Exercise on GARCH and SV models - Common Market...
Transcript of Exercise on GARCH and SV models - Common Market...
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
Sample (adjusted): 2 2515
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
S.E. of regression 0.013760 Akaike info criterion -5.838276
Sum squared resid 0.475805 Schwarz criterion -5.831320
Log likelihood 7341.713 Hannan-Quinn criter. -5.835751
Durbin-Watson stat 2.175792
ARCH(5)
Dependent Variable: RETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/03/14 Time: 16:17
Sample (adjusted): 2 2515
Included observations: 2514 after adjustments
Convergence achieved after 13 iterations
Presample variance: backcast (parameter = 0.7)
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
Adjusted R-squared -0.000990 S.D. dependent var 0.013758
S.E. of regression 0.013765 Akaike info criterion -6.189799
Sum squared resid 0.476140 Schwarz criterion -6.173567
Log likelihood 7787.578 Hannan-Quinn criter. -6.183908
Durbin-Watson stat 2.174259
GARCH(1,1)
Dependent Variable: RETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/03/14 Time: 16:17
Sample (adjusted): 2 2515
Included observations: 2514 after adjustments
Convergence achieved after 10 iterations
Presample variance: backcast (parameter = 0.7)
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
Adjusted R-squared -0.000995 S.D. dependent var 0.013758
S.E. of regression 0.013765 Akaike info criterion -6.246468
Sum squared resid 0.476142 Schwarz criterion -6.237192
Log likelihood 7855.810 Hannan-Quinn criter. -6.243101
Durbin-Watson stat 2.174250
EGARCH(1,1,1)
Dependent Variable: RETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/03/14 Time: 16:18
Sample (adjusted): 2 2515
Included observations: 2514 after adjustments
Convergence achieved after 16 iterations
Presample variance: backcast (parameter = 0.7)
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
Adjusted R-squared -0.000032 S.D. dependent var 0.013758
S.E. of regression 0.013758 Akaike info criterion -6.284429
Sum squared resid 0.475685 Schwarz criterion -6.272835
Log likelihood 7904.527 Hannan-Quinn criter. -6.280221
Durbin-Watson stat 2.176342
GARCH-M(1,1)
Dependent Variable: RETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/03/14 Time: 16:18
Sample (adjusted): 2 2515
Included observations: 2514 after adjustments
Convergence achieved after 17 iterations
Presample variance: backcast (parameter = 0.7)
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
Adjusted R-squared -0.003165 S.D. dependent var 0.013758
S.E. of regression 0.013780 Akaike info criterion -6.245947
Sum squared resid 0.476985 Schwarz criterion -6.234353
Log likelihood 7856.155 Hannan-Quinn criter. -6.241739
Durbin-Watson stat 2.169342
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)
Dependent Variable: RETURN
Method: ML - ARCH (Marquardt) - Student's t distribution
Date: 04/03/14 Time: 16:20
Sample (adjusted): 2 2515
Included observations: 2514 after adjustments
Convergence achieved after 15 iterations
Presample variance: backcast (parameter = 0.7)
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
Adjusted R-squared -0.001635 S.D. dependent var 0.013758
S.E. of regression 0.013769 Akaike info criterion -6.268695
Sum squared resid 0.476447 Schwarz criterion -6.257100
Log likelihood 7884.749 Hannan-Quinn criter. -6.264486
Durbin-Watson stat 2.172860
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?