RiskMetrics IGARCH and the Distribution of Multi-Period Returns
Summary
The document asks whether the sum of log returns over multiple periods is conditionally normal under the RiskMetrics IGARCH(1,1) specification. It outlines the model’s zero conditional mean, normal one-step innovations, and variance recursion, then explains the difficulty: the next return’s volatility is known at the forecast origin, but later conditional variances depend on intervening random innovations. Thus later returns do not each have a known, fixed conditional variance given the initial information set.
The cited claim says that the multi-period return is conditionally normal with mean zero and a forecast variance. The questioner doubts this follows from the stated equations and notes that some sources may simply repeat the result or present it as a model assumption. The document provides no resolution or proof, so it serves as a prompt to distinguish a normal one-step conditional distribution from the distribution of an aggregate whose future variance is itself random. It does not establish whether the quoted multi-period claim is valid under the stated model.
Key ideas
- The IGARCH specification uses normal one-step innovations and a recursively updated variance.
- At the forecast origin, the next-period variance is known, while later variances depend on future innovations.
- A sum of returns may not inherit normality just because each one-step innovation is normal.
- The document questions whether the cited multi-period normality result follows from the equations.
- It presents the issue without providing a proof or final resolution.
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Full text
# RiskMetrics VAR calculations and conditional distribution of sum of log returns
# RiskMetrics VAR calculations and conditional distribution of sum of log returns
According to Tsay's book in Chapter 7, for the Risk Metrics model:
> A nice property of such a special random-walk IGARCH model is that the conditional distribution of a multiperiod return is easily available. Specifically, for a k-period horizon, the log return from time t + 1 to time t + k (inclusive) is rt [k] = rt+1 + · · · + rt+k−1 + rt+k. We use the square bracket [k] to denote a k horizon return. Under the special IGARCH(1,1) model in Eq. (7.2), the conditional distribution $r_t[k]|F_t$ is normal with mean zero and variance $σ_t^2[k]$, where $σ_t^2[k]$ can be computed using the forecasting method discussed in Chapter 3.
The RiskMetric IGARCH model is with the assumption that $r_t|F_{t−1} ∼ N(µ_t, σ_t^2)$, where $µ_t = 0$ is the conditional mean and $σ_t^2$ is the conditional variance of $r_t$. The following equations are satisfied:
$µ_t = 0$ $σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2$ also written: $σ_t^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(\epsilon_{t-1}^2 - 1)$ for all t $1 > α > 0$ $r_t = σ_t * \epsilon_t$ is an IGARCH(1,1) process without drift $\epsilon_t ∼ N(0,1)$
I don't see from this how the sum of the log returns are conditional normally distributed. The $r_{t+1}$ term makes sense to be conditional normally distributed given $F_{t}$ since then the $σ_{t+1}$ term in $σ_{t+1}* \epsilon_{t+1}$ is known, and therefore $r_{t+1}$ is just a normal random variable. But for higher values $r_{t+2}$, etc, the $σ_{t+2}^2 =σ_{t+1}^2 + (1 − α)σ_{t+1}^2(\epsilon_{t+1}^2 - 1)$ is not known and is still a random variable. So I don't see how $r_{t+2}$ is conditionally normally distributed.
I've searched all over the internet and it seems that some people just state this without any details (seems like they just used Tsay's book as the source) and some places say that it's an assumption made by the RiskMetrics model. If it's just an assumption made by the model, I still don't see how the equations agree with the conditional distribution of sum of log returns though.
Any help would be greatly appreciated. Thanks!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.