1. GARCH Models Þættir í fjármálum Verkefni 1-f Bjartur Logi Ye Shen
Egill Helgi Lárusson

2. Data S&P Industrials Index, Daily from 1st Jan 1990 – 29th Sep 2003
3586 observations in total

3. Objectives & Procedures
The objective of this project is to estimate the volatility of S&P Industrials Index by using various GARCH (Generalised Autoregressive Conditional Heteroscedasticity) models.
The models used in this project are GARCH (1,1); ARCH(20); TGARCH (1,1,1); EGARCH (1,1) and GARCH (1,1) with Student’s T distributed residuals
The procedure is divided into three stages, i.e. Pre-estimation analysis; Estimation of parameters by various models; Post-estimation analysis

4. Pre-estimation Analysis
The objective of this procedure is to determine whether GARCH models are appropriate for the analysis of the data
The figures display the sample ACF of the returns based on the assumption that all autocorrelations are zero beyond lag zero.
ACF of squared returns show significant correlation and die out slowly.

5. Pre-estimation Analysis
Ljung-Box-Pierce Q test for autocorrelation: (at 95% confidence)
Under the null hypothesis of no serial correlation, the Q-test statistic is asymptotically Chi-Square distributed
Engle Arch test for heteroscedasticity : (at 95% confidence)
Under the null hypothesis that a time series is a random sequence of Gaussian disturbances (i.e., no heteroscedasticity). This test statistic is also asymptotically Chi-Square distributed

6. Estimation of Parameters
The model: GARCH (1,1)
Parameters estimated by GARCH (1,1)

7. Outcome of GARCH (1,1)
Return, Estimated Volatility, and Innovations (i.e. residuals)

8. Model: Garch (1,1) Forecast volatility for the next 5 days
Standardised residuals:
Unconditional Std of [ε]

9. Post-estimation Analysis
Both returns and residuals show volatility clustering. The sum of GARCH and ARCH parameters is close to 1, i.e the process remains stationary.
Standardised residuals shows no clustering, no correlation, and no heteroscedasticity, which indicates the model has sufficient explanatory power toward the data

10. Further Data Analysis
Looking more carefully at the data, it doesn’t seem to have the normal distribution property, which GARCH (1,1) assumes.
Mean
Variance
1.1102
Std
1.0537
Min
-7.448
Max
5.5728
Skewness
Kurtosis
6.7713

11. Outcome of Student-t GARCH
Model, parameters, estimated volatilty, standardised residuals,ACF of squared standardised residuals, & test results
JBP Q test
Lag
Test statistics
Critical Value
P-value
Test results 10
0.3039 H0 15
0.0776 20
0.0504
Engle ARCH test
Lag
Test statistics
Critical Value
P-value
Test results 10
0.2832 H0 15
0.0828 20
0.0514

12. TGARCH The volatility model of TGARCH Parameters Value v 0.014311
a (ARCH)
g (Leverage)
b (GARCH)

13. LBP Q test & ARCH test
Ljung-Box-Pierce Q test for autocorrelation: (at 95% confidence)
Under the null hypothesis of no serial correlation, the Q-test statistic is asymptotically Chi-Square distributed
Lag
Test statistics
Critical Value
P-value
Test results 10
5.5887
0.8486 H0 15
0.1328 H0 20
0.1308 H0
Engle ARCH test for heteroscedasticity : (at 95% confidence)
Under the null hypothesis that a time series is a random sequence of Gaussian disturbances (i.e., no heteroscedasticity). This test statistic is also asymptotically Chi-Square distributed
Lag
Test statistics
Critical Value
P-value
Test results 10
5.6764
0.8417 H0 15
0.1617 H0 20
0.1824 H0

14. Outcome of TGARCH
Return, Estimated Volatility, and Innovations (i.e. residuals)

15. EGARCH
Empirical studies on financial series have shown that they are characterised by increased conditional variance following negative shocks (bad news).
In EGARCH model, the volatility depends on the signs of lagged residuals
The difference between absolute residuals and expectation of the absolute residuals.
We do not have to restrict the parameters to be positive.

16. EGARCH The volatility model of EGARCH Parameters Value v -0.0668 a b
1.0182 g
0.0848

17. Outcome of EGARCH
Returns, Estimated Volatility, and Standardised residuals

18. ARCH (20) Model
The major problem with ARCH model is that it requires long lag. We experimented with ARCH (1) and ARCH (10). Both models fail to capture the property of serial correlation and heterscedasticity of the data.
Forecast the volatility of five next days (conditional mean of returns is )

19. Outcome of ARCH (20)
Estimated volatility, standardised residuals, ACF of squared standardised residuals, & test results
JBP Q test
Lag
Test statistics
Critical Value
P-value
Test results 10
4.3644
0.9294 H0 15
8.6931
0.893 20
0.8892
Engle ARCH test
Lag
Test statistics
Critical Value
P-value
Test results 10
4.3496
0.9302 H0 15
8.6468
0.8953 20
0.8768

20. Matlab codes (GARCH 1,1) %reading data for S&P industrial
SnP_industrial=xlsread('daily_data.xls','Data');
dates=SnP_industrial(:,1)+datenum('30-Dec-1899');
returns=diff(log(SnP_industrial(:,2)))*100;
returns(3586,1)=0;
plot (dates,SnP_industrial(:,2));
datetick('x',23);
ylabel('S&P Industrial Index');
title('Daily S&P Industrial Index 01 Jan Sep 2003');
figure;
plot (dates,returns);
datetick ('x',23);
ylabel ('S&P Industrial Returns');
title('Daily Return of S&P Industrial Index 01 Jan Sep 2003');
%Preestimation Analysis
autocorr(returns);
title('ACF with Bounds for Raw Return Series');
autocorr(returns.^2);
title('ACF of the Squared Returns');
%Ljung-box-pierce q test for ACF
[H,pValue,Stat,CriticalValue] = lbqtest(returns-mean(returns),[ ]',0.05);
[H pValue Stat CriticalValue]
%LBP q test squared ACF
[H,pValue,Stat,CriticalValue] = lbqtest(returns-mean(returns).^2,[ ]',0.05);
%Heteroscedasticity Engle ARCH test
[H,pValue,Stat,CriticalValue]=archtest(returns-mean(returns),[ ]',0.05);

21. Matlab codes (GARCH 1,1) %Estimating parameters GARCH (1,1)
[coeff,errors,LLF,innovations,sigmas,summary] = garchfit(returns);
garchdisp(coeff,errors);
%unconditional unconditional std as h0
h0 = sqrt(coeff.K/(1 - sum([coeff.GARCH(:);coeff.ARCH(:)])))
plot(sigmas), hold('on');
plot([0 size(sigmas,1)],[h0 h0],'red');
title('Fitted Conditional Standard Deviations');
hold('off');
%Forecast variance at time t
hsquared=coeff.K+coeff.GARCH*sigmas(3586,1)^2+coeff.ARCH*innovations(3586,1)^2
%Forecast volatility (std) at time t (single period)
h=sqrt(hsquared)
%Forecast volatililty (5 periods)
[sigmaForecast,meanForecast] = garchpred(coeff,returns,5);
[sigmaForecast,meanForecast]
%Postestimation Analysis
garchplot(innovations,sigmas,returns);
figure;
%standardised innovations
stdresiduals=returns./sigmas;
mean(stdresiduals);
std(stdresiduals);
hist(stdresiduals,100);
plot(stdresiduals);
ylabel('Innovation');
title('Standardized Innovations');
%ACF of standardised innovations
autocorr((stdresiduals).^2)
title('ACF of the Squared Standardized Innovations');
%LBP q test and Engle ARCH test on standardised innovations
[H, pValue,Stat,CriticalValue] = lbqtest((stdresiduals).^2,[ ]',0.05);
[H pValue Stat CriticalValue]
[H, pValue, Stat, CriticalValue] = archtest(stdresiduals,[ ]',0.05);

22. Matlab codes ARCH (20) %Estimating parameters ARCH (20)
spec=garchset('P',0,'Q',20)
[coeff,errors,LLF,innovations,sigmas,summary] = garchfit(spec,returns);
garchdisp(coeff,errors);
%Forecast volatililty (5 periods)
[sigmaForecast,meanForecast] = garchpred(coeff,returns,5);
[sigmaForecast,meanForecast]
%Postestimation Analysis
garchplot(innovations,sigmas,returns);
figure;
%standardised innovations
stdresiduals=returns./sigmas;
mean(stdresiduals);
std(stdresiduals);
hist(stdresiduals,100);
plot(stdresiduals);
ylabel('Innovation');
title('Standardized Innovations');
%ACF of standardised innovations
autocorr((stdresiduals).^2)
title('ACF of the Squared Standardized Innovations');
%LBP q test and Engle ARCH test on standardised innovations
[H, pValue,Stat,CriticalValue] = lbqtest((stdresiduals).^2,[ ]',0.05);
[H pValue Stat CriticalValue]
[H, pValue, Stat, CriticalValue] = archtest(stdresiduals,[ ]',0.05);

23. Matlab codes student-t GARCH
[parameters, likelihood, stderrors, robustSE, ht, scores] = fattailed_garch(returns, 1, 1,'STUDENTST');
subplot(3,1,1);plot(1:3586,returns);legend('Data');
subplot(3,1,2);plot(1:3586,sqrt(ht));legend('Estimated Std Dev');
subplot(3,1,3);plot(1:3586,returns./sqrt(ht));legend('Standardized Residuals');
stdresiduals=returns./sqrt(ht);
hist(stdresiduals,100);
plot(stdresiduals);
ylabel('Innovation');
title('Standardized Innovations');
figure;
%ACF of standardised innovations
autocorr((stdresiduals).^2)
title('ACF of the Squared Standardized Innovations');
%LBP q test and Engle ARCH test on standardised innovations
[H, pValue,Stat,CriticalValue] = lbqtest((stdresiduals).^2,[ ]',0.05);
[H pValue Stat CriticalValue]
[H, pValue, Stat, CriticalValue] = archtest(stdresiduals,[ ]',0.05);

24. Matlab codes TGARCH & EGARCH
[parameters, likelihood, stderrors, robustSE, ht, scores]=egarch(returns,1,1,'NORMAL');
subplot(3,1,1);plot(1:3586,returns);legend('Data');
subplot(3,1,2);plot(1:3586,sqrt(ht));legend('Estimated Std Dev');
subplot(3,1,3);plot(1:3586,returns./sqrt(ht));legend('Standardized Residuals');
stdresiduals=returns./sqrt(ht);
hist(stdresiduals,100);
plot(stdresiduals);
ylabel('Innovation');
title('Standardized Innovations');
figure;
%ACF of standardised innovations
autocorr((stdresiduals).^2)
title('ACF of the Squared Standardized Innovations');
%LBP q test and Engle ARCH test on standardised innovations
[H, pValue,Stat,CriticalValue] = lbqtest((stdresiduals).^2,[ ]',0.05);
[H pValue Stat CriticalValue]
[H, pValue, Stat, CriticalValue] = archtest(stdresiduals,[ ]',0.05);
*****************************************************************************************************************
TGARCH
[parameters, likelihood, ht, stderrors, robustSE, scores] = tarch(returns,1, 1, 1);