| title | author | subtitle | output | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Time Series Course 2020/2021 |
Emilia Wisnios, Piotr Rutkowski |
Tutorial 9 |
|
e.wisnios@student.uw.edu.pl |
Data loading:
www = "https://www.mimuw.edu.pl/~noble/courses/TimeSeries/data/m-gmsp5008.txt"
m_gmsp5008 <- read.table(www, header=T)
sp_ts = ts(m_gmsp5008$sp, start= c(1950, 01), end = c(2008,12), frequency = 12)
plot(sp_ts, type= 'l')
acf(sp_ts)
a) Build a Gaussian GARCH model for monthly log return of the S&P 500 index.
sp.garch <- garch(sp_ts, trace=FALSE)
sp.res <- sp.garch$res[-1]
acf(sp.res)
acf(sp.res^2)
Both correlograms suggest that the residuals of the fitted GARCH model behave like white noise, indicating a satisfactory fit has been obtained.
sp500fit = garchFit(~arma(3,0)+garch(1,1), data=sp_ts, trace=F)
summary(sp500fit)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(3, 0) + garch(1, 1), data = sp_ts, trace = F)
##
## Mean and Variance Equation:
## data ~ arma(3, 0) + garch(1, 1)
## <environment: 0x55efce4c5d50>
## [data = sp_ts]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3 omega alpha1
## 7.1284e-03 1.9290e-02 -4.2114e-02 1.6513e-02 7.9652e-05 1.1366e-01
## beta1
## 8.4816e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 7.128e-03 1.499e-03 4.755 1.99e-06 ***
## ar1 1.929e-02 4.107e-02 0.470 0.6385
## ar2 -4.211e-02 4.051e-02 -1.040 0.2985
## ar3 1.651e-02 3.936e-02 0.420 0.6748
## omega 7.965e-05 3.355e-05 2.374 0.0176 *
## alpha1 1.137e-01 2.659e-02 4.275 1.91e-05 ***
## beta1 8.482e-01 2.892e-02 29.330 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 1273.601 normalized: 1.798871
##
## Description:
## Thu May 27 22:00:31 2021 by user: emilia
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 69.75041 6.661338e-16
## Shapiro-Wilk Test R W 0.9843575 7.145384e-07
## Ljung-Box Test R Q(10) 8.939989 0.5378082
## Ljung-Box Test R Q(15) 13.87093 0.5353401
## Ljung-Box Test R Q(20) 17.76692 0.6027591
## Ljung-Box Test R^2 Q(10) 7.460676 0.6813552
## Ljung-Box Test R^2 Q(15) 8.389826 0.9072052
## Ljung-Box Test R^2 Q(20) 9.440223 0.9772248
## LM Arch Test R TR^2 7.598162 0.815692
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -3.577968 -3.532859 -3.578161 -3.560539
b) Is there a summer effect on the volatility of the index return? Make a variable [u_t = \begin{cases} 1 & \text{for months: June, July, August} \ 0 & \text{for other months.}\end{cases}] Try fitting the model: [\sigma^2(t) = \alpha X^2 (t-1) + \beta \sigma^2(t-1) + \gamma(1 - u_t)] Then the coefficients are ((\alpha, \beta, \gamma)) for September to May and ((\alpha, \beta, 0)) for June, July and August. Is (\gamma), which represents the difference, signinificant?
u_t <- function(var){
month = floor((var %% 10000)/100)
if(month == 6 | month == 7 | month == 8){
return (1)
}
else{
return (0)
}
}summer <- c(0,0,0,0,0,1,1,1,0,0,0,0)
Sum <- rep(summer, 59)
Sum <- as.matrix(Sum)
spec = ugarchspec(variance.model = list(model='sGARCH', garchOrder = c(1,1), external.regressors=Sum), mean.model = list(armaOrder=c(0,0), distribution.model='std'))## Warning: unidentified option(s) in mean.model:
## distribution.model
fit = ugarchfit(data=m_gmsp5008$sp, spec=spec)
round(fit@fit$matcoef, 6)## Estimate Std. Error t value Pr(>|t|)
## mu 0.007147 0.001425 5.017022 0.000001
## omega 0.000032 0.000038 0.826427 0.408562
## alpha1 0.101576 0.026392 3.848687 0.000119
## beta1 0.846108 0.031392 26.952826 0.000000
## vxreg1 0.000268 0.000115 2.338066 0.019384
The parameter lambda is vxreg1 in our model. T value is equal to 2.338066, which means it's statistically significant.
Are the lagged returns of GM stock useful for modelling the index volatility? USE your GARCH model as a baseline for comparison.
gm_ts = ts(m_gmsp5008$gm, start= c(1950, 01), end = c(2008,12), frequency = 12)
plot(gm_ts, type= 'l')
acf(gm_ts)
gm.garch <- garch(gm_ts, trace=FALSE)
gm.res <- gm.garch$res[-1]
acf(gm.res)
acf(gm.res^2)
gm500fit = garchFit(~arma(3,0)+garch(1,1), data=gm_ts, trace=F)
summary(gm500fit)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(3, 0) + garch(1, 1), data = gm_ts, trace = F)
##
## Mean and Variance Equation:
## data ~ arma(3, 0) + garch(1, 1)
## <environment: 0x55efcbb76648>
## [data = gm_ts]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3 omega alpha1
## 0.01083183 0.05711266 -0.04279013 0.01949961 0.00011125 0.10307557
## beta1
## 0.88582030
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.083e-02 2.489e-03 4.352 1.35e-05 ***
## ar1 5.711e-02 3.953e-02 1.445 0.1485
## ar2 -4.279e-02 4.019e-02 -1.065 0.2870
## ar3 1.950e-02 3.954e-02 0.493 0.6219
## omega 1.112e-04 6.012e-05 1.851 0.0642 .
## alpha1 1.031e-01 2.369e-02 4.351 1.35e-05 ***
## beta1 8.858e-01 2.632e-02 33.650 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 870.6694 normalized: 1.229759
##
## Description:
## Thu May 27 22:00:31 2021 by user: emilia
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 20.17247 4.164889e-05
## Shapiro-Wilk Test R W 0.9909099 0.000236162
## Ljung-Box Test R Q(10) 8.860885 0.5453552
## Ljung-Box Test R Q(15) 14.59028 0.4813123
## Ljung-Box Test R Q(20) 18.47383 0.556225
## Ljung-Box Test R^2 Q(10) 9.625037 0.4739824
## Ljung-Box Test R^2 Q(15) 20.90861 0.1397748
## Ljung-Box Test R^2 Q(20) 24.24148 0.232003
## LM Arch Test R TR^2 17.71585 0.1245948
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -2.439744 -2.394635 -2.439937 -2.422316
summary(sp500fit)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~arma(3, 0) + garch(1, 1), data = sp_ts, trace = F)
##
## Mean and Variance Equation:
## data ~ arma(3, 0) + garch(1, 1)
## <environment: 0x55efce4c5d50>
## [data = sp_ts]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu ar1 ar2 ar3 omega alpha1
## 7.1284e-03 1.9290e-02 -4.2114e-02 1.6513e-02 7.9652e-05 1.1366e-01
## beta1
## 8.4816e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 7.128e-03 1.499e-03 4.755 1.99e-06 ***
## ar1 1.929e-02 4.107e-02 0.470 0.6385
## ar2 -4.211e-02 4.051e-02 -1.040 0.2985
## ar3 1.651e-02 3.936e-02 0.420 0.6748
## omega 7.965e-05 3.355e-05 2.374 0.0176 *
## alpha1 1.137e-01 2.659e-02 4.275 1.91e-05 ***
## beta1 8.482e-01 2.892e-02 29.330 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 1273.601 normalized: 1.798871
##
## Description:
## Thu May 27 22:00:31 2021 by user: emilia
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 69.75041 6.661338e-16
## Shapiro-Wilk Test R W 0.9843575 7.145384e-07
## Ljung-Box Test R Q(10) 8.939989 0.5378082
## Ljung-Box Test R Q(15) 13.87093 0.5353401
## Ljung-Box Test R Q(20) 17.76692 0.6027591
## Ljung-Box Test R^2 Q(10) 7.460676 0.6813552
## Ljung-Box Test R^2 Q(15) 8.389826 0.9072052
## Ljung-Box Test R^2 Q(20) 9.440223 0.9772248
## LM Arch Test R TR^2 7.598162 0.815692
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -3.577968 -3.532859 -3.578161 -3.560539
Based on log likelihood, the lagged returns o GM stock is useful for modelling the index volatility.