GARCH_google_stock
jaspercastro
2025-04-24
Load Required Libraries
library(quantmod)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(rugarch)## Warning: package 'rugarch' was built under R version 4.4.3## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
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## sigmalibrary(tseries)## Warning: package 'tseries' was built under R version 4.4.3library(PerformanceAnalytics)## Warning: package 'PerformanceAnalytics' was built under R version 4.4.3##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
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## legendlibrary(FinTS)## Warning: package 'FinTS' was built under R version 4.4.3Get GOOGL Stock Data
getSymbols("GOOGL", src = "yahoo", from = "2018-01-01", to = "2025-01-01")## [1] "GOOGL"goog_data <- na.omit(Cl(GOOGL)) #Use closing prices only
plot(goog_data, main = "GOOGL Closing Prices")
Convert to Log Returns
You can also embed plots, for example:
goog_returns <- dailyReturn(goog_data, type = "log")
plot(goog_returns, main = "GOOGL Log Returns")
Test for ARCH Effects
Box.test(goog_returns, lag = 20, type = "Ljung-Box")##
## Box-Ljung test
##
## data: goog_returns
## X-squared = 74.216, df = 20, p-value = 3.679e-08ArchTest(goog_returns, lags = 12)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: goog_returns
## Chi-squared = 129.04, df = 12, p-value < 2.2e-16Specify and Fit GARCH(1,1) Model
Since p-value for ARCH LM-test is very small, we reject the null hypothesis
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)), mean.model = list(armaOrder = c(0,0)), distribution.model = "std") # Student-t distribution for heavy tails
fit <- ugarchfit(spec = spec, data = goog_returns)
fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001539 0.000351 4.3887 1.1e-05
## omega 0.000013 0.000002 6.5273 0.0e+00
## alpha1 0.089658 0.002685 33.3953 0.0e+00
## beta1 0.886031 0.013216 67.0398 0.0e+00
## shape 4.037919 0.331074 12.1964 0.0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001539 0.000295 5.2123 0.0e+00
## omega 0.000013 0.000003 4.0305 5.6e-05
## alpha1 0.089658 0.011869 7.5539 0.0e+00
## beta1 0.886031 0.017139 51.6958 0.0e+00
## shape 4.037919 0.525728 7.6806 0.0e+00
##
## LogLikelihood : 4637.547
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.2613
## Bayes -5.2457
## Shibata -5.2613
## Hannan-Quinn -5.2555
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.3666 0.5449
## Lag[2*(p+q)+(p+q)-1][2] 1.3846 0.3887
## Lag[4*(p+q)+(p+q)-1][5] 3.1197 0.3856
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.048e-08 0.9999
## Lag[2*(p+q)+(p+q)-1][5] 1.097e+00 0.8375
## Lag[4*(p+q)+(p+q)-1][9] 2.188e+00 0.8814
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.03834 0.500 2.000 0.8448
## ARCH Lag[5] 0.28664 1.440 1.667 0.9438
## ARCH Lag[7] 1.01153 2.315 1.543 0.9117
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 21.5902
## Individual Statistics:
## mu 0.04618
## omega 1.86210
## alpha1 0.21768
## beta1 0.36155
## shape 0.36039
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.14921 0.8814
## Negative Sign Bias 0.01854 0.9852
## Positive Sign Bias 0.32110 0.7482
## Joint Effect 0.10349 0.9914
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 36.47 0.009244
## 2 30 49.34 0.010626
## 3 40 56.17 0.036845
## 4 50 70.94 0.021842
##
##
## Elapsed time : 0.193414Plot GARCH Model Results
plot(fit, which = "all")##
## please wait...calculating quantiles...
Model Diagnostics
very small autocorrelation bars - this suggests that most serial correlation has been removed from the residuals.however, p-value for ljung-box is just slightly below 0.05 so we barely reject the null hypothesis. There’s a small amount of autocorrelation left in the residuals. Not perfect but not alarming either. ARMA can be tweaked for better results
for the ARCH test, p-value is very high so we fail to reject the null hypothesis. This suggests that the GARCH model has successfully captured the volatility clustering - no ARCH effects remain in the residuals
residuals <- residuals(fit, standardize = TRUE)
acf(residuals, main = "ACF of Standardized Residuals")
Box.test(residuals, lag = 20, type = "Ljung-Box")##
## Box-Ljung test
##
## data: residuals
## X-squared = 32.378, df = 20, p-value = 0.03944# ARCH test on residuals
ArchTest(residuals, lags = 12)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: residuals
## Chi-squared = 6.2963, df = 12, p-value = 0.9004Forecast Volatility
forecast <- ugarchforecast(fit, n.ahead = 10)
plot(forecast, which = 3)
Modify ARMA order to (1,1)
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(1, 1)),
distribution.model = "std") # Student-t distribution for heavy tails
fit <- ugarchfit(spec = spec, data = goog_returns)
fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001507 0.000249 6.0495 0
## ar1 0.840009 0.055583 15.1128 0
## ma1 -0.887317 0.046709 -18.9965 0
## omega 0.000013 0.000002 7.7498 0
## alpha1 0.087831 0.004139 21.2227 0
## beta1 0.889828 0.012973 68.5886 0
## shape 3.911377 0.320824 12.1916 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.001507 0.000247 6.0993 0
## ar1 0.840009 0.042531 19.7507 0
## ma1 -0.887317 0.036366 -24.3995 0
## omega 0.000013 0.000003 5.0781 0
## alpha1 0.087831 0.010046 8.7433 0
## beta1 0.889828 0.015659 56.8259 0
## shape 3.911377 0.466670 8.3815 0
##
## LogLikelihood : 4645.266
##
## Information Criteria
## ------------------------------------
##
## Akaike -5.2678
## Bayes -5.2460
## Shibata -5.2678
## Hannan-Quinn -5.2597
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.010 0.3150
## Lag[2*(p+q)+(p+q)-1][5] 1.310 0.9996
## Lag[4*(p+q)+(p+q)-1][9] 3.416 0.8181
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.007733 0.9299
## Lag[2*(p+q)+(p+q)-1][5] 1.088510 0.8393
## Lag[4*(p+q)+(p+q)-1][9] 2.200653 0.8797
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.01421 0.500 2.000 0.9051
## ARCH Lag[5] 0.24893 1.440 1.667 0.9535
## ARCH Lag[7] 1.01946 2.315 1.543 0.9104
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 24.7794
## Individual Statistics:
## mu 0.07970
## ar1 0.02795
## ma1 0.02549
## omega 2.35690
## alpha1 0.22678
## beta1 0.35083
## shape 0.34723
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.2024 0.2294
## Negative Sign Bias 0.4231 0.6723
## Positive Sign Bias 0.8783 0.3799
## Joint Effect 1.5647 0.6674
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 37.63 0.00662
## 2 30 42.32 0.05252
## 3 40 49.27 0.12552
## 4 50 56.80 0.20714
##
##
## Elapsed time : 0.2920611Plot GARCH Model Results again
plot(fit, which = "all")##
## please wait...calculating quantiles...
Model Diagnostics
p-value for ljung-box test is now well above 0.05, which suggests that there is no correlation left in the residuals. p-value for ARCH test is also slighty higher that the previous value.
residuals <- residuals(fit, standardize = TRUE)
acf(residuals, main = "ACF of Standardized Residuals")
Box.test(residuals, lag = 20, type = "Ljung-Box")##
## Box-Ljung test
##
## data: residuals
## X-squared = 26.73, df = 20, p-value = 0.143# ARCH test on residuals
ArchTest(residuals, lags = 12)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: residuals
## Chi-squared = 5.9293, df = 12, p-value = 0.9196Forecast Volatility Again
forecast <- ugarchforecast(fit, n.ahead=10)
plot(forecast, which = 3)