Chapter 14 Time-Varying Volatility and ARCH Models

rm(list=ls()) #Removes all items in Environment!
library(FinTS) #for function `ArchTest()`
library(rugarch) #for GARCH models
library(tseries) # for `adf.test()`
library(dynlm) #for function `dynlm()`
library(vars) # for function `VAR()`
library(nlWaldTest) # for the `nlWaldtest()` function
library(lmtest) #for `coeftest()` and `bptest()`.
library(broom) #for `glance(`) and `tidy()`
library(PoEdata) #for PoE4 datasets
library(car) #for `hccm()` robust standard errors
library(sandwich)
library(knitr) #for `kable()`
library(forecast) 

New packages: FinTS (Graves 2014) and rugarch (Ghalanos 2015).

The autoregressive conditional heteroskedasticity (ARCH) model concerns time series with time-varying heteroskedasticity, where variance is conditional on the information existing at a given point in time.

14.1 The ARCH Model

The ARCH model assumes that the conditional mean of the error term in a time series model is constant (zero), unlike the nonstationary series we have discussed so far), but its conditional variance is not. Such a model can be described as in Equations $\ref{eq:archdefA14}$, $\ref{eq:archdefB14}$ and $\ref{eq:archdefC14}$.

$$\begin{equation} y_{t}=\phi +e_{t} \label{eq:archdefA14} \end{equation}$$ $$\begin{equation} e_{t} | I_{t-1} \sim N(0,h_{t}) \label{eq:archdefB14} \end{equation}$$ $$\begin{equation} h_{t}=\alpha_{0}+\alpha_{1}e_{t-1}^2, \;\;\;\alpha_{0}>0, \;\; 0\leq \alpha_{1}<1 \label{eq:archdefC14} \end{equation}$$

Equations $\ref{eq:archeffectseqA14}$ and $\ref{eq:archeffectseqB14}$ give both the test model and the hypotheses to test for ARCH effects in a time series, where the residuals $\hat e_{t}$ come from regressing the variable $y_{t}$ on a constant, such as $\ref{eq:archdefA14}$, or on a constant plus other regressors; the test shown in Equation $\ref{eq:archeffectseqA14}$ may include several lag terms, in which case the null hypothesis (Equation $\ref{eq:archeffectseqB14}$) would be that all of them are jointly insignificant.

$$\begin{equation} \hat e_{t}^2 = \gamma_{0}+\gamma_{1}\hat e_{t-1}^2+...+\gamma_{q}e_{t-q}^2+\nu_{t} \label{eq:archeffectseqA14} \end{equation}$$ $$\begin{equation} H_{0}:\gamma_{1}=...=\gamma_{q}=0\;\;\;H_{A}:\gamma_{1}\neq 0\;or\;...\gamma_{q}\neq 0 \label{eq:archeffectseqB14} \end{equation}$$

The null hypothesis is that there are no ARCH effects. The test statistic is $$(T-q)R^2 \sim \chi ^2_{(1-\alpha,q)}$$ . The following example uses the dataset $byd$, which contains 500 generated observations on the returns to shares in BrightenYourDay Lighting. Figure 14.1 shows a time series plot of the data and histogram.

data("byd", package="PoEdata")
rTS <- ts(byd$r)
plot.ts(rTS)
hist(rTS, main="", breaks=20, freq=FALSE, col="grey")

Figure 14.1: Level and histogram of variable ‘byd’

Let us first perform, step by step, the ARCH test described in Equations $\ref{eq:archeffectseqA14}$ and $\ref{eq:archeffectseqB14}$, on the variable $r$ from dataset $byd$.

byd.mean <- dynlm(rTS~1)
summary(byd.mean)

## 
## Time series regression with "ts" data:
## Start = 1, End = 500
## 
## Call:
## dynlm(formula = rTS ~ 1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.847 -0.723 -0.049  0.669  5.931 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.078      0.053    20.4   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.19 on 499 degrees of freedom

ehatsq <- ts(resid(byd.mean)^2)
byd.ARCH <- dynlm(ehatsq~L(ehatsq))
summary(byd.ARCH)

## 
## Time series regression with "ts" data:
## Start = 2, End = 500
## 
## Call:
## dynlm(formula = ehatsq ~ L(ehatsq))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -10.802  -0.950  -0.705   0.320  31.347 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    0.908      0.124    7.30  1.1e-12 ***
## L(ehatsq)      0.353      0.042    8.41  4.4e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.45 on 497 degrees of freedom
## Multiple R-squared:  0.125,  Adjusted R-squared:  0.123 
## F-statistic: 70.7 on 1 and 497 DF,  p-value: 4.39e-16

T <- nobs(byd.mean)
q <- length(coef(byd.ARCH))-1
Rsq <- glance(byd.ARCH)[[1]]
LM <- (T-q)*Rsq
alpha <- 0.05
Chicr <- qchisq(1-alpha, q)

The result is the LM statistic, equal to $62.16$, which is to be compared to the critical chi-squared value with $\alpha =0.05$ and $q=1$ degrees of freedom; this value is $\chi^2 _{(0.95,1)}=3.84$; this indicates that the null hypothesis is rejected, concluding that the series has ARCH effects.

The same conclusion can be reached if, instead of the step-by-step procedure we use one of $R$’s ARCH test capabilities, the ArchTest() function in package FinTS.

bydArchTest <- ArchTest(byd, lags=1, demean=TRUE)
bydArchTest

## 
##  ARCH LM-test; Null hypothesis: no ARCH effects
## 
## data:  byd
## Chi-squared = 62.16, df = 1, p-value = 3.22e-15

Function garch() in the tseries package, becomes an ARCH model when used with the order= argument equal to c(0,1). This function can be used to estimate and plot the variance $h_{t}$ defined in Equation $\ref{eq:archdefC14}$, as shown in the following code and in Figure 14.2.

byd.arch <- garch(rTS,c(0,1))

## 
##  ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 
## 
## 
##      I     INITIAL X(I)        D(I)
## 
##      1     1.334069e+00     1.000e+00
##      2     5.000000e-02     1.000e+00
## 
##     IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
##      0    1  5.255e+02
##      1    2  5.087e+02  3.20e-02  7.13e-01  3.1e-01  3.8e+02  1.0e+00  1.34e+02
##      2    3  5.004e+02  1.62e-02  1.78e-02  1.2e-01  1.9e+00  5.0e-01  2.11e-01
##      3    5  4.803e+02  4.03e-02  4.07e-02  1.2e-01  2.1e+00  5.0e-01  1.42e-01
##      4    7  4.795e+02  1.60e-03  1.99e-03  1.3e-02  9.7e+00  5.0e-02  1.36e-02
##      5    8  4.793e+02  4.86e-04  6.54e-04  1.2e-02  2.3e+00  5.0e-02  2.31e-03
##      6    9  4.791e+02  4.16e-04  4.93e-04  1.2e-02  1.7e+00  5.0e-02  1.39e-03
##      7   10  4.789e+02  3.80e-04  4.95e-04  2.3e-02  4.6e-01  1.0e-01  5.36e-04
##      8   11  4.789e+02  6.55e-06  6.73e-06  9.0e-04  0.0e+00  5.1e-03  6.73e-06
##      9   12  4.789e+02  4.13e-08  3.97e-08  2.2e-04  0.0e+00  9.8e-04  3.97e-08
##     10   13  4.789e+02  6.67e-11  6.67e-11  9.3e-06  0.0e+00  4.2e-05  6.67e-11
## 
##  ***** RELATIVE FUNCTION CONVERGENCE *****
## 
##  FUNCTION     4.788831e+02   RELDX        9.327e-06
##  FUNC. EVALS      13         GRAD. EVALS      11
##  PRELDF       6.671e-11      NPRELDF      6.671e-11
## 
##      I      FINAL X(I)        D(I)          G(I)
## 
##      1    2.152304e+00     1.000e+00    -2.370e-06
##      2    1.592050e-01     1.000e+00    -7.896e-06

sbydarch <- summary(byd.arch)
sbydarch

## 
## Call:
## garch(x = rTS, order = c(0, 1))
## 
## Model:
## GARCH(0,1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -1.459  0.220  0.668  1.079  4.293 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)    
## a0    2.1523      0.1857    11.59   <2e-16 ***
## a1    0.1592      0.0674     2.36    0.018 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 48.57, df = 2, p-value = 2.85e-11
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.1224, df = 1, p-value = 0.726

hhat <- ts(2*byd.arch$fitted.values[-1,1]^2)
plot.ts(hhat)

Figure 14.2: Estimated ARCH(1) variance for the ‘byd’ dataset

14.2 The GARCH Model

# Using package `rugarch` for GARCH models
library(rugarch)
garchSpec <- ugarchspec(
           variance.model=list(model="sGARCH",
                               garchOrder=c(1,1)),
           mean.model=list(armaOrder=c(0,0)), 
           distribution.model="std")
garchFit <- ugarchfit(spec=garchSpec, data=rTS)
coef(garchFit)

##        mu     omega    alpha1     beta1     shape 
##  1.049280  0.396544  0.495161  0.240827 99.999225

rhat <- garchFit@fit$fitted.values
plot.ts(rhat)
hhat <- ts(garchFit@fit$sigma^2)
plot.ts(hhat)

Figure 14.3: Standard GARCH model (sGARCH) with dataset ‘byd’

# tGARCH 
garchMod <- ugarchspec(variance.model=list(model="fGARCH",
                               garchOrder=c(1,1),
                               submodel="TGARCH"),
           mean.model=list(armaOrder=c(0,0)), 
           distribution.model="std")
garchFit <- ugarchfit(spec=garchMod, data=rTS)
coef(garchFit)

##        mu     omega    alpha1     beta1     eta11     shape 
##  0.986682  0.352207  0.390636  0.375328  0.339432 99.999884

rhat <- garchFit@fit$fitted.values
plot.ts(rhat)
hhat <- ts(garchFit@fit$sigma^2)
plot.ts(hhat)

Figure 14.4: The tGARCH model with dataset ‘byd’

# GARCH-in-mean
garchMod <- ugarchspec(
          variance.model=list(model="fGARCH",
                               garchOrder=c(1,1),
                               submodel="APARCH"),
           mean.model=list(armaOrder=c(0,0),
                          include.mean=TRUE,
                          archm=TRUE,
                          archpow=2
                          ), 
           distribution.model="std"
                           )
garchFit <- ugarchfit(spec=garchMod, data=rTS)
coef(garchFit)

##         mu      archm      omega     alpha1      beta1      eta11 
##   0.820603   0.193476   0.368730   0.442930   0.286748   0.185531 
##     lambda      shape 
##   1.897586 100.000000

rhat <- garchFit@fit$fitted.values
plot.ts(rhat)
hhat <- ts(garchFit@fit$sigma^2)
plot.ts(hhat)

Figure 14.5: A version of the GARCH-in-mean model with dataset ‘byd’

Figures 14.3, 14.4, and 14.5 show a few versions of the GARCH model. Predictions can be obtained using the function ugarchboot() from the package ugarch.

References

Graves, Spencer. 2014. FinTS: Companion to Tsay (2005) Analysis of Financial Time Series. https://CRAN.R-project.org/package=FinTS.

Ghalanos, Alexios. 2015. Rugarch: Univariate Garch Models. https://CRAN.R-project.org/package=rugarch.