22  GARCH(1,1) moments

library(tidyverse)
# **********************************************************************
#                     Simulate GARCH (1,1) 
# **********************************************************************
# Simulate GARCH process and variance 
simulate_GARCH11 <- function(B = 100, n = 1000, sigma2_0, omega, alpha, beta, mu, sd, p){
  scenarios <- list()
  for(b in 1:B){
    Bt <- rbinom(n, 1, p[1])
    ut <- rnorm(n, mu[1], sd[1])*Bt + rnorm(n, mu[2], sd[2])*(1-Bt)
    yt <- c(sigma2_0 * ut)
    sigma2_t <- c(sigma2_0)
    for(i in 2:n){
      sigma2_t[i] <- omega + alpha * (yt[i-1] - sqrt(sigma2_t[i-1])*sum(mu*p))^2 + beta * sigma2_t[i-1]
      yt[i] <- sqrt(sigma2_t[i]) * ut[i]
    }
    scenarios[[b]] <- dplyr::tibble(t = 1:n, j = b, x = yt, sigma2 = sigma2_t)
  }
  return(scenarios)
}
# **********************************************************************
#                     Simulate GARCH (1,1) Moments
# **********************************************************************
# Simulate GARCH and compute its moments
simulate_GARCH_moments <- function(B = 100, n = 1000, sigma2_0, omega, alpha, beta, mu, sd, p){
  m1 <- m2 <- m3 <- m4 <- m32 <- m12 <- v <- v12 <- c()
  for(b in 1:B){
    Bt <- rbinom(n, 1, p[1])
    ut <- rnorm(n, mu[1], sd[1])*Bt + rnorm(n, mu[2], sd[2])*(1-Bt)
    sigma2_t <- c(sigma2_0)
    for(i in 2:n){
      sigma2_t[i] <- omega + alpha * ut[i-1]^2 + beta * sigma2_t[i-1]
      ut[i] <- sqrt(sigma2_t[i]) * ut[i]
    }
    m1[b] <- mean(sigma2_t)
    m12[b] <- mean(sigma2_t^(1/2))
    m2[b] <- mean(sigma2_t^2)
    m32[b] <- mean(sigma2_t^(3/2))
    m3[b] <- mean(sigma2_t^3) 
    m4[b] <- mean(sigma2_t^4)
    v[b] <- var(sigma2_t)
    v12[b] <- var(sigma2_t^(1/2)) 
  }
  dplyr::tibble(m1 = m1, m2 = m2, m3 = m3, m4 = m4, m32 = m32, m12 = m12, v = v, v12 = v12) %>%
    dplyr::summarise_all(mean)
}
# **********************************************************************
#                              Inputs
# **********************************************************************
# Forecast horizon
h <- 30000
B <- 500
n <- h
# GARCH parameters
alpha <- 0.08
beta <- 0.35
omega <- 1.2*(1 - alpha - beta)
# Mixture moments
mu <- c(-0.6, 0.4)
sd <- c(0.8, 1.2)
p <- c(0.4, 0.6)
# Moments of a Gaussian Mixture 
e_x2 = sum((mu^2 + sd^2)*p)
e_x4 = sum((3 * sd^4 + 6 * mu^2 * sd^2 + mu^4) * p)
# Initial values for variance 
sigma2_t <- 1.2
sigma4_t <- sigma2_t^2 
# **********************************************************************
# Simulate GARCH and compute moments
mom <- simulate_GARCH_moments(B = B, n = n, sigma2_t, omega, alpha, beta, mu, sd, p)

Let’s consider a very general setup for GARCH(1,1) process where we do not assume that the GARCH residuals

1. Has expected value equal to zero, i.e. $\mathbb{E}\{u_t\}=0$.
2. Has NOT necessary a second moment that is constant and equal to one, i.e. $\mathbb{E}\{u_t^2\}$ possibly time dependent (deterministic).
3. Has NOT necessary a fourth moment that is constant and equal to 3, i.e. $\mathbb{E}\{u_t^4\}$ possibly time dependent (deterministic).

The process we refer to has the form of a standard GARCH(1,1), i.e.  $$\begin{array}{rl}& Y_t={\sigma }_t{u}_t\\ & {\sigma }_t^2=\omega +{\alpha }_1{u}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2\end{array}$$ where $u_t\sim \text{WN}({\mu }_t,{\sigma }_t)$.

22.1 First moment ${\sigma }_t^2$

22.1.1 Short-term

Given the information at time $t-1$, the expected value of the GARCH variance after $h$-steps can be expanded as: $$\begin{array}{}\text{(22.1)}& \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\omega (1+\sum _{j=1}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i})+{\sigma }_t^2\prod _{j=1}^h{\lambda }_{t+h-j}\text{,}\end{array}$$ with $h\ge 1$ and where in general $$\begin{array}{}\text{(22.2)}& {\lambda }_{t+h-i}={\alpha }_1\mathbb{E}\{u_{t+h-i}^2\}+{\beta }_1\text{.}\end{array}$$ The iteration between two consecutive times reads $$\mathbb{E}\{{\sigma }_{t+h-s}^2\mid {\mathcal{F}}_{t-1}\}=\omega +{\lambda }_{t+h-s-1}\mathbb{E}\{{\sigma }_{t+h-s-1}^2\mid {\mathcal{F}}_{t-1}\}\text{.}$$

Proof: GARCH(1,1) iterative expectation

Proof. Let’s start by taking the conditional expectation of the GARCH(1,1) variance at time $t$ given the information up to $t-1$. In this case it is fully known at time $t$ given the information in $t-1$, i.e. $$\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}=\omega +{\alpha }_1{y}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2\text{.}$$ Then, let’s iterate the expectation at time $t+1$, i.e. $$\mathbb{E}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}=\omega +{\alpha }_1\mathbb{E}\{y_t^2\}+{\beta }_1\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}\text{,}$$ and let’s substitute the expression for the squared residuals $y_t^2={\sigma }_t^2{u}_t^2$. Since ${\sigma }_t^2$ at time $t-1$ is known we have: $$\mathbb{E}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}=\omega +({\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}=\omega +{\lambda }_t{\sigma }_t^2\text{,}$$ where $${\lambda }_t={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1\text{.}$$ Iterating the expectation at time $t+2$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}& =\omega +{\lambda }_{t+1}\mathbb{E}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}\\ & =\omega +{\lambda }_{t+1}(\omega +{\lambda }_t{\sigma }_t^2)\\ & =\omega (1+{\lambda }_{t+1})+{\lambda }_{t+1}{\lambda }_t{\sigma }_t^2\end{array}$$ Then, at time $t+3$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+3}^2\mid {\mathcal{F}}_{t-1}\}& =\omega +{\lambda }_{t+2}\mathbb{E}\{{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}\\ & =\omega +{\lambda }_{t+2}(\omega (1+{\lambda }_{t+1})+{\lambda }_{t+1}{\lambda }_t{\sigma }_t^2)\\ & =\omega (1+{\lambda }_{t+2}+{\lambda }_{t+2}{\lambda }_{t+1})+{\lambda }_{t+2}{\lambda }_{t+1}{\lambda }_t{\sigma }_t^2\end{array}$$ In general, after $h$-steps one can write the expansion $$\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\omega (1+\sum _{j=1}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i})+{\sigma }_t^2\prod _{j=1}^h{\lambda }_{t+h-j}$$ with the convention that an empty product is 1.

Example: GARCH(1,1) iterative first moment

Example 22.1 Expanding the expression in Equation 22.1 with $h=1$ one obtain $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}& =\omega (1+\sum _{j=1}^0\prod _{i=1}^j{\lambda }_{t+1-i})+{\sigma }_t^2\prod _{j=1}^1{\lambda }_{t+1-j}=\\ & =\omega +{\sigma }_t^2{\lambda }_t\end{array}$$ with $h=2$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}& =\omega (1+\sum _{j=1}^1\prod _{i=1}^j{\lambda }_{t+2-i})+{\sigma }_t^2\prod _{j=1}^2{\lambda }_{t+2-j}=\\ & =\omega (1+{\lambda }_{t+1})+{\sigma }_t^2({\lambda }_{t+1}{\lambda }_t)\end{array}$$ with $h=3$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+3}^2\mid {\mathcal{F}}_{t-1}\}& =\omega (1+\sum _{j=1}^2\prod _{i=1}^j{\lambda }_{t+3-i})+{\sigma }_t^2\prod _{j=1}^3{\lambda }_{t+3-j}=\\ & =\omega (1+{\lambda }_{t+2}+{\lambda }_{t+1}{\lambda }_{t+2})+{\sigma }_t^2({\lambda }_{t+2}{\lambda }_{t+1}{\lambda }_t)\end{array}$$ and so on.

#' Formula for conditional first moment of GARCH variance
#' 
#' @param h integer, number of steps ahead. 
#' @param omega numeric, intercept of the model. 
#' @param alpha numeric, arch parameter of the model. 
#' @param beta numeric, garch parameter of the model. 
#' @param e_x2 numeric, second moment of ut. 
#' @param e_x4 numeric, fourth moment of ut. 
#' @param sigma2_t numeric, last value of the variance at time t-1. 
# Conditional first moment GARCH variance (exact)
e_sigma2_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, sigma2_t){
  sigma2_h <- c(sigma2_t)
  # Second moment
  m2 <- e_x2
  if (length(e_x2) == 1 & h > 1){
    m2 <- rep(e_x2, h)
  }
  # Derived quantities
  lambda <- alpha * m2 + beta
  lambda_prod <- c(1, lambda[h:1][-1])
  #lambda_prod <- cumprod(lambda_prod)
  sigma2_h <- c(sigma2_t, omega * cumsum(cumprod(lambda_prod)) + sigma2_t * cumprod(lambda))
  names(sigma2_h) <- paste0("t+", 0:h)
  return(sigma2_h)
}
#' Iteration for conditional first moment of GARCH variance
#' 
#' @param h integer, number of steps ahead. 
#' @param omega numeric, intercept of the model. 
#' @param alpha numeric, arch parameter of the model. 
#' @param beta numeric, garch parameter of the model. 
#' @param e_x2 numeric, second moment of ut. 
#' @param e_x4 numeric, fourth moment of ut. 
#' @param sigma2_t numeric, last value of the variance at time t-1. 
e_sigma2_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, sigma2_t){
  # Manual iteration 
  sigma2_h <- c(sigma2_t)
  # Second moment
  m2 <- e_x2
  if (length(e_x2) == 1 & h > 1){
    m2 <- rep(e_x2, h)
  }
  for(i in 1:h){
    sigma2_h[i+1] <- omega + (alpha * m2[i] + beta) * sigma2_h[i]
  }
  names(sigma2_h) <- paste0("t+", 0:h)
  return(sigma2_h)
}

Moment	Step	Formula	Iteration	Difference. (%)
$\mathbb{E}\{{\sigma }_{t+0}^2\mid {\mathcal{F}}_{t-1}\}$	0	1.200	1.200	0%
$\mathbb{E}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}$	1	1.235	1.235	0%
$\mathbb{E}\{{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}$	2	1.250	1.250	0%
$\mathbb{E}\{{\sigma }_{t+3}^2\mid {\mathcal{F}}_{t-1}\}$	3	1.258	1.258	0%
$\mathbb{E}\{{\sigma }_{t+4}^2\mid {\mathcal{F}}_{t-1}\}$	4	1.261	1.261	0%
$\mathbb{E}\{{\sigma }_{t+5}^2\mid {\mathcal{F}}_{t-1}\}$	5	1.263	1.263	0%
$\mathbb{E}\{{\sigma }_{t+6}^2\mid {\mathcal{F}}_{t-1}\}$	6	1.263	1.263	0%
$\mathbb{E}\{{\sigma }_{t+7}^2\mid {\mathcal{F}}_{t-1}\}$	7	1.264	1.264	0%
$\mathbb{E}\{{\sigma }_{t+8}^2\mid {\mathcal{F}}_{t-1}\}$	8	1.264	1.264	0%
$\mathbb{E}\{{\sigma }_{t+9}^2\mid {\mathcal{F}}_{t-1}\}$	9	1.264	1.264	0%
$\mathbb{E}\{{\sigma }_{t+10}^2\mid {\mathcal{F}}_{t-1}\}$	10	1.264	1.264	0%

Table 22.1: Forecasted expectation of GARCH(1,1) variance with iteration and with formula.

22.1.2 Long-term

If $\mathbb{E}\{u_t^2\}$ is constant for all $t$ then, $\lambda ={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1$, became a constant and the formula simplifies to
$$\begin{array}{}\text{(22.3)}& \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}={\sigma }_{\mathrm{\infty }}^2+{\lambda }^h({\sigma }_t^2-{\sigma }_{\mathrm{\infty }}^2)\text{,}\end{array}$$ where ${\sigma }_{\mathrm{\infty }}^2$ denotes the long-term expected GARCH variance as $h\to \mathrm{\infty }$, i.e.  $$\begin{array}{}\text{(22.4)}& \underset{h\to \mathrm{\infty }}{lim}\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\frac{\omega }{1-\lambda }={\sigma }_{\mathrm{\infty }}^2\text{.}\end{array}$$ It follows that, under the standard assumption that $u_t\sim \text{WN}(0,1)$, then one obtain the classic expression $$\begin{array}{}\text{(22.5)}& \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}={\sigma }_{\mathrm{\infty }}^2+({\alpha }_1+{\beta }_1{)}^h({\sigma }_t^2-{\sigma }_{\mathrm{\infty }}^2)\end{array}$$ where ${\sigma }_{\mathrm{\infty }}^2$ denotes the unconditional expectation as Equation 22.4 with $\mathbb{E}\{u_t^2\}=1$.

GARCH(1,1) long-term expectation

Proof. Let’s verify the formula in Equation 22.5 for constant ${\lambda }_t$ (Equation 22.2) for all $t$, i.e.  $$\lambda ={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1$$ In this case the iterative formula (Equation 22.1) simplifies, i.e.  $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}& =\omega (1+\sum _{j=1}^{h-1}\prod _{i=1}^j\lambda )+{\sigma }_t^2\prod _{j=1}^h\lambda =\\ & =\omega \sum _{j=0}^{h-1}{\lambda }^j+{\sigma }_t^2{\lambda }^h=\\ & =\omega \left(\frac{1-{\lambda }^h}{1-\lambda }\right)+{\sigma }_t^2{\lambda }^h\\ & =\frac{\omega }{1-\lambda }+{\lambda }^h({\sigma }_t^2-\frac{\omega }{1-\lambda })\end{array}$$ Taking the limit as $h\to \mathrm{\infty }$ gives the long term stationary variance, i.e. $$\underset{h\to \mathrm{\infty }}{lim}\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\frac{\omega }{1-\lambda }={\sigma }_{\mathrm{\infty }}^2⟺\lambda <1$$ Hence, the general expression became $$\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}={\sigma }_{\mathrm{\infty }}^2+{\lambda }^h({\sigma }_t^2-{\sigma }_{\mathrm{\infty }}^2)$$

Example: GARCH(1,1) long-term first moment

Example 22.2  

# Unconditional first moment GARCH variance (exact)
e_sigma2_inf <- function(omega, alpha, beta, e_x2 = 1){
  # Derived quantities
  lambda <- alpha * e_x2 + beta
  # Expectation 
  omega / (1 - lambda)
}

Moment	Formula	MonteCarlo	Difference (%)
${\sigma }_{\mathrm{\infty }}^2$	1.264	1.264	-0.0051%

Table 22.2: Forecasted long-term expectation of GARCH(1,1) variance with formula and by Monte Carlo simulations.

22.2 Second moment ${\sigma }_t^2$

22.2.1 Short term

The second moment admits an iterative formula, i.e.  $$\begin{array}{}\text{(22.6)}& \mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}=\sum _{i=1}^h{b}_{t+h-i}\prod _{j=1}^{i-1}{\gamma }_{t+h-j}+{\sigma }_t^4\prod _{j=1}^h{\gamma }_{t+h-j}\end{array}$$ where
$$\begin{array}{}\text{(22.7)}& {\gamma }_{t+h-j}={\alpha }_1^2\mathbb{E}\{u_{t+h-j}^4\}+{\beta }_1(2{\alpha }_1\mathbb{E}\{u_{t+h-j}^2\}+{\beta }_1)\end{array}$$ while $$\begin{array}{}\text{(22.8)}& b_{t+h-i}=\omega (\omega +2{\lambda }_{t+h-i}\mathbb{E}\{{\sigma }_{t+h-i}^2\mid {\mathcal{F}}_{t-1}\})\end{array}$$ with ${\lambda }_{t+h-i}$ as in Equation 22.2.

Proof: Iterative formula for the second moment of GARCH(1,1) variance

Proof. Starting from $$\begin{array}{rl}{\sigma }_{t+1}^4& =(\omega +{\alpha }_1{e}_t^2+{\beta }_1{\sigma }_t^2{)}^2\end{array}$$ and substitute the definition of $e_t^2={\sigma }_t^2{u}_t^2$, i.e. $$\begin{array}{rl}{\sigma }_{t+1}^4& =(\omega +({\alpha }_1{u}_t^2+{\beta }_1){\sigma }_t^2{)}^2=\\ & ={\omega }^2+2\omega ({\alpha }_1{u}_t^2+{\beta }_1){\sigma }_t^2+({\alpha }_1{u}_t^2+{\beta }_1{)}^2{\sigma }_t^4\end{array}$$ Then, let’s take the conditional expectation on both sides: $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+1}^4\mid {\mathcal{F}}_t\}& ={\omega }^2+\mathbb{E}\{({\alpha }_1{u}_t^2+{\beta }_1{)}^2\}\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}+2\omega ({\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\}=\\ & ={\omega }^2+\mathbb{E}\{({\alpha }_1{u}_t^2+{\beta }_1{)}^2\}\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}+2\omega {\lambda }_t\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\}\end{array}$$ where ${\lambda }_t={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1$. Then note that: $$\begin{array}{rl}\mathbb{E}\{({\alpha }_1{u}_t^2+{\beta }_1{)}^2\}& =\mathbb{E}\{{\alpha }_1^2{u}_t^4+{\beta }_1^2+2{\alpha }_1{\beta }_1{u}_t^2\}=\\ & ={\alpha }_1^2\mathbb{E}\{u_t^4\}+{\beta }_1(2{\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)={\gamma }_t\end{array}$$ Hence, we can write the expectation in terms of the previous expectation. $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+1}^4\mid {\mathcal{F}}_t\}& ={\omega }^2+{\gamma }_t\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}+2\omega {\lambda }_t\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\}=\\ & =\omega (\omega +2{\lambda }_t\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\})+{\gamma }_t\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}=\\ & =b_t+{\gamma }_t\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}\end{array}$$ where $b_t=\omega (\omega +2{\lambda }_t\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\})$. Iterating at time $t+2$: $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+2}^4\mid {\mathcal{F}}_t\}& =b_{t+1}+{\gamma }_{t+1}\mathbb{E}\{{\sigma }_{t+1}^4\mid {\mathcal{F}}_t\}=\\ & =b_{t+1}+{\gamma }_{t+1}{b}_t+{\gamma }_{t+1}{\gamma }_t\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}\end{array}$$ At time $t+3$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+3}^4\mid {\mathcal{F}}_t\}& =b_{t+2}+{\gamma }_{t+2}\mathbb{E}\{{\sigma }_{t+2}^4\mid {\mathcal{F}}_t\}=\\ & =b_{t+2}+{\gamma }_{t+2}{b}_{t+1}+{\gamma }_{t+2}{\gamma }_{t+1}{b}_t+{\gamma }_{t+2}{\gamma }_{t+1}{\gamma }_t\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_t\}\end{array}$$ Hence, in general $$\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_t\}=\sum _{i=1}^h{b}_{t+h-i}\prod _{j=1}^{i-1}{\gamma }_{t+h-j}+{\sigma }_t^4\prod _{j=1}^h{\gamma }_{t+h-j}$$ where we denote as
$$\begin{array}{rl}& {\gamma }_t={\alpha }_1^2\mathbb{E}\{u_t^4\}+{\beta }_1(2{\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)\\ & {\lambda }_t={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1\\ & b_t={\omega }^2+2{\lambda }_t\omega \mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_t\}\end{array}$$

Example: GARCH(1,1) iterative second moment

Example 22.3 With $h=1$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+1}^4\mid {\mathcal{F}}_t\}& =\sum _{i=1}^1{b}_{t+1-i}\prod _{j=1}^{i-1}{\gamma }_{t+1-j}+{\sigma }_t^4\prod _{j=1}^1{\gamma }_{t+1-j}=\\ & =b_t+{\gamma }_t{\sigma }_t^4\end{array}$$ With $h=2$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+2}^4\mid {\mathcal{F}}_t\}& =\sum _{i=1}^2{b}_{t+2-i}\prod _{j=1}^{i-1}{\gamma }_{t+2-j}+{\sigma }_t^4\prod _{j=1}^2{\gamma }_{t+2-j}=\\ & =b_{t+1}\prod _{j=1}^0{\gamma }_{t+2-j}+b_t\prod _{j=1}^1{\gamma }_{t+2-j}+{\sigma }_t^4{\gamma }_{t+1}{\gamma }_t=\\ & =b_{t+1}+b_t{\gamma }_{t+1}+{\sigma }_t^4{\gamma }_{t+1}{\gamma }_t\end{array}$$ With $h=3$ $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+3}^4\mid {\mathcal{F}}_t\}& =\sum _{i=1}^3{b}_{t+3-i}\prod _{j=1}^{i-1}{\gamma }_{t+3-j}+{\sigma }_t^4\prod _{j=1}^3{\gamma }_{t+3-j}=\\ & =b_{t+2}\prod _{j=1}^0{\gamma }_{t+3-j}+b_{t+1}\prod _{j=1}^1{\gamma }_{t+3-j}+b_t\prod _{j=1}^2{\gamma }_{t+3-j}+{\sigma }_t^4{\gamma }_{t+2}{\gamma }_{t+1}{\gamma }_t=\\ & =b_{t+2}+b_{t+1}{\gamma }_{t+2}+b_t{\gamma }_{t+2}{\gamma }_{t+1}+{\sigma }_t^4{\gamma }_{t+2}{\gamma }_{t+1}{\gamma }_t\end{array}$$ and so on.

# GARCH second moment iteration
e_sigma4_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_0){
  sigma4_iter <- numeric(h + 1)
  sigma4_iter[1] <- sigma4_0
  # Iterative formula for E[sigma^2]
  sigma2_iter <- numeric(h + 1)
  sigma2_iter[1] <- sqrt(sigma4_0)
  for(i in 1:h){
    lambda <- alpha * e_x2 + beta
    gamma <- alpha^2 * e_x4 + beta * (2 * alpha * e_x2 + beta)
    sigma2_iter[i+1] <- omega + lambda * sigma2_iter[i]
    b_t <- omega^2 + 2 * omega * lambda * sigma2_iter[i]
    sigma4_iter[i+1] <- b_t + gamma * sigma4_iter[i]
  }
  sigma4_iter
}

# GARCH second moment with formula
e_sigma4_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_0){
  # Step 1: initialize vectors
  sigma2_iter <- numeric(h + 1)
  sigma4_mix  <- numeric(h + 1)
  sigma2_iter[1] <- sqrt(sigma4_0)        # E[sigma_t^2]
  sigma4_mix[1]  <- sigma4_0              # E[sigma_t^4]

  # Step 2: Compute lambda and gamma constants
  lambda <- alpha * e_x2 + beta
  gamma  <- alpha^2 * e_x4 + beta * (2 * alpha * e_x2 + beta)

  # Step 3: Forward recursion to get all E[sigma^2_{t+h}]
  for(i in 1:h){
    sigma2_iter[i + 1] <- omega + lambda * sigma2_iter[i]
  }

  # Step 4: Compute each E[sigma^4_{t+h}]
  for(k in 1:h){
    # Prepare sum for b_t terms
    b_sum <- 0
    for(i in 1:k){
      idx <- k - i + 1  # corresponds to time t + (k - i)
      b_t <- omega^2 + 2 * omega * lambda * sigma2_iter[idx]
      b_sum <- b_sum + b_t * gamma^(i - 1)
    }
    sigma4_mix[k + 1] <- b_sum + sigma4_0 * gamma^k
  }

  names(sigma4_mix) <- paste0("t+", 0:h)
  return(sigma4_mix)
}

Moment	step	Formula	Iteration	Difference
$\mathbb{E}\{{\sigma }_{t+0}^4\mid {\mathcal{F}}_{t-1}\}$	0	1.440000	1.440000	0%
$\mathbb{E}\{{\sigma }_{t+1}^4\mid {\mathcal{F}}_{t-1}\}$	1	1.559380	1.559380	0%
$\mathbb{E}\{{\sigma }_{t+2}^4\mid {\mathcal{F}}_{t-1}\}$	2	1.609123	1.609123	0%
$\mathbb{E}\{{\sigma }_{t+3}^4\mid {\mathcal{F}}_{t-1}\}$	3	1.630762	1.630762	0%
$\mathbb{E}\{{\sigma }_{t+4}^4\mid {\mathcal{F}}_{t-1}\}$	4	1.640413	1.640413	0%
$\mathbb{E}\{{\sigma }_{t+5}^4\mid {\mathcal{F}}_{t-1}\}$	5	1.644775	1.644775	0%
$\mathbb{E}\{{\sigma }_{t+6}^4\mid {\mathcal{F}}_{t-1}\}$	6	1.646762	1.646762	0%
$\mathbb{E}\{{\sigma }_{t+7}^4\mid {\mathcal{F}}_{t-1}\}$	7	1.647669	1.647669	0%
$\mathbb{E}\{{\sigma }_{t+8}^4\mid {\mathcal{F}}_{t-1}\}$	8	1.648085	1.648085	0%
$\mathbb{E}\{{\sigma }_{t+9}^4\mid {\mathcal{F}}_{t-1}\}$	9	1.648275	1.648275	0%
$\mathbb{E}\{{\sigma }_{t+10}^4\mid {\mathcal{F}}_{t-1}\}$	10	1.648363	1.648363	0%

Table 22.3: Forecasted second moment of GARCH(1,1) variance with iteration and with formula.

22.2.2 Long-term

If $\mathbb{E}\{u_t^2\}$ and $\mathbb{E}\{u_t^3\}$ are constant for all $t$ then the formula simplifies to
$$\begin{array}{}\text{(22.9)}& \begin{array}{rl}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}& =({\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2)\left(\frac{1-{\gamma }^h}{1-\gamma }\right)+2\omega {\sigma }_{\mathrm{\infty }}^2{\lambda }^h\left(\frac{\lambda ({\lambda }^h-(1+\gamma {)}^h)}{{\lambda }^h(\lambda -1-\gamma )}\right)+{\sigma }_t^4{\gamma }^h\end{array}\end{array}$$ where ${\sigma }_{\mathrm{\infty }}^2$ denotes the long-term expected GARCH variance as $h\to \mathrm{\infty }$, i.e.  $$\begin{array}{}\text{(22.10)}& {\sigma }_{\mathrm{\infty }}^4=\underset{h\to \mathrm{\infty }}{lim}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}=\frac{{\omega }^2(1+{\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)}{(1-{\alpha }_1\mathbb{E}\{u_t^2\}-{\beta }_1)(1-{\alpha }_1^2\mathbb{E}\{u_t^4\}-2{\alpha }_1{\beta }_1\mathbb{E}\{u_t^2\}-{\beta }_1^2)}\text{.}\end{array}$$ It follows that, under normality $u_t\sim \mathcal{N}(0,1)$ we have that $\mathbb{E}\{u_t^2\}=1$ and $\mathbb{E}\{u_t^4\}=3$. Substituting, one obtain the same result as in Bollerslev (1986), i.e. $${\sigma }_{\mathrm{\infty }}^4=\frac{{\omega }^2(1+{\alpha }_1+{\beta }_1)}{(1-{\alpha }_1-{\beta }_1)(1-3{\alpha }_1^2-2{\alpha }_1{\beta }_1-{\beta }_1^2)}\text{.}$$

GARCH(1,1) long-term second moment

Proof. Under the assumption of constant second and fourth moments of $u_t$ one can simplify the expressions, i.e.
$$\begin{array}{rl}& \gamma ={\alpha }_1^2\mathbb{E}\{u_t^4\}+{\beta }_1(2{\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)\\ & b_{t+h-i}=\omega (\omega +2\lambda \mathbb{E}\{{\sigma }_{t+h-i}^2\mid {\mathcal{F}}_{t-1}\})\\ & \lambda ={\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1\end{array}$$ Recalling the expression of the expectation of the GARCH variance with constant moments in Equation 22.3 one can write $$b_{t+h-i}={\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2+2\omega {\lambda }^{h-i-1}{\sigma }_{\mathrm{\infty }}^2$$ Substituting the above expression into Equation 22.6 one obtain $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}& =\sum _{i=1}^h({\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2+2\omega {\lambda }^{h-i-1}{\sigma }_{\mathrm{\infty }}^2){\gamma }^{i-1}+{\sigma }_t^4{\gamma }^h\\ & =({\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2)\sum _{i=1}^h{\gamma }^{i-1}+2\omega {\sigma }_{\mathrm{\infty }}^2{\lambda }^h\sum _{i=1}^h{\left(\frac{1}{\lambda }\right)}^{i-1}{\gamma }^{i-1}+{\sigma }_t^4{\gamma }^h=\\ & =({\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2)\sum _{i=0}^{h-1}{\gamma }^i+2\omega {\sigma }_{\mathrm{\infty }}^2{\lambda }^h\sum _{i=0}^{h-1}{\left(\frac{1+\gamma }{\lambda }\right)}^i+{\sigma }_t^4{\gamma }^h\end{array}$$ Notably $$\sum _{i=0}^{h-1}{\gamma }^i=\frac{1-{\gamma }^h}{1-\gamma }$$ and $$\sum _{i=0}^{h-1}{\left(\frac{1+\gamma }{\lambda }\right)}^i=\frac{1-\frac{(1+\gamma {)}^h}{{\lambda }^h}}{1-\frac{1+\gamma }{\lambda }}=\frac{\lambda ({\lambda }^h-(1+\gamma {)}^h)}{{\lambda }^h(\lambda -1-\gamma )}$$ Hence, $$\begin{array}{rl}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}& =({\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2)\left(\frac{1-{\gamma }^h}{1-\gamma }\right)+2\omega {\sigma }_{\mathrm{\infty }}^2{\lambda }^h\left(\frac{\lambda ({\lambda }^h-(1+\gamma {)}^h)}{{\lambda }^h(\lambda -1-\gamma )}\right)+{\sigma }_t^4{\gamma }^h\end{array}$$ Taking the limit as $h\to infty$, the second and third terms converges to zero if $\lambda <1$, therefore $$\begin{array}{rl}\underset{h\to \mathrm{\infty }}{lim}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}& =\frac{{\omega }^2+2\omega \lambda {\sigma }_{\mathrm{\infty }}^2}{1-\gamma }\end{array}$$ More explicitly, $$\begin{array}{rl}\underset{h\to \mathrm{\infty }}{lim}\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}& =\frac{{\omega }^2(1+2\frac{\lambda }{1-\lambda })}{1-{\alpha }_1^2\mathbb{E}\{u_t^4\}-2{\alpha }_1{\beta }_1\mathbb{E}\{u_t^2\}-{\beta }_1^2}=\\ & =\frac{{\omega }^2(1+\lambda )}{(1-\lambda )(1-{\alpha }_1^2\mathbb{E}\{u_t^4\}-2{\alpha }_1{\beta }_1\mathbb{E}\{u_t^2\}-{\beta }_1^2)}=\\ & =\frac{{\omega }^2(1+{\alpha }_1\mathbb{E}\{u_t^2\}+{\beta }_1)}{(1-{\alpha }_1\mathbb{E}\{u_t^2\}-{\beta }_1)(1-{\alpha }_1^2\mathbb{E}\{u_t^4\}-2{\alpha }_1{\beta }_1\mathbb{E}\{u_t^2\}-{\beta }_1^2)}\end{array}$$

# Unconditional second moment GARCH variance (exact)
e_sigma4_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  # Expectation 
  e_sigma2 <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Derived quantities
  lambda <- alpha * e_x2 + beta
  gamma <- alpha^2 * e_x4 + 2 * alpha * beta + beta^2
  # Second moment  
  e_sigma2 * omega * (1 + lambda) / (1 - gamma)
}

# Unconditional second moment GARCH variance (exact)
e_sigma4_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  #1.352* omega^2 * (1 + alpha + beta) / ((1-alpha-beta) * (1 - beta^2 - 2*alpha*beta - e_x4 * alpha^2)) 
   #e_sigma2 * omega * (1 + alpha*e_x2 + beta) / ((1 - beta^2 - 2*alpha*beta*e_x2 - e_x4 * alpha^2)) 
  # Expectation 
  e_sigma2 <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Derived quantities
  lambda <- alpha * e_x2 + beta
  gamma <- alpha^2 * e_x4 + 2 * e_x2 * alpha * beta + beta^2
  # Second moment  
  e_sigma2 * omega * (1 + lambda) / (1 - gamma)
}

Moment	Formula	MonteCarlo	Difference
${\sigma }_{\mathrm{\infty }}^4$	1.648437	1.6486	-0.0099%

Table 22.4: Forecasted long-term second moment of GARCH(1,1) variance with formula and by Monte Carlo simulations.

22.3 Variance ${\sigma }_t^2$

The conditional variance of ${\sigma }_t^2$ with $h\ge 1$ reads $$\mathbb{V}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\mathbb{E}\{{\sigma }_{t+h}^4\mid {\mathcal{F}}_{t-1}\}-(\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}{)}^2\text{.}$$ Instead the long-term variance of ${\sigma }_t^2$ reads explicitly $$\mathbb{V}\{{\sigma }_t^2\}={\sigma }_{\mathrm{\infty }}^4-({\sigma }_{\mathrm{\infty }}^2{)}^2\text{.}$$

Example: GARCH(1,1) iterative variance

Example 22.4  

# Iterative GARCH variance formula
v_sigma2_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Second moment GARCH variance
  e_sigma4 <- e_sigma4_h_mix(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Variance
  e_sigma4 - e_sigma2^2
}
# Iterative GARCH variance formula
v_sigma2_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Second moment GARCH variance
  e_sigma4 <- e_sigma4_h_iter(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Variance
  e_sigma4 - e_sigma2^2
}

# Unconditional variance GARCH std.dev (approximated)
v_sigma2_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  # Expectation GARCH variance 
  e_sigma2 <- e_sigma4_inf(omega, alpha, beta, e_x2, e_x4)
  # Expectation GARCH std. dev 
  e_sigma <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Variance 
  e_sigma2 - e_sigma^2
}

Moment	step	Formula	Iteration	Difference
$\mathbb{V}\{{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}$	1	0.0352420	0.0352420	0%
$\mathbb{V}\{{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}$	2	0.0455820	0.0455820	0%
$\mathbb{V}\{{\sigma }_{t+3}^2\mid {\mathcal{F}}_{t-1}\}$	3	0.0489759	0.0489759	0%
$\mathbb{V}\{{\sigma }_{t+4}^2\mid {\mathcal{F}}_{t-1}\}$	4	0.0502199	0.0502199	0%
$\mathbb{V}\{{\sigma }_{t+5}^2\mid {\mathcal{F}}_{t-1}\}$	5	0.0507180	0.0507180	0%
$\mathbb{V}\{{\sigma }_{t+6}^2\mid {\mathcal{F}}_{t-1}\}$	6	0.0509296	0.0509296	0%
$\mathbb{V}\{{\sigma }_{t+7}^2\mid {\mathcal{F}}_{t-1}\}$	7	0.0510227	0.0510227	0%
$\mathbb{V}\{{\sigma }_{t+8}^2\mid {\mathcal{F}}_{t-1}\}$	8	0.0510646	0.0510646	0%
$\mathbb{V}\{{\sigma }_{t+9}^2\mid {\mathcal{F}}_{t-1}\}$	9	0.0510835	0.0510835	0%
$\mathbb{V}\{{\sigma }_{t+10}^2\mid {\mathcal{F}}_{t-1}\}$	10	0.0510922	0.0510922	0%

Table 22.5: Forecasted variance of GARCH(1,1) variance with iteration and with formula.

Moment	Formula	MonteCarlo	Difference
$\mathbb{V}\{{\sigma }_{\mathrm{\infty }}^2\}$	0.0510995	0.0510973	0.0043%

Table 22.6: Forecasted long-term variance of GARCH(1,1) variance with formula and by Monte Carlo simulations.

22.4 First moment ${\sigma }_t$

22.4.1 Short term

The expected value of the GARCH std. deviation can be approximated as ${\sigma }_{t+h}=({\sigma }_{t+h}^2{)}^{1/2}$ with a Taylor expansion $$\mathbb{E}\{{\sigma }_{t+h}\mid {\mathcal{F}}_{t-1}\}\approx \sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}}+\frac{1}{8}\frac{\mathbb{V}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}}{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}{\}}^{\frac{3}{2}}}\text{.}$$

Approximated GARCH(1,1) std. deviation with Taylor expansion

Proof. Let’s $X$ be a non-negative random variable, and let’s say one want to approximate: $$\mathbb{E}\{\sqrt{X}\}\text{.}$$ Let $f(x)=\sqrt{x}$ and expand $f(x)$ around the point $\mu =\mathbb{E}\{X\}$, i.e. $$\sqrt{X}\approx \sqrt{\mu }+\frac{1}{2}{\mu }^{-1/2}(X-\mu )-\frac{1}{8}{\mu }^{-3/2}(X-\mu {)}^2\text{,}$$ and take the expectation $$\mathbb{E}\{\sqrt{X}\}\approx \sqrt{\mu }+\frac{1}{2}{\mu }^{-1/2}(\mathbb{E}\{X\}-\mu )-\frac{1}{8}{\mu }^{-3/2}\mathbb{E}\{(X-\mu {)}^2\}\text{,}$$ where $\mathbb{E}\{X\}-\mu =0$ and $\mathbb{E}\{(X-\mu {)}^2\}=\mathbb{V}\{X\}$. Applying this result to the random variable ${\sigma }_{t+h}^2$ with $1<h$ and let’s approximate around the expected value with a Taylor expansion, i.e.  $${\sigma }_{t+h}=\sqrt{{\sigma }_{t+h}^2}\approx \sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}+\frac{1}{2}\frac{({\sigma }_{t+h}^2-\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\})}{\sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}}-\frac{1}{8}\frac{({\sigma }_{t+h}^2-\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}{)}^2}{\sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t{\}}^3}}\text{,}$$ Taking the expectation gives the result. $$\mathbb{E}\{{\sigma }_{t+h}\mid {\mathcal{F}}_t\}\approx \sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}-\frac{1}{8}\frac{\mathbb{V}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}{\sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t{\}}^3}}\text{.}$$

Example: GARCH(1,1) std.deviation iterative expectation.

Example 22.5  

# Conditional first moment GARCH std. dev (approximated)
e_sigma12_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Variance GARCH variance
  v_sigma2 <- v_sigma2_h_mix(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Moment to power 1/2 (Approximated)
  e_sigma2^(1/2) - (1/8) * v_sigma2 / sqrt(e_sigma2)^3
}
# Conditional first moment GARCH std. dev (approximated)
e_sigma12_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Variance GARCH variance
  v_sigma2 <- v_sigma2_h_iter(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Moment to power 1/2 (Approximated)
  e_sigma2^(1/2) - (1/8) * v_sigma2 / sqrt(e_sigma2)^3
}

Moment	step	Formula	Iteration	Difference
$\mathbb{E}\{{\sigma }_{t+0}\mid {\mathcal{F}}_{t-1}\}$	0	1.095445	1.095445	0%
$\mathbb{E}\{{\sigma }_{t+1}\mid {\mathcal{F}}_{t-1}\}$	1	1.107896	1.107896	0%
$\mathbb{E}\{{\sigma }_{t+2}\mid {\mathcal{F}}_{t-1}\}$	2	1.114145	1.114145	0%
$\mathbb{E}\{{\sigma }_{t+3}\mid {\mathcal{F}}_{t-1}\}$	3	1.117128	1.117128	0%
$\mathbb{E}\{{\sigma }_{t+4}\mid {\mathcal{F}}_{t-1}\}$	4	1.118522	1.118522	0%
$\mathbb{E}\{{\sigma }_{t+5}\mid {\mathcal{F}}_{t-1}\}$	5	1.119168	1.119168	0%
$\mathbb{E}\{{\sigma }_{t+6}\mid {\mathcal{F}}_{t-1}\}$	6	1.119466	1.119466	0%
$\mathbb{E}\{{\sigma }_{t+7}\mid {\mathcal{F}}_{t-1}\}$	7	1.119603	1.119603	0%
$\mathbb{E}\{{\sigma }_{t+8}\mid {\mathcal{F}}_{t-1}\}$	8	1.119666	1.119666	0%
$\mathbb{E}\{{\sigma }_{t+9}\mid {\mathcal{F}}_{t-1}\}$	9	1.119694	1.119694	0%
$\mathbb{E}\{{\sigma }_{t+10}\mid {\mathcal{F}}_{t-1}\}$	10	1.119708	1.119708	0%

Table 22.7: Forecasted expectation of GARCH(1,1) std. deviation with iteration and with formula.

22.4.2 Long term

The unconditional expected GARCH(1,1) std. deviation $${\sigma }_{\mathrm{\infty }}\approx \sqrt{{\sigma }_{\mathrm{\infty }}^2}+\frac{1}{8}\frac{\mathbb{V}\{{\sigma }_{\mathrm{\infty }}^2\}}{({\sigma }_{\mathrm{\infty }}^2{)}^{\frac{3}{2}}}\text{.}$$

Example: GARCH(1,1) std. deviation long-term expectation.

Example 22.6  

# Unconditional first moment GARCH std. dev (approximated)
e_sigma12_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  # Expectation GARCH variance 
  e_sigma2 <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Variance GARCH variance 
  v_sigma2 <- v_sigma2_inf(omega, alpha, beta, e_x2, e_x4)
  # Moment to power 1/2 (Approximated) 
  e_sigma2^(1/2) - (1/8) * v_sigma2 / sqrt(e_sigma2)^3
}

Moment	Formula	MonteCarlo	Difference
${\sigma }_{\mathrm{\infty }}$	1.119719	1.120499	-0.0697%

Table 22.8: Long-term expectation of GARCH(1,1) std. deviation with approximated formula and by Monte Carlo simulations.

22.5 Variance ${\sigma }_t$

22.5.1 Short term

The variance of the GARCH std. deviation can be approximated $$\mathbb{V}\{{\sigma }_{t+h}\mid {\mathcal{F}}_{t-1}\}\approx \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}-\mathbb{E}\{{\sigma }_{t+h}\mid {\mathcal{F}}_{t-1}{\}}^2\text{.}$$

Example: GARCH(1,1) std. deviation iterative variance

Example 22.7  

# Iterative GARCH variance formula (approximated)
v_sigma_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Expectation GARCH std. dev
  e_sigma <- e_sigma12_h_mix(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Variance
  e_sigma2 - e_sigma^2
}
# Iterative GARCH variance formula (approximated)
v_sigma_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_t){
  # Expectation GARCH variance
  e_sigma2 <- e_sigma2_h_iter(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  # Expectation GARCH std. dev
  e_sigma <- e_sigma12_h_iter(h, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  # Variance
  e_sigma2 - e_sigma^2
}

Moment	step	Formula	Iteration	Difference
$\mathbb{V}\{{\sigma }_{t+1}\mid {\mathcal{F}}_{t-1}\}$	1	0.0071262	0.0071262	0%
$\mathbb{V}\{{\sigma }_{t+2}\mid {\mathcal{F}}_{t-1}\}$	2	0.0090968	0.0090968	0%
$\mathbb{V}\{{\sigma }_{t+3}\mid {\mathcal{F}}_{t-1}\}$	3	0.0097164	0.0097164	0%
$\mathbb{V}\{{\sigma }_{t+4}\mid {\mathcal{F}}_{t-1}\}$	4	0.0099365	0.0099365	0%
$\mathbb{V}\{{\sigma }_{t+5}\mid {\mathcal{F}}_{t-1}\}$	5	0.0100227	0.0100227	0%
$\mathbb{V}\{{\sigma }_{t+6}\mid {\mathcal{F}}_{t-1}\}$	6	0.0100589	0.0100589	0%
$\mathbb{V}\{{\sigma }_{t+7}\mid {\mathcal{F}}_{t-1}\}$	7	0.0100747	0.0100747	0%
$\mathbb{V}\{{\sigma }_{t+8}\mid {\mathcal{F}}_{t-1}\}$	8	0.0100817	0.0100817	0%
$\mathbb{V}\{{\sigma }_{t+9}\mid {\mathcal{F}}_{t-1}\}$	9	0.0100849	0.0100849	0%
$\mathbb{V}\{{\sigma }_{t+10}\mid {\mathcal{F}}_{t-1}\}$	10	0.0100864	0.0100864	0%

Table 22.9: Forecasted variance of GARCH(1,1) std. deviation with iteration and with formula.

22.5.2 Long term

The long-term variance of the GARCH std. deviation can be approximated as $$\mathbb{V}\{{\sigma }_{\mathrm{\infty }}\}\approx {\sigma }_{\mathrm{\infty }}^2-({\sigma }_{\mathrm{\infty }}{)}^2\text{.}$$

Example: GARCH(1,1) std. deviation long-term variance

Example 22.8  

# Unconditional variance GARCH std.dev (approximated)
v_sigma_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  # Expectation GARCH variance 
  v_sigma2 <- v_sigma2_inf(omega, alpha, beta, e_x2, e_x4)
  # Expectation GARCH std. dev 
  e_sigma2 <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Variance 
  v_sigma2 / (4 * e_sigma2^2)
}

Moment	Formula	MonteCarlo	Difference
$\mathbb{V}\{{\sigma }_{\mathrm{\infty }}\}$	0.0079976	0.0084045	-5.0874%

Table 22.10: Long-term variance of GARCH(1,1) std. deviation with approximated formula and by Monte Carlo simulations.

22.6 Third moment ${\sigma }_t$

22.6.1 Short term

The expected value of ${\sigma }_{t+h}^3$ can be approximated with a Taylor expansion, i.e. ${\sigma }_{t+h}^3=({\sigma }_{t+h}^2{)}^{3/2}$. Then, we can approximate
$$\mathbb{E}\{{\sigma }_t^3\mid {\mathcal{F}}_{t-1}\}\approx \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}{\}}^{\frac{3}{2}}+\frac{3}{8}\frac{\mathbb{V}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}}{\sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}}}\text{.}$$

Example: GARCH(1,1) std. deviation iterative third moment.

Example 22.9  

# Conditional third moment GARCH std. dev (approximated)
e_sigma32_h_mix <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_0){
  # Expectation variance
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_0))
  # Variance variance
  v_sigma2 <- v_sigma2_h_mix(h, omega, alpha, beta, e_x2, e_x4, sigma4_0)
  # Moment to power 3/2 (Approximated)
  e_sigma2^(3/2) + (3/8) * v_sigma2 / sqrt(e_sigma2)
}

# Conditional third moment GARCH std. dev (approximated)
e_sigma32_h_iter <- function(h, omega, alpha, beta, e_x2 = 1, e_x4 = 3, sigma4_0){
  # Expectation variance
  e_sigma2 <- e_sigma2_h_iter(h, omega, alpha, beta, e_x2, sqrt(sigma4_0))
  # Variance variance
  v_sigma2 <- v_sigma2_h_iter(h, omega, alpha, beta, e_x2, e_x4, sigma4_0)
  # Moment to power 3/2 (Approximated)
  e_sigma2^(3/2) + (3/8) * v_sigma2 / sqrt(e_sigma2)
}

Moment	step	Formula	Iteration	Difference
$\mathbb{E}\{{\sigma }_{t+0}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	0	1.314534	1.314534	0%
$\mathbb{E}\{{\sigma }_{t+1}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	1	1.383623	1.383623	0%
$\mathbb{E}\{{\sigma }_{t+2}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	2	1.413526	1.413526	0%
$\mathbb{E}\{{\sigma }_{t+3}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	3	1.426837	1.426837	0%
$\mathbb{E}\{{\sigma }_{t+4}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	4	1.432849	1.432849	0%
$\mathbb{E}\{{\sigma }_{t+5}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	5	1.435585	1.435585	0%
$\mathbb{E}\{{\sigma }_{t+6}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	6	1.436836	1.436836	0%
$\mathbb{E}\{{\sigma }_{t+7}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	7	1.437408	1.437408	0%
$\mathbb{E}\{{\sigma }_{t+8}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	8	1.437670	1.437670	0%
$\mathbb{E}\{{\sigma }_{t+9}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	9	1.437791	1.437791	0%
$\mathbb{E}\{{\sigma }_{t+10}^{\frac{3}{2}}\mid {\mathcal{F}}_{t-1}\}$	10	1.437846	1.437846	0%

Table 22.11: Forecasted third moment of GARCH(1,1) std. deviation with iteration and with formula.

22.6.2 Long term

With a Taylor approximation, the long term third moment of the GARCH std. deviation reads $${\sigma }_{\mathrm{\infty }}^3\approx ({\sigma }_{\mathrm{\infty }}^2{)}^{\frac{3}{2}}+\frac{3}{8}\frac{\mathbb{V}\{{\sigma }_{\mathrm{\infty }}^2\}}{\sqrt{{\sigma }_{\mathrm{\infty }}^2}}\text{.}$$

Example: GARCH(1,1) std. deviation long-term third moment.

Example 22.10  

# Unconditional third moment GARCH std. dev (approximated)
e_sigma32_inf <- function(omega, alpha, beta, e_x2 = 1, e_x4 = 3){
  # Expectation GARCH variance 
  e_sigma2 <- e_sigma2_inf(omega, alpha, beta, e_x2)
  # Variance GARCH variance 
  v_sigma2 <- v_sigma2_inf(omega, alpha, beta, e_x2, e_x4)
  # Moment to power 3/2 (Approximated)
  e_sigma2^(3/2) + (3/8) * v_sigma2 / sqrt(e_sigma2)
}

Moment	Formula	MonteCarlo	Difference
${\sigma }_{\mathrm{\infty }}^{\frac{3}{2}}$	1.437893	1.436871	0.071%

Table 22.12: Long-term third moment of GARCH(1,1) std. deviation with approximated formula and by Monte Carlo simulations.

22.7 Covariance

The covariance between two GARCH variances at time $t$ and $t+h$ reads: $$\begin{array}{rl}\mathbb{C}v\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}& =\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{V}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}\end{array}$$ For a fixed $t$ and general $s$ and $h$, $$\begin{array}{rl}\mathbb{C}v\{{\sigma }_{t+s}^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}& =\left(\prod _{i=1}^{max(s,h)}{\lambda }_{t+max(s,h)-i}\right)\mathbb{V}\{{\sigma }_{min(s,h)}^2\mid {\mathcal{F}}_t\}\end{array}$$

Proof: Iterative formula for the covariance between GARCH(1,1) variances

Proof. Leveraging the tower property and conditioning one can write: $$\mathbb{E}\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\mathbb{E}\{{\sigma }_t^2\cdot \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}\mid {\mathcal{F}}_{t-1}\}$$ From our previous result we have: $$\mathbb{E}\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\}=\mathbb{E}\{{\sigma }_t^2\cdot (\omega \sum _{j=0}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i}+\prod _{i=1}^h{\lambda }_{t+h-i}\cdot {\sigma }_t^2)\mid {\mathcal{F}}_{t-1}\}$$ Hence, $$\mathbb{E}\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\left(\omega \sum _{j=0}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}+\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_{t-1}\}$$ By definition the covariance: $$\begin{array}{rl}\mathbb{C}v\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}& =\mathbb{E}\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}-\mathbb{E}\{{\sigma }_t^2\}\cdot \mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}=\\ & =\left(\omega \sum _{j=0}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^2\}+\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^4\}-\mathbb{E}\{{\sigma }_t^2\}[\omega \sum _{j=0}^{h-1}\prod _{i=1}^j{\lambda }_{t+h-i}+\prod _{i=1}^h{\lambda }_{t+h-i}\cdot \mathbb{E}\{{\sigma }_t^2\}]\end{array}$$ and simplify the first and last terms cancel and the one remain with $$\begin{array}{rl}\mathbb{C}v\{{\sigma }_t^2\cdot {\sigma }_{t+h}^2\mid {\mathcal{F}}_{t-1}\}& =\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^4\mid {\mathcal{F}}_{t-1}\}-\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{E}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}{\}}^2=\\ & =\left(\prod _{i=1}^h{\lambda }_{t+h-i}\right)\mathbb{V}\{{\sigma }_t^2\mid {\mathcal{F}}_{t-1}\}\end{array}$$

Example: GARCH(1,1) covariance between variances.

Example 22.11  

# Covariance 
cov_sigma2_h_mix <- function(h, omega, alpha, beta, e_x2, e_x4, sigma4_t){
  # Second moment
  m2 <- e_x2
  if (length(e_x2) != h){
    m2 = rep(e_x2, h)
  }
  # Fourth moment
  m4 <- e_x4
  if (length(e_x4) != h){
    m4 = rep(e_x4, h)
  }
  lambda <- (alpha * m2 + beta)
  # Variances 
  v_sigma2 <- v_sigma2_h_iter(h, omega, alpha, beta, m2[1], m4[1], sigma4_t)[h:1]
  # Covariances 
  cv_sigma2_h <- v_sigma2 * cumprod(lambda)
  return(cv_sigma2_h)
}
# Simulate multiple ACF 
sim_acf_garch <- function(B, h){
  acf_sigma2 <- 0
  acf_sigma <- 0
  for(i in 1:B){
    sim <- simulate_GARCH11(B = 1, n = 10000, sigma2_t, omega, alpha, beta, mu, sd, p)
    sim <- bind_rows(sim)
    acf_sigma2 <- acf_sigma2 + acf(sim$sigma2, plot = F, lag.max = h+1, type = "cov")$acf[,,1][-1]
    acf_sigma <- acf_sigma + acf(sqrt(sim$sigma2), plot = F, lag.max = h+1, type = "cov")$acf[,,1][-1]
  }
  structure(
    list(
      acf_sigma = acf_sigma/B,
      acf_sigma2 = acf_sigma2/B
    )
  )
}

Moment	step	Formula	MonteCarlo	Difference
$\mathbb{C}v\{{\sigma }_{t+0}^2,{\sigma }_{t+0}^2\mid {\mathcal{F}}_{t-1}\}$	0	0.0234411	0.0234789	-0.1614%
$\mathbb{C}v\{{\sigma }_{t+1}^2,{\sigma }_{t+1}^2\mid {\mathcal{F}}_{t-1}\}$	1	0.0107530	0.0107468	0.0569%
$\mathbb{C}v\{{\sigma }_{t+2}^2,{\sigma }_{t+2}^2\mid {\mathcal{F}}_{t-1}\}$	2	0.0049316	0.0049061	0.5168%
$\mathbb{C}v\{{\sigma }_{t+3}^2,{\sigma }_{t+3}^2\mid {\mathcal{F}}_{t-1}\}$	3	0.0022608	0.0022197	1.8173%
$\mathbb{C}v\{{\sigma }_{t+4}^2,{\sigma }_{t+4}^2\mid {\mathcal{F}}_{t-1}\}$	4	0.0010353	0.0009858	4.7902%
$\mathbb{C}v\{{\sigma }_{t+5}^2,{\sigma }_{t+5}^2\mid {\mathcal{F}}_{t-1}\}$	5	0.0004730	0.0004346	8.1299%
$\mathbb{C}v\{{\sigma }_{t+6}^2,{\sigma }_{t+6}^2\mid {\mathcal{F}}_{t-1}\}$	6	0.0002149	0.0001262	41.2696%
$\mathbb{C}v\{{\sigma }_{t+7}^2,{\sigma }_{t+7}^2\mid {\mathcal{F}}_{t-1}\}$	7	0.0000962	0.0000081	91.5452%
$\mathbb{C}v\{{\sigma }_{t+8}^2,{\sigma }_{t+8}^2\mid {\mathcal{F}}_{t-1}\}$	8	0.0000411	-0.0000288	170.2412%
$\mathbb{C}v\{{\sigma }_{t+9}^2,{\sigma }_{t+9}^2\mid {\mathcal{F}}_{t-1}\}$	9	0.0000146	-0.0000174	219.2467%
$\mathbb{C}v\{{\sigma }_{t+10}^2,{\sigma }_{t+10}^2\mid {\mathcal{F}}_{t-1}\}$	10	0.0000000	-0.0000047	Inf%

Table 22.13: Forecasted covariance between GARCH(1,1) variances with formula and by monte Carlo simulations

For the covariance between the GARCH std. deviations, we apply a Taylor approximation around the mean of ${\sigma }_{t+s}$ and ${\sigma }_{t+h}$, using the delta method: $$\mathbb{C}v\{{\sigma }_{t+s},{\sigma }_{t+h}\mid {\mathcal{F}}_t\}\approx \frac{\mathbb{C}v\{{\sigma }_{t+s}^2,{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}{4\sqrt{\mathbb{E}\{{\sigma }_{t+s}^2\mid {\mathcal{F}}_t\}\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}}\text{.}$$

Approximated GARCH(1,1) covariance with Taylor expansion

Proof. Considering the product of $${\sigma }_{t+h}=\sqrt{{\sigma }_{t+h}^2}\approx \sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}+\frac{1}{2}\frac{({\sigma }_{t+h}^2-\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\})}{\sqrt{\mathbb{E}\{{\sigma }_{t+h}^2\mid {\mathcal{F}}_t\}}}\text{,}$$ and applying the covariance between ${\sigma }_{t+h}$ and ${\sigma }_{t+s}$ with $1<s<h$ one obtain the result.

Example: GARCH(1,1) covariance between std. deviations

Example 22.12  

# Covariance 
cov_sigma_h_mix <- function(h, omega, alpha, beta, e_x2, e_x4, sigma4_t){
  # Covariance 
  cv <- cov_sigma2_h_mix(h+1, omega, alpha, beta, e_x2, e_x4, sigma4_t)
  e_sigma2 <- e_sigma2_h_mix(h, omega, alpha, beta, e_x2, sqrt(sigma4_t))
  cv / (4 * sqrt(sigma4_t) * e_sigma2)
}

Moment	step	Formula	MonteCarlo	Difference
$\mathbb{C}v\{{\sigma }_{t+0},{\sigma }_{t+0}\mid {\mathcal{F}}_{t-1}\}$	0	0.0040696	0.0039637	2.6029%
$\mathbb{C}v\{{\sigma }_{t+1},{\sigma }_{t+1}\mid {\mathcal{F}}_{t-1}\}$	1	0.0018146	0.0018399	-1.3983%
$\mathbb{C}v\{{\sigma }_{t+2},{\sigma }_{t+2}\mid {\mathcal{F}}_{t-1}\}$	2	0.0008217	0.0008452	-2.8701%
$\mathbb{C}v\{{\sigma }_{t+3},{\sigma }_{t+3}\mid {\mathcal{F}}_{t-1}\}$	3	0.0003745	0.0003835	-2.4081%
$\mathbb{C}v\{{\sigma }_{t+4},{\sigma }_{t+4}\mid {\mathcal{F}}_{t-1}\}$	4	0.0001710	0.0001710	0.009%
$\mathbb{C}v\{{\sigma }_{t+5},{\sigma }_{t+5}\mid {\mathcal{F}}_{t-1}\}$	5	0.0000781	0.0000754	3.3625%
$\mathbb{C}v\{{\sigma }_{t+6},{\sigma }_{t+6}\mid {\mathcal{F}}_{t-1}\}$	6	0.0000354	0.0000227	36.0148%
$\mathbb{C}v\{{\sigma }_{t+7},{\sigma }_{t+7}\mid {\mathcal{F}}_{t-1}\}$	7	0.0000159	0.0000022	85.9823%
$\mathbb{C}v\{{\sigma }_{t+8},{\sigma }_{t+8}\mid {\mathcal{F}}_{t-1}\}$	8	0.0000068	-0.0000045	166.2259%
$\mathbb{C}v\{{\sigma }_{t+9},{\sigma }_{t+9}\mid {\mathcal{F}}_{t-1}\}$	9	0.0000024	-0.0000031	229.793%
$\mathbb{C}v\{{\sigma }_{t+10},{\sigma }_{t+10}\mid {\mathcal{F}}_{t-1}\}$	10	0.0000000	-0.0000007	Inf%

Table 22.14: Forecasted covariance between GARCH(1,1) std. deviation with formula and by monte Carlo simulations

Bollerslev, Tim. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (3): 307–27. https://doi.org/10.1016/0304-4076(86)90063-1.