Coca-Cola (KO) Stock Modeling¶
Hao Zhang, University of Washington¶
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(Adjust Plotting Settings)
sc = .75
options(repr.plot.width=16*sc,
repr.plot.height=7*sc,
repr.plot.pointsize = 13, # Text height in pt
repr.plot.bg = 'white',
repr.plot.antialias = 'gray',
#nice medium-res DPI
repr.plot.res = 300,
#jpeg quality bumped from default
repr.plot.quality = 90,
#vector font family
repr.plot.family = 'serif')
options(warn=-1)
options("getSymbols.warning4.0"=FALSE)
1. Acquring Data & Detecting Autocorrelation¶
library(quantmod)
getSymbols('KO', from='2007-01-03', to='2018-02-01', auto.assign=TRUE, Warnings=FALSE)
KO = Ad(KO)
head(KO)
dim(KO)
# Let r be percentage daily log returns
r = 100*diff(log(KO))[-1]
head(r)
# Plotting the acf and doing Box-Ljung test for autocorrelation
acf(r, main="sample ACF of KO daily log returns")
Box.test(r, lag=10, type = "Ljung-Box")
Sample ACF and Box-Ljung test show there is autocorrelation in $\{r_t\}$.
2. Detecting ARCH Effect¶
# Fit an ARIMA model
library(forecast)
fit=auto.arima(r, max.p = 20, max.q = 20, max.d = 2, ic="bic")
print(fit)
# Inspecting residual's square
par(mfrow=c(2,1));par(mar=c(3,3,3,3))
plot(resid(fit)**2, type="l", col=1, main = expression(residual^2))
smoother = loess((resid(fit)**2) ~ seq(1,length(resid(fit))), span=0.1)
lines(seq(1,length(resid(fit))),fitted(smoother),col=2)
acf((resid(fit)**2), main=expression("sample ACF of "~ residual^2))
# Box-Ljung test
Box.test(resid(fit)**2, lag = 5, type = "Ljung-Box")
There is existence of ARCH effect.
3. Fitting ARMA+GARCH (1, 1) Model With Normal Residuals¶
# Fitting
library(fGarch)
fit_magarch = garchFit(~arma(0,1)+garch(1,1), data=r, trace = F)
summary(fit_magarch)
We fit an ARMA(0,1)+GARCH(1,1) model, as follows, $\ $
$$ \left\{\begin{array}{}r_t = 0.057823+a_t-0.029829a_{t-1};\quad a_t=\sig_t\eps_t,\quad\eps_t\overset{i.i.d.}\sim N(0,1)\\\sig_t^2=0.038895+0.100440a_{t-1}^2+0.866927\sig_{t-1}^2\end{array}\right. $$# ACF plot of standardized residuals
plot(fit_magarch, which=10)
The ACF plot of the standardized residual show that the ARMA(0,1) specification is adequate. The Ljung-Box test of standardized residuals for 10 , 15, and 20 lags indicate the model is adequate.
# ACF plot of square standarized residuals
plot(fit_magarch, which=11)
The ACF plot of the squared standardized residual shows that the GARCH(1,1) specification is adequate. $\ $
So do the Ljung-Box tests of standardized residuals$^2$.
# QQ plot
plot(fit_magarch, which=13)
The normal probability plot of the standardized residuals indicates fat-tailed distribution, so we proceed with model modification, using t-distribution residuals.
4. Model Modification: Fitting An ARMA+GARCH (1, 1) Model With Student-t Residuals¶
# Fit the model
fit_magarch_tdist = garchFit(~arma(0,1)+garch(1,1), data=r,
cond.dist = "std",trace = F)
summary(fit_magarch_tdist)
We fit an ARMA(0,1)+GARCH(1,1) model with student-t residuals, as follows, $\ $
$$ \left\{\begin{array}{}r_t = 0.063830+a_t-0.020213a_{t-1};\quad a_t=\sig_t\eps_t,\quad\eps_t\overset{i.i.d.}\sim t_{4.817566}\\\sig_t^2=0.019006+0.076508a_{t-1}^2+0.911551\sig_{t-1}^2\end{array}\right. $$# QQ plot
plot(fit_magarch_tdist, which=13)
We have shown in (b) that ARMA(0,1)+GARCH(1,1) specification is adequate. $\\$
And the t-plot of the standardized residuals is a much better fit.
5. Detecting Skewness¶
To detect skewness, we fit an ARMA+GARCH(1,1) model with skew-t-distribution, and do model validation test.
# Fit the model
fit_magarch_stdist = garchFit(~arma(0,1)+garch(1,1), data=r,
cond.dist = "sstd",trace = F)
summary(fit_magarch_stdist)
We fit an ARMA(0,1)+GARCH(1,1) model with skewed student-t residuals, as follows, $\ $
$$ \left\{\begin{array}{}r_t = 0.052696+a_t-0.021578a_{t-1};\quad a_t=\sig_t\eps_t,\quad\eps_t\overset{i.i.d.}\sim t(\nu=4.833,\xi=0.958)\\\sig_t^2=0.019581+0.077416a_{t-1}^2+0.910055\sig_{t-1}^2\end{array}\right. $$# QQ plot
plot(fit_magarch_stdist, which=13)
The skewed t-plot of the standardized residuals is almost the same with the one without skewness. Both models are adequate. $\ $
However, based on the model, the skewness $\xi = 0.958$ is less than 1, which makes it not significant. We conclude that the returns of KO stock is not skewed.
6. Detecting Risk Premium: Fitting An ARMA+GARCH-M Model¶
# Fit the model
library('rugarch')
spec = ugarchspec(variance.model=list(garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,1),
archm=TRUE, archpow=2))
fit_garchm = ugarchfit(data=r, spec=spec)
show(fit_garchm)
We fit an ARMA(0,1)+GARCH(1,1)-M model, as follows, $\ $
$$ \left\{\begin{array}{}r_t = 0.03478+0.0276\sig_t^2+a_t-0.029738a_{t-1};\quad a_t=\sig_t\eps_t,\quad\eps_t\overset{i.i.d.}\sim N(0,1)\\\sig_t^2=0.039521+0.101143a_{t-1}^2+0.865663\sig_{t-1}^2\end{array}\right. $$The estimated risk-premium is 0.0276, which is statistically insignificant. So the return of KO stock has no risk premium.
# Model validation
par(mfrow=c(3,2));par(mar=c(3,3,3,3))
plot(fit_garchm,which=1)
plot(fit_garchm,which=3)
plot(fit_garchm,which=10)
plot(fit_garchm,which=11)
plot(fit_garchm,which=8)
plot(fit_garchm,which=9)
The model seems adequate, the residuals are not normal though.
7. Detecting Leverage Effect: Fitting An ARMA+EGARCH model¶
# Fit the model
spec = ugarchspec(variance.model=list(model="eGARCH", garchOrder=c(1,1)),
mean.model=list(armaOrder=c(0,1)))
fit_egarch = ugarchfit(data=r, spec=spec)
show(fit_egarch)
We fit an ARMA(0,1)+EGARCH(1,1) model, as follows, $\ $
$$ \left\{\begin{array}{}r_t = 0.041052+a_t-0.030159a_{t-1};\quad a_t=\sig_t\eps_t,\quad\eps_t\overset{i.i.d.}\sim N(0,1)\\\ln(\sig_t^2)=0.008262-0.063297\eps_{t-1}+0.189598(|\eps_{t-1}|-\sqrt{2/\pi})+0.966645\ln(\sig_{t-1}^2)\end{array}\right. $$The lag-1 leverage parameter $\alpha_1$=−0.063297 is significant (p-value is 0.000002). So, the model implies leverage effect in the KO stock returns.
# Model validation
par(mfrow=c(3,2));par(mar=c(3,3,3,3))
plot(fit_egarch,which=1)
plot(fit_egarch,which=3)
plot(fit_egarch,which=10)
plot(fit_egarch,which=11)
plot(fit_egarch,which=8)
plot(fit_egarch,which=9)
The model seems adequate, the residuals are not normal though.