observations: 422 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.003550 0.001715 2.069671 0.0391
RESID^2(-1) 0.525881 0.048799 10.77647 0.0000
RESID^2(-2) 0.014939 0.055171 0.270783 0.7867RESID^2(-3) -0.076649 0.055046 -1.392449 0.1645
RESID^2(-4) -0.031782 0.055166 -0.576113 0.5649
RESID^2(-5) 0.096629 0.048796 1.980285 0.0483
R-squared 0.275401 Mean dependent var 0.007528
Adjusted R-squared 0.266692 S.D. dependent var 0.039443
S.E. of regression 0.033777 Akaike info criterion -3.923970
Sum squared resid 0.474602 Schwarz criterion -3.866459
Log likelihood 833.9578 Hannan-Quinn criter. -3.901243
F-statistic 31.62209 Durbin-Watson stat 1.997073
Prob(F-statistic) 0.000000
Figure 5.2: The Test Result for ARCH effects
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Dependent Variable: LN_RETURNS
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 07/21/11 Time: 11:06
Sample (adjusted): 4 428
Included observations: 425 after adjustments
Failure to improve Likelihood after 28 iterations
MA Backcast: 2 3
Presample variance: backcast (parameter = 0.7)
LOG(GARCH) = C(6) + C(7)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(8)
*RESID(-1)/@SQRT(GARCH(-1)) + C(9)*LOG(GARCH(-1))
Variable Coefficient Std. Error z-Statistic Prob.
C 0.000141 0.001716 0.082457 0.9343
AR(1) -0.641460 0.070216 -9.135488 0.0000
AR(2) 0.313084 0.075499 4.146858 0.0000
MA(1) 0.517979 0.053567 9.669700 0.0000
MA(2) -0.428954 0.061709 -6.951287 0.0000
Variance Equation
C(6) -0.107045 0.025320 -4.227755 0.0000
C(7) 0.041223 0.022997 1.792504 0.0731
C(8) -0.162136 0.018338 -8.841542 0.0000
C(9) 0.987224 0.003585 275.3548 0.0000
R-squared 0.011197 Mean dependent var -0.000437
Adjusted R-squared 0.001779 S.D. dependent var 0.088910
S.E. of regression 0.088831 Akaike info criterion -3.119243
Sum squared resid 3.314185 Schwarz criterion -3.033434
Log likelihood 671.8391 Hannan-Quinn criter. -3.085344
F-statistic 0.594477 Durbin-Watson stat 1.900367
Prob(F-statistic) 0.782604
Inverted AR Roots .32 -.97
Inverted MA Roots .45 -.96
Figure 5.2: The Estimation of the E-GARCH model
Using the E-views result, the model for the log returns is given by the following:
1 2 1 2
2 2 11 2
1
0.6415 0.3131 0.5180 0.4290
ln 0.1070 0.9872ln 0.1621
t t t t t t
tt t
t
r r r u u uu
(1.17)
where the coefficients which are not statistically significant have been left out of the model.
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The ARMA(2,2) model seems to be a good fit for the conditional return equation since all the
parameters except the intercept are statistically significant. In estimated conditional variance
model , all coefficients except in equation (1.15) were found to be statistically significant. The
value for in (1.15) has a value of -0.1621 which is negative implying that the relationship
between returns and volatility is negative, thus, the impact of negative shocks on volatility is
higher than that of a positive shock.
Part f)
Figure 6.1 shows the Conditional Variance for the log returns. To comment on the graph,
we will compare it to the plot of the log returns which is shown in Figure 6.2. When there is a
large negative value for the returns, one can see that the volatility is very high and volatility has a
tendency to be higher for large negative shocks than for positive shocks. In addition, it is clear
that the volatility is dependent on the magnitude of the returns so that if returns are high, then
volatility will also be high. It also indicates that there is a certain persistence in volatility since
the volatility takes a time to die down after the shock has decreased.
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Figure 6.1: The Conditional Variance for the Log returns
Figure 6.2: The Log returns
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
50 100 150 200 250 300 350 400
Conditional variance
-.8
-.6
-.4
-.2
.0
.2
.4
.6
.8
50 100 150 200 250 300 350 400
LN_RETURNS
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References
Brooks, C. (2008). Introductory Econometrics for Finance. Cambridge University Press, UK.
Gujarati, D., Porter, D.C. (2009). Basic Econometrics. Mc Graw-Hill International Edition,
Singapore.
Fattouh., B. (2011). Financial Econometrics. University of London External Programme,
CEFIMS, SOAS.