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.



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