2011s4C359 TMA2 Camile a Rowe

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.


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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



.|** | .|** | 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


Date: 07/20/11 Time: 21:01




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






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



.|. | .|. | 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


Date: 08/01/11 Time: 23:00




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








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



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







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.

Documents

2011s4C359 TMA2 Camile a Rowe