GARCH calibration with overlapping time intervals

Question

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 1

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...