# Optimal bandwidth for Realized Kernel

If I want to estimate Realized Kernel for 1 min bins, is there a way to compute the optimal bandwidth? In the reference paper: Realised Kernels in Practice: Trades and Quotes (Ole Barndoff-Nielsen et al.), it provides the following formula for the optimal bandwidth $$H^*$$:

$$\hat{H}^* = c^* \hat{\xi}^{4/5} n^{3/5}$$ with $$\xi^2 \approx \frac{\omega^2}{IV}$$ and $$IV$$ is estimating by averaging 20 minutes realized variances.

What bothers me here is that $$IV$$ is calculated by using 20-min Realized Variances, does it mean that we can not determine the optimal bandwidth if we are interested in the Realized Kernel for time bins smaller than 20 minutes?

This issue with using a kernel to estimate a quantity for a one-minute bin is that you can write $$\xi^2$$ as $$\hat\xi^2 = \frac{\overline{RV}_{\text{dense}}}{\overline{RV}_{\text{sparse}}}.$$

The estimator $$\overline{RV}_{\text{sparse}} \approx\widehat{IV}\approx \sqrt{T\int_0^Y \sigma_u^4 du}$$ samples returns sparsely. The idea is that the sampling period is sufficiently long as to justify a claim that there are no "microstructure noise" (bid-ask bounce) effects at that time scale. Hence Barndorff-Nielsen, Hansen, Lunde and Shephard (2008) use 20-minute returns.

You are also likely to run into issues sampling at periods below five minutes since variance estimators explode with sampling periods that are shorter than five minutes, as shown in Andersen, Bollerslev, Diebold, and Labys (2003).

Perhaps the best advice would be to do some pre-averaging of your data, as advised in Podolskij and Vetter (2006), and then use kernels to give you a rolling estimation. Your best bet for that would be to look at Figueroa-López and Wu (2020, WP). Alternatively, you could just accept that your one-minute metrics include a quantity computed using data from the past few hours.

• Can we really use $\overline{RV}_{sparse}$ from the past few hours? From my understanding, if we are estimating the Realized Kernel for the bin $[0,T]$, we should use 1) $\overline{RV}_{sparse}$ in the bin $[0,T]$ and not for instance $[-T,0]$ and 2) even if it was possible, if $T=1$ min, we cannot use anymore the 20-min $\overline{RV}_{sparse}$ Aug 26 '20 at 23:28
• The idea of $\overline{RV}_{\text{sparse}}$ is a slow-moving variance estimate that doesn't include bid-ask bounce. You won't be able to get that at a 1-minute sampling period, so you will need to look beyond your 1-minute window in some manner. I suppose I revealed a bias in that I tend to do prediction, so I look backwards; you need not have that bias and could center your time period around the 1-minute bin. Also, the ABDL work suggests you could probably drop down to a 5-minute sampling frequency. That's not a lot better, but it helps get you closer to something more "local". Aug 26 '20 at 23:41
• Also, if you do like the Zhang, Mykland, and Aït-Sahalia (2005?) TSRV estimator, your 20-minute (or 5-minute) samples could be overlapping subsamples that you then average. This is sort of like what is done to estimate $\hat\omega^2$ in the BNHLS paper. Aug 26 '20 at 23:43
• So if I understood correctly, there seems to be two possible solutions if I want to estimate the de-noised volatility in a 1 min bin: 1) Use Realized Kernel, and estimating the optimal bandwidth with $\overline{RV}_{sparse}$ with a larger time bin than 1 min (at least 5min). Isn't there information leakage then? It seems that if I want to estimate the volatility for $[t, t+1min]$ using information from $[t,t+5min]$ doesn't seem very neat. Aug 26 '20 at 23:53
• I guess the look ahead is not so severe since we're only using it to determine the best bandwidth length, but it would be better if we didn't have to look ahead. Aug 27 '20 at 0:01