Daily GARCH Forecasts from Daily Returns Versus Intraday Return Recursion
Summary
The document compares two ways to forecast daily variance with a GARCH(1,1) model when five-minute returns are available. In the first approach, the model is fit or applied at the daily horizon, using a realized-variance estimate and a prior daily variance forecast. In the second, variance is updated recursively at the five-minute interval, with returns generated along simulated paths, and interval forecasts are summed to obtain a daily expectation.
The question focuses on how these approaches differ, including their time scales and the role of simulation. It specifies recursive equations but provides no answer, empirical comparison, or guidance on parameter estimation, aggregation assumptions, or forecast accuracy. Thus it identifies a modeling choice rather than establishing that either approach is superior; the appropriateness of each depends on the intended horizon and how the intraday dynamics are calibrated.
Key ideas
- A daily GARCH forecast can update variance using daily returns and realized-variance estimates.
- An intraday approach recursively updates five-minute variance using simulated returns.
- The intraday method aggregates interval forecasts across the day to estimate daily variance.
- The document poses a comparison but provides no evidence about relative accuracy or parameter calibration.
Tags
Full text
# GARCH(1,1) variance forecast in one-step or multi-step?
# 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?
- 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.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.