# IGARCH model property (conditional distribution) when used to model sum of log returns

I already asked this in the quant, crossvalidated, and math SEs, but no help there. I'm not sure many people are familiar with whatever I'm asking, and I tried rewording the question too, but seems like no one is coming up with any ideas. I was hoping someone here could help and that the question is acceptable.

According to Tsay's book (Analysis of Financial Time Series) 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!

I also formulated the question in any abbreviated form if anyone finds that clearer:

According to Tsay's book (Analysis of Financial Time Series) in Chapter 7, for the Risk Metrics model, the following sum, $$r_{t+1} + r_{t+2}$$, should be conditionally normal distributed.

$$σ_t^2 = ασ_{t-1}^2 + (1 − α)r_{t-1}^2 = σ_{t-1}^2 + (1 − α)σ_{t-1}^2(\epsilon_{t-1}^2 - 1)$$
$$r_t = σ_t * \epsilon_t$$
$$\epsilon_t ∼ N(0,1)$$

According to the book $$r_{t+1} + r_{t+2}$$ conditional on $$\epsilon_{t}$$ & $$σ_t$$ is normally distributed. The $$r_{t+1}$$ term makes sense to be conditional normally distributed given $$\epsilon_{t}$$ & $$σ_t$$ since then the $$σ_{t+1}$$ term in $$r_{t+1} = σ_{t+1}* \epsilon_{t+1}$$ is known, and therefore $$r_{t+1}$$ is just a normal random variable.

But for $$r_{t+2}$$, 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.