Generalised Autoregressive Conditional Heteroskedasticity model
Bollerslev (1986) proposed the GARCH model which estimates the conditional volatility of time series. Many textbook on time series analysis or econometrics deal with this important model at the neighborhood of Chapter 6 (in the middle part of that book).
GARCH model
The standard GARCH model (Bollerslev, 1986) formulates the conditional variance (\(\sigma_t^2\)) of a stochastic error term (\(\epsilon_t\)) as follows.
\[\begin{align} \epsilon_t &= \sigma_t Z_t \\ \sigma_t^2 &= \text{Var}[\epsilon_t| \mathcal{F}_{t-1}] \\ &= \omega + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^{p} \beta_i \sigma_{t-i}^2 \\ &= \text{GARCH}(p,q) \end{align}\]
where \(E[\epsilon_t| \mathcal{F}_{t-1}] = 0\) and \({F}_{t-1}\) means all the information available at time \(t\). \(Z_t\) is a white noise with mean zero and variance 1.
The conditional distribution of \(\epsilon_t| \mathcal{F}_{t-1}\) is assumed to be normal. Of course, other distributions can be applied.
The standard GARCH model is about an error term and this means a mean constant zero. However, the conditional mean is appropriate in many cases. For this purpose, ARMA-GARCH model is used.
\[\begin{align} &y_t - c - \sum_{i=1}^{r} \phi_i y_{t-i} - \sum_{i=1}^{m} \theta_i \epsilon_{t-i} = \epsilon_t \\ & \epsilon_t = \sigma_t Z_t \\ &\sigma_t^2 = \text{GARCH}(p,q) \end{align}\]
Beautiful Graphs from output of ugarchfit()
rugarch R package provides so nice looking figures when we plot just the results of the ugarchfit() fitting function. There are 12 figures, and we can select one by its number or all by "all" option.
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Data
Data is the S&P 500 index return (S&P 500 ETF Trust) as follows.
R code
The following R code uses rugarch R package to estimate ARMA(1,1)-GARCH(1,1)-Normal model using ugarchfit() function and performs return and conditional volatility forecasts using ugarchforecast() function.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | #========================================================# # Quantitative Financial Econometrics & Derivatives # Machine/Deep Learing, R, Python, Tensorflow # by Sang-Heon Lee # # https://shleeai.blogspot.com #--------------------------------------------------------# # GARCH model #========================================================# graphics.off(); rm(list = ls()) library(quantmod) library(rugarch) #=========================================== # Data #=========================================== # S&P 500 index getSymbols("SPY",from="2007/01/01") df <- data.frame( as.Date(index(SPY)), SPY$SPY.Adjusted, diff(log(SPY$SPY.Adjusted))) rownames(df) <- NULL colnames(df) <- c("dt", "pr", "rt") head(df) # zoo type zf <- read.zoo(df) # draw data x11(); plot(zf, xlab = "date", main = "S&P 500", ylab = c("Index", "Return"), col=c(4,3)) #=========================================== # ARMA(1,1)-GARCH(1,1)-Normal distribution #=========================================== # default specification : ARMA(1,1)-GARCH(1,1)-N # mod_spec = ugarchspec() # same specification : ARMA(1,1)-GARCH(1,1)-N mod_spec = ugarchspec( variance.model=list(model="sGARCH",garchOrder=c(1,1)), mean.model=list(armaOrder=c(1,1), include.mean=TRUE), distribution.model="norm" ) # GARCH Estimation mod_fit = ugarchfit(spec = mod_spec, data = zf$rt[-1]) mod_fit # draw all the estimated results x11(width = 9, height=7); plot(mod_fit, which = "all") # draw conditional variance x11(width = 7, height=5); plot(mod_fit, which = 3) # Extract estimated coefficients mod_fit@fit$coef # estimated conditional variances convar <- mod_fit@fit$var # estimated squared residuals res2 <- (mod_fit@fit$residuals)^2 #plot squared residuals and conditional variance x11(); plot(res2, type = "l",col="darkgray", xlab="Index", ylab ="Residuals & Variance", lwd=2) lines(convar, col = "blue", lwd=1) #=========================================== # Forecast conditional volatility #=========================================== # run forecasting mod_fcst <- ugarchforecast(mod_fit, n.ahead = 500) # read return forecast mod_fcst_ret <- as.vector(mod_fcst@forecast$seriesFor) temp <- data.frame( fcst_cvol = as.vector(merge(tail(zf$rt,100), head(mod_fcst_ret,100))) ) temp$fit_cvol = temp$fcst_cvol temp$fit_cvol[102:nrow(temp)] <- NA # draw return forecast with data x11(width = 7, height=5) matplot(temp, type = "l", lty = 1, lwd = 3, ylab="Return", col = c(4,2), main = paste0("ARMA(1,1)-GARCH(1,1)-Normal : ", "Conditional Mean Forecast")) # read sigma forecast mod_fcst_cvol <- as.vector(mod_fcst@forecast$sigmaFor) # read sigma fitted mod_fit_cvol <- as.vector(mod_fit@fit$sigma) temp <- data.frame( fcst_cvol = c(mod_fit_cvol, mod_fcst_cvol), fit_cvol = c(mod_fit_cvol, mod_fcst_cvol[1], mod_fcst_cvol[-1]*NA)) # draw conditional volatility forecast with fitted values x11(width = 7, height=5) matplot(temp, type = "l", lty = 1, lwd = 3, ylab="Volatility", col = c(4,2), main = paste0("ARMA(1,1)-GARCH(1,1)-Normal : ", "Conditional Volatility Forecast")) | cs |
After running the above R code, we can draw summary plots of GARCH estimation with which = "all" option. It's very nice.
we can also draw a plot of the estimated conditional volatility with which = 3 option.
Estimation results of GARCH model consist of parameter estimates, its statistical significances, and some test statistics including Ljung-Box, ARCH-LM test, and so on. These test statistics are academic so that you can learn how to interpret these statistics with the help of some famous financial time series textbooks such as Applied Econometric Time Series - Walter Enders.
Forecast
By using ugarchforecast() function, we can obtain two forecasting results of mean and conditional volatility. First, due to the small forecasted range of s, the return forecasts with recent period are shown as follows.
Second, as volatility range is not so small, we don't have to magnify some part of sample period. The conditional volatility forecasts with the whole sample are shown as follows.
Concluding Remarks
This post introduces rugarch R package to estimate the famous GARCH model. This package also provides a variety of extended GARCH models, and you can find them in the rugarch manual. \(\blacksquare\)
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