exercise on garch and sv models - comesa monetary … · s.e. of regression 0.013758 akaike info...

Exercise on GARCH and SV models

Introduction

This exercise illustrates the application of univariate (G)ARCH-type models in the analysis of

conditional stock return volatility. Models in this class assume that the conditional variance at time t

depends on past errors and past variances. As such, in these models, the (conditional) variance at

time t is expected to be higher when past errors and (conditional) variances are higher and vice

versa.

Data

Daily data on the SP500 Composite Index is used for the January 2001 – Dec 2010 period. The eviews

file GARCH_EXERCISE.wf1 contains the data including 2515 observations. The log price and return

data series are plotted below.

Question 1: Descriptive Analysis

a) Construct the return series from the prices series (Hint: take log-difference of prices)

b) Provide descriptive analysis of the return data with regard to (i) mean, (ii) historical volatility,

(iii) skewness, (iv) kurtosis, (v) normality.

c) How do the results change in the subsample 2007-2010?

0

200

400

600

800

1,000

-0.10 -0.05 0.00 0.05 0.10

Series: RETURN

Sample 1 2515

Observations 2514

Mean -8.02e-06

Median 0.000638

Maximum 0.109572

Minimum -0.094695

Std. Dev. 0.013758

Skewness -0.122743

Kurtosis 11.18776

Jarque-Bera 7028.697

Probability 0.000000

d)

Question 2: Test for ARCH(1) effects

a) Method 1: Run an AR(1) on the return series, save the residuals, define the squares of the

residuals as a new variable and run an AR(1) on this new series.

b) Method 2: Run an AR(1) on the return series, and test directly for heteroscedasticity in the

residuals by clicking View/Residual Diagnostics/Heteroskedasticity Tests/ARCH

Heteroskedasticity Test: ARCH F-statistic 83.05870 Prob. F(1,2510) 0.0000

Obs*R-squared 80.46230 Prob. Chi-Square(1) 0.0000

0

50

100

150

200

250

300

350

400

-0.10 -0.05 0.00 0.05 0.10

Series: RETURNSample 1509 2515Observations 1007

Mean -0.000118Median 0.000836Maximum 0.109572Minimum -0.094695Std. Dev. 0.017309Skewness -0.200235Kurtosis 9.849646

Jarque-Bera 1975.316Probability 0.000000

Test Equation:

Dependent Variable: RESID^2

Method: Least Squares

Date: 04/03/14 Time: 16:11

Sample (adjusted): 4 2515

Included observations: 2512 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 0.000153 1.23E-05 12.49612 0.0000

RESID^2(-1) 0.178974 0.019638 9.113655 0.0000 R-squared 0.032031 Mean dependent var 0.000187

Adjusted R-squared 0.031646 S.D. dependent var 0.000597

S.E. of regression 0.000587 Akaike info criterion -12.04195

Sum squared resid 0.000865 Schwarz criterion -12.03731

Log likelihood 15126.69 Hannan-Quinn criter. -12.04027

F-statistic 83.05870 Durbin-Watson stat 2.134020

Prob(F-statistic) 0.000000

Question 3: Modelling conditional volatility

a) Estimate the following models on the return data: ARCH(1), ARCH(5), GARCH(1,1),

EGARCH(1,1,1), GARCH-M(1,1)

ARCH(1)

Dependent Variable: RETURN

Method: ML - ARCH (Marquardt) - Normal distribution

Date: 04/03/14 Time: 16:16


Included observations: 2514 after adjustments

Convergence achieved after 8 iterations

Presample variance: backcast (parameter = 0.7)

GARCH = C(2) + C(3)*RESID(-1)^2 Variable Coefficient Std. Error z-Statistic Prob. C 0.000224 0.000204 1.099632 0.2715 Variance Equation C 0.000131 2.46E-06 53.32080 0.0000

RESID(-1)^2 0.342038 0.024495 13.96354 0.0000 R-squared -0.000285 Mean dependent var -8.02E-06

Adjusted R-squared -0.000285 S.D. dependent var 0.013758




Durbin-Watson stat 2.175792

ARCH(5)



Date: 04/03/14 Time: 16:17





GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*RESID(-2)^2 + C(5)*RESID(-3)^2

+ C(6)*RESID(-4)^2 + C(7)*RESID(-5)^2 Variable Coefficient Std. Error z-Statistic Prob. C 0.000425 0.000177 2.405071 0.0162 Variance Equation C 3.62E-05 1.87E-06 19.38984 0.0000

RESID(-1)^2 0.041527 0.010957 3.790140 0.0002

RESID(-2)^2 0.164288 0.023377 7.027743 0.0000

RESID(-3)^2 0.210796 0.023269 9.059021 0.0000

RESID(-4)^2 0.188423 0.022649 8.319397 0.0000

RESID(-5)^2 0.199958 0.022338 8.951442 0.0000 R-squared -0.000990 Mean dependent var -8.02E-06






GARCH(1,1)



Date: 04/03/14 Time: 16:17





GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob. C 0.000426 0.000182 2.334859 0.0196 Variance Equation C 1.34E-06 2.16E-07 6.202713 0.0000

RESID(-1)^2 0.082621 0.008413 9.820833 0.0000

GARCH(-1) 0.908036 0.008754 103.7240 0.0000 R-squared -0.000995 Mean dependent var -8.02E-06






EGARCH(1,1,1)



Date: 04/03/14 Time: 16:18





LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(4)

*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1)) Variable Coefficient Std. Error z-Statistic Prob. C 7.01E-05 0.000172 0.407499 0.6836

Variance Equation C(2) -0.221327 0.022741 -9.732645 0.0000

C(3) 0.100859 0.013183 7.650565 0.0000

C(4) -0.120621 0.009629 -12.52645 0.0000

C(5) 0.984456 0.001895 519.6292 0.0000 R-squared -0.000032 Mean dependent var -8.02E-06






GARCH-M(1,1)



Date: 04/03/14 Time: 16:18





GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob. @SQRT(GARCH) 0.044968 0.052918 0.849766 0.3955

C 4.38E-05 0.000494 0.088711 0.9293 Variance Equation C 1.34E-06 2.22E-07 6.045856 0.0000

RESID(-1)^2 0.082759 0.008465 9.776294 0.0000

GARCH(-1) 0.907894 0.008831 102.8046 0.0000 R-squared -0.002766 Mean dependent var -8.02E-06






b) Interpret the parameters

c) Do you find evidence for the leverage effect? (Hint: check the corresponding EGARCH

parameter sign)

d) Which model is supported by the data most based on the log-likelihood?

e) Calculate the static and dynamic forecast of conditional volatility for each of the models

Question 4: Departure from Normal distribution

a) Change the distributional assumptions from normal to student t and redo question 3.

GARCH(1,1)


Method: ML - ARCH (Marquardt) - Student's t distribution

Date: 04/03/14 Time: 16:20





GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob. C 0.000548 0.000171 3.198874 0.0014 Variance Equation C 9.25E-07 3.23E-07 2.861678 0.0042

RESID(-1)^2 0.084385 0.011247 7.503022 0.0000

GARCH(-1) 0.911630 0.010564 86.29198 0.0000 T-DIST. DOF 8.484451 1.361261 6.232786 0.0000 R-squared -0.001635 Mean dependent var -8.02E-06






b) How do the parameter estimates change?

c) Is the change of distribution supported by the data? (Hint: Check the log-likelihood)

Question 5: Stochastic volatility

a) Estimate the basic stochastic volatility model in eviews. (Hint: use the SV eviews program.)

b) How does the model compare to a GARCH(1,1) model?

exercise on garch and sv models - comesa monetary … · s.e. of regression 0.013758 akaike info...

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