17/01/2023, 19:13
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
09 November, 2022
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Statistical properties of nancial returns and conditional volatility models
Download prices for Tesla stock and convert into returns
Check for Fat Tails
1. QQ Plot
17/01/2023, 19:13
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
## [1] 1934 88
2. Histogram
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Statistical properties of financial returns and conditional volatility models
3. JB Test
## Test
## data: log_returns_demean
## X-squared = 2856.3, df = 2, p-value < 2.2e-16
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
## [1] "Critical Value = 5.99146454710798"
17/01/2023, 19:13 HW1
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
## [1] "H0: JB = 0 is rejected."
Statistical properties of financial returns and conditional volatility models
Check for Volatility Clusters
1. ACF plot on returns squared
2. LB Test
17/01/2023, 19:13 HW1
## Box-Ljung test
## data: log_returns_demean^2
## X-squared = 55.244, df = 1, p-value = 1.065e-13
## [1] "Critical Value is: 3.84145882069412"
## Box-Ljung test
## data: log_returns_demean^2
## X-squared = 276.29, df = 10, p-value < 2.2e-16
## [1] "Critical Value is: 18.3070380532751"
## Box-Ljung test
## data: log_returns_demean^2
## X-squared = 550.12, df = 30, p-value < 2.2e-16
## [1] "Critical Value is: 43.7729718257422"
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
The null H0: LB=0 is rejected at 1, 10 and 30 lags.
17/01/2023, 19:13
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
17/01/2023, 19:13
GARCH(1,1)
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
17/01/2023, 19:13
GJR-GARCH(1,1)
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
17/01/2023, 19:13 HW1
Compare the 3 models graphically
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
17/01/2023, 19:13 HW1
Statistical properties of financial returns and conditional volatility models
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
Graphical inspection indicates that EWMA produces more extreme volatility forecasts.
Statistical test to compare GARCH and GARCH-GJR
## [1] "Log-likelihood Ratio Test Statistic = 1.22187909254899"
## [1] "Critical Value = 3.84145882069412"
## [1] "We cannot reject H0: LR = 0"
17/01/2023, 19:13 HW1
GARCH-GJR is not statistically significantly better than the GARCH model. We could also reach this conclusion by looking at the -statistic of the Gamma coefficient.
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
Volatility forecast following observed negative return
Statistical properties of financial returns and conditional volatility models
Coeficient t-Value
omega 0.0000 6.9738
alpha1 0.0424 10.1861
beta1 0.9525 257.1235
gamma1 -0.0086 -1.1474
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Comparing the volatility forecasts of the 3 models around the date of extreme negative return 22-06-2016.
Statistical properties of financial returns and conditional volatility models
TSLA.Adjusted sigmaEWMA sigmaGARCH_1_1 sigmaGJR_GARCH_1_1
2016- 0.0175 0.0216 0.0255 0.0253 06-
2016- -0.0023 0.0214 0.0254 0.0252 06-21
2016- -0.1123 0.0208 0.0250 0.0249 06-
2016- -0.0033 0.0341 0.0333 0.0321 06-
2016- -0.0186 0.0331 0.0327 0.0315 06-
2016- 0.0256 0.0324 0.0323 0.0311 06-
2016- 0.0143 0.0320 0.0321 0.0310 06-
2016- 0.0389 0.0312 0.0316 0.0306 06-
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17/01/2023, 19:13 HW1
GARCH(1,1) estimated coefficients:
Statistical properties of financial returns and conditional volatility models
TSLA.Adjusted sigmaEWMA sigmaGARCH_1_1 sigmaGJR_GARCH_1_1
2016- 0.0080 0.0317 0.0320 0.0311 06-
Coeficient t-Value
omega 0.0000 5.8160
alpha1 0.0401 17.1432
beta1 0.9500 223.3908
GJR-GARCH(1,1) estimated coefficients:
Coeficient t-Value
omega 0.0000 6.9738
alpha1 0.0424 10.1861
beta1 0.9525 257.1235
gamma1 -0.0086 -1.1474
https://moodle.lse.ac.uk/pluginfile.php/2189837/mod_resource/content/2/HW1.html
On the 22-06-2016 we observed a very high negative return of -11.23% which was not followed by an increase in volatility. The EWMA model generates a higher volatility forecast compared to the GARCH and GJR models, since more weight is given to ¡°news¡±, alpha=0.6. There is no significant difference between GARCH and GJR, as expected, since the estimated leverage parameter is not statistically significant.
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