# Distributional forecast for $X_{t+1}$ in GARCH(1,1) with residuals student t distributed

Let's say I have time series of $n$ (simulated) data from the GARCH(1,1) process: $$X_{t+1,n} = \sigma_{t+1,n}(...)*Z_{t+1,n}$$ where $Z_{t,n}$ is iid student t-distributed with $v$ degree of freedom.
I want to create a distributional forecast for $X_{t+1,n}$. When working with $Z$ normally distributed I simply let $X_{t+1,n}$ have the conditional distribution: $N(0,\sigma_{t,n}^2)$.

Is it different when the distribution of $Z$ is student-t? In general, how do I create distributional forecast for $X_{t+1,n}$ when $(X_{1,n},...X_{t,n},\sigma_{1,n},...,\sigma_{t,n}$) is known?

(my purpose is application so a theoretical derivation is not necessary)

• I assumes $\sigma^2$ is a typo? You mean $X_{t+1}=\sigma_t(..)*Z_t$ Nov 5, 2017 at 13:49
• I believe it is standard notation to have $X_{t+1} = \sigma_{t+1}^2 Z_{t+1}$, where $\sigma_{t+1}^2 = \omega + \alpha \varepsilon_t^2 + \beta \sigma_t^2$. Jan 20, 2018 at 17:43
• @JohanStaxJakobsen you mean $X_{t+1} = \sigma_{t+1} Z_{t+1}$ and $\sigma^2_{t+1} = \omega + \alpha X_{t}^2 + \beta \sigma^2_t$ Jan 20, 2018 at 19:01
• @Taylor Yes - of course. Sorry. Jan 20, 2018 at 19:05

The distributional forecast of $X_{t+1,n}$ is a non-standardized Student's t-distribution with location parameter $\mu = 0$ and scale parameter $\sigma_{t+1,n}$.