1
$\begingroup$

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.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.