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?

• What exactly are you trying to accomplish with this? :-)
– Pleb
Jun 19 '21 at 21:01
• @Pleb: I just added the motivation. Does it make sense to you?
– Hans
Jun 20 '21 at 2:27

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...

• Good answer! Do you know any papers describing this procedure?
– Pleb
Jun 21 '21 at 17:27
• @Pleb, unfortunately, no. I have done some simulations for my own needs before but do not remember reading about it anywhere. Jun 21 '21 at 17:42
• Okay, if you are estimating each GARCH model separately and then average the estimates, then it sounds like the subsampling approach usually seen for realized measures (see eg. Zhang et al. (2005)). I have always wondered whether this approach was applicable for volatility models, but I have never found any papers about it.
– Pleb
Jun 21 '21 at 19:01
• @Pleb, I would estimate them jointly, but the results would probably be quite similar to averaging, at least asymptotically. However, I did not suggest to average across frequencies, though that is also an interesting possibility. Jun 21 '21 at 19:26
• @Pleb and Richard: I have added an estimation method which minimizes the negative sum of the two log-likelihood functions of the individual non-overlapping time interval sequences. Please comment on the legitimacy of this procedure.
– Hans
Jun 21 '21 at 22:13