Using GARCH Model in R
Using GARCH Model in R
Jasmin G. Geronaga
2025-04-24
A. Model Introduction
One fundamental assumption for the classic linear regression model is homoskedasticity, meaning the variance of the error term is constant. However, in financial data, we often observe time-varying variance (heteroskedasticity). This is where a GARCH model (Generalized Autoregressive Conditional Heteroskedasticity) comes into play.
The GARCH model describes the variance of the current error term as following an ARMA process, instead of being constant.
Model Formula:
yt=xt+ϵtWhere:
- \(y_t\) is a stock return
- \(x_t\) is a mean-reverting
process
- \(\epsilon_t = \sqrt{\delta_t}
z_t\), with \(z_t \sim
\mathcal{D}(0,1)\)
- \(\mathcal{D}\) is a specified
distribution
- \(\delta_t\) is time-varying
follows an ARIMA process:
δt=ω+α1ϵ2t−1+⋯+αqϵ2t−q+β1δ2t−1+⋯+βpδ2t−pB. Data Exploration
B.1 Import Data
library(quantmod)
library(xts)
library(zoo)
library(TTR)
google <- getSymbols("GOOGL", from="2010-01-01", to="2022-08-03", auto.assign = F)
head(google[, 1:5], 5)## GOOGL.Open GOOGL.High GOOGL.Low GOOGL.Close GOOGL.Volume
## 2010-01-04 15.68944 15.75350 15.62162 15.68443 78169752
## 2010-01-05 15.69520 15.71171 15.55405 15.61537 120067812
## 2010-01-06 15.66216 15.66216 15.17417 15.22172 158988852
## 2010-01-07 15.25025 15.26527 14.83108 14.86737 256315428
## 2010-01-08 14.81481 15.09635 14.74249 15.06557 188783028B.2 Calculate Daily Returns
daily_ret <- (google$GOOGL.Close - stats::lag(google$GOOGL.Close)) / stats::lag(google$GOOGL.Close)B.3 Convert to Data Frame
daily_ret <- data.frame(index(daily_ret), daily_ret)
colnames(daily_ret) <- c("date", "return")
rownames(daily_ret) <- 1:nrow(daily_ret)B.4 Plot Daily Return
library(ggplot2)
p1 <- ggplot(daily_ret, aes(x=date, y=return))
p1 + geom_line(colour="#C71585") + labs(title="Google Stock Daily Return", x="Date", y="Return")
B.5 Plot Histogram of Returns
p2 <- ggplot(daily_ret)
p2 + geom_histogram(aes(x=return, y=..density..), binwidth = 0.005, color="#C71585", fill="pink", size=1) +
stat_function(fun = dnorm, args = list(mean = mean(daily_ret$return, na.rm = T), sd = sd(daily_ret$return, na.rm = T)), size=1)
B.6 Calculate Monthly Volatility
library(PerformanceAnalytics)
library(xts)
daily_ret_xts <- xts(daily_ret[,-1], order.by=daily_ret[,1])
realizedvol <- rollapply(daily_ret_xts, width = 20, FUN=sd.annualized)
vol <- data.frame(index(realizedvol), realizedvol)
colnames(vol) <- c("date", "volatility")B.7 Plot Volatility
p3 <- ggplot(vol, aes(x=date, y=volatility))
p3 + geom_line(color="#C71585") + labs(title="Monthly Volatility", x="Date", y="Volatility")
C. Fit a GARCH Model
C.1 Model Specification
library(rugarch)
garch_spec <- ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0)))C.2 Fit the Model
fit_garch <- ugarchfit(spec = garch_spec, data = vol[-c(1:19),2])
fit_garch##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.183618 0.001297 141.605668 0
## omega 0.000276 0.000027 10.226852 0
## alpha1 0.999000 0.033681 29.660723 0
## beta1 0.000000 0.007195 0.000002 1
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.183618 0.010063 18.247385 0.000000
## omega 0.000276 0.000121 2.285627 0.022276
## alpha1 0.999000 0.084219 11.861995 0.000000
## beta1 0.000000 0.005298 0.000003 0.999998
##
## LogLikelihood : 4348.209
##
## Information Criteria
## ------------------------------------
##
## Akaike -2.7600
## Bayes -2.7523
## Shibata -2.7600
## Hannan-Quinn -2.7572
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1862 0
## Lag[2*(p+q)+(p+q)-1][2] 2695 0
## Lag[4*(p+q)+(p+q)-1][5] 4905 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1021 0.7493
## Lag[2*(p+q)+(p+q)-1][5] 0.1275 0.9969
## Lag[4*(p+q)+(p+q)-1][9] 0.2006 0.9999
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.004844 0.500 2.000 0.9445
## ARCH Lag[5] 0.040714 1.440 1.667 0.9963
## ARCH Lag[7] 0.062792 2.315 1.543 0.9998
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.2256
## Individual Statistics:
## mu 0.2599
## omega 0.4897
## alpha1 0.6476
## beta1 0.4083
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.2672 0.2052
## Negative Sign Bias 0.3633 0.7164
## Positive Sign Bias 1.0105 0.3123
## Joint Effect 2.3665 0.4999
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 7769 0
## 2 30 7843 0
## 3 40 9979 0
## 4 50 8988 0
##
##
## Elapsed time : 0.3232291D. Conclusion
The GARCH model results indicate that:
Volatility in Google stock returns is highly sensitive to past shocks, as evidenced by the very high 𝛼 1 α 1 value close to 1. This suggests that recent market movements have a large and persistent impact on future volatility.
The model fit is strong, with significant coefficients, a high log-likelihood value, and favorable information criteria (AIC and BIC).
The residual diagnostics, including the Ljung-Box and ARCH LM tests, indicate that the model has captured most of the volatility dynamics. However, there is some serial correlation in the standardized residuals, suggesting potential room for improvement.
The stability of the parameters is confirmed, which adds confidence in the reliability of the volatility forecasts generated by the model.