Using Overlapping Samples to Estimate GARCH Parameters
Summary
The document considers estimating a GARCH(1,1) model from overlapping return intervals to increase the number of calibration observations. It proposes fitting the variance parameters jointly across interval sequences offset in time, using a Gaussian likelihood contribution for each sequence. The answer confirms that this can produce point estimates and suggests using every possible set of consecutive nonoverlapping periods to make fuller use of the observations.
Overlapping sequences create dependence between the resulting observations, so conventional standard errors should not be treated as reliable without adjustment. The author also cautions that the increase in estimation accuracy may be small when the total sample is long relative to the model interval. With very few nonoverlapping periods, the estimates may remain unreliable even if the underlying data are sampled more finely. The discussion provides methodological guidance and illustrative examples, but no simulations or quantitative comparison of estimator performance.
Key ideas
- Offset nonoverlapping interval sequences can be combined to increase data used for GARCH point estimation.
- The likelihood can be optimized jointly over the variance parameters across those sequences.
- Overlap induces dependence, so ordinary standard errors may not be valid.
- The accuracy gain is likely limited when the sample span is large relative to the model period.
- A small number of nonoverlapping periods can still produce unreliable estimates.
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Full text
# GARCH calibration with overlapping time intervals
# GARCH calibration with overlapping time intervals
In constructing a GARCH(1,1) model over a time length $\delta$, I am considering the following procedure. The purpose of this procedure is to give more training (calibrating) samples than non-overlapping consecutive samples. \begin{align} \nu(k\delta,(k+1)\delta) &= \omega+\alpha r((k-1)\delta,k\delta)^2+\beta \nu((k-1)\delta,k\delta) \\ \nu\big((k+0.5)\delta,(k+1.5)\delta\big) &= \omega+\alpha\,r\big((k-0.5)\delta,(k+0.5)\delta\big)^2+\beta\,\nu\big((k-0.5)\delta,(k+0.5)\delta\big) \end{align} for $k\in \bf \bar{Z^-}$, where $\nu(s,t)$ stands for the variance between time $s$ and $t$, $r(s,t)$ the return between $s$ and $t$. I calibrate the parameter tuple $(\omega, \alpha, \beta)$ simultaneously with the above equations for overlapping time intervals.
I use the following method to estimate the parameters. Define $\displaystyle l(k):=\ln v(k\delta,(k+1)\delta)+\frac{r(k\delta,(k+1)\delta)^2}{v(k\delta,(k+1)\delta))}$ and $$-2\,\text{Log-Likelihood}=(k_1+k_2)\ln(2\pi)+\sum_{k}l(k)+\sum_{k}l(k-0.5)$$ where $k_1$ and $k_2$ are the numbers of intervals in the two non-overlapping sequences of intervals. This function $-2\,\text{Log-Likelihood}$ is then minimized for over $(\omega,\alpha,\beta)$.
Is this legitimate?
## Answer by Richard Hardy (score 3)
https://quant.stackexchange.com/a/65635
Yes, this is legitimate for obtaining point estimates of $\omega,\alpha,\beta$. To utilize the data fully, you would find all sets of consecutive nonoverlapping periods and use all of them in estimation. (In financial econometrics and time series analysis, estimation is the common word for what you mean here with calibration.) E.g. if you have daily data (weekdays only) and want a weekly GARCH model, you would fit it on five observation sets with time indices $(1,6,11,\dots)$, $(2,7,12,\dots)$, $\dots$, $(5,10,15,\dots)$ for the most efficient use of data. You would not take the standard errors at face value, though, since they would now be based on overlapping data.
However, I would not expect a great improvement in estimation accuracy unless the length of a single time period (a month) is a large fraction (say, 1/5 or larger) of the total sample span. E.g. if you have 240 months of daily data for a monthly GARCH model, you will not gain much from this approach. If you only had 5 months, that could have a noticeable effect. On the other hand, estimating a GARCH model given only 5 nonoverlapping periods of data (however finely sampled) would not give a reliable result anyway...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.