Let’s consider a very general setup for GARCH(1,1) process where we do not assume that the GARCH residuals
Has expected value equal to zero, i.e. \(\mathbb{E}\{u_t\} = 0\).
Has NOT necessary a second moment that is constant and equal to one, i.e. \(\mathbb{E}\{u_t^2\}\) possibly time dependent (deterministic).
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. \[
\begin{aligned}
{} & Y_t = \sigma_t u_t \\
& \sigma_t^2 = \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2
\end{aligned}
\] 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 \(h\)-steps ahead, with \(h\ge1\), can be expanded as: \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \omega\left(
1 + \sum_{j=1}^{h-1} \prod_{i=1}^{j} \lambda_{t+h-i}\right) + \sigma_t^2 \prod_{j=1}^{h} \lambda_{t+h-j}
\text{,}
\tag{22.1}\] with the convention that when \(h=1\) the summation is equal to zero and where \[
\lambda_{t+h-i} = \alpha_1 \mathbb{E}\{u_{t+h-i}^2\} + \beta_1
\text{.}
\tag{22.2}\] The iteration between two consecutive times \(t+s\) and \(t+s-1\) reads \[
\mathbb{E}\{\sigma_{t+s}^2 \mid \mathcal{F}_{t-1} \} = \omega + \lambda_{t+s-1} \mathbb{E}\{\sigma_{t+s-1}^2 \mid \mathcal{F}_{t-1}\}
\text{.}
\]
Proof: GARCH(1,1) iterative expectation
Proof. We start the proof by recalling that the conditional variance at time \(t\) is included in the information set \(\mathcal{F}_{t-1}\), i.e it is known, therefore at time \(t\) one obtain simply \[
\sigma_{t}^2 = \omega + \alpha_1 y_{t-1}^2 + \beta_1 \sigma_{t-1}^2
\text{.}
\] Then, the GARCH variance at time \(t+1\) reads: \[
\sigma_{t+1}^2 = \omega + \alpha_1 y_{t}^2 + \beta_1 \sigma_{t}^2
\text{.}
\] Taking the conditional expectation and recalling that \(y_{t} = \sigma_t u_t\)\[
\begin{aligned}
\mathbb{E}\{\sigma_{t+1}^2 \mid \mathcal{F}_{t-1} \} {} & = \omega + \alpha_1 \mathbb{E}\{y_{t}^2\} + \beta_1 \sigma_{t}^2 = \\
& = \omega + \alpha_1 \sigma_t^2 \mathbb{E}\{u_{t}^2\} + \beta_1 \sigma_{t}^2 = \\
& = \omega + (\alpha_1 \mathbb{E}\{u_{t}^2\} + \beta_1) \sigma_{t}^2
\end{aligned}
\] Therefore, denoting with \(\lambda_t = \alpha_1 \mathbb{E}\{u_t^2\} + \beta_1\) one obtain a recursive expression, i.e. \[
\mathbb{E}\{\sigma_{t+1}^2 \mid \mathcal{F}_{t-1} \} = \omega + \lambda_t \sigma_{t}^2
\text{.}
\] Iterating the expectation at time \(t+2\) gives \[
\begin{aligned}
\mathbb{E}\{\sigma_{t+2}^2 \mid \mathcal{F}_{t-1} \} &= \omega + \lambda_{t+1} \mathbb{E}\{\sigma_{t+1}^2 \mid \mathcal{F}_{t-1}\} \\
&= \omega + \lambda_{t+1}\left(\omega + \lambda_{t} \sigma_t^2 \right) \\
&= \omega (1 + \lambda_{t+1}) + \lambda_{t+1}\lambda_{t} \sigma_t^2
\end{aligned}
\] Then, at time \(t+3\)\[
\begin{aligned}
\mathbb{E}\{\sigma_{t+3}^2 \mid \mathcal{F}_{t-1} \} &= \omega + \lambda_{t+2}\,\mathbb{E}\{\sigma_{t+2}^2 \mid \mathcal{F}_{t-1} \} \\
&= \omega + \lambda_{t+2}\left(\omega (1+\lambda_{t+1}) + \lambda_{t+1} \lambda_t \sigma_t^2 \right) \\
&= \omega\left(1 + \lambda_{t+2} + \lambda_{t+2} \lambda_{t+1}\right) + \lambda_{t+2} \lambda_{t+1} \lambda_t \sigma_t^2
\end{aligned}
\]
#' 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_x2if (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_x2if (length(e_x2) ==1& h >1){ m2 <-rep(e_x2, h) }for(i in1:h){ sigma2_h[i+1] <- omega + (alpha * m2[i] + beta) * sigma2_h[i] }names(sigma2_h) <-paste0("t+", 0:h)return(sigma2_h)}
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 \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1} \} = \sigma_{\infty}^2 + \lambda^h (\sigma_t^2 - \sigma_{\infty}^2)
\text{,}
\tag{22.3}\] where \(\sigma_{\infty}^2\) denotes the long-term expected GARCH variance as \(h\to\infty\), i.e. \[
\lim_{h\to\infty} \mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1} \} = \frac{\omega}{1- \lambda} = \sigma_{\infty}^2
\text{.}
\tag{22.4}\] It follows that, under the standard assumption that \(u_t \sim \text{WN}(0, 1)\), then one obtain the classic expression \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \sigma_{\infty}^2 + (\alpha_1 + \beta_1)^h (\sigma_t^2 - \sigma_{\infty}^2)
\tag{22.5}\] where \(\sigma_{\infty}^2\) 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. \[
\begin{aligned}
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1} \} & {} =
\omega\left(1 + \sum_{j=1}^{h-1} \prod_{i=1}^{j}\lambda \right) + \sigma_t^2 \prod_{j=1}^{h} \lambda = \\
& = \omega \sum_{j=0}^{h-1} \lambda^j + \sigma_t^2 \lambda^h = \\
& = \omega \left(\frac{1- \lambda^h}{1 - \lambda} \right) + \sigma_t^2 \lambda^h \\
& = \frac{\omega}{1- \lambda} + \lambda^h \left(\sigma_t^2 - \frac{\omega}{1-\lambda} \right)
\end{aligned}
\] Taking the limit as \(h \to \infty\) gives the long term stationary variance, i.e. \[
\lim_{h\to\infty} \mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \frac{\omega}{1- \lambda} = \sigma_{\infty}^2 \iff \lambda < 1
\] Hence, the general expression became \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \sigma_{\infty}^2 + \lambda^h (\sigma_t^2 - \sigma_{\infty}^2)
\]
Example: GARCH(1,1) long-term first moment
Example 22.2
Unconditional expected value of GARCH(1,1) variance
Table 22.5: Forecasted variance of GARCH(1,1) variance with iteration and with formula.
Moment
Formula
MonteCarlo
Difference
\(\mathbb{V}\{\sigma_{\infty}^2\}\)
0.0510995
0.0511493
-0.0975%
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 \[
\mathbb{E}\{\sqrt{X}\} \approx
\sqrt{\mu} + \frac{1}{2} \mu^{-1/2} \left(\mathbb{E}\{X\} - \mu \right) - \frac{1}{8} \mu^{-3/2} \mathbb{E}\{(X - \mu)^2\}
\text{,}
\] 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. \[
\sigma_{t+h} = \sqrt{\sigma_{t+h}^2} \approx
\sqrt{\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}} + \frac{1}{2} \frac{(\sigma_{t+h}^2 -\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}) }{\sqrt{\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}}} - \frac{1}{8} \frac{(\sigma_{t+h}^2 -\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\})^2 }{\sqrt{\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}^3}}
\text{,}
\] Taking the expectation gives the result. \[
\mathbb{E}\{\sigma_{t+h} \mid \mathcal{F}_{t-1}\} \approx
\sqrt{\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}} - \frac{1}{8} \frac{\mathbb{V}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}}{\sqrt{\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}^3}}
\text{.}
\]
# 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
\(\sigma_{\infty}\)
1.119719
1.120488
-0.0687%
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 \[
\mathbb{V}^{\mathbb{P}}\{\sigma_{t+h} \mid \mathcal{F}_{t-1} \} \approx \frac{\mathbb{V}^{\mathbb{P}}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}}{4\mathbb{E}^{\mathbb{P}}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\}}
\text{,}
\]
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 \[
\mathbb{V}\{\sigma_{\infty}\} \approx \frac{\mathbb{V}\{\sigma_{\infty}^2\}}{\sigma_{\infty}^2}
\text{.}
\]
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
Iterative formula for third moment of GARCH std. deviation.
# 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)}
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 \[
\sigma^3_{\infty} \approx (\sigma_{\infty}^2)^{\frac{3}{2}} + \frac{3}{8} \frac{\mathbb{V}\{\sigma_{\infty}^2\}}{\sqrt{\sigma_{\infty}^2}}
\text{.}
\]
Example: GARCH(1,1) std. deviation long-term third moment.
Example 22.10
Long-term third moment of GARCH std. deviation.
# 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
\(\sigma_{\infty}^{\frac{3}{2}}\)
1.437893
1.436856
0.0721%
Table 22.12: Long-term third moment of GARCH(1,1) std. deviation with approximated formula and by Monte Carlo simulations.
22.7 Covariance
Given the information at time \(t-1\), the covariance between two GARCH variances at time \(t+h\) and \(t+s\) reads \[
\mathbb{C}v\{\sigma_{t+s}^2 \cdot \sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \left( \prod_{i=1}^{|h-s|} \lambda_{t+\max(s,h)-i} \right) \mathbb{V}\{\sigma_{\min(s,h)}^2 \mid \mathcal{F}_{t-1}\}
\] It follows that the covariance between any variance at time \(t+h\) with \(h>1\) and the variance at time \(t\), given the infomation at time \(t-1\), is zero, i.e. \[
\begin{aligned}
\mathbb{C}v\{\sigma_t^2 \cdot \sigma_{t+h}^2 \mid \mathcal{F}_{t-1} \} {} & = \left( \prod_{i=1}^{h} \lambda_{t+h-i} \right) \mathbb{V}\{\sigma_t^2 \mid \mathcal{F}_{t-1}\} = 0
\end{aligned}
\] for all \(h \ge 1\) since the variance of \(\sigma_t^2\), given \(\mathcal{F}_{t-1}\) is zero.
Proof: Iterative formula for the covariance between GARCH(1,1) variances
Proof. Without loss of generality, consider \(s < h\). Then, leveraging the tower property of the conditional expectation we can write: \[
\mathbb{E}\{\sigma_{t+s}^2 \cdot \sigma_{t+h}^2 \mid \mathcal{F}_{t-1} \} =
\mathbb{E}\{ \sigma_{t+s}^2 \cdot \mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t+s-1}\} \mid \mathcal{F}_{t-1}\}
\] From our previous result we have: \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t+s-1}\} = \omega\left(
1 + \sum_{j=1}^{h-s-1} \prod_{i=1}^{j} \lambda_{t+h-i}\right) + \sigma_{t+s}^2 \prod_{j=1}^{h-s} \lambda_{t+h-j}
\text{,}
\] Therefore, denoting as \[
\mathbb{E}\{\sigma_{t+h}^2 \mid \mathcal{F}_{t+s-1}\} = A_{h|s-1} + \sigma_{t+s}^2 B_{h|s}
\text{,}
\] We obtain that the covariance by definition is: \[
\mathbb{C}v\{\sigma_{t+s}^2, \sigma_{t+h}^2 \mid \mathcal{F}_{t-1}\} = \mathbb{V}\{\sigma_{t+s}^2 \mid \mathcal{F}_{t-1}\} \prod_{j=1}^{h-s} \lambda_{t+h-j}
\text{,}
\]
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