# GARCH(1,1) variance forecast in one-step or multi-step?

I would like to forecast the daily variance of a stock using GARCH(1,1) model while I have high frequency data of 5 minute returns. What is the difference between applying GARCH(1,1) in one-step forecast on the daily returns and applying GARCH(1,1) in multiple-step forecast on the 5 minute returns?

More specifically, let $$r(s,t)$$ and $$h(s,t)$$ denote the return and variance respectively over the time interval $$[s,t]$$. What is the difference between the following two approaches?

1. One step forecast. A GARCH(1,1) model for subsamples at time intervals length $$k\delta$$ on a stock return time series $$\big(r(i\delta,(i+1)\delta)\big)_{i=0}^{kq-1}$$ each element of which is the return between time $$i\delta$$ and $$(i+1)\delta$$. $$h(t,t+k\delta) = c_1+a_1\,u(t-k\delta,t)^2 +b_1\,h(t-k\delta,t) \tag1$$ where $$h(t-k\delta,t)$$ is the forecast variance for time interval $$[t-k\delta,t]$$ and $$u(t-k\delta,t)$$ is the variance estimation for time interval $$(t-k\delta,t)$$. It is computed in the framework of the realized variance as described in this question.

2. Multi-step forecast. We recurse over $$i$$ for $$h\big(t+i\delta,t+(i+1)\delta\big) = c_2+a_2\,r\big(t+(i-1)\delta,t+i\delta\big)^2 +b_2\,h\big(t+(i-1)\delta,t+i\delta\big) \tag2$$ where $$r\big(t+(i-1)\delta,t+i\delta\big)$$ is computed by simulation according to the distribution of the returns (Gaussian, student-t, etc). Then $$h(t,t+k\delta)=\mathbf E_{\text{simulation paths}}\sum_{i=0}^{k-1}h\big(t+i\delta,t+(i+1)\delta\big)$$ where $$\mathbf E_{\text{simulation paths}}$$ denote taking the expectation or just the average over the simulation paths.